Mini-BFSS in Silico
Tarek Anous Cameron Cogburn




MIT-CTP-4877



 
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 Mini-BFSS in Silico



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Tarek Anous^1,2 and Cameron Cogburn^3

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^1Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, B.C. V6T 1Z1, Canada

^2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

^3Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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 tarek@phas.ubc.ca, ccogburn@mit.edu






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We study a mass-deformed 𝒩=4 version of the BFSS matrix model with three matrices and gauge group SU(2). This model has zero Witten index. Despite this, we give numerical evidence for the existence of four supersymmetric ground states, two bosonic and two fermionic, in the limit where the mass deformation is tuned to zero.






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§ INTRODUCTION

This paper concerns itself with the supersymmetric quantum mechanics of three bosonic SU(N) matrices and their fermionic superpartners. The model in question, introduced in <cit.>, has four supercharges and describes the low energy effective dynamics of a stack of N wrapped D-branes in a string compactification down to 3+1 dimensions. When the compactification manifold has curvature and carries magnetic fluxes, the bosonic matrices obtain masses<cit.>. When the compact manifold is Calabi-Yau and carries no fluxes, the matrices are massless. 

This theory has flat directions whenever the matrices are massless, and hence is a simplified version of the BFSS matrix model <cit.>, which, for the sake of comparison, has nine bosonic SU(N) matrices and 16 supercharges and describes the non-Abelian geometry felt by D-particles in a non-compact 9+1 dimensional spacetime. We hence dub the model studied here: mini-BFSS (or mini-BMN <cit.> in the massive case). The Witten index W_I has been computed for mini-BFSS  <cit.> and vanishes, meaning that the existence of supersymmetric ground states is still an open question. Even the refined index, twisted by a combination of global symmetries and calculated in <cit.>, gives us little information about the set of ground states due to the subtleties associated with computing such indices in the presence of flat directions in the potential. This is in stark contrast with the full BFSS model, whose Witten index W_I=1, implying beyond doubt the existence of at least one supersymmetric ground state. The zero index result for mini-BFSS has led to the interpretation that it may not have any zero energy ground states <cit.>, and hence no holographic interpretation. The logic being that, without a rich low energy spectrum, scattering in mini-BFSS would not mimic supergraviton scattering in a putative supersymmetric holographic dual <cit.>.
Of course 
a vanishing W_I does not confirm the absence of supersymmetric ground states—as there may potentially be an exact degeneracy between the bosonic and fermionic states at zero energy.

We weigh in on the existence of supersymmetric states in mini-BFSS by solving the Schrödinger equation numerically for the low-lying spectrum of the N=2 model, in the in silico spirit of <cit.>. To deal with the flat directions we numerically diagonalize the Hamiltonian of the mass-deformed mini-BMN matrix model, for which the flat directions are absent, and study the bound state energies as a function of the mass. A numerical analysis of mini-BFSS can also be found in <cit.> which use different methods.

What we uncover is quite surprising. As we tune the mass parameter m to zero, we find evidence for four supersymmetric ground states, two bosonic and two fermionic, which cancel in the evaluation of W_I. This result seems to agree with plots found in <cit.>. It must be said that our result does not constitute an existence proof for supersymmetric threshold bound states in the massless limit, but certainly motivates a further study of the low-lying spectrum of these theories. 

The organization of the paper is as follows: in section <ref> we present the supercharges, Hamiltonian and symmetry generators of the mini-BMN model for arbitrary N. In section <ref> we restrict to N=2 and give coordinates in which the Schrödinger equation becomes separable. In section <ref> we provide our numerical results and in section <ref> we derive the one-loop effective theory on the moduli space in the massless theory. We conclude with implications for the large-N mini-BFSS model in section <ref>. We collect formulae for the Schrödinger operators maximally reduced via symmetries in appendix <ref> and compute the one-loop metric on the Coulomb branch moduli space in appendix <ref>.



§ SETUP



 §.§ Supercharges and Hamiltonian

Let us consider a supersymmetric quantum mechanics of SU(N) bosonic matrices X^i_A and their superpartners λ_Aα. The quantum mechanics we have in mind has four supercharges:[Spinors and their conjugates transform respectively in the 2 and  2̅ of of SO(3). Spinor indices are raised and lowered using the Levi-Civita symbol ϵ^αβ=-ϵ_αβ with ϵ^12=1. Thus in our conventions:

    (ψ̅ϵ)_α=ψ̅^γϵ_γα ,        (ϵψ)^α=ϵ^αγψ_γ ,        ϵ_αωϵ^ωβ=δ_α^  β .
]

    Q_α=(-i∂_X^i_A-i m X^i_A-i W^i_A)σ^i γ_αλ_Aγ ,    Q̅^β=λ̅^ γ_Aσ^i β_γ(-i∂_X^i_A+i m X^i_A+i W^i_A) .

The parameter m is simply the mass of X^i_A. The massless version of this model was introduced in <cit.> and can be derived by dimensionally reducing 𝒩=1, d=4 super Yang-Mills to the quantum mechanics of its zero-modes. The mass deformation was introduced in <cit.>, and can be obtained from a dimensional reduction of the same gauge theory on R× S^3. We direct the reader to <cit.> for an introduction to these models. This quantum mechanics should be thought of as a simplified version of the BMN matrix model <cit.> (mini-BMN for brevity). The massless limit should then be thought of as a mini-BFSS matrix model <cit.>. The lowercase index i=1,…,3 runs over the spatial dimensions (in the language of the original gauge theory), and the uppercase index A=1,…,N^2-1, runs over the generators of the gauge group SU(N).  The σ^i are the Pauli matrices and greek indices run over α=1,2. In keeping with <cit.>, we have defined W^i_A≡∂ W/∂ X^i_A where


    W≡g/6f_ABC ϵ_ijk X^i_A X^j_B X^k_C,

and f_ABC are the structure constants of SU(N). 
The gauginos obey the canonical fermionic commutation relations {λ_Aα,λ̅^β_B}=δ_ABδ_α^ β, and hence the algebra generated by these supercharges is <cit.>

    {Q_α,Q̅^β}=2(δ_α^ β H-g σ^k β_α X^k_A G_A+ m σ^k β_α J^k) ,    {Q̅^α,Q̅^β}={Q_α,Q_β}=0 ,

with Hamiltonian:

    H≡ -1/2∂_X^i_A∂_X^i_A+1/2m^2 (X^i_A)^2+m X^i_A W^i_A+g^2/4(f_ABC X^i_B X^j_C)^2-3/4m[λ̅_A,λ_A]+ig f_ABCλ̅_A X^k_B σ^kλ_C .

The operators G_A and J^k appearing in the algebra are, respectively, the generators of gauge transformations and SO(3) rotations. These are given by:

    G_A≡ -i f_ABC(X^i_B ∂_X^i_C+λ̅_Bλ_C) ,      J^i≡-i ϵ_ijk X^j_A ∂_X^k_A+1/2λ̅_Aσ^iλ_A .

In solving for the spectrum of this theory, we must impose the constraint G_A|ψ⟩=0 , ∀ A. In the above expressions, whenever fermionic indices are suppressed, it implies that they are being summed over.

Let us briefly note the dimensions of the fields and parameters in units of the energy [ℰ]=1. These are [X]=-1/2, [λ]=0, [g]=3/2 and [m]=1. Therefore, an important role will be played by the dimensionless quantity

    ν≡m/g^2/3 .


We consider here the mass deformed gauge quantum mechanics because, in the absence of the mass parameter m, the classical potential has flat directions (see figure <ref>). Turning on this mass deformation gives us a dimensionless parameter ν, to tune in studying the spectrum of this theory, and allows us to approach the massless limit from above. 



 §.§ Symmetry algebra


Let us now give the symmetry algebra of the theory. The components of J⃗ satisfy:


    [J^i,J^j]=i ϵ_ijk J^k ,       [J^i,Q_α]=-1/2σ^i γ_α Q_γ ,
       [J⃗^ 2,J^i]=0 ,       [J^i,Q̅^α]=1/2Q̅^β σ^i α_β .

