Microstate Counting of $AdS_4$ Hyperbolic Black Hole Entropy via the Topologically Twisted Index
Alejandro Cabo-Bizet Victor I. Giraldo-Rivera Leopoldo A. Pando Zayas





































 
 


ḍ
∏_i=1^n⟨i ,i+1⟩





























    [ α^' e.g.  i.e.  Tr tr⟩⟨αβ̱γ̧δϵϕφφ̃γηıιȷψκ̨łλμνωπθρ̊στῠξζΔΦΓΨŁΛΩΠΘΥΞ∇∂α̇β̇γ̇δ̇L̂M̂N̂ĴX̂ĜL̂̂̂Ĵ̂Ĝ̂superconformal algebrał̂ł̂̂̂Q̃higher spin black holeequationsection MCTP-17-01
                
                
                
                 .7 cm
                
                
                
                  Microstate Counting of AdS_4 Hyperbolic Black Hole Entropy .6 cm
                
                   via the Topologically Twisted Index
                 1 cm
                
                 Alejandro Cabo-Bizet^a, Victor I. Giraldo-Rivera^b and Leopoldo A. Pando Zayas^c
                 .4cm 
                ^a Instituto de Astronomía y Física  del Espacio (CONICET-UBA)  Ciudad Universitaria, C.P. 1428
                Buenos Aires, Argentina .4cm 
                ^b International Centre for Theoretical Sciences (ICTS-TIFR) Shivakote, Hesaraghatta Hobli,
                Bengaluru 560089, India .4cm^bThe Abdus Salam International Centre for Theoretical Physics Strada Costiera 11, 34014 Trieste, Italy .4cm ^c Michigan Center for Theoretical
                Physics Randall Laboratory of Physics, The University of
                Michigan Ann Arbor, MI 48109-1120 1.5 cm
                
                We compute the topologically twisted index for general N=2 supersymmetric field theories on ℍ_2× S^1. 
                We also discuss asymptotically AdS_4 magnetically charged  black holes with hyperbolic horizon, in four-dimensional N=2 gauged supergravity. With certain assumptions, put forward by Benini, Hristov and Zaffaroni, we find precise agreement between the black hole entropy and the topologically twisted index, for ABJM theories.
                
                
                
                
                § INTRODUCTION
                
                
                
                
                Black holes have an entropy that fits neatly in a thermodynamics framework as originally established in the works of Bekenstein and Hawking in the early 1970's. The microscopic origin, that is, the nature of the degrees of freedom that this entropy counts, has been an outstanding challenge for many decades. Any candidate to a theory of quantum gravity must provide an answer to this fundamental question. String theory, in the works of Strominger and Vafa,  has successfully passed this test for a particular type of black holes <cit.>.  In the context of the AdS/CFT correspondence, the original work of Strominger and Vafa can be interpreted as an instance of  AdS_3/CFT_2.  A natural question pertains higher dimensional versions of the AdS/CFT correspondence.  Recent work by Benini, Hristov and Zaffaroni addresses  the microscopic counting of the entropy of certain black holes from the point of view of AdS_4/CFT_3<cit.>.
                
                
                In this manuscript we explore the topologically twisted index, originally introduced by Benini and Zaffaroni in the framework of N=2 supersymmetric  three-dimensional field theories in S^2× S^1<cit.>  (see also  <cit.>),  for the case of supersymmetric theories in ℍ_2× S^1, where ℍ_2 is the hyperbolic plane.   Although we provide the ingredients for arbitrary N=2 supersymmetric theories, we will particularize our results for a specific deformation of ABJM theory. The holographic dual of such deformation is thought to be a hyperbolic black hole. In this work, our main motivation comes from the prospect of understanding the D=3 SCFT representation of the appropriate AdS_4 black hole microstates. With this aim we are driven to explore four dimensional N=2 gauged supergravity and find black hole solutions with ℍ_2 horizon. Hyperbolic black holes have been discussed in the context of AdS/CFT in, for example, <cit.>. 
                
                Asymptotically AdS_4 black holes in 𝒩=2 gauged supergravity, which are sourced by magnetic fluxes, have been widely studied  <cit.>.  Roughly speaking, from the bulk perspective, the presence of fluxes allows to define the black hole as interpolating  from the UV AdS_4 to the near horizon AdS_2× S^2. As a result of our study we are able to identify the role of such fluxes from the dual SCFT perspective. These flavor fluxes, together with a continuous of color fluxes, generate a one-parameter hierarchy of Landau levels on ℍ_2, that determines the value of the ABJM index. What we are set to explore in this paper, is whether the leading behavior in the large N limit of the topologically twisted index of a specific deformation of ABJM, evaluated on the Hilbert space composed by the aforementioned Landau levels, coincides with the Bekenstein-Hawking expression for the semiclassical entropy of the black holes in question.  We will find that indeed both results coincide.
                
                Another important motivation for our work, is the intrinsically interesting field theory problem of localization of supersymmetric field theories in non-compact spaces. This problem naturally appears in the context of localization of supergravity theories, for an understanding of exact black hole entropy counting <cit.>. The same problem appears in holographic approaches  to Wilson loops where the world volume of the classical configuration contains an AdS_2 factor. For example, the excitations on a D3 brane which is dual to a Wilson loop in the totally  symmetric rank k  representation <cit.>  were identified to correspond, to an N=4 vector multiplet in ℍ_2 × S^2<cit.>.  Localization in non-compact spaces has recently been addressed in <cit.> and  <cit.>,  our work constitutes an extension to the topologically twisted case. 
                
                The manuscript is organized as follows. In section <ref> we discuss the preliminary ingredients we need, for example, our guidance principle on the field theory side: supersymmetric localization <cit.>, the background metric, spin connection, and supersymmetric structure of the actions needed to compute BPS observables in a generic three-dimensional 𝒩=2 Chern-Simons-Matter theory on ℍ_2× S^1. To complete section  <ref>, we discuss the boundary conditions to be used in the manuscript. In section <ref> we present the space of square and delta-normalizable functions  that will be used to integrate upon, and their respective discrete and continuous spectrum. In section <ref> we compute the one loop super-determinants. In section <ref> we assemble our results to write down the ABJM index on ℍ_2× S^1, and then move on to compute its leading contribution in the large N expansion, by following the procedure pioneered in <cit.>. In section <ref> we find what we believe to be the dual AdS_4 black holes and compare its Bekenstein-Hawking entropy to the leading contribution in the large N expansion of the ABJM index on ℍ_2× S^1. In section <ref> we conclude with a short summary of our results and comment on interesting open and related problems.  In a series of appendices we discuss more technical aspects such as, for instance, the construction of square integrable modes in appendix <ref>. 
                
                § TOWARDS THE INDEX ON ℍ_2× S^1
                
                
                In this section we summarize the building blocks that will be needed in order to compute the topologically twisted index of a generic 𝒩=2 Chern-Simons-Matter theory on ℍ_2× S^1. The zero locus will be parametrized by a continuous of color fluxes and holonomies. On ℍ_2, these flux BPS configurations are non-normalizable but they are part of the zero locus: The localizing term Q_ϵ V, which is constructed to be semi-positive definite, will vanish at them. 
                
                First, we review the SUSY localization method to compute the partition function of 3d Chern-Simons-Matter defined over a Euclidean space ℳ with off-shell supersymmetry charge 𝒬.   The space ℳ is usually taken to be compact. The localization principle is well known  and has been elegantly summarized and interpreted in various reviews, for example,   <cit.>. However, given some of the intricacies we face, for the case we discuss, we review it here, with the goal of setting up our guiding principle, notation and to highlight some of the points on which we will make a particular emphasis. 
                
                 To close this section we elaborate on the specific set of boundary conditions that we shall use for background and fluctuations.
                
                
                
                
                
                
                
                
                
                
                
                
                 §.§ SUSY localization principle
                
                
                
                The SUSY localization method is summarized in the following steps
                
                  *  Select a “middle dimensional" section Γ in the space of complex fields, such as a 3d vector multiplet {A_μ,σ,D,λ,λ̅} of your theory. The path integral defining the SUSY partition function of a classical action S_cl, Z[Γ] is to be performed over Γ. The path Γ must be a consistent path of integration of S_cl.
                 
                  * 
                The contour Γ intersects a set of Q_ϵ-BPS configurations that will be denoted as BPS[Γ] and that is better known as: The localization locus.
                
                  * 
                For each Γ there should exist a Q_ϵ V local functional of fields whose bosonic part is semi-positive definite at Γ and vanishes at BPS[Γ]. 
                
                  * 
                Given the previous conditions, the strict limit τ→∞ can be taken in such a way that the final result for the partition function is guaranteed to be
                
                    Z[Γ]   =   ∑_X^(0)∈ BPS[Γ] e^-S_cl[X^(0)]Z_X^(0)[Γ],
                    
                     Z_X^(0)[Γ]   :=   ∫_Γ e^- δ^(2)( Q_ϵ V,  X^(0))  ,
                
                where δ^(2)( Q_ϵ V,X^(0)) is the quadratic expansion of Q_ϵ V about X^(0).
                We have omitted the integration over the space ℳ to ease the reading, but remember it is there.
                Let us review the semiclassical reduction (<ref>). 
                
                 The starting point, is to notice that the partition function Z[Γ] does not change if the initial classical action S_cl is deformed by an arbitrary Q_ϵ-exact deformation τ Q_ϵ V∂_τ(Z[Γ]   :=   ∫_Γ e^-S_cl[X]-τ Q_ϵ V )
                
                   =   ∫_ΓQ_ϵ( V e^-S_cl[X]-τ Q_ϵ V)=0,
                
                provided the measure of integration in field configuration space is Q_ϵ invariant and that there are not contributions from the boundary of the latter.
                
                Under the aforementioned conditions, we can choose a deformation term Q_ϵ V with semi-positive definite bosonic part and thereafter perform a field redefinition
                
                    X→ X^(0)+1/√(τ) X^(1).
                
                As Z[Γ] is independent of τ we are free to take the limit τ→∞ and proceed as follows
                
                    ∫_X e^-S_cl[X^(0)]- τ QV   =   e^-S_cl[X^(0)]∫_X^(1)  e^-τ QV
                       →   e^-S_cl[X^(0)]-τ QV[X^(0)]∫_X^(1) e^- δ^(2)( Q_ϵ V ,X^(0))  .
                
                
                Because of the suppression factor e^- τ Q V[X^(0)] and semi-positive definiteness of the bosonic part of Q_ϵ V only classical configurations X^(0)∈Γ, namely X^(0)∈ BPS[Γ], solutions of the zero locus of Q_ϵ V (Q_ϵ V_bos=0) contribute in this limit and (<ref>) is recovered.
                
                
                
                
                 §.§ Background geometry and supersymmetry
                
                
                In this subsection we introduce the basic elements needed for the evaluation of the localization formula for the topologically twisted index of a generic 𝒩=2 Chern-Simons-Matter theory. Specifically, we are interested in U(N)_k× U(N)_-k Chern-Simons theory coupled to matter in the bi-fundamental representation: ABJM<cit.>,  living in the non-compact space ℳ=ℍ_2× S^1 whose metric we will represent as
                
                ds^2   =    -dt^2+ds_2d^2,  
                 
                 ds_2d^2   :=   - dθ^2-sinh^2( θ)  dφ^2,  
                       φ∼φ+2π,     t ∼ t+1.
                We shall use in this paper the following signature (-,-,-) on the 3d boundary theory. The flat space metric is η=diag(-1,-1,-1). 
                 
                In the conventions used in this section, the non trivial spin connection component is
                
                    ω^21_φ= -coshθ.
                
                 The 2d space ℍ_2 has infinite volume. When dealing with extensive quantities on ℍ_2 we will use a cut-off at large θ and drop out the dependence on such cut-off in the very end. More precisely, this recipe has been used in the context of black hole entropy in  <cit.> and, in the context of holographic computations for Wilson loops it was discussed in  <cit.>; it amounts to  defining the volume of ℍ_2 as:
                 
                 vol_ℍ_2=-2π.
                 
                 As general principle, we will consider background configurations that grow asymptotically as the volume element, or slower. As for extensive quantities constructed out of such non normalizable backgrounds, we shall apply the previous  regularization recipe  [
                
                
                 For example, to work with boundary objects - like the boundary action (<ref>)- with finite limit in the cut-off θ_0 →∞, we follow <cit.>. The idea is to use coordinates (:=_0-,  φ̃=1/2e^_0φ), in such a way the metric
                
                d^2+sinh^2 dφ^2,
                
                transforms to
                
                d^2+(e^--e^-2_0+)^2 dφ̃^2,
                
                where φ̃ is a periodic coordinate with period =π e^_0 and 0<<_0.
                ]. 
                
                 
                
                
                 
                 The results of these sections allow to compute the topologically twisted index of any 𝒩=2 Chern-Simons theory coupled to matter.  As mentioned before we are interested in the particular case of ABJM. The ABJM theory is composed by two vector multiplets and four matter multiplets in the bi-fundamental of the gauge group. Specifically
                 
                    Chern-Simons  ± k   :   {A_μ, σ, D, λ_q=1, λ̅_q=1}_± k,   
                    
                    matter   :   {ϕ_q^a,ϕ̅_q^a, ψ^a_q-1, ψ̅^a_q-1, F^a, F̅^a},     a=1,2,3,4.
                
                where [The supersymmetry transformation rules are defined over the complex conjugated of (,j̅,F̅), which are denoted as (^†,j̅^†,F̅^†).]q is the charge of the corresponding field under the R-symmetry flux (<ref>).
                
                We can represent ABJM theories by the following standard quiver diagram:
                
                [every node/.style=circle,draw,thick,minimum size=1.5cm]
                  (NL) at (0,0)N__k;
                  (NR) at (3,0)N__-k;
                  [->>](NL) to [bend left=40] node[draw=none,minimum size=0.1cm,above]^1,^2(NR);
                    [->>](NR) to[bend left=40]node[draw=none,minimum size=0.1cm,below]^3,^4(NL);
                
                
                ABJM theories have = 8 superconformal symmetry for level k=1,2 and =6 for  level k≥3, the global symmetry that is manifest in the =2 notation is SU(2)_1,2× SU(2)_3,4× U(1)_T× U(1)_R, where each SU(2) acts upon the doublets composed by the corresponding labels  <cit.>. 
                
                
                  We are interested in a specific deformation of ABJM. Part of such deformation is a classical background for the R-symmetry potential
                
                V_μ dx^μ=1/2coshθ dφ. 
                The background (<ref>) is non normalizable. However,  V goes like the volume element of ℍ_2 for large θ. The deformation (<ref>) has non trivial consequences in the final result of the localization technic.
                
                
                
                 The R-symmetry background allows for the presence of a Killing spinor ϵ with R-charge q=1. The Killing spinor equation (KSE) being
                 (∂_μ +1/4ω^a b_  μσ_a b- i V_μ) ϵ   =   0, 
                 with σ_a b:=[σ_a, σ_b]/2 and σ_a, σ_b being  Pauli matrices.  As we are using negative signature it is important to keep in mind that
                 σ^a=-σ_a. 
                 In fact, the algebrae and actions that will be defined later on, are obtained out of the results in <cit.> by the appropriate change of signature, and (<ref>).
                 
                 The most general normalized solution  to the KSE  (<ref>), is proportional to
                 
                    ϵ= ([ 1; 0 ]).
                
                 Out of ϵ we can construct an off-shell supercharge Q_ϵ.  Before dealing with the construction of Q_ϵ, it is convenient to perform the following field redefinition 
                 Â_3:=A_3+i σ.
                 
                In terms of the new variables, the offshell algebra takes the following form for the vector
                 
                    Q_ϵÂ_θ, φ   =    -i/2(
                     -λ̅^†σ_θ, φϵ),    Q_ϵÂ_t=0, 
                    
                    Q_ϵσ   =   1/2(
                    -λ̅^†ϵ),     
                    Q_ϵλ = -1/2σ^μνϵF̂_μν+D ϵ
                    +i σ^3 ϵ𝒟̂_3 σ,
                    
                    Q_ϵλ̅^†   =   0,
                           
                     Q_ϵ D= 
                     
                     +i/2  (𝒟̂_μλ̅)^†σ^μϵ,
                
                 and matter multiplet [Where ϵ^C:=C ϵ^* and C=-i σ_2. Notice, that the C conjugation matrix is real.]
                    Q_ϵϕ   =   
                    0,        Q_ϵϕ̅^† = - ψ̅^†ϵ,
                    
                    Q_ϵψ   =   
                    +i σ^μϵ𝒟̂_μϕ,      
                    Q_ϵψ̅^†= 
                    
                    F̅^†ϵ^c^†, 
                    
                    Q_ϵ F   =   
                    +i (ϵ^c)^†σ^μ𝒟̂_μψ+i (ϵ^c)^†λ ϕ,       
                    Q_ϵF̅^†= 0.
                
                The action of the gauge covariant derivative being
                 𝒟̂_μ:={[ ( ∂_μ+1/4σ_a bω^a b_  μ-i Â_μ -   i q_sp V_μ)                                   on spinors,;                  ( ∂_μ-i Â_μ  -   i q_sc V_μ)                                   on scalars. ]. 
                It can be shown, that  the Chern-Simons theory
                ℒ_CS= -i k/4 π( ϵ^μνβ(Â_μ∂_νÂ_̂β̂ -2 i/3Â_μÂ_νÂ_β)  -λ̅^† 1-σ_3/2λ),  
                is annihilated by (<ref>), up to a total derivative
                 
                -i k/4 π𝒟̂_μ( ϵ^μνβ(Q_ϵÂ_ν) Â_β).
                There can also be a mixed CS term whenever we have several Abelian factors:
                _mCS=-i k_ij/4π( ^Â^(i)__Â^(j)_+̱ł̅^(i)†(1-_3/2)ł^(j)).
                
