K-theoretic Global Symmetry in String-constructed QFT and T-duality
Hao Y. Zhang









Kavli Institute for the Physics and Mathematics of the Universe (WPI),

University of Tokyo, Kashiwa, Chiba 277-8583, Japan




We propose that generalized symmetries in some string-constructed QFTs are given by twisted K-theory, so as to be manifestly consistent under T-duality. We thus have even-form and odd-form symmetries determined by K_N( X), the twisted K-theory as D-brane charges on the asymptotic boundary X of internal geometry X with twist class N. For these QFTs, “p-form symmetries" are no longer separately well-defined for individual p, but instead mixed up. We discuss 6D ADE-type (2,0) SCFTs and some 6d (1,0) LSTs as examples, and demonstrate their twisted K-theoretic symmetries to be consistent with T-duality.


K-theoretic Global Symmetry in String-constructed QFT and T-duality
    Hao Y. Zhang
    April 29, 2024
===================================================================





§ INTRODUCTION


Generalized symmetries <cit.> are symmetries that act on not only point objects but also extended objects. Such symmetries are defined to be the topological operators that, after crossing a charged object, exert phase shift on them. The initial generalization is p-form symmetry acting on spacetime p-dimensional operators or spatially p-dimensional excitations. Later, a further generalization is higher-group symmetries <cit.> which can be seen as a non-trivial extension of different p-form symmetries. 

The usual assumption is that p-form symmetries are separately well-defined for each p, and charged objects with different dimensionalities are also distinct. However, in this letter, we propose yet another fundamental generalization by considering symmetries whose symmetry operators and charged objects are not restricted to a specific dimensionality. Indeed, in strongly coupled QFTs constructed from string theory, sometimes non-dynamical defects of different dimensionality can be charged under the same symmetry group. Therefore, the symmetry operators themselves no longer wrap ordinary (co)homology of the spacetime manifold, but instead wrap some generalized (co)homology of the spacetime. In addition, the algebraic object characterizing such symmetry should also be given by suitable generalized (co)homology of the internal manifold. 

We take a preliminary step by focusing on a family of compactifications of type II string theory that admit no type I/heterotic constructions, where we argue that the generalized symmetries are described by K-theory. Our key evidence comes by placing string duality at a central role. Specifically, we insist that the generalized global symmetry in the compactified theories must be compatible with T-duality. We take the viewpoint that in such string-constructed QFTs, generalized symmetry operators also admit natural top-down constructions (see <cit.> in the context of holography and <cit.> in the context of geometric engineering). The innocent-looking consistency requirement under T-duality between pairs of type-II string theories leads to a dramatic consequence that one can no longer describe generalized symmetries as branes wrapping ordinary homological cycles. Instead, we must take the twisted K-theory charge of branes and reduce it on twisted K-theory cocycles of the boundary of internal manifolds. The fact that brane charges and RR-fields are described in K-theory was discovered in the 90s in <cit.>. But in the literature of generalized symmetries, the usage of K-theory remains at a conceptual level <cit.> or in specific examples (see <cit.> where twisted K-theory was introduced in an example to obtain a non-trivial homology in 6D = (2,0) and thus 4D = 4 SYM, and <cit.> proposing to use a KO-theoretic field for anomaly cancellation). In particular, the process of open-string tachyon condensation was built into the K-theory description of D-brane charges, whose counterpart for topological operators is carefully analyzed in <cit.>. The terminology of K-theoretic symmetry also appeared in the review <cit.>. 

It might appear that the notion of mixing p-form symmetries by twisted K-theory is not new, since 2-group symmetries also admit a somewhat similar mixing <cit.>. We now emphasize the distinction between K-theoretic global symmetries against (discrete) 2-group global symmetries. Physically, discrete 2-group symmetry is the statement that the F-move of symmetry operators of the same form degree produces a higher-codimensional symmetry operator. On the other hand, K-theoretic global involves examining symmetry operators of different dimensionalities modulo an equivalence relation. Mathematically, discrete higher group symmetries generically involve the interplay between p-form symmetry and (p+1)-form symmetry by an extension, whereas K-theoretic global symmetries only mix a set of q-form symmetries via the Chern Character, where q are either all even or all odd. From the UV point of view, 2-Group symmetry also admits a different string-theory construction <cit.> using Mayer-Vietoris sequence on the asymptotic boundary of the internal manifold.