There is an additional U(1)_R generator R≡λ̅_Aλ_A which counts the number of fermions. It satisfies

    [R,Q_α]=-Q_α ,        [R,Q̅^α]=+Q̅^α ,        [R,J^i]=0 .

The Hamiltonian also has a particle-hole symmetry:

    λ̅^α_A→ϵ^αγλ_Aγ ,    λ_Aα→λ̅_A^γϵ_γα ,    ϵ^12=-ϵ_12=1 ,

where ϵ^αβ is the Levi-Civita symbol. This transformation leaves the Hamiltonian invariant but takes R→ 2(N^2-1)-R and effectively cuts our problem in half.

One peculiar feature of the mass deformed theory is that the supercharges do not commute with the Hamiltonian as a result of the vector J⃗ appearing in (<ref>). It is easy to show that

    [H,Q_α]=m/2Q_α ,      [H,Q̅^β]=-m/2Q̅^β .

Thus, acting with a supercharge increases/decreases the energy of a state by ±m/2. This is a question of R-frames, as discussed in <cit.>. Essentially we can choose to measure energies with respect to the shifted Hamiltonian H_m≡ H+m/2R, which commutes with the supercharges, and write the algebra as:

    {Q_α,Q̅^β}=2{δ_α^ β (H_m-m/2R)-g σ^k β_α X^k_A G_A+ m σ^k β_α J^k} .







 §.§ Interpretation as D-particles

The ν→ 0 limit of this model can be thought of as the worldvolume theory of a stack of N D-branes compactified along a special Lagrangian cycle of a Calabi-Yau three-fold <cit.>. The X^i_A then parametrize the non-Abelian geometry felt by the compactified D-particles in the remaining non-compact 3+1 dimensional asymptotically flat spacetime. The addition of the mass parameter corresponds to adding curvature and magnetic fluxes to the compact manifold <cit.> 
changing the asymptotics of the non-compact spacetime to AdS_4. This interpretation was argued in <cit.> and passes several consistency checks. Hence we should think of the mass deformed theory as describing the non-relativistic dynamics of D-particles in an asymptotically AdS_4 spacetime and the massless limit as taking the AdS radius to infinity in units of the string length.

To be more specific, it will be useful to translate between our conventions and the conventions of <cit.>. One identifies m=Ω, g^2=1/m_v, {X,λ}_ us= m_v^1/2{X,λ}_ them in units where the string length l_s=1. Reintroducing l_s, this dictionary implies that g^2=g_s /l_s^3√(2π), with g_s the string coupling, gets set by a combination of the magnetic fluxes threading the compact manifold and similarly ℓ_ AdS≡ 1/m gets set by a combination of these magnetic fluxes and the string length. For AdS_4×CP^3 compactifications dual to ABJM this was worked out in detail in <cit.> and they identify

    g_s=(32π^2N/k^5)^1/4 ,    ℓ_ AdS=(N/8π^2 k)^1/4l_s ,

where k and N are, respectively, integrally quantized magnetic 2-form and 6-form flux. In this example taking ν=√(2π)(k^2/N)^1/3→ 0 while keeping g_s fixed takes the AdS radius to infinity in units of l_s.

The main focus of the next sections is on whether this stack of D-particles forms a supersymmetric bound state, particularly in the ν→ 0 limit. There the Witten index W_I≡Tr_ℋ{ (-1)^R e^-β H} has been computed <cit.> and evaluates to zero. This is in contrast with the full BFSS matrix model, whose index is W_I=1, confirming the existence of a supersymmetric ground state. We will use the numerical approach of <cit.> and verify if supersymmetry is preserved or broken in the SU(2) case. We find evidence that supersymmetry is preserved in the ν→ 0 limit, and that there are precisely 4 ground states contributing to the vanishing Witten index.



§ QUANTIZING THE SU(2) THEORY




 §.§ Polar representation of the matrices


We are aiming to solve the Schrödinger problem H_m|ψ⟩=ℰ_m|ψ⟩. We will not be able to do this for arbitrary N and from here on we will restrict to gauge group SU(2) for which the structure constants f_ABC=ϵ_ABC. In this case the wavefunctions depend on 9 bosonic degrees of freedom tensored into a 64-dimensional fermionic Hilbert space.  It is thus incumbent upon us to reduce this problem maximally via symmetry. In order to do so, we exploit the fact that the matrices X^i_A admit a polar decomposition as follows

    X_A^i=L_AB Λ_B^j  M^T ji

with

    L≡ e^-i φ_1 ℒ^3e^-i φ_2 ℒ^2e^-i φ_3 ℒ^3 ,       M≡ e^-i ϑ_1 ℒ^3e^-i ϑ_2 ℒ^2e^-i ϑ_3 ℒ^3 ,

and  [ℒ^i]_jk≡-iϵ_ijk are the generators of SO(3). The diagonal matrix

    Λ≡diag(𝐱_1,𝐱_2,𝐱_3)

represents the spatial separation between the pair of D-branes in the stack. The φ_i and ϑ_i represent the (respectively gauge-dependent and gauge-independent) Euler-angle rigid body rotations of the configuration space. This parametrization is useful because the Schrödinger equation is separable in these variables, as we show in appendix <ref>.

The metric on configuration space can be re-expressed as:

    ∑_A,idX_A^i dX_A^i   =∑_a=1^3 d𝐱^2_a+ I_a(dΩ_a^2+dω_a^2)-2K_a dΩ_a dω_a ,
    
    I_a   ≡𝐱_b 𝐱_b-𝐱_a^2 ,       K_a≡|ϵ_abc| 𝐱_b 𝐱_c .

The angular differentials are the usual SU(2) Cartan-Maurer differential forms defined as follows:

    dω_a=-1/2ϵ_abc [L^T· dL]_bc ,       dΩ_a=-1/2ϵ_abc[M^T· dM]_bc .





The volume element used to compute the norm of the wavefunction is

    ∏_i,A dX_A^idX_A^i=Δ(𝐱_a)∏_i=1^3 d𝐱_isinφ_2∏_j=1^3 dφ_j sinϑ_2∏_k=1^3 dϑ_k  ,

where Δ(𝐱_a)≡(𝐱_1^2-𝐱_2^2)(𝐱_3^2-𝐱_2^2)(𝐱_3^2-𝐱_1^2) is the Vandermonde determinant with squared eigenvalues. To cover the configuration space correctly, we take the new coordinates to lie in the range <cit.>:

    𝐱_3≥𝐱_1≥|𝐱_2|≥0 ,    π≥φ_2 ,ϑ_2≥0 ,     2π≥φ_i≠2 ,ϑ_i≠2≥0 .

The generators of gauge-transformations G_A and rotations  J^i are given in (<ref>). These satisfy

    [J^i,J^j]=i ϵ_ijk J^k ,       [G_A,G_B]=i ϵ_ABC G_C ,
       [J⃗^ 2,J^i]=0 ,       [J^i,G_A]=0  .

To label the SU(2)_gauge× SO(3)_J representations of the wavefunctions, it is useful to define the “body fixed" angular momentum and gauge operators P⃗≡ M^-1·J⃗ and S⃗≡ L^-1·G⃗, which satisfy

    P⃗^2=J⃗^2 ,       S⃗^2=G⃗^2 ,
       [P^i,P^j]=-i ϵ_ijk P^k ,       [S_A,S_B]=-i ϵ_ABC S_C ,
       [P^i,J^j]=0 ,       [S_A,G_B]=0 .

Unlike the generators of angular momentum, P⃗ is not conserved. However, as we explain in appendix <ref>, it is still useful for separating variables.