                Where k_ij is symmetric and i≠ j, in this case one similarly gets boundary pieces
                      _mCS=-i k_ij/4π(_( ^ (Q_Â^(i)_) Â^(j)_). 
                The discussion of the topological current in <cit.> is valid for any Chern-Simons theory. In the case of ABJM, the topological U(1)_T global symmetry is generated by the conserved current J^_T= tr (*F̂-*F̂̃̂)^. One can couple background U(1)_T gauge potentials Â^T_μ, to the current J^_T. The supersymmetry completion of such term,  is a particularization of the action (<ref>).  Such particularization, is given by picking k_ij=k_j i=1 
                and regarding just a couple of indices (i=1,2). The index “1” labels a background Q_ϵ-spurion vector multiplet, and the index “2” labels a U(1) dynamical vector multiplet. In such a way, we obtain the corresponding mixed supersymmetric Chern-Simons action, out of (<ref>). For instance, in the case of gauge group U(N), there is a unique dynamical U(1), and the bosonic term of the latter action is
                ^Bos_T=-i/4 π^(Â^T__[ Â_]̱+[Â_] _Â^T_).
                
                In the very end, we will fix the v.e.v of the spurion vector supermultiplet to specific Q_ϵ-BPS values
                [Which is the family Â_3^T=u^T, D^T= iF^T_12, ^T=0.].
                
                 
                 At this point, we must select a “middle dimensional”  contour of integration in field space. Let us introduce a contour Γ consistent with the one of  <cit.>Γ_vector:     Bosonic  Fields=(Bosonic  Fields)^* ,  e. g.  D=-(D)^*. 
                 The contour Γ_vector will cross a specific family of  Q_ϵ-BPS configurations. 
                 
                 BPS[Γ_vector] : {[ F_12=-𝔪/2,   D=-i 𝔪/2,   fermions=0.;                                     ;                   Â_3=u=u^*∈ [0,2π), ]. 
                
                
                
                 
                
                where 𝔪 and u are Cartan valued arbitrary constants. The u are the Coulomb moduli and parametrize the Coulomb branch of the theory. Expression (<ref>) is the most general solution - single valued at the S^1 factor and without fermionic zero modes- to the BPS equation
                
                Q_ϵλ = -1/2σ^μνϵF̂_μν+D ϵ+i σ^3 ϵ𝒟̂_3 σ=0,
                 
                along the contour (<ref>) . 
                
                As for the matter multiplet we define
                Γ_matter:   ϕ̅=ϕ,  F̅=F. 
                In our case, the zero locus of matter is 
                
                BPS[Γ_matter]:   ϕ̅=ϕ=F̅=F=fermions=0.  
                
                Finally, we define Q_ϵ exact terms. The Q_ϵ exact terms must be semi-positive definite along Γ, as already stressed. In the case of the vector multiplet  and the choice of Γ(<ref>), such a term is
                
                    Q_ϵ V^vector   :=   -Q_ϵ((∙Q_ϵλ) λ), 
                    (∙Q_ϵλ)   :=   (Q_ϵλ)^* |_Â^*→Â,  σ^*→σ,   D^* →- D.
                
                The bosonic and fermionic part of (<ref>) are
                
                    Q_ϵ V^vector_B   :=   (F_1 2+ 𝒟̂_3 σ +i D )^2+(F̂_13)^2+ (F̂_2 3)^2, 
                    
                     Q_ϵ V^vector_F    :=   - i  λ̅^†_2 𝒟̂_t  λ_2.
                
                where λ_2 is the lower component of the gaugino λ=([ λ_1; λ_2 ]).
                
                 For the matter multiplet, and given the choice of Γ in (<ref>) and (<ref>), such a term is
                
                    Q_ϵ V^matter   :=   
                    -Q_ϵ( -i ϵσ^μψ𝒟̂_μϕ̅^†+ F ψ̅^†ϵ^c + i ϕ̅^†ϵλϕ).
                
                The bosonic and fermionic part of (<ref>) are
                
                    -Q_ϵ V^matter_B   =    (𝒟̂^μϕ̅)^† 𝒟̂_μϕ +   ϕ̅^†(𝒟̂_3σ+i D- ϵ^μν_  βv^β(q V_μν+W_μν) )ϕ
                            +F̅^† F+ 𝒟̂_μ( i  ϵ^μν_  βv^βϕ̅^†𝒟̂_νϕ)
                    , 
                          
                    
                    -Q_ϵ V^matter_F   =   -i  ψ̅^†σ^μ𝒟̂_μψ- i ψ̅^†λ ϕ- i  ϕ̅^† λ̅^† P^- ψ
                    + i 𝒟̂_μ( ψ̅^† P^+ σ^μψ),
                
                where P^∓:=1∓σ_3/2 and V_μν, W_μν are the field strengths of R- and flavor symmetry backgrounds, respectively. 
                
                The term Q_ϵ V^matter_B is semi-positive definite when expanded around BPS[Γ_vector] and over Γ_matter. As shall be shown in due time, this last statement is implied by the requirement of square integrability over ℍ_2. Square integrability over ℍ_2, imposes bounds on the spectrum of eigenvalues of the relevant magnetic Laplacian. The aforementioned bounds imply the convergence of the Gaussian path integral ∫_X^(1) e^- δ^(2)( Q_ϵ V ,X^(0)) in (<ref>). 
                
                
                Chern-Simons, being a gauge theory, requires gauge fixing, which we choose to be the axial condition 
                Â_3=const. 
                In contradistinction to 3d pure Yang-Mills theory, in 3d Chern-Simons coupled to Yang-Mills and/or matter, the constraint (<ref>), fixes the gauge degeneracy completely. In the latter theory there are 3-1=2 physical off-shell  vector degrees of freedom (DoF) meanwhile in 3d pure Yang-Mills there is 3-2=1  massless vector offshell DoF. For a nice review on the canonical quantization of 3d Chern-Simons theory, see for instance <cit.>.
                
                To implement the gauge fixing, we use BRST method <cit.> and enlarge the vector multiplet, by adding the ghost fields (c,c̅,b̅). We enlarge the algebra (<ref>), by the following transformation rules
                
                Q_ϵ c=0,   Q_ϵc̅=0,   Q_ϵb̅=0. 
                
                Any gauge invariant functional of the physical fields is BRST invariant. The BRST transformations Q_B are 
                
                    Q_BÂ_μ= 𝒟̂_μ  c,    Q_Bc̅= b̅,    Q_B c =i/2{c,c} ,   Q_Bλ=i {c, λ}, 
                     
                    Q_Bλ̅^†= i {c,λ̅^†},    Q_Bσ= i[c,σ],    Q_B D=i {c,σ}, 
                    
                    Q_Bϕ=i[c,σ],    Q_Bϕ̅^† =i [c,ϕ̅^†]
                    ,  Q_Bψ=i{c,σ},    Q_Bψ̅^† =i {c,ψ̅^†},
                    
                     Q_B F= i [c,F],    Q_BF̅^†= i [c,F̅^†],
                
                from (<ref>), (<ref>) and (<ref>) it can be shown that
                (Q_ϵ+Q_B)^2={Q_ϵ,Q_B}=0.  
                As the V's in the Q_ϵ V's localizing terms, (<ref>) and (<ref>), are gauge invariant objects, then from the corresponding algebra (<ref>) is easy to check that the V's in (<ref>) and (<ref>), are Q_B invariant and consequently (<ref>) and (<ref>) are (Q_ϵ+Q_B)-exact. 
                
                 On top of the localizing actions  (<ref>) and (<ref>), a gauge fixing term must be added. To our purposes the most convenient choice is the following  (Q_ϵ+Q_B)-exact term 
                
                Q_BTr(c̅(Â_t-const))= c̅𝒟̂_t c+  b̅(Â_t-const). 
                From (<ref>) and Q_ϵÂ_3=0, it follows that (<ref>) is (Q_ϵ+Q_B)-exact. In (<ref>) we wrote the gauge index trace Tr only on the LHS, but the reader should keep in mind that by default we are working with gauge invariant density Lagrangians.
                 
                Our BRST construction is conceptually  that of Pestun <cit.> and it has been previously presented in the 3d case by Kapustin, Willet and Yaakov <cit.>.
                
                
                
                
                 §.§ Boundary conditions
                
                In  non-compact manifolds like ℍ_2× S^1  or manifolds with boundary, appropriate boundary conditions must be imposed in order to have a well defined variational -Lagrangian- problem. Once a proper classical theory has been defined, quantization is in order. Let
                
                X^(0) = {A^(0)_μ, σ^(0), D^0,…}∈ BPS[Γ]
                
                and
                
                X^(1)={δ A_μ, δσ, δ D, δλ, δλ̅, δ c, δc̅, b̅,  δϕ, δϕ̅,  δψ, δψ̅, δ F, δF̅},
                
                be the non trivial zero locus background fields and offshell fluctuations respectively. As for the X^(0) we define the following boundary condition
                
                e^μ_a A^(0)_μ,   D^(0),  σ^(0)   θ→∞∼    O(1). 
                
                
                
                As for offshell fluctuations X^(1), we define Dirichlet boundary conditions
                
                 e^μ_a δ A_μ, δσ, δ D, δλ, δλ̅, δ c, δc̅, δb̅, δϕ, δϕ̅,  δψ, δψ̅, δ F, δF̅   θ→∞∼    O(e^-κθ), 
                
                 e^μ_a δ A_μ, δσ, δ D, δλ, δλ̅, δ c, δc̅, δb̅,  δϕ, δϕ̅,  δψ, δψ̅, δ F, δF̅   θ→ 0∼   O(1), 
                with κ≥1/2. The value of κ defines important features of the spectrum of the associated S^1 quantum mechanics, if a sort of dimensional reduction is possible to perform in this case.  
                
                The following table sketches the relation between the boundary conditions (<ref>) and the results reported in the next section:
                
                κ    Spectrum     Norm 
                1/2    Continuous: λ∈ [0,∞)                  Delta-Square Integrable 
                >1/2    Discrete: j=|s|-1,|s|-2, … >-1/2.     Square Integrable 
                
                
                
                
                The total derivative part of the offshell variation of the Chern-Simons Lagrangian (<ref>), multiplied by the volume element √(-g) is
                
                -i k/4 π𝒟̂_μ(√(-g)ϵ^μνβ(δÂ_ν) Â_β),    with    ϵ^θφ t:= 1/√(-g).
                After integration and imposition of gauge fixing condition δÂ_3=0 - see (<ref>)-, the total derivative (<ref>) becomes a boundary term
                
                -i k/4 π∫_φ=0^2πdφ∫_t=0^1dt (δÂ_φ)Â_t |^θ=∞_θ=0. 
                Boundary conditions (<ref>), do not imply the vanishing of (<ref>) at θ=∞, due to the non-compactness of ℍ_2 - specifically because  e^φ_2θ→∞→ 0-. The contribution from θ=0 vanishes.
                
                
                To have a well-defined variational principle, we redefine the classical action from Chern-Simons to
                ∫√(-g)ℒ_cl= ∫√(-g)ℒ_CS + S_bdy,    S_bdy=+i k/4π∫_φ=0^2πdφ∫_t=0^1dt  Tr ( Â_φÂ_t )   at θ=∞. 
                Note that S_bdy is gauge invariant, provided we restrict the derivatives of gauge transformations parameters to vanish at θ→ ∞. In this paper, we will assume the latter condition.
                
                 It is immediate to check that the supersymmetric transformation of ℒ_cl is trivial by construction:  the supersymmetry variation of S_bdy cancels the integration of the total derivative term (<ref>), as it should. 
                 
                 The classical action evaluated on the zero locus is
                 ∫√(-g)ℒ_cl[BPS[Γ]]
                 =-ik/2  u ·𝔪
                 Where u ·𝔪:=u_i 𝔪_i= 1/2Tr(u 𝔪). In our conventions h_i and h_j are Cartan generators in the Chevalley basis, and consequently Tr[h_i h_j]=2δ_i j. 
                
                The contributions proportional to
                coshθ_Max cancel out, θ_Max being the large cut off in θ. The divergent terms coming from the integral over ℍ_2 and the boundary term (<ref>) cancel each other.
                 Whenever we have contributions which diverge like the volume, we regulate them as we regulate the volume in (<ref>), and boundary terms are regulated as  explained in footnote in page 4. 
                
                 It is convenient to write down the exponential of -(<ref>)
                x_i^k 𝔪_i/2, 
                with x_i=e^i u_i. Expression (<ref>), is the contribution to the index of a Chern-Simons term with level k.
                
                The total derivative part of the variation with respect to ϕ̅^† of the bosonic localizing action of matter is
                
                +∫_ℳ𝒟̂_μ(√(-g)(δϕ̅^† 𝒟̂^μϕ+i  ϵ^μν_  βv^βδϕ̅^†𝒟̂_νϕ) )  
                Under off-shell boundary conditions (<ref>) and (<ref>), the integration of (<ref>) gives
                
                +i ∫^2π_0 dt ∫^2π_0 dφ[ (δϕ̅ ^†𝒟̂_φϕ) ]_θ=0. 
                 
                 The term (<ref>) vanishes, when evaluated in the functional space we are going to integrate over. The explanation of the latter fact, shall be given in the beginning of subsection <ref>. 
                 
                Notice, that the ghosts vanish at the boundary,  due to (<ref>). Consequently, BRST gauge transformations do not affect the boundary. 
                 
                 
                 
                
                
                
                
                 
                
                
                
                Having established the localization locus, the next step is to compute one loop determinant contributions Z_X^(0).  In order to do that, we need to define an appropriate functional space to integrate upon. That, will be the scope of the next section. Thereafter, we can compute Z_X^(0) and use equation (<ref>) to evaluate our final result for the topologically twisted index.
                
                
                
                
                
                
                
                
                § THE SPECTRUM ON ℍ_2 WITH FLUX
                
                
                The spectrum of the Laplace operator  on ℍ_2 has a long history (<cit.> and references therein).  
                Even though this section might seem just a technical remark, the result of its analysis is very  relevant in reaching our conclusions.  We therefore choose to include it in the main body of the text and provide more details in an appendix.
                 
                 The eigenvalue problem solved in appendix <ref>, is related to the propagation  of a scalar particle in the presence of a flux on ℍ_2. The hierarchy of modes to be reported in this section, can be interpreted as a series of Landau levels on ℍ_2, that emerge due to the presence of a flux s<cit.>. These alternative viewpoints deserve further attention and we are keen to pay them so in forthcoming works. 
                
                In this section, we will present the outcome of the analysis that shall be reported in appendix <ref>. We encourage the reader looking for a detailed understanding, to go through that appendix. 
                
                The Laplacian in the presence of a flux s, coming from a potential
                
                A=s coshθ dφ,  
                is given by
                □_s:=-∂_θ^2-^2θ∂_θ+1/sinh^2θ(j_3-scoshθ)^2,
                
                with j_3=-i∂_φ.
                
                
                The equation that defines the functional space upon which we will compute
                determinants is
                (  □_s+Δ)  f_Δ,j_3=0. 
                The boundary conditions that will define our functional space are
                (<ref>)  and  (<ref>) with  κ > 1/2. 
                
                
                 §.§ The discrete spectrum
                
                
                First, we parametrize the eigenvalue as:
                Δ=j(j+1)-s^2.
                The quantization conditions
                j_3-s,   j-|s|∈ℤ, together with equation (<ref>), define a finite (resp. infinite) dimensional space of square integrable functions on ℍ_2, that we denote asΞ_j^(1)
                (s):={  f_Δ,j_3^(1)}  _j_3,
                respectively
                Ξ_j^(2)(s):={  f_Δ,j_3^(2)}  _j_3.
                
                The explicit form of the eigenfunctions f_Δ,j_3^(1) and f_Δ,j_3^(2), is defined in appendix <ref> and <ref>, respectively, for the case s>1/2. The case s<-1/2 can be worked out analogously.
                
                The range of j_3 is given by the relations
                c]lll
                s≥ j_3≥max(|j|,|j+1|)    if    s>+1/2,
                
                -j_3≥-s≥max(|j|,|j+1|)    if    s<-1/2, 
                respectively
                c]lll
                j_3≥ s≥max(|j|,|j+1|)     if     s>+1/2,
                
                -s≥-j_3≥max(|j|,|j+1|)     if    s<-1/2. 
                
                There are additional constraints to the value of j. Indeed, the eigenfunctions
                f^(1,2)_Δ,j_3 and henceforth the spectrum Δ=j(j+1)-s^2,  are invariant under the transformation
                    j→-(  j+1).
                
                Thenceforth we must restrict j to be either
                [ j>-1/2     or j<-1/2 ]
                .
                
                In order to have square integrable functions
                
                j≠-1/2. 
                 For the choice j>-1/2, restrictions (<ref>) and (<ref>), imply an upper bound for j
                j=|s|-1, |s|-2,... > -1/2.
                
                Interestingly, for
                
                0≤|s| ≤1/2,
                
                there are no square integrable modes. A particular conclusion of this last statement, is the known fact that in ℍ_2 there are no square integrable scalar modes. In the presence of flux  s: |s|> 1/2, square integrable modes emerge. 
                 
                 In appendix <ref>, it is proven that in the case |s|=1, our square integrable eigenmodes match those well known discrete modes, of the vector Laplace-Beltrami operator in ℍ_2, with helicity s=±1. This last statement, suggests to explore the possibility that our spectrum encodes the full tower of higher spin square integrable eigenmodes of the Laplace-Beltrami operator on ℍ_2.  We hope to come back to this point in the future.
                
                The relevant scalar product is<𝑓,𝑔>:=∫_0^∞𝑑θ∫_0^2π
                𝑑φsinh(  θ)  𝑓^∗(θ
                ,φ)𝑔(θ,φ). 
                As already mentioned, and proven in appendix <ref>, square integrability off_Δ,j_3
                ^(1) (resp. f_Δ,j_3^(2)) is interrelated to the
                specific bounds on j_3 and j  that were previously written. 
                
                 Different states f_Δ,j_3^(1,2)e^ij_3φ in Ξ_j^(1,2)(s) are orthogonal with
                respect to the scalar product (<ref>).  Spaces Ξ_j^(1,2)(s) with
                different label j>-1/2(orj<-1/2) are orthogonal.
                This is because □_s is Hermitian in Ξ_j(s)
                and spaces with different label j>- 1/2(orj<-1/2), have
                different eigenvalues Δ  under □_s.
                
                Summarizing, the space of square integrable modes for a given s is
                Ξ(s)= ⊕^|s|-1_j>-1/2(Ξ^(1)_j(s) ⊕Ξ^(2)_j(s)).
                