The rest of this letter is organized as follows. After giving backgrounds for defect groups and higher-form symmetries in geometric engineering, we explain that the same answer cannot be reproduced in the dual brane constructions. We conclude that characterizing generalized symmetries with (twisted) ordinary cohomology is incompatible with T-duality. We then propose that the generalized symmetries in type II compactifications should be described by twisted K-theory, whose compatibility with T-duality is well-established. We illustrate our proposal by giving examples of 6D (2,0) SCFTs and some (1,0) LSTs that admit no type I or heterotic constructions. In some (2,0) cases, we point out that the K-theoretic symmetries have been explicitly reproduced from the worldsheet. We proceed to the physical implication of such a K-theoretic description of generalized symmetries in terms of symmetry operators and charged states. [We relegate some technical material to the Supplemental Material, including a careful treatment of K-theoretic global symmetries in 6D (2,0) of DE-type, and the way to see the twisted K-theoretic symmetry from the string worldsheet.]




§ REVIEW AND MOTIVATION



We start from a relative QFT which only has a partition vector as opposed to a partition function <cit.>, which contains non-dynamical (“heavy") objects charged under the defect group <cit.>. On the defect group, there is a fractional valued Dirac pairing encoding the Heisenberg flux non-commutativity of these non-dynamical defect operators. We then pick a maximal non-commuting subset of flux operators to make it an absolute QFT, whenever possible. See <cit.> for the first set of such examples in 4D = 4 SYM, and <cit.> for the explicit procedure of producing absolute theories from the defect group. 

In geometric engineering of QFT via string theory, the defect group is constructed by wrapping branes on relative homological cycles of the internal manifold γ_rel∈ H_*(X_D-d, ∂ X_D-d) modulo 't Hooft screening by (extended) dynamical objects <cit.>:
    𝔻 = ⊕_nH_n(X_D-d, ∂ X_D-d)/H_n(X_D-d),
where we sum over all available branes in string theory (e.g., Dp for p even in IIA, p odd in IIB; M2 and M5 in M-theory). The procedure of picking a polarization has been explained in the context of type IIB string compactification in <cit.>. We remark that having the same defect group in the relative QFT is a necessary condition for having the same higher-form symmetries in the absolute QFT (whenever possible) after picking polarizations. 

The above screening argument provides us with the equivalence classes of charges of the heavy defects, and H_*( X_D-d) appeared from the charged defect perspective as a consequence of the long exact sequence of relative (co)homology. But one can understand the result more physically: H_*( X_D-d) label the charge of topological operators, which are directly obtained by branes wrapping the cycles of the asymptotic boundary ∂ X_D-d of the internal manifold X_D-d (<cit.>).



 §.§ T-duality Rules Out Ordinary (co)Homology


For QFTs admitting any type II string constructions, they always come in T-dual pairs. Such pairs of constructions should be treated on equal footing. Specifically, a pure geometry configuration is generically dual to a brane configuration, therefore focusing on either pure geometric constructions or pure brane constructions would be incomplete under T-duality. In principle, we should also include heterotic, and type I compactifications, but we defer those to future study. To stay away from those string frames, we choose to focus on QFTs without those type I or heterotic constructions for the moment, with examples including 6D (2,0) SCFTs (which descend to 4D = 4 SYM by T^2 compactification) and some 6D (1,0) LSTs. By insisting that the generalized symmetry be manifestly consistent with T-duality, we will see that the formulation of generalized symmetries in terms of ordinary (co)homology no longer holds.



  
6D (2,0) Examples We find it helpful to start from 6D (2,0), which admits simple configurations of internal space. Take the 6D (2,0) theories of A-type as a family of examples. These theories admit various dual constructions: type IIB on a ^2/_N orbifold singularity, and the worldvolume theory of a stack of N NS5 branes in type IIA. From the IIB side, the 2-from defect group (controlling the 2-form symmetry) is known to be ^(2) (= Ab[_N]) = _N. Take ^2/_N with N=4, so it has a _4-valued “2-form defect group". The theory can descend to an absolute theory whose charge lattice is isomorphic to the character lattice of SU(4)/_2 so that it admits a _2-valued “2-form symmetry". On the other hand, the brane construction from IIA involving NS5 brane worldvolume has very different behaviors. Investigating the generalized symmetries of such theories from type IIA side has not been pursued in the literature, but this is a natural case to examine the compatibility with T-duality. In fact, we will run into an apparent puzzle until we eventually promote ordinary cohomology to twisted K-theory. 