 Let us give expressions for the bosonic parts of J⃗ and P⃗, which we call 𝒥⃗ and 𝒫⃗ respectively, in terms of the angular coordinates. These are:

    𝒥^1   =-i(-cosϑ_1ϑ_2 ∂_ϑ_1-sinϑ_1∂_ϑ_2+cosϑ_1/sinϑ_2 ∂_ϑ_3) ,
    𝒥^2   =-i(-sinϑ_1ϑ_2 ∂_ϑ_1+cosϑ_1∂_ϑ_2+sinϑ_1/sinϑ_2 ∂_ϑ_3) ,
    𝒥^3   =-i ∂_ϑ_1 ,

and

    𝒫^1   =-i(-cosϑ_3/sinϑ_2 ∂_ϑ_1+sinϑ_3∂_ϑ_2+ϑ_2 cosϑ_3 ∂_ϑ_3) ,
    𝒫^2   =-i(  sinϑ_3/sinϑ_2 ∂_ϑ_1+cosϑ_3∂_ϑ_2-ϑ_2 sinϑ_3 ∂_ϑ_3) ,
    𝒫^3   =-i ∂_ϑ_3 .

Similarly let us define 𝒢_A and 𝒮_A as the bosonic parts of the the G_A and S_A operators. The 𝒢_A are related to the 𝒥^i by replacing ϑ_i→φ_i. It is easy to guess that the 𝒮_A are then related to the 𝒫^i via the same replacement.

We are now ready to give expressions for the momentum operators and the kinetic energy operator in terms of the new variables. These are <cit.>:

    -i∂_X^i_A   =-iL_AaM^ib{δ_ab ∂_𝐱_a+iϵ_abc/𝐱_a^2-𝐱_b^2(𝐱_a 𝒫^c+𝐱_b 𝒮_c)} ,
    
    -1/2∂_X^i_A∂_X^i_A   =-1/2Δ∂_𝐱_aΔ∂_𝐱_a+1/2∑_a=1^3I_a(𝒫^a2+𝒮_a^2)+2K_a 𝒫^a𝒮_a/I_a^2-K_a^2 .

It is also straightforward to write down the bosonic potential V in terms of the new variables:

    V=1/2m^2 𝐱_a 𝐱_a+3g m 𝐱_1𝐱_2𝐱_3+g^2/2(𝐱_1^2𝐱_2^2+𝐱_1^2𝐱_3^2+𝐱_2^2𝐱_3^2) .

As expected it is independent of the angular variables. We have depicted constant potential surfaces in figure <ref>.

Apart from the coordinates 𝐱_a the following non-linear coordinates will often appear in the equations below:

    𝐲_a≡I_a/I_a^2-K_a^2=1/2|ϵ_abc|𝐱_b^2+𝐱_c^2/(𝐱_b^2-𝐱_c^2)^2 ,        𝐳_a≡K_a/I_a^2-K_a^2=|ϵ_abc|𝐱_b 𝐱_c/(𝐱_b^2-𝐱_c^2)^2 .

With these definitions the kinetic term can be written as:

    -1/2∂_X^i_A∂_X^i_A=-1/2Δ∂_𝐱_aΔ∂_𝐱_a+1/2[𝐲_a(𝒫^a2+𝒮_a^2)+2 𝐳_a 𝒫^a𝒮_a] .

Notice that the term ∑_a=1^3𝐲_a 𝒫^a2 is the kinetic energy of a rigid rotor with principal moments of inertial 𝐲_a^-1.

Unlike the c=1 matrix model, the angular-independent piece of the kinetic term can not be trivialized by absorbing a factor of √(Δ) into the wavefunction <cit.>. Instead we have:

    -1/2Δ∂_𝐱_aΔ∂_𝐱_a=-1/2(1/√(Δ)∂_𝐱_a^2√(Δ)+T) ,

where

    T≡∑_a=1^3𝐲_a=𝐱_1^2+𝐱_2^2/(𝐱_1^2-𝐱_2^2)^2+𝐱_1^2+𝐱_3^2/(𝐱_1^2-𝐱_3^2)^2+𝐱_2^2+𝐱_3^2/(𝐱_3^2-𝐱_3^2)^2 ,

and its appearance in the Schrödinger equation acts as an attractive effective potential between the 𝐱_a.



 §.§ Gauge-invariant fermions

Because the operators G_A in (<ref>) have a nontrivial dependence on the gauginos λ_Aα it is not sufficient to suppress the wavefunction's dependence on gauge angles φ_i entirely. Instead we can write down a set of gauge-invariant fermions that will contain the entire dependence on the gauge angles <cit.>:

    χ_Aα≡ L_BAλ_Bα ,      χ̅_A^β≡ L_BAλ̅_B^β .






These satisfy {χ_Aα,χ̅_B^β}=δ_ABδ_α^ β, but no longer commute with bosonic derivatives.
Defining  σ̃^i β_α≡ M^jiσ^j β_α, we can now write the supercharges in terms of the new parametrization. These are:

    Q_α=-i σ̃_α^b γχ_aγ(δ_ab{∂_𝐱_b+m 𝐱_b+g/2|ϵ_bst| 𝐱_s𝐱_t}+iϵ_abc/𝐱_a^2-𝐱_b^2(𝐱_a 𝒫^c+𝐱_b 𝒮_c)) ,
    Q̅^β=-i χ̅_a^γ σ̃_γ^b β(δ_ab{∂_𝐱_b-m 𝐱_b-g/2|ϵ_bst| 𝐱_s𝐱_t}+iϵ_abc/𝐱_a^2-𝐱_b^2(𝐱_a 𝒫^c+𝐱_b 𝒮_c)) ,

where we have put the gauge-invariant fermions to the left so as to remind the reader that the bosonic derivatives are not meant to act on them in the supercharges. The Hamiltonian H (not H_m) in the new parametrization is:

    H=-1/2Δ∂_𝐱_aΔ∂_𝐱_a+1/2[𝐲_a(𝒫^a2+𝒮_a^2)+2 𝐳_a 𝒫^a𝒮_a]
    +1/2m^2 𝐱_a 𝐱_a+3g m 𝐱_1𝐱_2𝐱_3+g^2/2(𝐱_1^2𝐱_2^2+𝐱_1^2𝐱_3^2+𝐱_2^2𝐱_3^2)-3/4m[χ̅_A,χ_A]+ig ϵ_AkCχ̅_A 𝐱_k σ̃^kχ_C .




§ NUMERICAL RESULTS


In order to calculate the spectrum of the Hamiltonian (<ref>), we must reduce our problem using symmetry, that is we should  label our states via the maximal commuting set of conserved quantities: H_m, J^3 ,J⃗^ 2, R. Because of the discrete particle-hole symmetry (<ref>) we need only consider R=0,…,3. In appendix <ref> we construct gauge-invariant highest-weight representations of SO(3)_J in each R-charge sector. This means we fix the wavefunctions' dependence on the angles ϑ_i and φ_i and provide the reduced Schrödinger operators that depend only on 𝐱_a.[We only provide a small set of these reduced Schrödinger operators, as they increase in size with increasing SO(3)_J eigenvalue j.]

Our numerical results for the lowest energy states of H_m for each R and j are presented in Table <ref> and were obtained by inputting the restricted Schrödinger equations of appendix <ref> into Mathematica's NDEigenvalues command, which uses a finite element approach to solve for the eigenfunctions of a coupled differential operator on a restricted domain. We have labeled each row by the fermion number R and each column by the SO(3)_J highest weight eigenvalue j (i.e. J⃗^2|ψ⟩=j(j+1)|ψ⟩ and J^3|ψ⟩=j|ψ⟩).

A few comments are in order:


  * The most striking feature of these plots is the seeming appearance of zero energy states for (R,j)=(2,0) and (R,j)=(3,1/2) as ν→0. Since the Witten index W_I=0, and since the states in the (2,0) and (3,1/2) sectors seem have nonzero energy for any finite ν, it must be the case that these states are elements of the same supersymmetry multiplet. This must be so for the deformation invariance of W_I.