                 In the next section, we will refer to the following spaces
                 Ξ_j(s):=
                
                
                
                (  Ξ_j^(1)(s) 
                
                
                ⊕
                 Ξ_j^(2)(s)). 
                The spaces (<ref>) are subspaces of (<ref>).
                
                
                
                 §.§ The continuous spectrum
                
                
                The continuous spectrum is a direct generalization of the spectrum reported by Higuchi and Camporesi <cit.> to the case where there is  a constant flux on ℍ_2. The corresponding eigenmodes  solve the defining equation 
                (
                □_s+Δ_(λ,s))  f_Δ_(λ, s),j_3=0,
                 
                with
                
                    Δ_(λ,s):=-λ^2-s^2-1/4, λ∈ℝ,λ≥0,
                
                and boundary conditions 
                
                f_Δ_(λ,s),j_3(x)    x→ -∞∼      c_1(λ
                ,j_3,s)x^+iλ/x^1/2+c_2(λ,j_3,s)
                x^-iλ/x^1/2,  
                 f_Δ_(λ,s),j_3(x)    x→ 0∼   O(1). 
                Conditions (<ref>) and (<ref>) are given in coordinates x, but they are equivalent to the particularization κ=1/2  of (<ref>).
                
                The final solution to the boundary problem just presented, is obtained by imposing (<ref>) on the most general solution (<ref>). The result is
                
                f^(1)_Δ_(ł,s)(x)       if        j_3 ≥ s,  
                
                f^(2)_Δ_(ł,s)(x)       if        j_3 < s.
                
                
                The norm of f_Δ_λ j_3 under the scalar product (<ref>) is infinite. By choosing appropriately the remaining 
                integration constant one can set 
                ⟨ f_Δ_(λ,s), j_3, f_Δ_(λ^',s), j^'_3⟩ =δ(ł-ł')_j_3 j_3'. 
                
                  
                Comments:
                Our thermal cycle is not the S^1 inside the ℍ_2, but the trivially fibered one. The latter fact, is related to an important conceptual difference between the physical framework of our approach and the one of, for instance, <cit.>. In physical terms, our ℍ_2 modes are not probing the near horizon limit of a black hole in the presence of electric flux <cit.>, but the boundary dynamics of a magnetically charged hyperbolic AdS_4 black hole. 
                
                That said, if we interpret our φ-cycle as the thermal one, our hierarchy of square integrable modes is certainly probing Euclidean AdS_2, in the presence of an electric flux deformation. That is closely related to the problem addressed in <cit.>. To have a self-consistent approach, coming from supersymmetric localization, to the problem studied in <cit.>, one should try to localize an appropriate off-shell supercharge on the quantum gravity side. In that spirit, in <cit.>, AdS_2× S^2 was shown to be the unique ungauged BPS localizing solution, to 4d 𝒩=2 Super-Conformal gravity in a convenient gauge fixing.  It is plausible, that other electric or magnetic AdS_2×Σ localizing solutions can be found, by relaxing some of the conditions used in <cit.>, as suggested by the results in <cit.>. If that is the case, it would be quite interesting to explore what can supersymmetric localization say about the problems addressed in <cit.>.
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                § ONE-LOOP DETERMINANTS
                
                
                Having clarified the structure of the spectrum for the flux Laplacian on ℍ_2 (see appendix <ref>), we have all the ingredients to address the computation of one-loop determinants. 
                
                
                
                
                
                 §.§ Bosonic localizing operator
                
                
                For square integrable modes, the total derivatives of the quadratic expansion in the localizing
                terms, integrate to zero. In the case of the  matter multiplet, the total derivate term is 
                
                +∫_ℳ𝒟̂_μ(√(-g)(ϕ̅^† 𝒟̂^μϕ+i  ϵ^μν_  βv^βϕ̅^†𝒟̂_νϕ) ) 
                
                =+i ∫ d t d φ[ ϕ̅^†𝒟̂_φϕ]_ At θ=0, 
                where the result in the second line follows from the asymptotic behavior (<ref>) and (<ref>). This boundary term vanishes because, as proven in appendix <ref>, the only modes of ϕ that do not vanish at the contractible cycle θ=0, have the following angular dependence
                
                e^i  s  φ,    with      s   :=   -ρ(𝔪)-q_R/2, 
                and they are annihilated by
                𝒟̂_φ:=∂_φ -i s.
                
                 Due to a careful choice of boundary conditions, total derivatives are irrelevant for the current
                discussion, as they do not contribute to the 1-loop determinant.
                
                
                
                
                
                
                After integration by parts, the quadratic expansion of the bosonic part of the Lagrangian density of the matter localizing term (<ref>), takes the form
                
                
                ϕ̅^† O_Bϕ   
                
                
                := 
                ϕ̅^†(  (ρ(u)+i∂_t)^2+( □_s-s)  ) ϕ. 
                
                Notice that the operator O_B is 
                positive definite on “ representations”
                 Ξ_j(s) labeled by j running at step 1 down from |s|-1 but
                larger than -1/2.  For “ representations” labeled by j:-1/2 <
                j≤|s|-1 (or -|s|≤ j<-1/2), the operator (  □_s-s)
                has eigenvalues that obey the following inequality-j(j+1)+s(s-1)≥0 and
                hence is semi-positive definite. Consequently, O_B is positive definite if (ρ(u)+i∂_t)^2>0. This last condition is guaranteed, provided we avoid points in the Coulomb branch such that ρ(u) ∈ℤ.
                
                 Having in mind the particular case
                
                    j, j_3, s ∈ℤ,
                
                at some stages, we will denote the aforementioned set of
                j^'s as follows
                
                    j:0≤ j≤|s|-1.
                
                The union of the aforementioned Ξ_j(s), is the maximal space of square integrable modes (<ref>).
                
                For latter convenience, let us define
                □_s^±   :=   □_s± s.
                
                
                 Should we select ϕ in the vector space spanned by Ξ_j(s),
                thence for s>1/2
                
                    _Ξ_j^(1)(s)(O_B)       =
                    
                    
                    ∏_j_3=j+1^s
                    
                    
                    ∏_
                    k
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))  , 
                    _Ξ_j^(2)(s)(O_B)       =
                    
                    
                    ∏_j_3=s+1^∞
                    
                    
                    ∏_
                    k
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))  ,
                
                where k=i ∂_t. The result for s<-1/2 is obtained analogously.
                
                As ϕ is a complex scalar, the functional integration
                    ∫[Dϕ^† Dϕ]  
                    Exp[-
                    
                    
                    ∫_ℍ_2× S^1
                    ϕ^†·(O_B)·ϕ
                    ],
                
                is proportional to 1/(O_B), where
                    _Ξ(s)(O_B)={c]lll
                    ∏_j=0^s-1
                    
                    
                    ∏_j_3=j+1^∞
                    
                    
                    ∏_ρ^∗,k
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))      if    s>1/2
                    ∏_j=0^-s-1
                    
                    
                    ∏_j_3=-∞^-j-1
                    
                    
                    ∏_ρ^∗,k
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))     if    s<-1/2
                     .  .
                
                
                 §.§ Fermionic localizing operator 
                
                
                To compute the fermionic determinant,  we used the square of the kinetic operator that appears in the quadratic expansion of the fermionic part (<ref>) of the localizing term,  we also specify the space of functions on which each component acts
                
                    (
                    [                O_B                  0;                  0 (u+i∂_t)^2+□_s-1^+ ])  ψ^+∈ Span(Ξ(s))ψ^-∈ Span(Ξ(s-1)).
                
                
                While reproducing the computations that will be reported in section <ref>, it will be convenient  to use the following identity
                
                    □_s-1^+f^(1,2)_Δ(s-1),j_3=(
                    -j(j+1)+s(s-1))  f^(1,2)_Δ(s-1),j_3.
                
                
                
                
                 §.§ ζ-function regularization: s> 1/2
                
                We use ζ-function regularization to compute the determinants of O_B
                upon the functional space
                
                    Ξ_j(s):=Ξ_j^(1)(s)
                    
                    
                     ⊕ 
                    Ξ_j^(2)(s),  j=s-1  (or  -s).
                
                which is the space of zero modes of  □_s^-
                 (the space of eigenstates of O_B with eigenvalue (ρ(u)+k)^2). We stress  that in the case s>1/2 and after
                cohomological cancellations, these zero modes are the only ones that contribute
                to the one loop super-determinant.
                
                In order to compute the heat kernel K(0,0) associated to the eigenspaces in
                question, we need to analize the relevant case j_3=s. In the
                latter case and after particularizing to  j= s-1 or -s, the square
                integrable modes f^(1) and f^(2) drastically simplify to
                
                    f^(1,2)_Δ(s), j_3=s=χ(  x-1)  ^-s.
                
                The constant χ is determined from the normalization condition
                
                |χ|^22π vol_S_1∫_-∞^0dx(2) |  x-1|  ^-2s   
                =|χ|^2 2π vol_S^1/s-1/2=1.
                The 2 in the LHS of (<ref>) is the line element in coordinates x.
                From the value of |χ|^2, we obtain the heat kernel at origin
                
                K(t;0,0)=1/2π vol_S^1(s-1/2)e^-t(ρ(u)+k)^2.
                From (<ref>) we obtain the zeta function 
                ζ(z)       =vol_ℍ_2vol_S^1/2π vol_S^1(s-1/2
                )/((ρ(u)+k)^2)^z=-(s-1/2)/((ρ(u)+k)^2)^z
                ,
                
                with
                
                 vol_ℍ_2=-2π,    vol_S^1=1.
                
                
                
                
                
                
                
                
                
                
                
                
                After using ζ-function method, we obtain the desired determinant_Ξ_j=-s(s)(O_B)=_Ξ_j=s-1(s)(O_B
                )=e^-ζ^'(0)   =   ((ρ(u)+k)^2)^-s+1/2
                   =   |(ρ(u)+k)|^-2s+1. 
                In appendix <ref> we obtain the same result, by using an alternative procedure. Notice that in (<ref>) we could have also written 
                   =   (-|(ρ(u)+k)|)^-2s+1.
                
                It is possible that such a change in the election of sign, changes the value of the partition function. From now on, we will ignore this second choice, except for specific steps where having it in mind will be useful.
                
                
                
                
                
                
                 §.§ ζ-function regularization: s<1/2
                
                
                In the case s<1/2, the contribution to the super-determinant is coming from
                the following set of eigenfunctions
                Ξ_j(s-1):=Ξ_j^(1)(s-1)
                
                
                ⊕
                 Ξ_j^(2)(s-1),  j=-s  (or  s-1),
                
                which is the space of zero modes of 
                □_s-1^+.
                
                In the case s<1/2, and after cohomological cancellations, these zero modes are the only modes that contribute
                to the one loop super-determinant.
                
                Following the very same steps described in the previous section, we focus on the
                solutions obtained for j_3=s-1 and j=-s  (or  s-1). In this
                case, the zero mode solutions reduce to
                
                    f_Δ(s-1),j_3=s-1^(1,2)=χ(  x-1)  ^s-1,
                
                from where it is straigthforward to compute the ζ-function.
                
                The constant χ is determined from the normalization condition
                
                |χ|^22π vol_S_1∫_-∞^0dx(2)  |x-1|  ^2(s-1)   
                =|χ|^22 π vol_S^1/-s+1/2=1.
                
                From the value of |χ|^2, we obtain the heat kernel at origin
                
                K(t;0,0)=1/2 π vol_S^1(-s+1/2)e^-t(ρ(u)+k)^2. 
                From (<ref>), we obtain the zeta function 
                ζ(z)       =vol_ℍ_2vol_S^1/2 π vol_S^1(-s+1/2
                )/((ρ(u)+k)^2)^z=-(-s+1/2)/((ρ(u)+k)^2)^z.
                
                
                
                
                
                
                
                
                
                Finally, we obtain the desired determinant
                _Ξ_j=-s(s-1)(  (u+i∂_t)^2+□_s-1^+)
                   =   _Ξ_j=s-1(s-1)(  (u+i∂_t)^2+□_s-1
                ^+) 
                    =   e^-ζ^'(0)=((ρ(u)+k)^2)^s-1/2
                   =   |(ρ(u)+k)|^2s-1.
                
                
                
                
                
                 §.§ Super-determinant
                
                In the computation presented in this section, we will assume j,j_3,s∈ℤ, the other cases can be worked out in complete analogy. The super-determinant in the case s>1/2 is
                    √(_Ξ(s)(O_B)_Ξ(s-1)((ρ(u)+k)^2+□_s-1^+))/_Ξ
                    (s)(O_B)     =√(_Ξ(s-1)((ρ
                    (u)+k)^2+□_s-1^+)/_Ξ(s)(O_B))
                         =√(
                    
                    
                    ∏_j=0^s-2
                    
                    
                    ∏_j_3=j+1^∞
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))  /
                    
                    
                    ∏_j=0^s-2
                    
                    
                    ∏_j_3=j+1^∞
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))  )
                       ×√(1/_Ξ_j=s-1(s)(O_B)),
                         =√(1/_Ξ_j=s-1(s)(O_B))=|(ρ(u)+k)|^s-1/2.
                
                
                
                Notice that in the RHS of the second line, we have a quotient of two identical infinite products. This cancellation, occurs due to the supersymmetric pairing of eigenmodes: cohomological cancellations.
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                Let us comment about the particular case s=1. In that case, there are not cohomological cancellations. The reason is, that when s=1, only the
                space of zero modes j=s-1 (or j=-s), for the scalar ϕ, and the
                “ chiral” spinor ψ^+ exist. In the
                case s=1, there is not “ anti-chiral”
                 square integrable mode ψ^- on ℍ_2, because for such spinors the
                effective flux is s-1=0.
                
                Next, let us compute the super-determinant in the case s<1/2
                
                    √(_Ξ(s)(O_B)_Ξ(s-1)
                    ((ρ(u)+k)^2+□_s-1^+))/_Ξ(s)(O_B)   
                    =√(_Ξ(s-1)((ρ(u)+k)^2+□_s-1^+)/_Ξ(s)(O_B))
                         =√(
                    
                    
                    ∏_j=0^-s-1
                    
                    
                    ∏_j_3=-∞^-j-1
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))  /
                    
                    
                    ∏_j=0^-s-1
                    
                    
                    ∏_j_3=-∞^-j-1
                    (  (ρ(u)+k)^2-j(j+1)+s(s-1))  )
                       ×√(_Ξ_j=-s(s-1)(  (u+i∂_t)^2
                    +□_s-1^+)  ),
                         =√(_Ξ_j=-s(s-1)(  (u+i∂_t)^2+□
                    _s-1^+)  )=|(ρ(u)+k)|^s-1/2.
                
                
                
                The final expression coincides with the one of the case s>1/2. However, in the case
                of s<1/2 the unpaired modes are the “
                anti-chiral” square integrable modes ψ^-, labeled by a
                radial number j=-s, and perceiving a flux s-1 on ℍ_2.
                
                Let us treat separately the case s=0. In that case,
                there are not square integrable “chiral” modes ψ^+, neither scalar
                ones ϕ. However, there exist “
                anti-chiral” square integrable modes ψ^-, perceiving
                an effective magnetic flux of -1 and labeled by radial number j=0. In this
                case the regularized super-determinant is given by
                
                    √(
                    
                    
                    ∏_j=0^0
                    _Ξ(-1)((ρ(u)+k)^2+□_-1^+))=|(ρ(u)+k)|^-1/2.
                
                
                
                Collecting partial results, not only for the case j,j_3,s ∈ℤ, but for the most general spectrum  j,j_3 and s=-ρ(𝔪)-q_R/2 obeying “discreteness" conditions (<ref>),
                is straigthforward to obtain the final result for the one loop super-determinant part of the index on ℍ_2
                    Z_1-loop^matter(ℍ_2,𝔪,q_R)=
                    
                    
                    ∏_ρ:s≠1/2[
                    
                     |C_regsin( ρ(u)/2)|]  ^-ρ(𝔪)+q_R-1/2,
                
                where C_reg=-2 i. 
                
                The restriction to  s ≠1/2, is a necessary
                condition to have square integrable modes. However, |(ρ(u)+k)|^s-1/2=1 and consequently  s≠1/2 can very well be ignored and the result for Z_1-loop^matter(ℍ_2,𝔪,q_R) will be the same.
                
                Interestingly, under GNO conditions 
                
                    s∈ℤ   or    s∈ℤ+1/2,
                
                the one loop result for the index on ℍ_2× S^1
                    Z_1-loop^matter(ℍ_2,𝔪,q_R)=
                    
                    
                    ∏_ρ
                    [ |C_regsin(  ρ(u)/2)|]  ^-ρ(𝔪)+q_R-1/2,
                
                coincides with the square root of the analog result on S^2× S^1, under the identification of s_ℍ_2 with
                s_𝕊^2, for each mode (ρ,k). 
                 
                 Namely, under GNO quantization conditions
                
                Z_1-loop^matter(ℍ_2,𝔪,q_R)   =   
                
                
                ∏_ρ
                [ |C_regsin( ρ(u)/2)|  ]^-ρ(𝔪)+q_R-1/2
                
                   =   √(Z_1-loop^matter(S^2,𝔪,q_R)). 
                
                Let us remark that we are not forced to impose GNO conditions on ℍ_2. Consequently, we shall not impose GNO quantization conditions 
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                The one loop determinant of a matter multiplet in the adjoint representation of the gauge group, with R-charge q_R=2 (which coincides with
                the vector multiplet super-determinant, see appendix <ref>) in the presence of flux is
                
                Z_1-loop^vector(ℍ_2,𝔪)   =   
                
                
                ∏_α
                [| C_regsin(  α(u)/2)| ] ^-α(𝔪)+1/2
                
                   ∼   [∏_α>0sin(  α(u)/2)^2 ]^1/2
                   =   √(Z_1-loop^vector(S^2,𝔪)). 
                
                
                
                
                Notice that the result for the vector multiplet (<ref>)– which is independent on 𝔪–  , matches with the result of <cit.> in their
                flat space limit L→∞ and under the transformation u → i u. Notice that their transformation of A_μ under ϵ, δ_ϵ A_μ matches with ours, if only if A_μ is substituted by i A_μ.
                