Concretely, the IIA construction has no geometric singularity, so the boundary of the internal geometry has cohomology H^*(S^3) = {, 0, 0, }. Therefore, there are no torsional (co)cycles that could produce this _N 2-form defect group as in the dual IIB side, nor can any torsional 2-form symmetry be detected when the relative theory descends to an absolute theory. One then realizes that D-brane charge in such a IIA background takes value in twisted K-theory <cit.>. However, to demonstrate the necessity of promoting ordinary (co)homology to twisted K-theory, let us take a purposeful detour by forcing ourselves to only do the mildest modification, namely only using twisted cohomology, and see how far we can get.



  §.§.§ Failed Fix: Twisted (Co)homology
 

In the presence of non-trivial RR flux, the gauge transformation rule of RR potential C_p (and thus their flux G_p+1) needs to be modified as:

    C_p    → C_p + dΛ_p-1 - H ∧Λ_p-3 ( p ≥ 3) 
    
            G_p+1   → G_p+1 + H ∧ d Λ_p-3,

so that G_p+1 is no longer gauge-invariant. One can cancel this gauge variation by adding a compensating term

    F_p+1 = G_p+1 - H ∧ C_p-2 = d C_p - H ∧ C_p-2.

However, this gauge-invariant F_p+1 is no longer co-closed, but instead:

    dF_p+1 = H ∧ F_p-1.

thus F_p+1 is no longer a representative of ordinary (de Rham) cohomology of the spacetime. To relate to some version of cohomology, we need to consider the formal sum of all these gauge-invariant field strengths. Take type IIA for example:

    F_even = F_2 + F_4 + F_6 + F_8 + F_10.

This F_even is a representative of a d-H-twisted de Rham cohomology with a modified closedness condition:
    (d-H) F_even = 0,  [F_even] ∈ H_N( X),

where N = [H] ∈ H^3( X, ℤ) is the cohomology class of H_3 viewed as the twist class, and X asymptotic boundary of the internal manifold X. A different _2-valued twist is also needed for D-type, which can be found in the Supplemental Material. The H_3-twisted cohomology on S^3 with twist class N = k ξ_3 can be computed by definition (ξ_3 generates H_3(S^3) =):
    H^*_kξ_3(S^3) = {_N, 0},
so we attempt to construct generalized symmetry by reducing brane charges on this “twisted" cocycle. This produces a torsional symmetry ℤ_N, which coincides with the _N “defect group" obtained from type IIB compactified on ℂ^2/_N as groups.

However, we are suddenly facing a physical mismatch between two _N's: the twisted homology only has two elements for any X, so the notion of dimensionality of the cycles does not make sense unless we reduce it modulo 2. Therefore, the _N “2-form defect group" on the IIB side is fundamentally different from the _N “even-form defect group" on the IIA side. In general, for dual pairs between pure geometric constructions and brane constructions involving NS5 branes, such a physical mismatch between untwisted ordinary cohomology and twisted ones will persist as a fundamental issue. We are thus inevitably led to our main proposal of using twisted K-theory to describe the generalized symmetry in such type II compactifications. 



§ FULL TREATMENT: TWISTED K-THEORY


Now, we make use of the well-known fact that RR charges and RR fields are classified by the twisted K-theory of the spacetime manifold. From this we propose that, the generalized symmetry of the QFT coming from type II compactification is governed by twisted K-theory of asymptotic boundary X of internal manifold X and that of the external spacetime M, promoting the result in <cit.> to generalized cohomology.

K-theory is defined as the equivalence class of a pair of vector bundles (E, E') modulo the simultaneous direct sum of a third bundle F:
    (E, E') ≅ (E ⊕ F, E' ⊕ F),
where in string theory, E, E' are interpreted as the gauge bundles on D-branes and anti-D-branes, respectively. Twisted K-theory imposes extra structure on the bundle, and sometimes one needs to formally think of both E and E' as infinite-dimensional vector bundles. [See Supplemental Material for references containing the definition of twisted K-theory.] The equivalence relation of F' captures the dynamical process of open string tachyon condensation, with the tachyonic modes living on open strings from D-brane to anti-D-brane. In string-engineered QFTs, such tachyon condensations underlie the fusion rules of symmetry operators from branes wrapping ordinary homological cycles <cit.>.