  * Since we know, by construction, that the lowest energy (R,j)=(2,0) and (R,j)=(3,1/2) states are related by supersymmetry, we can use the difference in their numerically-obtained energies as a benchmark of our numerical errors. Obtaining the (R,j)=(2,0) ground state energy required solving a coupled Schrödinger equation involving 15 functions in 3 variables. For the (R,j)=(3,1/2) state, the number of functions one is numerically solving for jumps to 40. In the latter case, it was difficult to reduce our error (either by refining the finite element mesh, or increasing the size of the domain) in a significant way without Mathematica crashing. This is despite the fact that we had 12 cores and 64 Gb of RAM at our disposal. In figure <ref> we plot the percentage error in the H_m energy difference between these two states as a function of ν. We find that the energy difference between these states is around 13% of the total energy as a function of ν. For comparison, we also do this for the lowest (R,j)=(0,0) and (R,j)=(1,1/2) states, where the numerics are more reliable as a result of solving a much simpler set of equations. There the difference between the computed energies is at most 2%.
 


  * Our results suggest that there are 4 supersymmetric states, two of which are bosonic and two which are fermionic, which would cancel in the evaluation of the index. Explicitly, the two bosonic states are j=0 singlets in the R=2 and R=4 sectors (recall the discrete particle hole symmetry of the theory) and the two fermionic states are the j=1/2 doublet in the R=3 sector. It is interesting to note that there aren't more states in this multiplet, for example numerically studying the (R,j)=(0,1) sector reveals no evidence for a supersymmetric state in the ν→ 0 limit.   


  * The massless SU(2) model was studied using a different numerical approach in <cit.> and their plots for the ground state energies seem to approach ours, particularly figures 2 and 5 of <cit.>.


  * Our numerical evidence for these supersymmetric states does not constitute a proof since we will never be able to numerically resolve if this state has exactly zero energy. However, the result is highly suggestive of a supersymmetry preserving set of states at ν=0 and there is no contradiction with the analytically obtained Witten index result W_I=0. It would be interesting to analyze the existence of these states analytically in future work.








§ EFFECTIVE THEORY ON THE MODULI SPACE OF THE SU(2) MODEL


In order to get a better handle on the previous section's numerical results, we will now study the ν→0 limit of the matrix model analytically. Since the full problem is clearly quite difficult even for N=2, we will study the massless model in some parametric limit. This is possible because the theory has a moduli space[Also sometimes called a Coulomb branch.]—a flat direction where the D-branes can become well separated, and along this moduli space certain fields become massive and can be integrated out. We will parametrize this moduli space by the coordinates (𝐱_3,ϑ_2,ϑ_1) and will henceforth label them (𝐱_3,ϑ_2,ϑ_1)→(r,θ,ϕ) for the remainder of this section. The parametric limit we will take is the limit of large r.

To derive the effective theory along the moduli space we will first take (r,θ,ϕ) to be slowly varying and expand H=H^(0)+H^(1)+… in inverse powers of the dimensionless quantity g r^3. We will compute the effective Hamiltonian in perturbation theory by integrating out the other fields in their ground state, in which (r,θ,ϕ) appear as parameters. Similar analysis to this was performed in <cit.>.  Defining ∂⃗≡(∂_𝐱_1,∂_𝐱_2), the Hamiltonian, to lowest order, is

    H^(0)≡-1/2(𝐱_1^2-𝐱_2^2)∂⃗·(𝐱_1^2-𝐱_2^2)∂⃗-1/2(𝐱_1^2-𝐱_2^2)^2[(𝐱_1^2+𝐱_2^2)(∂_ϑ_3^2+∂_φ_3^2)+4 𝐱_1 𝐱_2 ∂_ϑ_3∂_φ_3]
    +g^2/2r^2(𝐱_1^2+𝐱_2^2)-i g r ϵ_3DE χ̅_D σ̃^3 χ_E ,

where σ̃^i β_α≡ M^jiσ^j β_α depends explicitly on (ϑ_1,ϑ_2,ϑ_3). It is straightforward to show that H^(0) admits a zero energy ground state given by:

    Ψ^(0)=g r/π√(32) e^-g/2 r (𝐱_1^2+𝐱_2^2)∑_B=1^2{χ̅_B ϵ χ̅_B-i∑_C=1^2ϵ_3BC χ̅_B σ̃^3(χ̅_Cϵ)}|0⟩ ,

where |0⟩ is the fermionic vacuum and we have normalized Ψ^(0) with respect to

    ∫_0^∞ d𝐱_1∫_-𝐱_1^𝐱_1d𝐱_2∫_0^2πdϑ_3∫_0^2πdφ_3 (𝐱_1^2-𝐱_2^2) .

Similarly we can expand the supercharges Q_α= Q_α^(0)+Q_α^(1)+…, where

    Q^(0)_α   ≡ -i∑_a,b=1^2σ̃_α^b γχ_aγ(δ_ab{∂_𝐱_b+g/2|ϵ_bst| 𝐱_s𝐱_t}+iϵ_abc/𝐱_a^2-𝐱_b^2(𝐱_a 𝒫^c+𝐱_b 𝒮_c)) ,
    
    Q^(1)_α   ≡-i σ̃_α^b γχ_3γ(δ_3b{∂_r+g/2|ϵ_3st| 𝐱_s𝐱_t}+iϵ_3bc/r(𝒫^c+𝒮_c)) .

It is easy to check that Q^(0)_αΨ^(0)=Q̅^(0)βΨ^(0)=0. We are now tasked with finding the effective supercharges Q^ eff_α= ⟨ Q^(1)_α⟩_Ψ^(0)+… that act on the massless degrees of freedom (r,θ,ϕ) along the moduli space. At lowest order we find the supercharges (acting on gauge-invariant wavefunctions) are those of a free particle in R^3 and its fermionic superpartner:

    Q_α^ eff=-i ∇_𝐱⃗·σ⃗_α^  γ ψ_γ ,      Q̅^β_ eff=-i ψ̅^γ∇_𝐱⃗·σ⃗_γ^  β ,

where we have labeled (r,θ,ϕ) in cartesian coordinates as well as defined (ψ_α,ψ̅^β)≡(χ_3 α,χ̅_3^β). Since the remaining gauge angles (φ_1,φ_2) have no kinetic terms in the effective theory along the moduli space, we need not consider them as dynamical variables and can treat ψ_α as a fundamental field.

Let us now compute the effective theory to next order in perturbation theory. Instead of computing this in the operator formalism, let us first invoke symmetry arguments to constrain what the answer should look like. The low energy effective theory on the moduli space should be a supersymmetric theory with four supercharges and an SO(3) R-symmetry, therefore it should fall in the class discovered in <cit.>:

    ℒ=1/2f(𝐱̇⃗̇^2+i(ψ̅ψ̇-ψ̇̅̇ψ) +D^2)+1/2(∇_k f) ϵ_klm 𝐱̇^l ψ̅ σ^m ψ
    -D/2 (∇_𝐱⃗f)·ψ̅ σ⃗ ψ+1/4(∇_i∇_j f)(ψ̅ σ^i ψ)(ψ̅ σ^j ψ) ,

which is invariant under

    δ𝐱⃗   =iψ̅ σ⃗ ξ-iξ̅ σ⃗ ψ
    δψ_α   =𝐱̇⃗̇·σ⃗_α^  βξ_β+iD ξ_α
    δψ̅^β   =𝐱̇⃗̇·ξ̅^ασ⃗_α^  β-iD ξ̅^β
    δ D   =-ψ̇̅̇ ξ-ξ̅ ψ̇ .

In order to preserve the SO(3) symmetry f should be a function of r≡ |𝐱⃗|. Notice that (<ref>) reduces to the theory of a free particle and its superpartner when f=1. Therefore we should find that at 1-loop order f=1+c/g r^3, since (g r^3)^-1 is our expansion parameter, with c to be determined. A calculation <cit.> reproduced in appendix <ref> gives c=-3/2 or

    f=1-3/2g r^3 .

Analytic evidence for the numerically found supersymmetric ground states can be obtained by studying the Schrödinger problem associated with (<ref>). We do not do this here, but we can gain some intuition by studying the existence of normalizable zero-modes of the Laplacian on moduli-space <cit.>:

    ds^2=(1-3/2g r^3)(dr^2+r^2 dΩ_2^2) .