                
                We tried to obtain the cohomological cancellations from ζ-regularisation
                of the non zero modes. However, the heat kernel method for spinors of
                <cit.>, definitely, does not respect supersymmetry
                (in the case of the discrete spectrum when the absolute value of the total flux felt by the mode must be larger than 1/2) and breaks the cancellations between
                fermions and bosons, unless the normalization associated to the heat kernel of
                antichiral modes ψ^- is modified in a non elegant way.
                
                
                
                 §.§ What about the continuous spectrum?
                
                
                So far, we have focused on the contribution of the discrete spectrum to the index. It is time to find out what is the contribution of the continuous spectrum.
                The eigenvalues of the relevant differential operators when acting upon the  eigenfunctions of the continuous spectrum are
                
                O_B f_Δ_(λ,s),j_3   =   (  (ρ(u)+k)^2+λ^2+1/4+s(s-1)) f_Δ_(λ,s),j_3
                ( (ρ(u)+k)^2+□_s-1^+f_Δ_(λ,s-1),j_3)   =   (  (ρ(u)+k)^2+λ^2+1/4+s(s-1)) f_Δ_(λ,s-1),j_3. 
                
                
                The super-determinant to compute is
                √((O_B)
                ((ρ(u)+k)^2+□_s-1^+))/(O_B).  
                
                The determinants in (<ref>) are computed by the method of Heat Kernel. Once the f_Δ_(λ,s), j_3 are normalized
                as in (<ref>) the Heat Kernel of an operator  with eigenvalues E[ł,s] when acting upon f__(ł,s),j_3 is defined as
                
                K^(k,)[p,p',]=∫_0^∞ dł∑_j_3 ∈ℤ f^*__(ł,s),j_3(p) f__(ł,s),j_3(p') e^-E[ł,s] ,
                
                where p={θ,φ,t} and p' labels the set of coordinates of a given point.  
                
                We do not need the full heat kernel, since for the -function  all we need is its value at the origin p=p'=0.
                As in the case of square integrable modes, all the eigenmodes f__(ł,s),j_3 vanish at the origin, except for those with j_3=s. After some work, the spectral function μ^(s)(λ) is found to be
                1/vol_ℍ_2 vol_S^1μ^(s)(ł)   :=   ∑_j_3 ∈ℤ f^*__(ł,s),j_3(0) f__(ł,s),j_3(0)=f^*__(ł,s),s(0) f__(ł,s),s(0) 
                   =   1/(2π)^2λsinh (2 πλ )/cosh (2 πλ )+cos (2 π  s). 
                Having the spectral function in (<ref>) we are ready to compute the -functions by using the following definition
                ^(z;s)= ∫_0^∞dł^(s)(ł)/(E_[ł,s])^z.
                
                From the definition of -function (<ref>) we compute the relevant determinants
                =e^-∂_z (0,s). 
                
                Notice that, the spectral function obeys the following property
                ^(s)=^(s-1), .
                
                and the eigenvalues of the operators O_B and ((ρ(u)+k)^2+□_s-1^+) upon the respective eigenfunctions f_Δ_(λ,s),j_3 and f_Δ_(λ,s-1),j_3 are the same, see equations (<ref>). From the latter facts and after the definitions  (<ref>) and (<ref>) it follows the triviality of (<ref>)√((O_B)
                ((ρ(u)+k)^2+□_s-1^+))/(O_B)=1.  
                In conclusion, the continuous spectrum provides a trivial contribution to the topologically twisted index.
                
                
                
                
                
                 §.§ GNO condition?
                
                
                The gauge potential representative (<ref>) is singular at the contractible cycle θ=0. There are ways to solve this issue. One of them is to impose the holonomy of the gauge potential θ=0 to be in the centre of the group, we shall not resort to this way.   A second way is to simply perform a non trivial gauge transformation with parameter
                Ł(φ)= -s φ.
                
                The new potential
                
                A= s(coshθ -1 )d φ, 
                 is regular at θ=0 and has the same behavior at the boundary θ→∞ as (<ref>). In fact (<ref>) is the analytic continuation of the section on the north chart of the magnetic monopole bundle on S^2 to the single chart that covers ℍ_2.
                 
                  To appreciate the consequence of the non triviality of (<ref>) at θ=0, let us comment on its effect on matter. Out of the hierarchy of eigenmodes, 
                the only modes that do not vanish at the contractible cycle, are those with j_3=s, see equation (<ref>). When cycling around the contractible cycle, these modes exhibit an Aharonov-Bohm phase of
                
                2 π s, 
                 due to the non triviality of (<ref>) at θ=0. Should we impose the scalars to be periodic at θ=0, the phase (<ref>) must be an integer multiplet of 2π. In consequence, periodicity of scalars is not implying GNO quantization conditions. 
                  The GNO conditions <cit.> are
                 
                    α(𝔪)∈ℤ,
                
                where α is any element of the root lattice of the gauge group 𝒢. Condition (<ref>) states that
                𝔪 is in the co-root lattice of 𝒢.
                
                 If we particularize (<ref>) to ρ=α and q_R=0 then (<ref>) implies
                 
                 s ∈ℤ or ℤ+1/2. 
                As already explained, (<ref>) is consistent with square integrability. However, if s∈ℤ+1/2, the Aharonov-Bohm phase is an odd multiple of π and the scalar modes are multivalued at θ=0. Notice that with the smooth representative (<ref>) such issue is no longer present. In this representation the only modes that do not vanish at θ=0 are those with j_3=0. This is because by performing a gauge transformation to a regular gauge potential the j_3 gets substituted by j_3-s. In fact, in this smooth representation, the Aharonov-Bohm phase at any φ-cycle becomes an integer multiple of 2π in virtue of our quantization conditions (<ref>).
                
                 
                Conditions (<ref>) are less restrictive than GNO conditions as they include not just (<ref>), but a continuous family of flux configurations. Consequently, we must not restrict our zero locus BPS[Γ] by the GNO quantization conditions.  
                
                Notice that in the case of S^2, the monopole bundle consists of two charts. In that case, the GNO condition comes from imposing single-valuedness of the structure group transformation that relates the sections at north and south <cit.>. In the case of ℍ_2, there is not such a feature.
                
                
                
                
                
                
                 
                 We must say, however, that in our case, relaxing GNO conditions has consequences on global gauge invariance  —see for instance section 2.1 of <cit.>, to appreciate a related discussion— . Let us analyze the case of ABJM. In that case, indeed, under a large gauge transformation u→ u+2π and ũ→ũ+2 π, the Chern-Simons terms x^i k 𝔪_i/2 and x̃^-i k 𝔪̃_i/2, change by a phase of e^π i k 𝔪_i and e^- π i k 𝔪̃_i, respectively. Those phases can be absorbed by a couple of topological U(1)_T holonomies [These terms arise from a couple of mixed Chern-Simons terms of the form (<ref>). Specifically, from the coupling of the U(1)_L,R dynamical vector multiplets and Q_ϵ-spurion vector multiplets: U(1)_L-spurion_L,  U(1)_R- spurion_R . To obtain (<ref>) we have considered the following non trivial v.e.v's Â_L 3^T=π p and Â^T_R 3=-πp̃ for the L and R spurion multiplets. Allover our discussion, we will fix the spurion fluxes to zero 𝔱_L=𝔱̃_R=0.
                 ] of the form
                
                    ∏^N_i=1ξ _p(𝔪_i/2)    :=   e^π i  p  ∑^N_i=1𝔪_i, 
                    ∏^N_i=1ξ_p(𝔪_i/2)    :=   e^-π i p∑^N_i=1 
                    𝔪_i,
                
                after the change of labels
                
                p     →    p-k, 
                p̃   →   p̃-k. 
                Transformation (<ref>), is a symmetry of the measure ∑_p, 
                p̃∈ℤ, if k∈ℤ, and consequently, symmetry under large gauge transformations is restored, if we perform an average over p and p̃.  The one loop contributions do not spoil the previous procedure, because the determinants in the case of ABJM are invariant under u→ u+2π. 
                
                
                
                
                
                  
                Comment In this section, it was proven that if we discard the discrete modes, and consider only the continuous spectrum, the index is trivial. In contradistinction,  the index becomes non trivial when evaluated on square integrable eigenfunctions. The index is somehow encoding information about the tower of normalizable modes.
                 Indeed, its one loop contribution is determined by the zeta regularized number of zero modes <cit.> of the operators □_s^-— if s>1/2—  and □_s-1^+— if s<-1/2—.
                
                
                
                 
                
                
                
                
                
                 
                 
                
                 
                
                § THE ABJM INDEX ON ℍ_2× S^1
                
                
                It is time to analyze the ABJM index on ℍ_2× S^1. In this section, we borrow notation and strategy from section 2.1 of <cit.>. The final scope is to obtain the leading large-N behavior of the index, in terms of flavor fluxes and holonomies. In order to do that, we will show that the corresponding large-N Bethe Ansatz equations (BAE), are equivalent to the ones defined in <cit.>. In fact, the leading large-N solution presented in <cit.> will be a solution to our BAE, and consequently, can be used to evaluate the leading large-N result for the ABJM index on ℍ_2× S^1. 
                
                Let us start by writing down the localization formula for the ABJM index. The Chern-Simons plus boundary term contribution is
                
                    i=1N∏x_i^1/2k 𝔪_ix
                    _i^-1/2k 𝔪_i.
                
                
                After collecting classical and 1-loop contributions, we can write down the expression of the ABJM index on ℍ_2× S^1
                    Z_ℍ_2× S^1   :=     c ∑_p,p∈ℤ( 
                    1/N!) ^2∫_|x|=|x̃|=1∏_i=1^N
                    dx_i/2π ix_idx_i/2π ix_i∫^M_+_-M_- ∫^M̃_+_-M̃_- ∏_i=1^N d𝔪_i d𝔪̃_i 
                    
                          ×∏_i=1^Nx_i^k 𝔪_i/2x_i^-k
                    𝔪_i/2ξ _p(𝔪_i/2) ξ_p(𝔪_i/2)  i≠ jN∏√(( 1-x_i/x_j) ( 1-
                    x_i/x_j))
                     ×
                          ∏_i,j=1^N∏_a=1,2( ±| √(
                    x_i/x_jy_a)/1-x_i/x_jy_a|
                    ) ^𝔪_i-𝔪_j-𝔫_a+1/2∏_b=3,4(±| 
                    √(x_j/x_iy_b)/1-x_j/x_i
                    y_b| ) ^𝔪_j-𝔪_i-𝔫_b+1/2,
                
                where c:=1/∑_p, p̃∈ℤ1 and 𝔫_a are flavor fluxes. Notice that we have written back the sign degeneracy, mentioned below equation (<ref>). If we suppose
                
                N ∈ 2 ℕ, 
                the former signs and the absolute values between parenthesis, become spurious in the contour of integration to be defined below, and consequently we drop them from now on. As we are interested in the large-N limit, (<ref>) is enough to our purposes.  
                
                The integration over fluxes and eigenvalues is dictated by the localization principle: they are the zero locus BPS[Γ] associated to our contour of field-integration Γ and their values are not fixed by boundary conditions. The color holonomies x_i=e^i u_i, x̃_j=e^i ũ_̃j̃ are integrated along S^1 as follows from our reality conditions on Γ:  Im[u_i]=0 and 2π-periodicity of the integrand dependence on u and ũ.
                
                 The general idea is to pick up certain residues in the large-N limit. In order to find out the position of the relevant simple poles,  we will need to compute the large-N  solution to the very same Bethe ansatz equation of <cit.>. In <cit.> it was suggested that in the large-N limit, the set of Bethe ansatz eigenvalues, which is the set of simple poles enclosed by our contour, condense to a support included in the region [Even though the evidence presented in <cit.> is quite convincing, it would be nice to have a proof of the absence of extra eigenvalues outside the region 𝕏(Δ_a). We believe this is a point that deserves further understanding, but we will leave that analysis for future work.   ]𝕏(Δ_a): ={(ũ_j-u_i)∈ℂ: {0<-ℝe [ũ_j -u_i]+Δ_a<2π      if     a=1,2
                0<  ℝe[ũ_j -u_i]+Δ_a<2π      if     a=3,4}}. 
                The region 𝕏(Δ_a) is the union of the regions covered by angles of the (i,j)-complex planes with coordinate x̃_ji:=e^i (ũ_j -u_i). Each (i,j)-angle is defined as
                max( -Δ_3,-Δ_4, max(Δ_1,Δ_2)-2π)<(x̃_ji)<min(Δ_1,Δ_2 ,2π -max(Δ_3,Δ_4)).
                
                 To compute our residues we have two possibilities. Either we deform the (ij)-S^1: |x̃_ji|=1, to the perimeter of the inner region of the (i,j)-angle including the origin, or we deform it to the perimeter of the outer region, and close the contour at infinity x̃_ji=∞[By inner (resp. outer) region we mean the region of the (i,j)-angle that is in (resp. out) of the (i,j)-S_1.]. Both choices are equivalent, provided we include the respective “boundary contribution” at x̃_ji=0 or x̃_ji=∞, depending on the case. The “inner” choice corresponds to selecting poles with 𝕀m(ũ_j-u_i)>0. The “outer” choice corresponds to poles with 𝕀m(ũ_j-u_i)<0. Notice that for the large N solution of our interest, an outer (i,j)-pole implies the presence of an inner (j,i)-pole and vice versa (see equation (2.39) of <cit.>). In fact in the limit N→∞ all of such poles will condense either at 0 or ∞, except for the case i=j. In the particular case i=j, the poles condense to an arc of (i,i)-S^1. For each (i,j), we will choose to deform the (i,j)-S^1 contour in the way that encloses the poles at the “bulk” (these, are the poles whose positions are the eigenvalues in equation (2.39) of <cit.>). 
                  
                
                
                
                
                
                The projection of the domain 𝕏(Δ_a) upon our integration contour, which is obtained by demanding 𝕀m[u]=𝕀m[ũ]=0 on the former, is a parametrization of the maximally connected region without Coulomb branch singularities [In fact in integrating x_i over S_1 we should avoid colliding with such singularities, either by slightly deforming the contour, or by turning on an infinitesimal mass regulator in the matter one loop determinant.].
                Actually, the intersection of the boundary of the complex region 𝕏(Δ_a), ∂𝕏(Δ_a) with our contour of integration S^1, is a parametrization of the domain of such singularities. By a singularity of the Coulomb branch we mean a point (u_i,ũ_j) such that the quantity that defines the one loop contributions
                
                    ∏_a=1,2( √(
                    x_i/x_jy_a)/1-x_i/x_jy_a)^-1∏_b=3,4( 
                    √(x_j/x_iy_b)/1-x_j/x_i
                    y_b)   =   sin(-ũ_j+ u_i+Δ_1)/2sin(-ũ_j+u_i+Δ_2)/2/sin(ũ_j-u_i+Δ_3)/2sin(ũ_j-u_i+Δ_4)/2,
                
                becomes 0 or ∞.
                
                Notice that, as we are not imposing the GNO conditions, we have to integrate over the values of fluxes 𝔪_i and 𝔪̃_i along the Cartan directions. As we have already stated, the fluxes 𝔪 and 𝔪̃ are non-normalizable modes, even though they are in BPS[Γ]. In that respect, our approach is reminiscent of the one advocated in <cit.>. In <cit.>, integration over non-normalizable modes belonging to the zero locus of the relevant supercharge, was suggested for the localization on ℍ_2× S^1. Although the localization  performed there, was on the branched sphere, the integration over the Coulomb branch parameter completes the nice picture suggested by (4.15) of <cit.>[A second possibility we will not explore in this work, is to fix the values of non normalizable modes in Q_ϵ[Γ] to specific values. However, in order to match the final result to the supergravity dual, an extremization procedure should be engineered for those values. In some sense, integration over these non normalizable color - not flavor- modes is such sort of extremization. In this second, more open, line of thought, perhaps one could relax our reality condition on u_i and simply define the latter extremization as integration over the complex and more abstract Jeffrey-Kirwan (JK) contour. It would be quite remarkable  if with such an alternative approach, one obtains the same result for the index. In that case, the approach followed in this manuscript, would provide a less abstract viewpoint of the JK contour. We suspect this is indeed the case, but as it is not the final goal of our study, we shall not check so in this manuscript. ].
                
                
                We use a couple of very large cut-offs M_±,  M̃_±>0, because the volume of the moduli space of fluxes (𝔪, 𝔪̃) is infinite. After computing the integral over the holonomies for fixed values of M_± and M̃_±, we are free to send one and only one, of either M_+(resp. M̃_+) or M_-(resp. M̃_-), to infinity. The other one, remains as a regulator that we will redefine as M (resp. M̃). Thereafter, we pick up the residues of the analytical continuation of the regulated integrand, the final result will be independent on M and M̃, and we are free to take M,  M̃→∞ on such residues 
                 [The consequences of using this regularization procedure are somehow reminiscent of the consequences of applying the Jeffrey-Kirwan recipe in <cit.>.  ].
                
                Next, we shall show independence of the regulated expression on the cutoffs M and M̃, but before proceeding, let us remind a couple of conditions that we have implicitly used so far. The topological twisting condition is
                
                    ∑_a 𝔫_a=2.
                
                To understand how (<ref>) is the topological twisting condition, please, refer to section 2.1 of <cit.>.
                From conservation of flavor symmetry, it follows that
                
                    ∏_a y_a=1.
                