Unlike ordinary cohomology which admits a -valued degree, twisted K-theory only admits a _2-valued degree. So we only get to talk about “even-form" and “odd-form" symmetries. As we have seen, RR fields of different degrees can transform into each other via gauge transformation. On the other hand, only the K-theory class x of the total spacetime X_tot is the physical degree of freedom. Its image under the Chern character is represented by the formal sum of RR field strengths F_p+1. Concretely,

    [ IIA: ch(x) = F_2 + F_4 + F_6 + F_8 + F_10,  x ∈ K^0(X_tot),;   IIB: ch(x) = F_1 + F_3 + F_5 + F_7 + F_9, x ∈ K^1(X_tot). ]


The object charged under the formal sum is the collection of all BPS branes. D-branes in IIB are charged under K^0_N(X_tot) by consider D9 branes and D9-D9 pairs, and then D-branes in IIA are classified by K^1_Ñ(X̃_tot) which is known via T-duality <cit.>.



  
K-theory of Spacetime and Internal Space

It is instructive to examine the Künneth formula of decomposing K-theory in the asymptotic boundary of the total spacetime X_tot =M × X, with M is the external spacetime and X the internal manifold. The formula can be found in <cit.>, but we first examine the special case when M has no torsion:


    [ K^0(X_tot) = K^0(M) ⊗ K^0( X) ⊕ K^1(M) ⊗ K^1( X),; K^1(X_tot) = K^0(M) ⊗ K^1( X) ⊕ K^1(M) ⊗ K^0( X). ]


When M has torsion, mathematically K^i(X_tot) will be an extension of the right-hand side over a direct sum of Tor(K^i(M), K^j( X)) terms for certain sets of (i, j). 

In general, we expect to take the right-hand side of the above equation as the definition of even-form and odd-form symmetries, so there is no mixing between K-theory of M and that of X. The last ingredient we need is to include the topological twist, which we now introduce. 

Twisted K-theory and Topological T-duality When we have a (trivial or non-trivial) twist x_3 ∈ H^3(X_tot; ) which decomposes as x_3 = y_0 ∪ z_3 for y_0 ∈ H^0(M; ) and z_3 ∈ H^3( X, ). It is natural to consider the following form of K-theoretic symmetries


    [                                          IIB K^0(M) ⊗ K^0_z_3( X) ⊕ K^1(M) ⊗ K^1_z_3( X),;                                          IIA K^0(M) ⊗ K^1_z_3( X) ⊕ K^1(M) ⊗ K^0_z_3( X),;                                              ]


where we need the twisted K-theory of X but untwisted K-theory in M along which there is no H_3 flux. 

With this formulation of generalized global symmetry via twisted K-theory, the matching across T-duality has exactly been covered in a well-studied subject known as topological T-duality, which has been investigated with considerable generality and mathematical rigor. See <cit.> for some early works on topological T-duality, and <cit.> for a comprehensive review. In particular, even for a trivial twist class, the twisted K-theory class is different from the untwisted K-theory, as we will soon see in an example. Physically, the B_2 field is part of the string theory spacetime, so we stick to the mathematical object of twisted K-theory even for a trivial twist class.



 §.§ Examples


We proceed by computing the even-form and odd-form global symmetries from the twisted K-theory group of X in some 6D examples.



  
6D (2,0) SCFTs

A 6D (2,0) SCFT can be constructed by compactifying type IIB on ^2/Γ with Γ⊂ SU(2) a finite subgroup. The K-theory of the internal manifold has been computed as (see, e.g., <cit.>):

    K^0(S^3/Γ) = ⊕Ab(Γ)    K^1(S^3/Γ) = ,

where Γ⊂ SU(2) is the ADE-type finite subgroup of SU(2). For type IIA, on the other hand, the A, D, and E families require a case-by-case analysis, and we only discuss A-type constructed by k NS5 branes in the main text[See Supplemental Material for the study of D-type (2,0) from type IIA and discussion about E-type (2,0).]. There, the IIA charge group from twisted K-theory is K^1_k ξ_3(S^3) = _k, which does not contain the  factor as in the ordinary K-theory from IIB side. Indeed, this  mismatch is removed only if we consider trivially twisted K-theory K_0ξ_3^*(S^3/_k) for IIB (the computations can be found in <cit.>) as stated above. Specifically, the relevant twisted K-theoretic symmetries are given by:

    [             K^0(M) : K_0ξ_3^0(S^3/_k)   = K^1_k ξ_3(S^3)              = _k,;             K^1(M) : K_0ξ_3^1(S^3/_k)   = K^0_k ξ_3(S^3)               = 0. ]

We see that twisted K^0 and K^1 of X are exchanged under T-duality, so the even-form (resp., odd-form) symmetry in M stays the same. Restricting to the 2-form sector would reproduce the result in <cit.>, but the K-theoretic symmetry also involves the 0-form and the 4-form sector.