We can construct two normalizable zero-modes as follows. The zero-form

    ω_0≡∫^r dr' 1/r'^2(1-3/2g r'^3)^-1/2

is a zero-mode of the Laplacian, but is not normalizable. To construct normalizable forms, we take

    ω_1≡ dω_0 ,    ω_2=⋆ ω_1 .

These are normalizable within the domain r∈[(3/2g)^1/3,∞]. Since there exists zero-modes in this toy-moduli-space approximation, it would be interesting to study the set of ground states of (<ref>) in more detail.



§ DISCUSSION

In this paper we have studied the mini-BFSS/BMN model with gauge group SU(2) and uncovered numerical evidence for a set of supersymmetric ground states in the massless limit of the theory. In the massless limit the matrices can become widely separated. The effective theory on the moduli space has non-trivial interactions governed by a metric that gets generated on this moduli space at one loop. Let us now discuss what may happen in the SU(N) case at large N. The quartic interaction in (<ref>) can be rewritten as a commutator-squared interaction (f_ABC X^i_B X^j_C)^2∼Tr([X^i,X^j])^2, where X^i≡ X^i_A τ_A and τ_A are the generators in the fundamental of SU(N). Therefore, at tree-level, along the moduli space there will be a set of N-1 massless, non-interacting, point particles in R^3 (and their superpartners), each one corresponding to an element of the Cartan of SU(N). At one-loop there will be a correction to the moduli space-metric, depending on the relative distances between these particles. Just like in the SU(2) case these corrections will come at order |r_a-r_b|^-3. One difference, however, is that there may be an enhancement of order N to this correction. It would certainly be interesting to see if we can isolate the |r_a-r_b|^-3 corrections to the moduli space metric by taking a large N limit, as can be done in the D0-D4 system <cit.> and in the three-node Abelian quiver <cit.>. Perhaps we can adapt the methods in <cit.> for these purposes. The analysis in <cit.> seems to suggest that such a decoupling limit at large N is possible .

Interestingly, it was shown in <cit.> that the one-loop effective action on the Coulomb branch of the three-node Abelian quiver exhibits an emergent conformal symmetry at large N. This conformal symmetry depends on the delicate balance between the form of the interaction potential and the metric on the moduli space, which has a similar |r_a-r_b|^-3 form as in (<ref>). It would be interesting to establishing whether the SU(N) generalization of the model studied in this paper also has a non-trivial conformal symmetry at infinite N, broken by finite N effects. We save this problem for future work, but list here some reasons why this would be worth studying:


  * The BFSS matrix model has a holographic interpretation <cit.>. At large N it is dual to a background of D0 branes in type IIA supergravity. In BFSS there is no correction to the moduli space metric and neither side of this duality is conformal. The BFSS matrix model is thus a theory of the 10d flat space S-matrix. It would be interesting to understand the large N version of mini-BFSS in the context of holography along similar lines. Because of the large number of coupled degrees of freedom at large N and the reduced supersymmetry, the effective theory along the moduli space of mini-BFSS has a non-trivial metric and may potentially exhibit a non-trivial conformal fixed point along this moduli space, as happens for quiver quantum mechanics models with vector rather than matrix interactions <cit.>. To answer this question definitively we will need to compute the effective theory along the Coulomb branch for N≫ 2 and check whether it is conformal. 


  * New results have shown that a certain class of disordered quantum mechanics models, known as SYK for Sachdev-Ye-Kitaev, exhibits phenomenology of interest for near-extremal black holes (see <cit.> and references therein as well was <cit.> for models without disorder).  These phenomena include an emergent conformal symmetry in the IR, maximal chaos <cit.>, and a linear in T specific heat. Despite the successes of these models, they are not dual to weakly coupled gravity. BFSS is a large N gauged matrix quantum mechanics dual to weakly-coupled Einstein gravity but, as we previously mentioned, it does not have an emergent conformal symmetry and remains a model of D-particles in flat space. It would certainly be interesting if mini-BFSS fell in the universality class of quantum mechanics models with emergent conformal symmetry in the IR and maximal chaos, such as the SYK model and its non-disordered cousins, but remains dual to weakly coupled gravity. Recently <cit.> advocated the study of such matrix models for similar reasons. In the same vein <cit.> studies classical chaos in BFSS numerically.


  * If this model, like SYK, is at all related to the holography of near-extremal black holes, then we can try to study its S-matrix to gain some insight into the real time dynamics of black hole microstates. A numerical implementation of such a study in the context of similar supersymmetric quantum mechanics models with flat directions can be found in <cit.>.


  * The slow moving dynamics of a class of BPS multi-black hole solutions in supergravity is a superconformal quantum mechanics <cit.> with no potential, provided a near horizon limit is taken. It would be interesting to understand if there is some limit in which the multi-black hole moduli space quantum mechanics and the large N matrix quantum mechanics on the moduli space coincide. Perhaps as a consequence of non-renormalization theorems as in <cit.>.












§ ACKNOWLEDGEMENTS

It is a pleasure to thank Dionysios Anninos, Frederik Denef, Felix Haehl, Rachel Hathaway, Eliot Hijano, Jaehoon Lee, Eric Mintun, Edgar Shaghoulian, Benson Way and Mark Van Raamsdonk for helpful discussions. We are particularly indebted to Dionysios Anninos, Frederik Denef, and Edgar Shaghoulian for their comments on an early draft. We made heavy use of Matthew Headrick's grassmann.m package. C.C. would like to thank David and Gay Cogburn for their support. T.A. is supported in part by the U.S. Department of Energy under grant Contract Number DE-SC0012567, by the Natural Sciences and Engineering Research Council of Canada, and by grant 376206 from the Simons Foundation.





§ REDUCED SCHRÖDINGER EQUATION

In this appendix we construct gauge-invariant highest-weight wavefunctions of SO(3)_J in each R-charge sector (up to 3) and use these to maximally reduce the Schrödinger equation via symmetries.


 §.§ R=0

This sector of the theory was studied in <cit.>, although without access to numerics. We repeat their analysis here. We wish to separate variables using the SO(3)_J symmetry. We therefore want to write down the highest weight state satisfying J^3|ψ⟩_0=j|ψ⟩_0 and J^+|ψ⟩_0=0 , with J^±≡ J^1± i J^2. The rest of the spin multiplet can be obtained by acting on |ψ⟩_0 with J^- up to 2j times. This however doesn't entirely fix the angular dependence of the wavefunction, as these two conditions only fix the dependence on up to two angles. Recall, however, that  the operators P⃗ commute with J⃗ and P⃗^2=J⃗^2, but [H,P⃗]≠ 0. We will then write |ψ⟩_0 as a sum of terms with definite P^3 eigenvalue. That is, we write |ψ⟩_0 as:

    |ψ⟩_0=e^i j ϑ_1sin^jϑ_2∑_p=-j^j e^i p ϑ_3^p(ϑ_2/2) f^p(𝐱_a) .

Since the number of terms in the wavefunction grows with j it will be cumbersome to give the reduced radial Schrödinger equation for arbitrary j. Instead we will give the expressions for j=0,1/2,1.
Before giving the reduced Schrödinger equations it is worth noting that it has long been known that there exists no supersymmetric states in this sector <cit.>. The reason is that the supersymmetry equations Q_α|ψ⟩_0=Q̅^β|ψ⟩_0=0 are easy to solve and give

    |ψ⟩_0^ SUSY∼exp{ g 𝐱_1 𝐱_2 𝐱_3+m/2𝐱_a 𝐱_a} ,

which is non-normalizable. It is also known that the spectrum in this sector is discrete <cit.>.

For parsimony let us define

    ℋ̂≡-1/2Δ∂_𝐱_aΔ∂_𝐱_a+V

with V defined in (<ref>).
Then for j=0 the reduced Schrödinger equation, obtained from H_m|ψ⟩_0=ℰ_m|ψ⟩_0, is simply

    (ℋ̂+9/2m)f^0(𝐱_a)=ℰ_m f^0(𝐱_a) .