                For clarity, it is convenient to re-organize the RHS of (<ref>) as follows
                
                    c ∑_p,p∈ℤ( 1/N!) ∫i=1N∏
                    dx_i/2π ix_idx_i/2π ix_i
                    ∫_-M_-^M_+∫_-M̃_-^M̃_+i=1N
                    ∏d𝔪_id𝔪_i( i≠ jN∏√(( 1-x_i/
                    x_j) ( 1-x_i/x_j)) A
                    )
                           ×( i=1N∏exp[Υ _i(x,
                    x)𝔪_i]) ( j=1N∏exp[
                    Υ _j(x,x)𝔪_j]),
                
                where
                
                    Υ _i(x,x)    =   log( x_i^k e^2 π i p∏
                    _j=1^N√(x_i/x_jy_1x_i
                    /x_jy_2)/( 1-x_i/x_j
                    y_1) ( 1-x_i/x_jy_2) 
                    ( 1-x_j/x_iy_3) ( 1-
                    x_j/x_iy_4) /√(x_j/x_i
                    y_3x_j/x_iy_4))^1/2
                       =   log( x_i^ke^2 π i pNj=1∏
                    ( 1-y_3x_j/x_i) ( 1-y_4
                    x_j/x_i) /( 1-y_1^-1x_j
                    /x_i) ( 1-y_2^-1x_j/x_i) 
                    )^1/2
                       :=   1/2log( e^i B_i)
                
                and
                
                    Υ_j(x,x)=1/2log( e^i B
                    _j),
                
                with
                
                
                
                
                e^i B_j   :=   x_j^ke^2 π i p̃Ni=1∏
                . ( 1-y_3x_j/x_i) ( 1-y_4
                x_j/x_i) /( 1-y_1^-1
                x_j/x_i) ( 1-y_2^-1x_j/x_i
                ) . , 
                
                A   :=   ∏_i,j=1^N(∏_a=1,2( . √(
                x_i/x_jy_a)/1-x_i/x_jy_a
                .) ^-𝔫_a+1/2∏_b=3,4(.
                √(x_j/x_iy_b)/1-x_j/x_i
                y_b.) ^-𝔫_b+1/2).  
                Definitions (<ref>), (<ref>) and (<ref>) are the ones in equations (2.21) and (2.20) of <cit.>. 
                
                
                 After evaluating the integral over fluxes, we obtain
                 
                    Z_ℍ^2×S^1(𝔫,y,M)   =   c ∑_p,p∈ℤ( 1/N!)^2 ∫_|x|=|x̃|=1i=1
                    N∏dx_i/2π ix_idx_i/
                    2π ix_i( i≠ jN∏
                    √(( 1-x_i/x_j) ( 1-x_i/
                    x_j)) A)
                       ×   i=1N∏s_Υ_iexp[ s_Υ_iΥ _i(x,x) M ]
                    /1/2log( e^i B_i) ×j=1N∏s_Υ̃_jexp[s_Υ̃_j
                    Υ _j(x,x) M̃]/1/2log( e^iB_j) .
                
                where
                
                s_Υ_i:= sign ℝeΥ_i(x,x̃),    s_Υ̃_j:= sign ℝe Υ̃_j(x,x̃). 
                In (<ref>) we have already taken M_-  or  +→∞ and M̃_-  or  +→∞.
                
                 The next step is to evaluate the residues of the analytical continuation of the regulated integrand in (<ref>), at the simple poles enclosed by our contour.  Such poles are located at positions x_* and x̃_*, defined by the eigenvalues of the BAE
                
                 e^i
                 B_i(x_∗,x_∗)=1,    e^i
                B_i(x_∗,x_∗)=1. 
                 Solutions of (<ref>) are not onto our integration contour [This is because the imaginary part of both eigenvalues u and ũ is different from 0 (see equation (2.39) of <cit.>).], but are enclosed by it.
                In this way we solve the remaining integrals in (<ref>).
                 Notice that we are naively focusing on the contribution coming from simple poles generated by the BAE (<ref>). Next, we shall see how the solution to (<ref>) is independent on the topological holonomies p and p̃. The integrand of (<ref>) is also independent on p and p̃. Consequently, the average over p and p̃ will be trivial.
                
                The final result for the index is
                
                
                
                
                
                
                
                
                
                    Z_ℍ_2 × S^1(𝔫,y)=∏_a=1^4y_a^-
                    N^2n_a/4∑_I ∈ BAE 2^2N/𝔹(Ni=1∏x_*i^Nx_*i^N
                    
                    i≠ jN∏( 1-x_*i/x_*j) ( 1-
                    x_*i/x_*j) /i≠ jN
                    ∏∏_a=1^2( x_*j-y_ax_*i)
                    ^1-n_a∏_a=3^4( x_*i-y_ax_*j)
                    ^1-n_a)^1/2,
                
                where
                
                    𝔹:=∂( e^i B_j,e^iB_j) /
                    ∂( log x_l,logx_l) =
                    [ x_l∂ e^i B_j/∂ x_l  x_l
                    ∂ e^iB_j/∂x_l;  x_l∂ e^iB_j/∂ x_l   x_l∂ e^iB_j/∂x_l ]
                    _2N× 2N.
                
                Notice that the cut-off dependence disappeared in (<ref>) due to the BAE. 
                
                
                
                
                
                
                 
                 
                 
                 
                 
                 
                 
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                 We are interested in the large-N limit of Z_ℍ_2× S^1. In that limit,
                there is a unique BAE solution thence the summation over the label I becomes spurious. 
                
                It is convenient to define
                
                    D(z):=( 1-z y_3) ( 1-z y_4) /(
                    1-z y_1^-1) ( 1-z y_2^-1) .
                
                
                In terms of D(z), the LHS of the BAE are
                
                    e^i B_i(x_∗,x_∗)=x_i^k e^2 π pN
                    j=1∏. D( x_j/x_i).
                    ,  e^i B_j(x_∗,x_∗)=
                    x_j^ke^2πp̃Ni=1∏. D( 
                    x_j/x_i). .
                
                
                
                
                
                
                
                
                
                In terms of the quantity
                
                    G_ij:=. ∂log D/∂log z| _z=
                    x_j/x_i,
                
                
                
                
                
                
                the matrix 𝔹 takes the form
                
                    . 𝔹| _BAE=
                    [ δ _jl[k-Nm=1∑G_jm]               G_jl;              -G_lj δ _jl[k+Nm=1∑G_mj] ].
                
                The next step is to write down the BAE in “angular” coordinates u_i, ũ_i, and Δ_a, which are defined from
                
                    x_i=e^iu_i, x_j=e^iu_j, y_a=e^i
                    Δ _a.
                
                In these coordinates, the constraint ∏_ay_a=1 looks like ∑_aΔ _a=0  (
                mod 2π ). 
                
                In “angular” coordinates, the BAE (<ref>) are
                
                    0    =   ku_i+i∑_j=1^N[
                    ∑_a=3,4Li_1( e^i( u_j-u_i+Δ
                    _a) ) -∑_a=1,2Li_1( e^i( u
                    _j-u_i-Δ _a) ) ] -2π(n_i -p ), 
                    
                    0    =   k
                    u_j+i∑_j=1^N[ ∑_a=3,4Li_1( e^i( u
                    _j-u_i+Δ _a) ) -∑_a=1,2Li_1( e^i( 
                    u_j-u_i-Δ _a) ) ] -2π( 
                    n_j-p̃),
                
                with n_i,  ñ_j ∈ℤ.  Equations (<ref>) are the same BAE given in (2.32) of <cit.>.
                
                To fix the values of n_i and ñ_j, we use the identity
                
                    Li_1( e^iu) -Li_1( e^-iu) =-iu+iπ
                
                together with the assumption of absence of “long range interaction”<cit.>. The latter condition, implies
                
                    2π n_i   =   2 π p +( ∑_aΔ _a-4π)
                    ∑_jΘ(𝕀m( u_i-u_j) ), 
                    
                     2πn_i   =   2 πp̃+( ∑_aΔ
                    _a-4π) ∑_iΘ(𝕀m( u_i-u
                    _j) ).
                
                We can solve the constraints (<ref>) and (<ref>), for n_i and ñ_j, with the choice
                
                    ∑_aΔ _a=2π .
                
                It is key to observe that (<ref>) and (<ref>), imply that solutions to the BAE are independent of the topological holonomies p, p̃. Additionally, (<ref>) is also independent of p and p̃ and we conclude that the average on p and p̃ can be substituted by one. 
                
                The solution of (<ref>), obeying (<ref>), is precisely the one used in <cit.>, to obtain their quite nice result, (2.89), in the large-N limit. In the next section, for completeness, we evaluate the aforementioned solution. We have tried other choices, such as ∑_a Δ_a=0. However, as pointed out in <cit.>, there are always some issues with the potential solutions. 
                
                
                
                 §.§ Large-N behavior of the index
                 
                
                From now on, we take the limit N→∞, assume the Chern-Simons level k=1,  introduce the density of eigenvalues ρ (t)=1/N
                ∑_i=1^Nδ (u_i-t) and the quantity δ v(t), as precisely done in
                section 2.3 of <cit.>. In this continuous limit, the BAE arise from the
                variations of the auxiliary Lagrangian
                
                    𝒱/iN^3/2   =   ∫ dt[ tρ (t)δ v(t)+ρ
                    (t)^2( ∑_a=3,4g_+(δ v(t)+Δ
                    _a)-∑_a=1,2g_-(δ v(t)-Δ _a)) ]  
                          -μ[ ∫ dtρ (t)-1] -i/N^1/2∫ dtρ (t)
                    [ ∑_a=3,4Li_2( e^i( δ v(t)+Δ
                    _a) ) -∑_a=1,2Li_2( e^i( δ
                    v(t)-Δ _a) ) ] ,
                
                where
                
                g_±(u):=u^3/6∓π/2 u^2+π^2/3u.
                
                It is easy to follow the steps in <cit.>. Indeed
                for ∑_aΔ _a=2π and under the assumptions 
                μ >0, ∃t̃:δ v(t̃)=0, Δ _1<Δ _2 < Δ _3<Δ _4, 
                together with (<ref>)
                0<-δ v(t)+Δ_a<2π      if        a=1,2, 
                
                0<  δ v(t)+Δ_a<2π      if        a=3,4,
                
                the large-N relevant part of the  solution to the continuous limit of the BAE, coming from
                Lagrangian (<ref>) is
                
                
                    ρ (t):={[                         -μ +Δ_3 t/(Δ_1+Δ_3) (Δ_2+Δ_3) (Δ_3-Δ_4)                                                              if                                                       t_0<t<t_1; 2 πμ +t (Δ_3Δ_4-Δ_1Δ_2)/(Δ_1+Δ_3) (Δ_1+Δ_4) (Δ_2+Δ_3) (Δ_2+Δ_4)                                                              if                                                       t_1<t<t_2;                            Δ_1 t-μ/(Δ_1-Δ_2) (Δ1+Δ_3) (Δ_1+Δ_4)                                                              if                                                       t_2<t<t_3 ]
                    . .
                
                    δ v(t):={[                                                                           -Δ_3 + e^-N^1/2 Y_3(t)                                                                                               if                                                                                        t_0<t<t_1; μ  (Δ_1Δ_2-Δ_3Δ_4)+t
                       (Δ_1Δ_2Δ_3+Δ_1Δ_2Δ_4+Δ_1Δ_3Δ_4+Δ_2Δ_3Δ_4)/2 πμ
                       +Δ_1Δ_2 (-t)+Δ_3Δ_4  t                                                                                               if                                                                                        t_1<t<t_2;                                                                              Δ_1-e^-N^1/2 Y_1(t)                                                                                               if                                                                                        t_2<t<t_3 ]
                    . .
                
                    Y_1(t)    =   {[ (Δ_1+Δ_4) (μ +Δ_3 t)/(Δ_2+Δ_3) (Δ_3-Δ_4)+t                                         if                                  t_0<t<t_1;                                          0                                         if                                  t_1<t<t_2;                           μ -Δ_2 t/Δ_1-Δ_2                                         if                                  t_2<t<t_3 ]
                    .  
                    
                    Y_2(t)   =   {[                        (Δ_2+Δ_4) (μ +Δ_3 t)/(Δ_1+Δ_3) (Δ_3-Δ_4)+t                                                                if                                                         t_0<t<t_1;                                                                 0                                                                if                                                         t_1<t<t_2; t-(Δ_2+Δ_3) (Δ_2+Δ_4)
                       (Δ_1 t-μ )/(Δ_1-Δ_2) (Δ_1+Δ_3) (Δ_1+Δ_4)                                                                if                                                         t_2<t<t_3 ]
                    . 
                    
                    Y_3(t)    =   {[                           μ +Δ_4 t/Δ_3-Δ_4                                         if                                  t_0<t<t_1;                                          0                                         if                                  t_1<t<t_2; -(Δ_2+Δ_3) (Δ_1 t-μ )/(Δ1-Δ_2) (Δ_1+Δ_4)-t                                         if                                  t_2<t<t_3 ]
                    . 
                    
                    Y_4(t)    =   {[ (Δ_1+Δ_4) (Δ_2+Δ_4) (μ
                       +Δ_3 t)/(Δ_1+Δ_3) (Δ_2+Δ_3) (Δ_3-Δ_4)-t                                                                if                                                         t_0<t<t_1;                                                                 0                                                                if                                                         t_1<t<t_2;                       -(Δ_2+Δ_4) (Δ_1 t-μ )/(Δ_1-Δ_2) (Δ_1+Δ_3)-t                                                                if                                                         t_2<t<t_3 ]
                    . ,
                
                with
                
                    t_0=-μ/Δ_3, t_1=-μ/Δ_4 ,t_2=μ/Δ_2,   t_3= μ/Δ_1 .
                
                From (<ref>) it follows the ordering of transition times
                
                
                    t_0<t_1<t_2<t_3, ρ >0.
                
                
                From the normalization condition ∫_t_0^t_3dtρ (t)=1 it follows that
                
                    μ =√(2Δ _1Δ _2Δ _3Δ _4).
                
                
                To obtain the leading free energy in the limit N →∞,
                one evaluates (<ref>) at the BAE solution (<ref>). The final result can be easily inferred, given the fact that our BAE
                solution is the same one found in <cit.>, for the case k=1,  ∑_aΔ _a=2π . In the latter case, and from the fact that the
                summand in (<ref>) is the square root 
                of the one in eq
                (2.24) of <cit.>, it results that
                
                    ℝelog. Z_ℍ_2×S^1| _Large N BAE
                    solution=1/2ℝelog. Z_S^2×S^1| _
                    Large N BAE solution+ sub.  terms
                
                 Finally, one can arrive to the result
                
                    ℝelog Z_ℍ_2× S^1^k=1   =   -F_ℍ_2×
                    S^1^k=1(𝔫,Δ )
                       =   -( 1/2) ×N^3/2
                    /3√(2 Δ _1Δ _2Δ _3Δ _4)∑
                    _a=1^4n_a/Δ _a+sub. terms. 
                     ∑_aΔ _a   =   2π .
                
                
                
                
                
                After extremizing with respect to Δ_1, Δ_2 and Δ_3, we obtain the following relation between fluxes and holonomies  
                
                    𝔫_1   =   Δ_1 (Δ_1-π )/Δ_1^2+Δ_1
                       (Δ_2+Δ_3)-2 π  (Δ_1+Δ_2+Δ_3)+Δ_2^2+Δ_2Δ_3+Δ_3^2+π ^2, 
                    𝔫_2    =   Δ_2 (Δ_2-π )/Δ_1^2+Δ_1
                       (Δ_2+Δ_3)-2 π  (Δ_1+Δ_2+Δ_3)+Δ_2^2+Δ_2Δ_3+Δ_3^2+π ^2, 
                    𝔫_3   =   Δ_3 (Δ_3-π )/Δ_1^2+Δ_1
                       (Δ_2+Δ_3)-2 π  (Δ_1+Δ_2+Δ_3)+Δ_2^2+Δ_2Δ_3+Δ_3^2+π ^2 ,
                
                that will prove to be useful later on, when comparing with the conjectured AdS/CFT dual quantity.
                
                
                
                
                
                 §.§ Comments on the index
                 
                
                The index computed in this work, which follows closely <cit.>, has the canonical interpretation of a Witten index counting ground states according to Z(𝔫_a, Δ_a)= Tr (-1)^F e^-β H e^iJ_a Δ_a. From the 3d perspective, we are simply counting operators with the corresponding relation among  quantum numbers. Now, assuming that the deformed ABJM theory flows to an effective quantum mechanics in the IR, then the index computes the degeneracy of ground states in  the quantum mechanics. Since the index is an invariant of the flow, we connect directly the 3d and 1d perspective. 
                
                On the gravity side, we have, similarly, the possibility of viewing the counting from the 4d or 2d perspectives. The better formulated one, at the moment, turns out to be the 2d perspective, which Sen has developed in the framework of AdS_2/CFT_1<cit.>. In this context, the ground state degeneracy in the quantum mechanics is computed by an AdS_2 partition function  with specific boundary conditions, which leads precisely to the quantum black hole entropy. There are, of course, some open issues with the application of Sen's proposal, in the context of asymptotically AdS black holes, but it certainly provides a solid starting point. 
                
                
                
                
                
                
                § THE HYPERBOLIC ADS_4 BLACK HOLE
                
                
                
                
                In this section we construct what we believe are the holographic dual to the ABJM configuration discussed thus far. Namely, we construct magnetically charged, asymptotically AdS_4 black holes with non-compact ℍ_2 horizon that are embedded in M-theory. Our construction follows similar spherical solutions in N=2 gauged supergravity, see, for example, <cit.>. We will comment on some similarities and differences with the solutions with spherical horizon in subsection <ref>. We shall focus on the case of n_V=3 vector multiplets. In this way the n_V+1 vector fields– counting also the one in the graviphoton vector multiplet – are set to be identified as dual to the global charges of ABJM. 
                
                
                
                
                 §.§ A brief summary of 4d 𝒩=2 gauged SUGRA with n_V=3
                
                
                For completeness, let us briefly introduce the concepts that we shall use. The central object is the pre-potential
                ℱ= ℱ(X^),
                
                which is a holomorphic function of the holomorphic sections X^(z^i), =1,2,3  and 4. The symplectic sections are functions of the physical scalars z^i with i=1,2,3. 
                
                Another important object is the Kähler potential 
                𝒦=- logi (X̅^Λℱ_Λ-X^Λℱ̅_Λ),    ℱ_Λ:=∂ℱ/∂
                X^Λ.
                
                 We will also need to use the period matrix
                
                    N_ΛΣ:=ℱ_ΛΣ+2i𝕀m(
                    ℱ_ΛΓ) X^Γ𝕀m( ℱ_ΣΔ) X^Δ/X^Γ𝕀m( ℱ_ΓΔ)
                    X^Δ,   ℱ_ΓΔ:=∂ℱ_Γ/∂
                    X^Δ.
                
                and the following auxiliary variables
                
                    ( L^Λ,M_Λ) :=e^K/2( X^Λ,ℱ_Λ) ,  ( f_i^Λ,h_Λ ,i)
                    :=e^K/2( D_iX^Λ,D_iℱ_Λ),
                
                where the covariant derivative D_i is defined as D_i:=∂ _z^i+K_i.
                