  
6D (2,0) SCFTs: Worldsheet Perspective

The analysis of D-branes on su(2) Wess-Zumino-Witten (WZW) model with ADE-type modular invariants can be found in, e.g., <cit.>. In our context, the relevant model should be thought of as the asymptotic boundary on type IIA side, and the resulting charge group for A-brane on S^3 with H_3-flux is computed from the boundary state formalism of the worldsheet WZW model. The result matches with the expectation from twisted K-theory, which coincides with the center of the ADE Lie group. [We summarize the details of the worldsheet analysis in the Supplemental Material <ref> for the convenience of the readers.]

This statement is in line with <cit.>, the study of gauged generalized symmetries in string spacetime from the worldsheet perspective. Recall that the global symmetry on the string worldsheet with boundary should be the gauge symmetry in the target manifold. If we instead view the target space as the asymptotic boundary of the internal manifold, then we will get the twisted K-theoretic global symmetry in the external spacetime dimensions. The author hopes to investigate this delicate match in more generality in the future.




  
6D (1,0) LST from type II

Now we go down to 8 supercharges. Some of them can be constructed from heterotic string on a ^2/Γ orbifold with suitable flat connection of gauge bundles at infinity <cit.>, for which we would need to go beyond T-duality. To avoid such an ambitious move, we focus on theories that thus do not admit heterotic or type I constructions, such as 6D (1,0) theories with non-trivial defect groups. 

To get the richer behavior of generalized symmetries, we examine 6D (1,0) Little String Theories (LSTs) <cit.>. LSTs are non-gravitational theories. They are also non-local since they admit dynamical strings with finite tension (hence “little string"). They can be thought of as intermediate theories between QFTs and quantum theories of gravity. As a remark, we do not have access to well-formulated concepts of relative LSTs and absolute LSTs, since they are not local QFTs. Nonetheless, we can still meaningfully discuss the twisted K-theoretic Defect group of LSTs, and track their behavior across T-duality. In addition to the “2-form symmetries" that are more common for 6D theories, LSTs can have discrete “1-form symmetries" as identified in <cit.>. 

We focus on LSTs constructed from IIA on a ^2/_k_1 singularity with k_2 NS5 branes. The T-dual construction of the same set of LSTs is given by IIB on ^2/_k_2 with k_1 NS5 branes, note again the exchange k_1 ↔ k_2. We express the LST with tensor branch notation introduced in <cit.> as:

    𝔰𝔲(k_1)2 …𝔰𝔲(k_1)2    (k_2 tensors).

Suppressing the details of string geometry, “// 2 2 … 2//" encodes the Dirac pairing of k_2 tensor multiplets in 6D that form an A-type affine Dynkin diagram, while (k_1) stands for the gauge algebra paired with each tensor multiplet on the tensor branch. The match of the twisted K-theory symmetry can be seen on both sides as:


    [                                             IIB                     IIA                        ;                  K^0(M) : K^0_k_2 ξ_3(S^3/_k_1) = K^1_k_1 ξ_3(S^3/_k_2)                  = _k_1;                  K^1(M) : K^1_k_2 ξ_3(S^3/_k_1) = K^0_k_1 ξ_3(S^3/_k_2)                 = _k_2. ]


We get a new odd-form symmetry _k_2, which is an uplift of the 1-form symmetry in <cit.>. The K-theoretic symmetries are indeed consistent under T-duality.



§ SYMMETRY OPERATOR & CHARGED OBJECT


Having obtained the description of generalized global symmetries by twisted K-theory from T-duality, we now turn to their physical interpretations. Compared to generalized symmetries by ordinary cohomology, K-theory mixes up p_i form and p_j-form symmetries for p_i - p_j even. In such a theory, the notion of p-form symmetries no longer makes sense for individual p, in that there is a single group controlling the spacetime symmetry operators of all odd (or even) dimensionalities. 