For j=1/2 there is no mixing between the f^±1/2(𝐱_a) and each satisfies

    (ℋ̂+9/2m+T/8)f^±1/2(𝐱_a)=ℰ_m f^±1/2(𝐱_a) ,

where T was defined in (<ref>). Finally, for j=1 we have

    {ℋ̂+9/2m+T/4+1/4[     𝐲_3       0 𝐲_1-𝐲_2;       0 T-2 𝐲_3       0; 𝐲_1-𝐲_2       0     𝐲_3 ]}[ f^-1;  f^0; f^+1 ]=ℰ_m[ f^-1;  f^0; f^+1 ] .




 §.§ R=1

Continuing on from the last section, we want to write down wavefunctions in the R=1 sector that are gauge invariant, and satisfy J^3|ψ⟩_1=j|ψ⟩_1 and J^+|ψ⟩_1=0. To do so, we will write our wavefuntions as a

    |ψ⟩_1=e^i j ϑ_1sin^jϑ_2∑_p=-j^j e^i p ϑ_3^p(ϑ_2/2) f^p_Aα χ̅_A^α|0⟩ ,

where |0⟩ is the fermionic vacuum and each term in the sum has definite P^3 eigenvalue.
The functions f^p_Aα that satisfy these conditions are:

    f^p_A1   =e^-iϑ_1/2{ e^-iϑ_3/2cos(ϑ_2/2)L^p_2A-1(𝐱_a)- e^iϑ_3/2sin(ϑ_2/2)L^p_2A(𝐱_a)} , 
    
     f^p_A2   =  e^iϑ_1/2{ e^-iϑ_3/2sin(ϑ_2/2)L^p_2A-1(𝐱_a)+ e^iϑ_3/2cos(ϑ_2/2)L^p_2A(𝐱_a)} .







We remind the reader that the χ̅_A^α are the gauge-invariant fermions defined in (<ref>). The reduced Schrödinger equation for j=0 (and hence p=0) is

    {ℋ̂+7/2m+5/8T+𝐀
    }[ L_1^0;     ⋮; L_6^0 ]=ℰ_m[ L_1^0;     ⋮; L_6^0 ]

where 𝐀 is a 6× 6 matrix that can be written in terms of 2× 2 blocks as follows

    𝐀≡i/2[            i 𝐲_11 -(2 g 𝐱_3+𝐳_3)σ^3  (2 g 𝐱_2+𝐳_2)σ^2;  (2 g 𝐱_3+𝐳_3)σ^3            i 𝐲_21 -(2 g 𝐱_1+𝐳_1)σ^1; -(2 g 𝐱_2+𝐳_2)σ^2  (2 g 𝐱_1+𝐳_1)σ^1            i 𝐲_31 ] ,

where the coordinates 𝐲_a and 𝐳_a (nonlinearly related to 𝐱_a) were defined in (<ref>).

































Using the above definitions it is straightforward to write down the equations for j=1/2. These are

    {ℋ̂+7/2m+3/4T+
    ([ 𝐀+𝐁   𝐂; 𝐂^† 𝐀-𝐁 ])
    }[ L_1^-12;       ⋮; L_6^-12;  L_1^12;       ⋮;  L_6^12 ]=ℰ_m[ L_1^-12;       ⋮; L_6^-12;  L_1^12;       ⋮;  L_6^12 ]

with

    𝐁≡1/4[   𝐲_3 σ^3 -2i 𝐳_3 1         0;  2i 𝐳_3 1   𝐲_3 σ^3         0;         0         0   𝐲_3 σ^3 ]

and

    𝐂≡1/4[ 𝐲_1 σ^1-i 𝐲_2 σ^2                 0           2 𝐳_2 1;                 0 𝐲_1 σ^1-i 𝐲_2 σ^2         -2i 𝐳_1 1;          -2 𝐳_2 1          2i 𝐳_1 1 𝐲_1 σ^1-i 𝐲_2 σ^2 ] .

























 §.§ R=2

As we can see, the number of equations keeps increasing with fermion number and spin. Therefore in this section and the next, we will only give the reduced Schrödinger equations for j=0. As before the general highest weight R=2 wavefunction admits a decomposition:

    |ψ⟩_2=e^i j ϑ_1sin^jϑ_2∑_p=-j^j e^i p ϑ_3^p(ϑ_2/2) f^p_ABαβ χ̅_A^αχ̅_B^β|0⟩ .

In order to avoid over-counting let us set f^p_ABαβ=0 whenever B<A and similarly f^p_AAαβ=0 (no sum on indices) whenever β≤α. Imposing that J^3|ψ⟩_2=j|ψ⟩_2, J^+|ψ⟩_2=0 and that each term in the sum have definite P^3 eigenvalue imposes that the functions f^p_ABαβ take on a particular form. These are (no sum on indices and A<B):

    f_AA12^p   =L^p_A(𝐱_a) ,
    
    f^p_AB12   =e^-iϑ_3/2sinϑ_2 Y^p_AB(𝐱_a)+cos^2(ϑ_2/2)  R_AB^p(𝐱_a)-sin^2(ϑ_2/2)  S_AB^p(𝐱_a)-e^iϑ_3/2sinϑ_2  U^p_AB(𝐱_a)
    
    f^p_AB21   =e^-iϑ_3/2sinϑ_2 Y^p_AB(𝐱_a)-sin^2(ϑ_2/2)  R_AB^p(𝐱_a)+cos^2(ϑ_2/2)  S_AB^p(𝐱_a)-e^iϑ_3/2sinϑ_2  U^p_AB(𝐱_a)
    
    f^p_AB11   =e^-iϑ_1{ e^-iϑ_3cos^2(ϑ_2/2)Y^p_AB(𝐱_a)-1/2sinϑ_2(R_AB^p(𝐱_a)+S_AB^p(𝐱_a))+e^iϑ_3sin^2(ϑ_2/2)U^p_AB(𝐱_a)}
    
    f^p_AB22   =  e^iϑ_1{ e^-iϑ_3sin^2(ϑ_2/2)Y^p_AB(𝐱_a)+1/2sinϑ_2(R_AB^p(𝐱_a)+S_AB^p(𝐱_a))+e^iϑ_3cos^2(ϑ_2/2)U^p_AB(𝐱_a)} .

Notice that even for j=0, determining the spectrum will involve solving a set of 15 coupled partial differential equations. We will label the set of functions Y_AB^p≡ Y_6-A-B^p and so on for the remaining functions. We also define the following vector of functions:

    Ψ^0_R=2≡(L_1^0,…,R_1^0,…,S_1^0,…,U_1^0,…,Y_1^0,…)^T .

The j=0 Schrödinger equation is then:

    {ℋ̂+5/2m+3/4T+𝐃+𝐋+g𝐌}Ψ^0_R=2
    
    
    
    
    
    
    
    
    
    
    
    =ℰ_m Ψ^0_R=2

where 𝐃, 𝐋 and 𝐌 are 15× 15 matrices that can be written in terms of 3× 3 blocks as follows

    𝐃≡[           𝐝^1             0             0             0             0;             0           𝐝^3    𝐝^1-𝐲_3/41             0             0;             0    𝐝^1-𝐲_3/41           𝐝^3             0             0;             0             0             0          -𝐝^3 1/4(𝐲_1-𝐲_2)1;             0             0             0 1/4(𝐲_1-𝐲_2)1          -𝐝^3 ]


    𝐋≡-1/2[     2∑_a𝐲_a |ℒ^a|                 0                 0                 0                 0;                 0                 0                 0 𝐳_1 ℒ^1+i 𝐳_2 ℒ^2 𝐳_1 ℒ^1-i 𝐳_2 ℒ^2;                 0                 0                 0 𝐳_1 ℒ^1+i 𝐳_2 ℒ^2 𝐳_1 ℒ^1-i 𝐳_2 ℒ^2;                 0 𝐳_1 ℒ^1-i 𝐳_2 ℒ^2 𝐳_1 ℒ^1-i 𝐳_2 ℒ^2        -2 𝐳_3 ℒ^3                 0;                 0 𝐳_1 ℒ^1+i 𝐳_2 ℒ^2 𝐳_1 ℒ^1+i 𝐳_2 ℒ^2                 0         2 𝐳_3 ℒ^3;                   ]