                In our case we will be interested in real holomorphic sections
                X̅^=X^,   z̅^i=z^i.
                
                
                
                To construct black holes, we shall set the fermions, and fermionic variations, to zero. The supersymmetry variation of the gravitino is
                
                    δψ _μ A:=∇ _με _A+2iF_μν^-ΛI_ΛΣL^Λγ^νϵ
                    _ABε ^B-g/2σ _AB^3ξ _ΛL^Λγ _με ^B,
                
                where the covariant derivative of the Killing spinor ε_A is
                
                    ∇ _με _A=( ∂ _μ-1/4ω
                    _μ^abγ _ab) ε _A+1/4(
                    K_i∂ z^i-K_i∂z^i
                    ) ε _A+i/2gξ _ΛA_μ^Λσ _A^3 Bε _B.
                
                The supersymmetry variation of the gaugino 
                
                    δλ ^iA=i∂ _μz^iγ ^με
                    ^A-g^i jf_j^ΛI_ΛΣF_μν^Σ -γ ^μνϵ ^ABε _B+igg^i
                    jf_j^Λξ _Λσ
                    ^3,ABε _B,
                
                will be used too. In equations (<ref>) and (<ref>) we are discarding higher order terms in fermions. These terms are not relevant to our discussion.
                
                To reproduce our results it will be useful to have the following definitions <cit.>
                    F_Λ^-   :=   1/2( F_Λ-i∗ F_Λ) , 
                    ∗ F_Λ_μν   :=   1/2ϵ _μναβF^αβ,
                
                
                    1+γ _5/2ε _A=1-γ _5/2ε
                    ^A=0.
                
                
                 §.§ Hyperbolic black holes
                
                
                
                To avoid confusion, the index Λ={1,2,3,4} is equivalent to the index a={1,2,3,4} that will be introduced in the next subsection.
                
                We are interested in the STU model. Thence, we fix the Fayet-Iliopoulus parameters in an isotropic manner 
                
                    ξ_0=ξ_1=ξ_2=ξ_3=ξ_V.
                
                The relevant pre-potential will be
                
                    ℱ(X)=-2 i √(X_1 X_2 X_3 X_4 ).
                
                We consider real sections, with the following parametrization
                
                X^Λ=X̅^Λ={-z^1/z^1+z^2+z^3+3,-z^2/z^1+z^2+z^3+3,-z^3/z^1+z^2+z^3+3
                   ,-1/z^1+z^2+z^3+3},
                
                and propose the following static, spherically symmetric ansatz, for the metric and sections
                
                    ds^2   =   -U^-2(r)dr^2-h^2(r)dθ ^2-h^2(r)sinh ^2θ dφ
                    ^2+U^2(r)dt^2,  
                    
                    X^Λ   =   α+ β_Ł/r.
                
                The non trivial components of the spin connection are [In this section we used different conventions than in section <ref>. We have used standard conventions on four-dimensional 𝒩=2 gauged supergravity, which are the ones given in <cit.> (see also <cit.>). For instance, the definition of spin connection is minus the one used in section <ref>. Consequently, in the covariant derivatives there is a relative minus sign in front of the term proportional to the spin-connection between this section and section 2. ]
                    ω _t^14   =   -U(r)U^'(r), ω _θ^12=-U(r)h^'(r), 
                    ω _φ^13   =   -U(r)h^'(r)sinhθ , ω _φ^23=-coshθ .
                
                In this section
                
                    ϵ _4123   =   1, 
                    η _ab   =   (-1,-1,-1,1), 
                    γ _5   =   iγ ^4γ^1γ ^2γ ^3.
                
                We use the following Ansätze for the functions U(r) and h(r)
                    U(r)    :=   e^K/2(g r-c/2g r),
                    
                    h(r)    :=   d e^-K/2r,
                
                where g, c and d are constants. 
                
                The corresponding black holes, are sourced by magnetic fluxes p_Ł:
                
                    A_Λφ=-p_Λcoshθ , F_Λθφ=-
                    p_Λ/2sinhθ .
                
                The non trivial components of the anti-selfdual field strength are
                
                    F_Λ _θφ^-=-F_Λ _θφ^-=-
                    p_Λ/4sinhθ ,  F_Λ _rt^-=-F_Λ
                    _tr^-=ip_Λ/4h^2(r)·
                
                The chiral and anti-chiral Killing spinors ϵ_A and ϵ^B, have to obey the following relation - these conditions are obtained from the vanishing of the gravitino supersymmetric transformation-
                
                    ε _A=ϵ _ABγ ^4ε ^B, ε
                    _A=±σ _AB^3γ ^1ε ^B.
                
                The most general solution to (<ref>) is
                
                    ε _A=
                    [    ±ϰ (r)  ∓ i ϰ(r); ± i ϰ (r)     ±ϰ(r);  -i ϰ (r)      ϰ(r);    -ϰ (r)   -i ϰ(r) ]
                    , ε ^B=
                    [     ϰ(r)   iϰ (r);   -iϰ(r)    ϰ (r);  ± iϰ(r)   ±ϰ (r);    ∓ϰ(r) ± iϰ (r) ].
                
                Solving the BPS conditions, leads to relations 
                
                    α   =   ∓1/4ξ _V, 
                    
                    c    =   1+8  d^2g^2ξ _V^2( β _4^2+β
                    _1^2+β _2^2+β _3^2) /d^2, 
                    
                    0    =   β _4+β _1+β _2+β _3,
                    
                    1    =   ± gξ _V( p_4+p_1+p_2+p_3).
                
                Notice that the constant c is positive.
                The relation between fluxes and the parameters β_a is also obtained from the BPS conditions
                
                    p_1   =   ±1+16 d^2g^2ξ _V^2 ( -β _1^2+β
                    _2^2+β _3^2+β _1β _2+β _2β _3+β
                    _1β _3) /4gξ _V,
                    
                    p_2   =   ± 1+16 d^2g^2ξ _V^2 ( +β _1^2-β
                    _2^2+β _3^2+β _1β _2+β _2β _3+β
                    _1β _3) /4gξ _V, 
                    
                    p_3   =   ±1+16 d^2g^2ξ _V^2 ( +β _1^2+β
                    _2^2-β _3^2+β _1β _2+β _2β _3+β
                    _1β _3) /4gξ _V.
                
                The warping of the Killing spinor is also fixed by the BPS conditions
                
                    ϰ =ϰ _0√(U(r)), ϰ=
                    ϰ_0√(U(r)).
                
                We have, finally, completely solved the BPS conditions and constructed our hyperbolic AdS_4 black holes.
                
                
                
                
                 §.§ Spherical black holes
                
                
                A prevalent intution in the context of supergravity states that changing the horizon from spherical to hyperbolic, leads  from black holes to naked singularities and  vice versa<cit.>.  In this section we explore the details of this intuition in the context of the magnetically charged black holes. 
                
                Let us first solve the BPS equations for the spherical black hole ansatz
                
                    ds^2   =   -U^-2(r)dr^2-h^2(r)dθ ^2-h^2(r)sin^2θ dφ
                    ^2+U^2(r)dt^2,  
                    
                    X^Λ   =   α+ β_Ł/r.
                
                with U(r) and h(r) defined in (<ref>) and (<ref>).
                
                The non vanishing components of the spin connection are
                
                    ω _t^14   =   -U(r)U^'(r), ω _θ^12=-U(r)h^'(r), 
                    ω _φ^13   =   -U(r)h^'(r)sinθ , ω _φ^23=-cosθ .
                
                
                For technical convenience, let us parametrize the gauge potential as follows [The definition of field strength used in this section differs from the one used in section <ref>.]
                    A_Λφ=-p_Λcosθ , F_Λθφ=
                    p_Λ/2sinθ .
                
                The non trivial components of the antiselfdual potential are
                
                    F_Λ _θφ^-=-F_Λ _θφ^-=
                    p_Λ/4sinθ ,  F_Λ _rt^-=-F_Λ
                    _tr^-=ip_Λ/4h^2(r).
                
                
                After solving the BPS equations, we arrive to
                
                    α   =   ∓1/4ξ _V, 
                    
                    c    =   -1+8  d^2g^2ξ _V^2( β _4^2+β
                    _1^2+β _2^2+β _3^2) /d^2,  
                    
                    0    =   β _4+β _1+β _2+β _3, 
                    
                    -1    =   ± gξ _V( p_4+p_1+p_2+p_3).
                
                and 
                
                    p_1   =   ∓ 1+16 d^2g^2ξ _V^2 ( -β _1^2+β
                    _2^2+β _3^2+β _1β _2+β _2β _3+β
                    _1β _3) /4gξ _V,
                    
                    p_2   =   ∓ 1+16 d^2g^2ξ _V^2 ( +β _1^2-β
                    _2^2+β _3^2+β _1β _2+β _2β _3+β
                    _1β _3) /4gξ _V, 
                    
                    p_3   =   ∓ 1+16 d^2g^2ξ _V^2 ( +β _1^2+β
                    _2^2-β _3^2+β _1β _2+β _2β _3+β
                    _1β _3) /4gξ _V.
                
                The warping of Killing spinor is also fixed by the BPS conditions
                
                    ϰ =ϰ _0√(U(r)), ϰ=
                    ϰ_0√(U(r)).
                
                In contradistinction to the hyperbolic solution, in the spherical case, c can be negative, see equation (<ref>). 
                
                The position of the curvature singularity is
                
                r_s= {-1/αmax{β_1, β_2 ,β_3 ,β_4 }>0    if    α<0
                
                -1/αmin{β_1, β_2 ,β_3 ,β_4 }>0    if    α>0. .
                
                If β_a=0, the position of the curvature singularity is r_s=0. In the case β_a=0, one encounters a hyperbolic black hole since c>0 and there is a horizon. However for the spherical solutions, c<0, and we encounter a naked singularity.  It is straightforward to check that, when β_a=0, the change 
                (r,  θ,  t,  p_a,  c ) →(i r,   i θ,  i t,  -p_a,  -c)
                
                transforms the hyperbolic black hole BPS solutions of the previous subsection, into the spherical BPS solution of this subsection, particularized to β_a=0. The latter has a naked singularity. Actually, the exchange p_a → -p_a can be cancelled by an exchange of Killing spinor — only because β_a=0—. By an exchange of Killing spinor, we mean to change the choice of sign in the constraint (<ref>). Such a change, has physical meaning, as it leads to a configuration that is BPS with respect to a different supercharge.
                
                We  emphasize that the intuition emanating from <cit.>, is restricted to the case of β_a=0, that is, the case of constant sections; it is, therefore, relaxed in the case β_a≠ 0.
                
                
                
                
                
                 §.§ The Bekenstein-Hawking entropy: ℍ_2 vs S^2
                
                In this subsection, we compare the entropy of hyperbolic  and spherical black holes. We first consider the case
                of isotropic fluxes. In the end, we will find that the entropy density of the hyperbolic solution coincides with the entropy density of the spherical one which were discussed in <cit.>. 
                
                
                 
                
                From now on, we particularize our hyperbolic solutions to the following case
                 α =-1/4, ξ _V=1, g=1/√(2). 
                
                
                A sufficient – not necessary – condition for the existence of hyperbolic AdS_4 black holes is
                
                    β_1,β_2, β_3>0,   r_h=√(c)> r_s=  4 max(β_1,β _2,β_3),
                
                where the domain of the radial coordinate r is r > r_s. The constant r_s is the radial position of the singularity, which is covered by the horizon at r_h=√(c).
                
                A particular solution to these conditions is
                
                    β _1= β _2= β _3=β >0.
                
                In that case, the fluxes are
                
                    p_1=p_2=p_3=p=1+32 d^2 β ^2/2√(2)>0.
                
                
                The regularized area density of the horizon is
                
                    A_ℍ_2(β)/vol_ℍ_2   =   1/2√(1+512 d^3β ^3(-6 d β +√(
                    1+48 d^2β ^2)))
                       =   √(-3 (1-2 √(2) p)^2+2 (2 √(2) p-1)^3/2√(6 √(2) p-1)+1)/2
                    
                    A_ℍ_2(p)/vol_ℍ_2   p→ +∞∼   √((4√(3)-
                    6))  p,          p>0.
                 
                
                 Next, we compare the entropy density (<ref>) to the one of spherical black holes used in <cit.>. The isotropic solution presented in section 4.1 of <cit.> is
                
                    𝔫_1=𝔫_2=𝔫_3=√(2)p^' ,  𝔫_4=2-3√(2)p^',    p^'<0.
                
                From the following quantities <cit.>
                    F_2(p)    :   =-( 12p^'  2-6√(2)p^' +1) ,  
                    Θ (p)    :   =192 p^'  4-160√(2)p^'  3+96p^'  2-12
                    √(2)p^' +1,
                
                one arrives to the following expression for the area density, of the spherical horizon in terms of the flux p^'
                    A_S^2(p^')/vol_S^2   :=   √(F_2+√(Θ))/√(2)
                       =   √(√(4 p^'(8 p^'(6 p^'  2-5 √(2) p^'+3)-3
                       √(2))+1)+6 (√(2)-2 p^') p^'-1)/√(2)
                    
                    A_S^2(p^')/vol_S^2   p^'→ -∞∼   -√((4√(3)-
                    6))  p^',             p^'<0.
                
                
                We can do the comparison with the result obtained from scratch, with our spherical solutions, however, in order to match our results with the ones in <cit.>, we report the comparison by using theirs. Notice that the large flux limits (<ref>) and (<ref>), do coincide. In fact, it can be checked that for any value of u the entropy density as a function of flux p and p^' coincide, as
                
                    A_ℍ_2(|u|)/vol_ℍ_2=A_S^2(|u|)/vol_S^2.
                
                Equation (<ref>) can be checked by comparing (<ref>) and (<ref>), order by order in Taylor expansions in u about 0 and ∞, or by simply working out the expressions. 
                
                
                
                
                 §.§ Matching results
                
                
                In this final section, we compare  the AdS/CFT dual results. On one side, we have the result for the ABJM index on ℍ_2× S^1(<ref>). On the other side, we have the entropy of the hyperbolic magnetic AdS_4 black holes (<ref>). The first thing to do, is to compute the Bekenstein-Hawking entropy of (<ref>). Thereafter, we check the relation between the classical entropy and the value of the holomorphic  sections X_a– or as were denoted in the previous subsection X^Ł– at the horizon r_h. The aforementioned relation is identical, up to a relabelling of variables, to the relation between the logarithm of the ABJM index and the holonomies Δ_a(<ref>). Finally, we prove that under the appropriate relabeling of variables and extremization of the logarithm of the ABJM index  (<ref>) with respect to the holonomies Δ_a, the bulk and SCFT results coincide, as was the case in <cit.>.
                
                 For generic p_1, p_2 and p_3, we have checked that the classical entropy
                
                    S_BH   =   A_ℍ_2/4 G_4d
                       =   -π/4 G_4 D√((Ψ -4 d β_1) (Ψ -4 d β_2) (Ψ -4 d
                       β_3) (4 d (β_1+β_2+β_3)+Ψ )),
                
                where
                
                    Ψ(β_1,β_2,β_3):=√(8 d^2 (β_1^2+β_1β_2+β_1β_3+β_2β_3+β_2^2+β_3^2)+1),
                
                coincides with the expression
                
                    +2 π/4G_4d√(X_1(r_h) X_2(r_h) X_3(r_h) X_4(r_h))∑_a=1^4 √(2) p_a /X_a(r_h),
                
                where
                
                    X_a(r_h)=d  β _a/√(4  d^2 (β_1^2+β_2^2+β_3^2+β_4^2)+1)-1/4.
                
                The value of β_4 is determined by (<ref>). The p_a's as function of β_a's have been given in equation (<ref>) which follows from the BPS equations. 
                 
                We will prove next, that equation (<ref>)– that comes from the analysis in the bulk–  is equal to the extremal value of the SCFT topologically twisted index (<ref>), under the specific dictionary
                √(2) p_a    ↔   𝔫_a,  
                
                -2π X_a(r_h)   ↔   Δ̅_a,    a=1,2,3,4,
                
                where the  Δ̅_a are the solutions to the variables Δ_a that come out of the inversion of equation (<ref>).
                
                There are many ways to prove that (<ref>) is equivalent to the extreme value of (<ref>), the simplest one is to evaluate (<ref>) on 
                
                    Δ_a=Δ_a(β_1,β_2,β_3)=-2π X_a(r_h),
                
                to obtain
                
                    𝔫_1=1/2+4 d^2 (-β_1^2+β_2^2+β_3^2+β_1β_2+β_1β_3+β_2β_3), 
                    𝔫_2=1/2+4 d^2 (+β_1^2-β_2^2+β_3^2+β_1β_2+β_1β_3+β_2β_3), 
                    𝔫_3=1/2+4 d^2 (+β_1^2+β_2^2-β_3^2+β_1β_2+β_1β_3+β_2β_3).
                
                Notice that
                
                    Equation (<ref>)= √(2) Equation (<ref>),
                 
                under (<ref>) and identification (<ref>). 
                
                The relation (<ref>) implies that the positions Δ̅_a of the saddle points of (<ref>) coincide with the values of the  sections X_a at the horizon (<ref>), under  identification (<ref>). As the logarithm of the ABJM index (<ref>) and the Bekenstein-Hawking entropy (<ref>) are the same under identification (<ref>) and 
                1/G_4d= 2 √(2)/3 N^3/2,
                
                we have thence proven that under the aforementioned identifications, boundary “degeneracy of states” and bulk black hole entropy coincide.
                
                As a final comment, we notice that the identifications (<ref>) are not directly obtained from the AdS/CFT dictionary. The AdS/CFT dictionary is naturally formulated in the UV, the UV value of the holomorphic sections X_a is -1/4. To obtain agreement, the use of extremization principle of the result of the SCFT side  is crucial <cit.>. We believe there is a proper way to clarify some of these  ad hoc issues but we leave the discussion  for future work. 
                