    𝐌≡[                 0           𝐱_3 𝐦^3           𝐱_3 𝐦^3   𝐱_1 𝐦^1+𝐱_2 𝐦^2   𝐱_2 𝐦^2-𝐱_1 𝐦^1;          𝐱_3 𝐦^3†          -𝐱_3 ℒ^3                 0           𝐱_2 𝐝^2  𝐱_2 𝐝^2†-𝐱_1 ℒ^1;          𝐱_3 𝐦^3†                 0           𝐱_3 ℒ^3 -𝐱_1 ℒ^1-𝐱_2 𝐝^2†          -𝐱_2 𝐝^2; 𝐱_1 𝐦^1†+𝐱_2 𝐦^2†          𝐱_2 𝐝^2†  -𝐱_1 ℒ^1-𝐱_2 𝐝^2           𝐱_3 ℒ^3                 0; 𝐱_2 𝐦^2†-𝐱_1 𝐦^1†  𝐱_2 𝐝^2 -𝐱_1 ℒ^1         -𝐱_2 𝐝^2†                 0          -𝐱_3 ℒ^3;                   ] .

In these definitions, the ℒ^i are the 3×3 generators of SO(3) defined below (<ref>). The 𝐝^i are

    𝐝^1≡(T/4-𝐲_a)δ_ab ,    𝐝^2≡[  0  0  0;  0  0  0; -1  0  0 ] ,    𝐝^3≡1/2(𝐲_a-1/2𝐲_3)δ_ab ,

and the 𝐦^i are

    𝐦^1≡[ 0 0 0; i 0 0; i 0 0 ] ,    𝐦^2≡[  0 -1  0;  0  0  0;  0 -1  0 ] ,    𝐦^3≡[ 0 0 i; 0 0 i; 0 0 0 ] .

Whenever a matrix appears in an absolute value symbol |·|, the absolute value is to be applied to the entries of the matrix.

The Schrödinger operator for j=1/2 will be a generalization of the above operator to one acting on 30 functions. We do not provide expressions for it here, but analyze its spectrum in the main text.



 §.§ R=3

The highest weight R=3 wavefunctions take the form

    |ψ⟩_3=e^i j ϑ_1sin^jϑ_2∑_p=-j^j e^i p ϑ_3^p(ϑ_2/2) f^p_ABCαβγ χ̅_A^α χ̅_B^β χ̅_C^γ|0⟩ .

To avoid overcounting we set

    f^p_ABCαβγ   =0    if C<B  or  B<A
    
    f^p_AABαβγ   =0    if β≤α
    
    f^p_ABBαβγ   =0    if γ≤β .

Because of the fermionic statistics f^p_AAAαβγ=0 identically. Imposing the highest weight condition forces f^p_123αβγ to take the following form

    f^p_123αβγ=∑_a,b, c=1^2 F^p_abc(𝐱_a)u_α a(ϑ⃗)u_β b(ϑ⃗)u_γ c(ϑ⃗) ,

with

    u_α a(ϑ⃗)≡ e^i/2((-1)^αϑ_1+(-1)^aϑ_3){(1-|α-a|)cos(ϑ_2/2)+(α-a)sin(ϑ_2/2)} .




Furthermore

    f_AABαβγ^p    =U^p_AAB(𝐱_a)y^1_αβγ(ϑ⃗)+Y^p_AAB(𝐱_a)y^2_αβγ(ϑ⃗)
    
    f_ABBαβγ^p    =U^p_ABB(𝐱_a)y^1_αβγ(ϑ⃗)+Y^p_ABB(𝐱_a)y^2_αβγ(ϑ⃗)

where

    y^1_αβγ(ϑ⃗)   ≡e^i/2{((-1)^α+(-1)^β+(-1)^γ)ϑ_1-ϑ_3}/2[(4-α β γ)cos(ϑ_2/2)+(α β γ-2)sin(ϑ_2/2)]
    
    y^2_αβγ(ϑ⃗)   ≡e^i/2{((-1)^α+(-1)^β+(-1)^γ)ϑ_1+ϑ_3}/2[(α β γ-4)sin(ϑ_2/2)+(α β γ-2)cos(ϑ_2/2)] .

Notice that for j=0 the reduced Schrödinger equation is a set of 20 coupled partial differential equations. We will give the Schrödinger operator acting on the following vector of functions

    Ψ_R=3^0≡(F^0_122,F^0_211,F^0_121,F^0_212,F^0_221,F^0_112,F^0_111,F^0_222,.
    . U^0_113,U^0_223,Y^0_113,Y^0_223,U_112^0,U^0_233,Y^0_112,Y^0_233,U^0_122,U^0_133Y^0_122,Y^0_133)^T .

With Ψ_R=3^0 defined, we are tasked with solving the following set of differential equations:

    {ℋ̂+3/2m+1/4(11/2T-𝐲_3)-𝐈+𝐉+𝐉^†+𝐊}Ψ_R=3^0=ℰ_m Ψ_R=3^0
     ,

where 𝐈, 𝐉, and 𝐊 are 20× 20 matrices that can be written in block form as follows:

    𝐈≡[              𝐲_11_2× 2;                                     𝐲_21_2× 2                      0;                                     𝐲_31_2× 2;                               (𝐲_1+𝐲_2)1_2× 2;                            3/4(𝐲_1+𝐲_2)1_4× 4;                                             0 1/2(T+𝐲_1-𝐲_2/2)1_4× 4;                        1/2(T-𝐲_1-𝐲_2/2)1_4× 4 ] ,

and 𝐉 and 𝐊 can be written in terms of 4×4 blocks as follows:


    𝐉≡[          0          0     𝐱_3 zz    𝐱_2  yy    𝐱_1  xx;          0          0          0   𝐱_2  yy2   𝐱_1  xx2;  𝐱_3  zz^†          0          0   𝐱_1  xx3   𝐱_2  yy3;  𝐱_2  yy^† 𝐱_2  yy2^†  𝐱_1 xx3^†          0    𝐱_3 zz2;  𝐱_1  xx^†  𝐱_1 xx2^†  𝐱_2 yy3^† 𝐱_3  zz2^†          0 ]


    𝐊≡1/4[        (T-5 𝐲_3) i1    𝐲_1  i4 +𝐲_2  i5           2𝐳_3  zz3          2 𝐳_2  yy4          2 𝐳_1  xx4; 𝐲_1  i4^†+𝐲_2  i5^†         (𝐲_1-𝐲_2)i2                   0           -2𝐳_2 yy4            2𝐳_1 xx5;         2 𝐳_3 zz3^†                   0           -4 𝐲_3 i3       2 i 𝐳_1 |xx3|          2 𝐳_2  yy5;         2 𝐳_2 yy4^†        -2 𝐳_2 yy4^†     -2i 𝐳_2 |xx3|^†           -4 𝐲_2 i3            2𝐳_3 zz4;         2 𝐳_1 xx4^†         2 𝐳_1 xx5^†         2 𝐳_2 yy5^†         2 𝐳_3 zz4^†           -4 𝐲_1 i3 ]



    𝐉≡i/2[                           0                           0          𝐳_3 𝐚^3+2g 𝐱_3 𝐛^3          𝐳_2 𝐚^2+2g 𝐱_2 𝐛^2          𝐳_1 𝐚^1+2g 𝐱_1 𝐛^1;                           0                           0                           0         -𝐳_2 𝐚^2+2g 𝐱_2 𝐜^2          𝐳_1 𝐞^1+2g 𝐱_1 𝐜^1;                           0                           0                           0      σ^1⊗(𝐳_1 1+2g 𝐱_1 σ^3) -σ^2⊗(𝐳_2 σ^1-2i g 𝐱_2 σ^2);                           0                           0                           0                           0      σ^3⊗(𝐳_3 1-2g 𝐱_3 σ^3);                           0                           0                           0                           0                           0 ] ,


    𝐊≡[ 1/4(T-5 𝐲_3)σ^1⊗σ^1    1/4(𝐲_1 𝐬+𝐲_2 𝐭)                   0                   0                   0;  1/4(𝐲_1 𝐬+𝐲_2 𝐭)^†   1/4(𝐲_1-𝐲_2)σ^1⊗1                   0                   0                   0;                   0                   0          -𝐲_3 1⊗σ^1                   0                   0;                   0                   0                   0          -𝐲_2 1⊗σ^1                   0;                   0                   0                   0                   0          -𝐲_1 1⊗σ^1;                     ] ,

where we have implicitly defined

    𝐚^1≡ -1/2[ 0 0; 1 1 ]⊗(1-σ^1)+1/2[  0  0;  1 -1 ]⊗(-i σ^2+σ^3) ,      𝐛^1≡1/2[ -1  1;  0  0 ]⊗(1+σ^1)-1/2[ 1 1; 0 0 ]⊗(i σ^2+σ^3) ,   
       𝐚^2≡ -i/2[ -1  1;  0  0 ]⊗(1-σ^1)+i/2[ 1 1; 0 0 ]⊗(-i σ^2+σ^3) ,      𝐛^2≡ -i/2[ 0 0; 1 1 ]⊗(1+σ^1)+i/2[  0  0;  1 -1 ]⊗(i σ^2+σ^3) ,   
       𝐚^3≡ i σ^2⊗[ -1  1;  0  0 ]+σ^3⊗[  0  0; -1  1 ] ,      𝐛^3≡ -1⊗[ 0 0; 1 1 ]-σ^1⊗[ 1 1; 0 0 ] ,



    𝐚^1≡ -1/2[ 0 0; 1 1 ]⊗(1-σ^1)+1/2[  0  0;  1 -1 ]⊗(-i σ^2+σ^3) ,
       𝐚^2≡ -i/2[ -1  1;  0  0 ]⊗(1-σ^1)+i/2[ 1 1; 0 0 ]⊗(-i σ^2+σ^3) ,
       𝐚^3≡ i σ^2⊗[ -1  1;  0  0 ]+σ^3⊗[  0  0; -1  1 ] ,
       𝐛^1≡1/2[ -1  1;  0  0 ]⊗(1+σ^1)-1/2[ 1 1; 0 0 ]⊗(i σ^2+σ^3) , 
       𝐛^2≡ -i/2[ 0 0; 1 1 ]⊗(1+σ^1)+i/2[  0  0;  1 -1 ]⊗(i σ^2+σ^3) , 
       𝐛^3≡ -1⊗[ 0 0; 1 1 ]-σ^1⊗[ 1 1; 0 0 ] ,


as well as

    𝐜^1≡1/2[  0  0;  1 -1 ]⊗(1+σ^1)+1/2[ 0 0; 1 1 ]⊗(i σ^2+σ^3) , 
       𝐜^2≡ -i/2[ 0 0; 1 1 ]⊗(1+σ^1)-i/2[  0  0;  1 -1 ]⊗(i σ^2+σ^3) , 
       𝐞^1≡ -1/2[ 1 1; 0 0 ]⊗(1-σ^1)+1/2[ -1  1;  0  0 ]⊗(-i σ^2+σ^3) ,

and finally

    𝐬≡[ 1 1; 0 0 ]⊗1+[  0  0; -3  1 ]⊗σ^1      and      𝐭≡[ -3  1;  0  0 ]⊗1+[  0  0;  1 -1 ]⊗σ^1 .

The Hamiltonian acting on the R=3, j=1/2 wavefunction will be a generalization of the above operator to one acting on 40 functions. We will not give the expression here, but we analyze the spectrum of the R=3, j=1/2 sector numerically in the main text.



§ METRIC ON THE MODULI SPACE

In order to determine the one-loop effective action for the ν=0 theory, we follow <cit.> and pass to the Lagrangian formulation of our gauge-quantum mechanics, including gauge-fixing terms and ghosts. We will use the background field method <cit.>—that is we will expand the fields X^i_A=B^i_A+X̃^i_A where B^i_A is a fixed background field configuration and X̃^i_A are the fluctuating degree of freedom. We choose B^i_A=δ_A3 𝐱⃗ such that it parametrizes motion along the moduli space.

The gauge-fixed Lagrangian is:

    ℒ=ℒ_ bos.+ℒ_ ferm.+ℒ_ g.f.+ℒ_ ghost

with

    ℒ_ bos.   =1/2(𝒟_t X^i_A)^2-g^2/4(f_ABC X^i_B X^j_C)^2
    ℒ_ ferm.   =i(λ̅_A 𝒟_tλ_A-g f_ABCλ̅_A X^k_B σ^kλ_C)
    ℒ_ g.f.   =-1/2ξ(𝒟_t^ bg A_A)^2
    
    ℒ_ ghost   =c̅_A(-δ_AB ∂_t^2-g f_ACB ∂_t(A_C·)+g^2 f_ACDf_DEBB^i_C X^i_E)c_B

and

    𝒟_t X^i_A≡Ẋ^i_A+g f_ABC A_B X^i_C ,    𝒟_t λ_Aα≡λ̇_Aα+g f_ABC A_B λ_Cα ,    𝒟_t^ bg A_A≡-Ȧ_A+g f_ABC B^i_B X^i_C .

 We further set ξ=1, corresponding to Feynman gauge.
 We can obtain the correction to the metric on moduli space by choosing a background field 𝐱⃗ as follows <cit.>

    𝐱⃗=(b,vt,0)

where b is to be thought of as an impact parameter for a particle moving at speed v. We now Wick rotate t→ -iτ, v→ iγ and A_A→ i A_A and expand the action to quadratic order in fluctuating fields about the background field B^i_A=δ_A3 𝐱⃗. The idea is to integrate out all fields that obtain a mass, through interaction with the background field, at one loop.

Following this procedure, it is easy to show that all fields with color index A=3 remain massless, while the rest obtain time dependent masses. After diagonalizing the mass-matrix for the bosonic fields we find the contribution to the Euclidean effective action coming respectively from the bosonic, fermionic and ghost determinants are:

    δ S_E^ bos.=   -2 Tr log(-∂_τ^2+g^2(b^2+γ^2τ^2))- Tr log(-∂_τ^2+g^2(b^2+γ^2τ^2)-2g γ)
    
                                                                           - Tr log(-∂_τ^2+g^2(b^2+γ^2τ^2)+2g γ)
    δ S_E^ ferm.=    Tr log[        ∂_τ -g(γ τ+ib); -g(γ τ-ib)        ∂_τ ]+Tr log[       ∂_τ g(γ τ-ib); g(γ τ+ib)       ∂_τ ] 
    δ S_E^ ghost=    2 Tr log(-∂_τ^2+g^2(b^2+γ^2τ^2)) .

Note that the ghost determinant cancels against the contribution coming from four of the eight massive bosons. Up to a diverging constant, which will cancel between the bosonic and fermionic terms, we can replace log(λ)=-∫_0^∞ds/se^-s λ and, summing over the spectra of the above differential operators, we find

    δ S_E   =∫_0^∞ds/s e^-b^2g^2s(cosh(2g γ s) csch(g γ s)-(g γ s))
       =∫_0^∞ds/s e^-b^2g^2ssech(g γ s/2) sinh(3g γ s/2) .

Let us now wick rotate back to Lorentzian time and use:

    e^-b^2g^2s/s=∫dt/√(π s)g v e^-s g^2 r^2 ,       r^2=b^2+v^2t^2 ,

to write down the Lorentzian action to O(v^2):

    i S_L   =i∫ dt[ v^2/2-g v ∫ds/√(π s)e^-s g^2 r^2sec(g v s/2) sin(3g v s/2)]
       =i∫ dt 1/2(1-3/2 g r^3)v^2+O(v^4) ,

which is the same correction as found in <cit.>. It also resembles the correction to the moduli space metric in the D0-D4 system <cit.>, albeit with a different coefficient and sign.

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