                
                
                
                § CONCLUSIONS
                
                
                
                In this manuscript we have first studied topologically twisted localization of N=2 supersymmetric field theories in ℍ_2× S^1. Our work differs in various important points from  recent work on localization on this space by  <cit.>. In particular, we have crucially considered topologically twisted theories and extended the type of theories under consideration beyond vector multiplets, to include, for example, matter multiplets. 
                
                
                
                At a technical level, we have also discussed explicitly subtle aspects of the eigenvalue problem corresponding to the Laplacian in the presence of a background magnetic field and we expect that such results could have wide application in the general context of localization. Quite interestingly, we have found a hierarchy of normalizable modes and its corresponding discrete spectrum. A particular sub-family of the aforementioned hierarchy,  corresponds to the vector zero modes of the Laplace-Beltrami operator on ℍ_2, that were introduced in <cit.> and figure prominently in, for example, <cit.> and more recently in <cit.>. The full hierarchy of normalizable modes exists due to the presence of magnetic fluxes s over a specific threshold:  |s|>1/2. We strongly suspect, that the discrete spectrum is encoding the full hierarchy of higher spin normalizable modes of the Laplace-Beltrami operator on ℍ_2.  If this is indeed the case, it would be very interesting to pursue a study of the potential traces of  2d higher spin symmetry, on the set of black holes microstates <cit.>.  As a first step toward formulating  such a problem, it would be useful to start by identifying the square integrable modes in the language of <cit.>. 
                
                
                
                
                
                We have also studied  N=2 gauged supergravity and found magnetically charged supersymmetric solutions with hyperbolic horizon. We have shown that under assumptions similar to those advanced in <cit.> the entropy of these solutions coincides with the real part of the logarithm of the topologically twisted index of the dual field theory. In conclusion, we have provided evidence in favor of identifying the set of square integrable modes in the presence of a constant flux on ℍ_2(× S^1), precisely speaking a very restricted set of zero modes out of the maximal set, as the boundary microstates responsible for the Bekenstein-Hawking entropy of the AdS_4 hyperbolic black holes presented in section <ref>. One important further test for this identification, would  be to  compute quantum corrections to the Bekenstein-Hawking entropy, on both sides of the duality.  
                
                
                
                On the 3d SCFT side we have made crucial use of the extremization approach advocated in <cit.>. The result of this approach is consistent with the constraints from the BPS equations on the gravity side and has been argued to be equivalent to the attractor mechanism. Under these conditions we  have found precise agreement between the leading large-N results on the two sides.  However, it would be important to elucidate the role of extremization intrinsically in the field theory  but also from the gravity perspective. This is particularly important because in some cases the attractor mechanism has been shown to apply away from the strictly supersymmetric context. 
                
                
                
                Another natural generalization of this work, following <cit.>, is to extend the analysis to dyonic black holes. More generally, it would be interesting to consider mapping the full space of deformations on both sides of the correspondence and, in particular, its modifications on the free energy and the entropy.  Another interesting direction concerns potential factorizations of the index on S^2× S^1, introduced in  <cit.>, in terms of blocks  given by the partition functions in ℍ_2× S^1. A similar factorization principle has been uncovered in various theories and in different dimensions, see, for example, <cit.>.  In this manuscript we have found a particular relation but it should be pointed out that we have set  all fermionic zero modes to zero and have integrated over a particular set of modes.  Clearly, to achieve a  bona fide factorization formula we will need to consider more general  boundary conditions and contemplate retracing some of the steps suggested in  <cit.>.  Indeed, such an approach with general boundary conditions has been implemented for GLSM's  in  <cit.>. 
                
                Finally, it would be interesting to discuss the microstate counting of magnetically charged strings in asymptotically AdS_5 spacetimes. Such magnetically charged solutions have a long history in supergravity dating back to explorations in <cit.>. It is logical to expect that the microscopic explanation should be found within 4d topologically twisted field theories on S^2× T^2 or possibly ℍ_2× T^2. Indeed, as a natural starting point along these lines,  the topologically twisted index introduced by Benini and Zaffaroni in <cit.> for supersymmetric field theories on S^2× S^1, was briefly discussed for 4d theories in S^2× T^2 in their original work, and was also addressed in <cit.>. It has recently been shown that, in the high temperature limit, the index produces a central charge that matches the supergravity answer <cit.> therefore providing  a strong argument in favor of the identification. We hope to report on some of these interesting directions soon. 
                
                
                
                
                
                
                § ACKNOWLEDGMENTS
                
                We are first and foremost grateful to the Abdus Salam Centre for Theoretical Physics where this work originated and was largely conducted; L. PZ., in particular, acknowledges sabbatical support. The work of ACB is supported by CONICET.
                
                We would like to thank Junya Yagi for collaboration in the early stages of this project. We  are especially thankful to E. Gava and K. Narain for discussions and clarifications at various  stages of this work. We are thankful to G. Bonelli and A. Tanzini for comments on a preliminary presentation by V. G-R. of parts of this work in summer 2016. We are also  grateful to  R.K. Gupta, K. Intriligator, U. Kol, J.  Liu, S. J. Rey, A. Sen and V. Rathee for discussions. ACB is very grateful to Carmen Núñez for her support and encouragement.
                
                
                
                 
                 
                
                § COMMENTS ON THE DISCRETE SPECTRUM
                
                
                In this appendix we report details on the construction of the square integrable modes defined in section <ref>.
                
                 The general solution to the defining equation (<ref>) is
                
                f      =   χ_1(  (  coshθ-1)  ^s/sinh^j_3
                +sθ_2F_1(a_1,b_1,c_1;-sinh^2θ/2)) 
                              +χ_2(  (  coshθ-1)  ^j_3/sinh^j_3
                +sθ_2 F_1(a_2,b_2,c_2;-sinh^2θ/2)),
                where χ_1,2 are arbitrary integration constants. It is very useful to work in the following coordinates
                
                x       :=-sinh^2θ/2,    -∞<x≤0.
                
                In  x coordinate, (<ref>) takes the form
                
                f     =   χ_1(  x^s(  x(  x-1)  )
                ^-j_3+s/2_2 F_1(a_1,b_1,c_1;x)) 
                             +χ_2(  x^j_3(  x(  x-1)  )
                ^-j_3+s/2_2 F_1(a_2,b_2,c_2;x))  , 
                
                
                
                
                
                
                
                
                
                
                
                where the parameters a_1,2, b_1,2 and c_1,2 are
                
                    a_1     =1/2-j_3-√(1/4+Δ+s^2),  
                    b_1=1/2-j_3+√(1/4+Δ+s^2,)
                    
                    c_1     =1-j_3+s, 
                    
                    a_2     =1/2-s-√(1/4+Δ+s^2),  
                    b_2=1/2-s+√(1/4+Δ+s^2),
                    
                    c_2     =1-s+j_3.
                
                Let us particularize Δ to 
                Δ= j(j+1)-s^2,     with   j ∈ℝ. 
                The choice (<ref>) completes perfect square inside the square roots in (<ref>).
                
                The asymptotic behavior of the linear solutions proportional to χ_1 and χ_2— from now on f_1 and f_2—,
                is 
                
                f_1(x)    x→ -∞∼   χ_1^-   x^-1-j(  1+O(1/x))  +χ_1^+ 
                x^j(  1+O(1/x)),  
                 f_1(x)    x→ 0∼   x^s-j_3/2( O(0)), 
                and
                
                f_2(x)    x→ -∞∼   χ_2^- x^-1-j(  1+O(1/x))  +χ_2^+
                 x^j(  1+O(1/x)),  
                 f_2(x)    x→ 0∼   x^j_3-s/2( O(0)). 
                
                Regularity at the contractible cycle x=0, conditions to pick up
                
                f_1      if        j_3 ≤ s,    
                
                f_2      if        j_3 > s.         
                
                We demand 𝒞^∞-differentiability at the contractible cycle x=0. In the vicinity of x=0, f_1 and
                f_2 go like x^s-j_3/2 and x^j_3-s/2,
                respectively.  Thus  𝒞^∞-differentiability at the contractible cycle x=0
                implies
                
                    j_3 -s ∈ℤ,
                
                and the appropriate choice among (<ref>) and (<ref>). Notice that from (<ref>), it follows that: j_3∈ℤ  implies s∈ℤ.. However, we are not forced to impose integrality of j_3, j or s.
                
                
                
                
                
                
                For the time being, let us assume s≥0. In due time, we extend the analysis to the case of generic s.
                
                It is important to stress that square integrability condition is equivalent to impose the conditions
                χ^+_1=0    if       j_3 ≤ s   and   j> -1/2
                χ^-_1=0     if     j_3 ≤ s   and   j< -1/2
                χ^+_2=0    if       j_3>s   and  j > -1/2
                χ^-_2=0     if     j_3 ≤ s   and   j< -1/2 . 
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                 §.§ The quantization conditions: f_1
                
                
                 Let us find out the quantization conditions that guarantee (<ref>). Our starting point in this subsection is
                 
                 j_3≤ s     and    s≥ 0 . 
                 For pedagogical reasons, let us assume for the time being
                 
                    j, j_3,  s ∈ℤ or ℤ+1/2.
                
                We shall see in due time that assumption (<ref>) is not necessary. As for j, let us not assume nothing else at this point. On the track, we will comment on the restrictions that arise for j.
                  
                 Condition (<ref>) selects the solution f_1
                
                    f_Δ(s),j_3^(1)
                    
                    
                         :=χ_1(  x^s(  x(  x-1)  )
                    ^-j_3+s/2_2 F_1(-j_3-j,1-j_3+j,1-j_3+s;x)).
                
                Notice that this solution is invariant under the transformation
                
                    j→-(  j+1)
                
                and in consequence we have to restrict j to be either
                
                j>- 1/2      or         j<- 1/2, 
                as preferred. 
                
                 Notice that j=-1/2 is left invariant by the transformation above. For j=-1/2 both independent solutions have
                the same asymptotic behavior x^j and x^-j-1, and they are not
                square integrable. 
                
                In order to have square integrability it is necessary to have
                
                    j≠ -1/2.
                
                We exclude the particular case  j=-1/2. 
                 
                
                
                
                Before writing down the quantization conditions, let us comment on the strategy. 
                It turns out that the quantization conditions are the conditions for which the hypergeometric factor
                in (<ref>) truncates to a specific polynomial. The sum of the degree of such polynomial with the degree of the leading power of the prefactor in (<ref>) in the limit x→-∞ must equate to
                
                -1-j       if      j>-1/2, 
                
                j      if      j<-1/2.
                
                The conditions to obtain the previously mentioned goal are
                
                1-j_3+j≤0        if       j> -1/2, 
                
                -j_3-j ≤0     if      j<-1/2.
                
                Together with (<ref>) these conditions are compactly written in the following one
                max(|j|,|j+1|)≤ j_3≤ s. 
                
                
                
                
                
                
                It is straightforward to check that, if we assume (<ref>) together with (<ref>) the desired truncation holds:
                
                
                    _2 F_1(-j_3-j,1-j_3+j,1-j_3+s;x)       =
                         =1+
                    
                    
                    ∑_n=0^∞
                    (a)_n(b)_n/(c)_nx^n+1/(n+1)!
                         =1+{c]lll
                    0    if    d^(1)=0
                    ∑_n=0^d^(1)-1
                    (a)_n+1(b)_n+1/(c)_n+1x^n+1/(n+1)!    if    d^(1)>0
                    .  ,
                    
                    (a)_n+1     :=
                    
                    
                    ∏_i=0^n
                    (a+i),
                
                The degree of the polynomial d^(1) being
                
                    d^(1):={c]lll
                    j_3-j-1     if     -1/2< j<j_3
                    j_3+j     if    -j_3≤ j<-1/2
                    .  .
                
                At this point is easy to check  that indeed the aforementioned asymptotic
                behavior of (<ref>) about x=0 and x=-∞ holds.
                
                
                
                 §.§ The quantization conditions: f_2
                
                
                In this subsection we analyze the case
                
                j_3 > s     and    s≥ 0. 
                We assume again (<ref>). Condition (<ref>) selects the solution f_2
                    f_Δ(s),j_3^(2)
                    
                    
                         :=χ_2(  x^j_3(  x(  x-1)  )
                    ^-j_3+s/2_2 F_1(-s-j,1-s+j,1-s+j_3;x))
                
                As the previous case, this solution is invariant under the transformation
                
                    j→-(  j+1)
                
                and in consequence at some stage we shall be forced to assume (<ref>). Let us, however, not assume
                the latter restriction on j yet. Let us just assume (<ref>).
                 
                 The quantization conditions are
                
                1-s+j≤0        if       j>-1/2, 
                
                -s-j ≤0     if      j<-1/2.
                
                Together with (<ref>), these conditions are compacted in the following one
                max(|j|,|j+1|)≤ s < j_3. 
                It is straightforward to check that, if we assume (<ref>) together with (<ref>) the desired truncation holds:
                
                
                
                
                
                
                
                    _2F_1(-s-j,1-s+j,1-s+j_3;x)       =
                         =1+{c]lll
                    0    if     d^(2)=0
                    ∑_n=0^d^(2)-1
                    (a)_n+1(b)_n+1/(c)_n+1x^n+1/(n+1)!    if    d^(2)>0
                    .  .
                
                The degree of the polynomial d^(2) being
                
                    d^(2):={c]lll
                    s-j-1     if     -1/2≤ j<s
                    s+j     if    -s≤ j<-1/2
                     .  ,
                
                and f_Δ(s),j_3^(2) is square integrable. Notice that, there are not
                square integrable modes for s=0, as well known.
                
                
                
                
                 §.§ The case of negative flux s<0
                
                
                So far, we have focused on the case of positive magnetic flux s>0, or being more specific on the case s>1/2. However, there are square integrable
                modes when s<0 too –as parity preservation dictates–. To find those, it is convenient to use the identity
                
                    _2F_1(a,b,c;x)=(1-x)^c-a-b_2F_1(c-a,c-b,c;x)
                
                upon the previously written solutions f_Δ(s),j_3^(1) and f_Δ
                (s),j_3^(2), to obtain
                
                    f_Δ(s),j_3^(1)     =χ_1 (  x^-j_3(  x(  x-1)  )  ^j_3+s/2_2F_1(s-j,1+s+j,1+s-j_3
                    ;x))  .
                    
                    f_Δ(s),j_3^(2)     =χ_2 (  x^-s(  x(  x-1)  )  ^j_3+s/2_2F_1(j_3-j,1+j_3
                    +j,1+j_3-s;x))  .
                
                Again, these eigenfunctions are invariant under the change (<ref>) and in consequence
                
                j>-1/2    or    j<-1/2.
                
                
                The hypergeometric factors written above, truncate to polynomials —and in consequence
                f_Δ(s),j_3^(1) and f_Δ(s),j_3^(2) square integrable— provided the following quantization conditions hold
                
                    max(|j|,|j+1|)    ≤-s≤-j_3,
                    max(|j|,|j+1|)    ≤-j_3≤-s,
                — and for the time being (<ref>)—, for f_Δ(s),j_3^(1) and f_Δ(s),j_3^(2), respectively. The explicit form of these square integrable
                modes can be obtained by repeating the analysis done for the case s>1/2, and they exist if and only if
                
                s< -1/2.
                
                
                 §.§ Generalized conditions
                
                
                So far we have been assuming 
                
                j_3,  j,  s ∈ℤ or ℤ+1/2.
                
                
                However, the aforementioned GNO conditions – see subsection <ref>– can be relaxed.
                
                 As already stated, to achieve regularity at the contractible cycle
                x=0, the following necessary condition
                
                    j_3-s∈ℤ,
                
                must hold. To have discrete spectrum there are necessary conditions too:
                
                    c]lll
                    f^(2)→-s+j∈ℤ, f^(1)→-j_3+j∈ℤ      if    s>+1/2, 
                    
                    f^(1)→ s+j∈ℤ,   f^(2)→ j_3+j∈ℤ      if    s<-1/2.
                
                Notice that the conditions in the right (resp. left) side, follow from a linear
                combination of the condition of regularity at the contractible cycle, and the respective conditions
                in the left (resp. right) side. Hence, we can write down the more compact and equivalent statement
                
                j_3-s ∈ℤ  and   j-|s| ∈ℤ.  
                
                In the table below, we write down the explicit form of the spectrum. For simplicity of presentation but without lack of generality, let us take
                j>-1/2. In that case, the relevant spectrum is
                    c]|l|l|l|∀ s such that    s>1/2   s<-1/2
                    j_3   j+1,j+2,...,j+k,...,∞   -∞,...,-k-j,...,-2-j,-1-j
                    
                    j   s-1,s-2,...,s-k,...> -1/2   -s-1,-s-2,...,-s-k,...>-1/2
                    .
                
                A particular case is when j,j_3,s∈ℤ+1/2. In that case, the table
                above reduces to
                    c]|l|l|l|∀ s such that    s >1/2   s<-1/2
                    j_3   j+1,j+2,...,j+k,...,∞   -∞,...,-k-j,...,-2-j,-1-j
                    
                    j   s-1,s-2,...,s-k,...,1/2.   -s-1,-s-2,...,-s-k,...,1/2.
                    .
                
                The corresponding eigenfunctions can be recovered from the summary that shall
                be presented next, and the results in previous sections.
                
                
                
                
                 §.§ Collecting the eigenfunctions
                
                
                The maximal functional space of square integrable modes isΞ(s):=
                
                
                ⊕_ -1/2<j<|s|
                (  Ξ_j^(1)(s)
                
                
                ⊕
                Ξ_j^(2)(s))  , 
                where the subspace Ξ_j^(1)(s) is defined as
                
                    Ξ_j^(1)(s):={f_Δ(s),j_3^(1)}_j_3, with 
                    Δ:=j(j+1)-s^2,
                
                together with conditions (<ref>) and
                
                
                c]lll
                max(|j|,|j+1|)≤ j_3≤ s,    if    s>1/2,
                
                max(|j|,|j+1|)≤-s≤-j_3    if    s<-1/2.
                
                
                
                The subspace Ξ_j^(2)(s) is defined as
                
                    Ξ_j^(2)(s):={f_Δ(s),j_3^(2)}_j_3  , with 
                    Δ:=j(j+1)-s^2,
                
                together with conditions (<ref>) and
                
                
                
                
                    c]lll
                    max(|j|,|j+1|)≤ s≤ j_3    if    s > 1/2,
                    
                    max(|j|,|j+1|)≤-j_3≤-s    if    s <-1/2.
                
                Of special interest will be the following limiting spaces
                
                    Ξ_s-1 (or -s)^(1,2)(s):={f_Δ,j_3^(1,2)
                    }, with j:=s-1 (or  -s) and Δ=-s.
                
                These spaces are the ones that contribute to the super-determinant that concerns
                us, when s> 1/2 and cohomological cancellations are performed.
                
                It will be useful to keep in mind that for every j, in the direct sum space
                Ξ_j(s):=Ξ_j^(1)(s)⊕Ξ_j^(2)(s) the angular number
                j_3 will range at step 1, departing from the lower (resp. upper) bound
                given below
                    c]lll
                    max(|j|,|j+1|)<j_3<∞    if    s>  1/2,
                    
                    -∞<j_3<-max(|j|,|j+1|)    if    s < -1/2.
                Comment: Notice that upon square integrable “
                representations” Ξ_j(s), which are labeled by j
                running at step 1 down from |s|-1 and greater than -1/2, namely
                
                    j:-1/2< j≤|s|-1,
                
                the bosonic operator O_B(<ref>), is positive definite if (ρ(u)+k )^2>0.
                Indeed, that operator needs to be positive definite in order to have
                convergence of the functional integral of the exponential of the quadratic
                expansion of the bosonic localizing term.
                 
                
                
                 
                
                 §.§ Normalizable modes from asymptotics
                
                
                One can also find the discrete spectrum by looking at  the asymptotic expansion of the general solutions (<ref>) (one can repeat the procedure for the other solution). We choose to focus on f_1 and for values s>1/2,  from regularity  and  smoothness at x=0 it follows that j_3≤ s and the difference s-j_3∈ℤ_+.
                
                As before, we define Δ:=j(j+1)-s^2.
                
                At x=-∞f_1  is
                
                f_1∼χ_1^- x^-1-j(  1+O(1/x))  +χ_1^+ 
                x^j(  1+O(1/x)).
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                The coefficients above are:
                
                
                    χ_1^-   ∼Γ1+s-j_3]Γ-1-2j]/Γ-j_3-j
                    ]Γ s-j],
                    χ_1^+   ∼Γ1+s-j_3]Γ1+2j]/Γ1-j_3+j
                    ]Γ s+j].
                
                Suppose j>-1/2, to cancel out the x^j behavior of f_1 while preserving the x^-1-j, we need to have that χ_1^+ vanishes, this is achieved when either of the arguments of the 's in the denominator is 0 or a negative integer. Then
                
                1-j_3+j=-n
                  or   s+j=-n  with   n∈ℤ_+ ,
                
                the second choice is out of order given our assumptions (s>1/2, j>-1/2)
                therefore 1-j_3+j=-n when replacing this value in χ_1^- one has to be careful, since the arguments of 's in the denominator might be also a negative integer, 
                χ_1^-   ∼Γ1+s-j_3]Γ1+2n-2j_3]/Γ-2j_3+n+1
                ]Γ s-j]   ∼Γ1+2n-2j_3]/Γ-2j_3+n+1
                ],
                
                We have not replaced the value of j in terms of n and j_3, these  functions, given our assumptions will be a number different from zero and finite and will not play a role. Looking at the denominator in the last expression one naively conclude that there are values of j_3 and  n for which the argument is negative integer (since j_3>n+1), and therefore χ_1^- is also  zero, but it is not the case since, for each of this values, the argument of  in the numerator is also a negative integer, and the divergences cancel.  One can recast the ratio above as:
                Γ1+2n-2j_3]/Γ-2j_3+n+1]=[-m+n]/[-m]  where   m>n, 
                  
                using :
                Γ [ϵ -m]=(-1)^m-1Γ [-ϵ ] Γ [ϵ +1]/Γ [m-ϵ +1]  where    is very small.
                Applying this relation on both numerator an denominator and taking →0[-m+n]/[-m]→(-1)^n [m+1]/[m-n+1],
                  
                therefore  for j>-1/2 we  conclude:
                
                    χ_1^-     =O(1),
                    χ_1^+     =0,
                
                We can then proceed analogously for j<-1/2 to get:
                
                    χ_1^-     =0,
                    χ_1^+     =O(1),
                
                 We then have:
                
                    for   s>1/2     {c]lll
                    j<j_3≤ s    if    j> -1/2
                    -s≤-j_3≤ j    if    j<-1/2
                    
                    ..
                
                
                
                
                
                
                
                
                
                
                
                 
                 
                 
                
                 §.§ The relation between spin-1 discrete modes and ours  
                
                
                Let ∇_μ be the covariant derivative of diffeomorphisms. The
                Laplace-Beltrami operator is defined as ∇^μ∇_μ and acting
                upon a covariant vector field X of components (X_θ,X_φ) has the explicit form
                
                    ∇^μ∇_μX:=(
                    [        □_s=0+^2(θ) 2(θ)/sinh^2(θ
                    )∂_φ;           -2(θ)∂_φ    □_s=0-2(θ)∂_θ+1 ])  (
                    [ X_θ; X_φ ]),
                
                where □_s=0 is the scalar Laplacian. We have added the subscript
                s=0 to remind that it can be obtained from the magnetic Laplacian previously
                defined, by particularising to s=0. The eigenvector
                
                    X_0=∇Φ, with  Φ:=(  sinh(θ)/1+cosh(θ))  ^|j_3|e^ij_3φ,  j_3=±1,±2,... .
                
                One can check that indeed
                
                    □_s=0Φ=0,
                
                and second that
                
                    ∇^μ∇_μX_0=X_0.
                
                In words, X_0 is an eigen-tensor of rank one of the
                Laplace-Beltrami operator ∇^μ∇_μ with eigenvalue 1.
                
                More important to our purpose, we have checked that
                
                    (  -∂_θ^2-^2θ∂_θ+1/sinh^2θ(|j_3|-coshθ)^2)  X_0θ=X_0θ.
                
                Notice that the operator in the LHS equation above coincides with our
                □_s=1if and only if 
                
                    j_3>0.
                
                In fact, the equation above for X_0θ implies that X_0θ obeys
                our defining equation
                
                    (  □_s+Δ)  X_0θ=0,
                if and only if
                
                    0=j (or-1=j)<s=1≤ j_3∈ℤ.
                
                It is then consequence that
                {  X_0θ}  _j_3∈ℕ   =   {(  coshθ-1)  ^j_3/sinh^j_3+1θe^ij_3φ=1/sinhθ(  tanhθ/2)
                ^j_3e^ij_3φ}  _j_3∈ℕ ,
                   =   Ξ_j=0^(2)(s=1).
                
                The remaining θ-components of the vector discrete modes, {
                X_0θ}  _j_3∈-ℕ, do solve our defining equation (
                □_s+Δ)  X_0θ=0 if and only if
                    j_3≤-1=s<j=0∈ Z.
                
                We have thence proven that {  X_0θ}  _j_3≠0∈ Z
                are included in our set of square integrable modes
                {  X_0θ}  _j_3∈-ℕ   =   {(  coshθ-1)  ^-j_3/sinh^-j_3+1θe^ij_3φ=1/sinhθ(  tanhθ/2)  ^-j_3
                e^ij_3φ}  _j_3∈-ℕ, 
                   =   Ξ_j=0^(1)(s=-1).
                
                and correspond to the two possible unit flux (spin one) “helicities”s=±1.
                
                § ON 1 LOOP DETERMINANTS
                
                
                 §.§ Alternative regularization
                
                In this appendix we report a second approach to regularize the determinant of O_B in the subspace _j=s-1(s)– We present the case s> 1/2, the s < -1/2 is analogous –:
                __j=s-1(s)O_B   =   ∏_k∈ℤ∏_j_3=s^∞((̊u)+k)^2,
                   =   ∏_k∈ℤ(((̊u)+k)^2)^∑_j_3=s1=∏_k∈ℤ(((̊u)+k)^2)^∑_j_3=1^∞1-∑_j_3=1^s-11,
                   =   ∏_k∈ℤ(((̊u)+k)^2)^(0)-(s-1)=∏_k∈ℤ|((̊u)+k)|^-2s+1.
                
                Where we use the basic definition of Riemann  function (t)=∑_n=1^∞1/n^t and the value (0)=-1/2.
                
                
                
                 §.§ Vector multiplet
                
                In this appendix, we prove that the index of a vector multiplet coincides with the index of matter multiplet with R-charge q_R=2.
                
                The quadratic actions coming out of the localizing terms (<ref>) and (<ref>), along the complex path (<ref>) and after imposing gauge fixing condition (<ref>), are
                
                    ℒ^B_quadratic   :=   (i δD̃ )^2+(𝒟_t δ A_1)^2+ (𝒟_t δ A_2)^2, 
                    ℒ^F_quadratic   :=   i  δλ̅^†_2 𝒟̂_t  δλ_2,
                
                where
                
                    i δD̃   :=   i δ D+δ F_1 2+ δ𝒟̂_3 σ .
                
                The δD̃ integrates trivially. The functional spaces to integrate over the vector and ghost degrees of freedom are
                
                    (δ A_1,  δ A_2,  δσ,  δc̅,  δ c) → (Ξ_(s-1), Ξ_(s-1),  Ξ_(s), Ξ_(s), Ξ_(s)).
                
                The integration of δ A_1 and δ A_2 is
                
                    ∏_k∏^|s-1|-1_j=0∏^∞_j_3=j+1 |(ρ(u)+k )| ×∏_k∏^|s-1|-1_j=0∏^∞_j_3=j+1 |(ρ(u)+k )|.
                
                 In obtaining  we have used √((ρ(u)+k )^2)=|(ρ(u)+k )|. In our contour of integration (ρ(u)+k ) is real.
                 The functional space to integrate the gaugini degres of freedom is
                 
                    (
                    δλ_2, δλ̅_2^† )
                    → (
                     Ξ_(s-1),  Ξ_(s-1)).
                
                
                The integration of δλ_2 and δλ̅^†_2, multiplied by the integration of δc̅^† and δ c, following from the BRST action (<ref>), gives
                
                    ∏_n∏^|s-1|-1_j=0∏^∞_j_3=j+1 |(ρ(u)+k )|  ×∏_n∏^|s|-1_j=0∏^∞_j_3=j+1 |(ρ(u)+k )| .
                
                As already mentioned, we will not integrate over the zero modes δλ_1 and δλ̅_1 in order not to obtain vanishing results. 
                
                The super-determinant to compute is
                (<ref>)/(<ref>). 
                The result (<ref>) is a divergent quantity. To regularize these objects we use the zeta-regularization procedure but only after co-homological cancellations are performed.
                 
                 
                  If s>1/2 the only contribution to the quotient (<ref>), comes from the integration of the ghosts degrees of freedom  (c̅,c) with quantum number j=s-1. The divergent contribution from this functional space is
                 ∏^s-1_j=s-1∏^∞_j_3=j+1 |(ρ(u)+k )|. 
                  Those degrees of freedom  live in Ξ_j=s-1(s) and are coupled to s units of flux. As a formal object (<ref>) is equal to
                √(Ξ_j=s-1(s)(O_B)),
                   
                as can be straightforwardly checked from taking the product between equations (<ref>) and (<ref>) and particularizing the result to j=s-1.   
                
                   We have already computed the zeta regularized determinant of the operator O_B on Ξ_j=s-1(s). In this space and for a given S^1 KK mode k, O_B has a unique eigenvalue: (ρ(u)+k )^2 and the square root of its zeta-regularized determinant is
                √(Ξ_j=s-1(s)(O_B))=|(ρ(u)+k)|^-s+1/2. 
                The value of the parameter s is
                
                s=-ρ(𝔪)/2,
                
                because the ghosts have q_R=0.
                
                From (<ref>), after taking the products over roots and KK modes and after regularization we obtain
                
                Z_1-loop^vector(ℍ_2,𝔪)   =   ∏_ρ^*[  |C_regsin(  ρ(u)/2)| ]  ^ρ(𝔪)+1/2, 
                   =   ∏_ρ^*[
                 |C_regsin(  ρ(u)/2)|]  ^-ρ(𝔪)+1/2,
                 
                where
                
                C_reg=-2 i. 
                Equality (<ref>) proves the statement made below equation (<ref>), that is, independence of the GNO conditions. In the second equality in (<ref>), we have
                performed the inversion of roots ρ→ -ρ.
                
                 For s < 1/2 the same result (<ref>) is obtained by following analog steps.
                
                
                
                
                § CONVENTIONS: 4D 𝒩=2 GAUGED SUPERGRAVITY
                
                In this appendix we summarize our conventions for 4d 𝒩=2 gauged supergravity. The construction of black holes  reported in section <ref>, was implemented in a Mathematica file. If the reader is interested in the file, please write an email to us. If there is interest, we are more than happy to share it.
                
                The 4d gamma matrices
                γ^1=(
                [  i  0  0  0;  0 -i  0  0;  0  0  i  0;  0  0  0 -i ]),  γ^2=(
                [  0  0  0  i;  0  0 -i  0;  0 -i  0  0;  i  0  0  0 ]),  γ^3=(
                [  0 -i  0  0; -i  0  0  0;  0  0  0 -i;  0  0 -i  0 ]), 
                γ^4=(
                [  0  0  0 -i;  0  0  i  0;  0 -i  0  0;  i  0  0  0 ]).   γ_a b=1/2[γ_a,γ_b],   γ_5=i γ^4γ^1γ^2γ^3.
                
                
                The SU(2)_RR-symmetry invariant tensors
                ϵ_AB=ϵ^A B=(
                [  0  1; -1  0 ]).
                
                
                The SU(2)_R  generators
                σ^1_AB   =   (
                [  1  0;  0 -1 ]),  σ^2_AB=(
                [ -i  0;  0 -i ]),  σ^3_AB=(
                [  0 -1; -1  0 ]), 
                σ^1 AB   =   (
                [ -1  0;  0  1 ]),  σ^2 AB=(
                [ -i  0;  0 -i ]),  σ^3 AB=(
                [ 0 1; 1 0 ]).
                
                
                The σ^I   B_A with I=1,2,3 are the Pauli matrices.
                
                The ordering of coordinates is
                (1,2,3,4)  ↔ (r, θ, φ,  t).
                
                
                For hyperbolic solutions:
                
                F^Λ_μν=(
                [                 0                 0                 0                 0;                 0                 0  1/2sinh (θ ) p_Λ                 0;                 0 -1/2sinh (θ ) p_Λ                 0                 0;                 0                 0                 0                 0 ]),
                F_μ,ν^- Λ=(
                [                 0                 0                 0    i p_Λ/4 h(r)^2;                 0                 0  1/4sinh (θ ) p_Λ                 0;                 0 -1/4sinh (θ ) p_Λ                 0                 0;   -i p_Λ/4 h(r)^2                 0                 0                 0 ]).
                
                
                 §.§ Parametrization in terms of scalar
                
                
                
                In this subsection we post  a series of useful parametrizations in terms of physical scalars z^1, z^2, z^3. The results reported in this subsection, are consistent with the BPS equations obtained for the choice + in equation (<ref>).
                𝒦=-log(8 √(z^1 z^2 z^3)/(z^1+z^2+z^3+3)^2),
                ℱ=(ℱ̅)^*=-2 i √(z^1 z^2 z^3/(z^1+z^2+z^3+3)^4),
                ℱ_Λ=(i √(z^1 z^2 z^3)/z^1+z^2+z^3+3,i √(z^2
                   z^3/z^1)/z^1+z^2+z^3+3,i √(z^1
                   z^3/z^2)/z^1+z^2+z^3+3,i √(z^1
                   z^2/z^3)/z^1+z^2+z^3+3),
                ℱ_ΛΣ=(ℱ̅_ΛΣ)^*=(
                [     1/2 i √(z^1 z^2 z^3)    -1/2 i √(z^2 z^3/z^1)    -1/2 i √(z^1 z^3/z^2) -1/2 i √(z^1
                   z^2/z^3);    -1/2 i √(z^2 z^3/z^1)   1/2 i √(z^2 z^3/z^1^3)  -i z^3/2 √(z^1 z^2 z^3)  -i z^2/2 √(z^1 z^2 z^3);    -1/2 i √(z^1 z^3/z^2)  -i z^3/2 √(z^1 z^2 z^3)   1/2 i √(z^1 z^3/z^2^3)  -i z^1/2 √(z^1 z^2 z^3);    -1/2 i √(z^1 z^2/z^3)  -i z^2/2 √(z^1 z^2 z^3)  -i z^1/2 √(z^1 z^2 z^3)   1/2 i √(z^1 z^2/z^3^3) ]),
                f̅^Λ_j=(
                [     1/8 √(2)√(z^1^ 5 z^2 z^3)    -3/8 √(2)√(z^1 z^2
                   z^3)   z^2/8 √(2)√(z^1^ 5 z^2 z^3)    z^3/8 √(2)√(z^1^5 z^2 z^3);      1/8 √(2)√(z^1 z^2^5 z^3) z^1/8 √(2)√(z^1
                   z^2^5 z^3)       -3/8 √(2)√(z^1 z^2 z^3)    z^3/8 √(2)√(z^1 z^2^5 z^3);      1/8 √(2)√(z^1 z^2 z^3^5) z^1/8 √(2)√(z^1 z^2
                   z^3^5)    z^2/8 √(2)√(z^1 z^2 z^3^5)       -3/8 √(2)√(z^1 z^2 z^3) ]),
                𝒩_ΛΣ= i (
                [   -√(z^1 z^2 z^3)                 0                 0                 0;                 0 -√(z^2 z^3/z^1^3)                 0                 0;                 0                 0 -√(z^1 z^3/z^2^3)                 0;                 0                 0                 0 -√(z^1 z^2/z^3^3) ]),
                
                g_z z̅=(
                [     3/161/z^1^2 -1/161/ z^1 z^2 -1/16 1/z^1 z^3; -1/161/ z^1 z^2     3/161/z^2^2 -1/161/ z^2 z^3; -1/161/ z^1 z^3 -1/161/ z^2 z^3     3/161/z^3^2 ]).
                
                
                
                
                
                
                 
                
                
                
                
                
                 
                
                 
                
                 
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                 
                
                
                
                
                
                
                
                
                JHEP ]