  
K-theoretic Symmetry as Equivalence Classes of Symmetry Operators

Indeed, Dp-branes for different p can wrap the same boundary homology cycle. They become symmetry operators of different dimensionality charged under the same (even- or odd-form) symmetry in the non-compact spacetime. This is consistent with the fact that individual components of RR potentials, C_i, are not independent physical objects, but only the K-theory class represented by their formal sums are gauge-invariant objects (see the ^7 example in <cit.>). Therefore, the symmetry operators that one obtains by integrating these C_i over submanifolds of the spacetime are not independent objects, either. In particular, different components of the RR fields C_0, C_2, … in the Wess-Zumino action on the D-brane worldvolume WZ(X) = 2π i ∫_X e^F_2 - B_2√(A) (C_0 + C_2 + ⋯) should not be thought of as independent objects, but instead as objects that are correlated according to the Chern character from a K-theory class x ∈ K^0(X) to ch(x) ∈ H^even(X) (we took IIA as an example). In addition, <cit.> analyzed the K-theoretic RR fluxes and RR charges in IIB compactification, with 6d theories as a family of examples. This suggests that the notions of defect group, absolute theories, and polarizations, should all be phrased in terms of such K-theoretic symmetry of the spacetime manifold, which is in line with our proposal.



  
6D (2,0) SCFTs From IIB, the “2-form _N symmetry" was originally constructed by D3 branes wrapping on relative two cycles of ^2/Γ <cit.>. However, on the same cycle, we can also wrap D1 branes or D5 branes, which will result in 0-dimensional or 4-dimensional (in addition to the 2-dimensional) non-dynamical defects in the spacetime. The apparently independent symmetries _N^(0), _N^(2), _N^(4) actually uplifts to a single _N-valued even-form K-theoretic symmetry. Wrapping D5 brane on the resolution divisor of ^2/Γ should produce for us Gukov-Witten operators <cit.>, that are charged under the same _N^(even) as the surface defects in <cit.>.

From IIA side, due to the presence of k units of H_3 flux, the combination of Dp-brane will detect twisted relative cycles γ_odd∈ H^*_kξ_3(S^3), as specified in equation (<ref>). But these twisted cycles are already living in a mod-2 cohomology, so it does not have a -valued degree. If we take the wrapped D3-brane configuration and attempt to T-dualize it to the IIA side, then we will get either a D2 brane or D4 brane on the IIA side depending on the specific duality circle we pick, i.e., a double T-duality would exchange D2 with D4. These apparent difficulties in using wrapped D-branes all support the proposal of describing the generalized symmetries by twisted K-theory, which is naturally compatible with T-duality.



§ ACKNOWLEDGMENTS

The author thanks Ibrahima Bah, Saghar Hosseini, Yuji Tachikawa, Xingyang Yu for discussions, and Yuji Tachikawa for helpful comments on an earlier version of the draft. The author thanks Jonathan Heckman, Max Hübner, Ethan Torres, and Xingyang Yu for collaboration on the role of K-theory in generalized symmetries and geometric engineering, together with Justin Kaidi and Yuji Tachikawa for collaboration on the spacetime implication of generalized symmetries on the string worldsheet. The author is supported in part by WPI Initiative, MEXT, Japan at Kavli IPMU, the University of Tokyo. 





§ SUPPLEMENTAL MATERIAL




 §.§ Twisted K-theories: Definition and Computations
 

As we reviewed in the main text, ordinary K-theory has a topological definition based on a pair of vector bundles on a manifold. In contrast, twisted K-theory often needs to be defined using analytic objects. See <cit.> for a version of definition, which involves projective unitary operators, Fredholm operators, and C^*-algebras on a Hilbert space. Possible twistings are classified by an element in

    (a_1, x_3) ∈ H^1(X; _2) × H^3(X, ),

and we denote the resulting twisted K-theory on the topological space X as:

    ^a_1K_x_3(X).


Here x_3 = k ξ_3 ∈ H^3(X, ) is the NS-NS H_3 flux (with ξ_3 a generator of H^3(X, ), for X a 3-manifold, H^3(X, ) =), and H^1(X; _2) captures the effect of an orientifold that “flips the pair of bundles" when going around an element of the fundamental group of X. Here the quotation marks account for the fact that, the the definition of (E, F) ∼ (E ⊕ H, F ⊕ H) does not always have a clear analog when the twist takes a non-trivial value in H^3(X, ).  Only for x_3 torsional, it is possible to find finite-dimensional twisted vector bundles <cit.>. But when ξ_3 is a free element, we would need to morally replace E, F with infinite-dimensional vector bundles. In these cases, the analytic definition in the reference above still enables us to take the most general twist that has both a degree-1 component and a degree-3 component.

In the algebraic formulation, the twist of K-theory is phrased as a Dixmier-Douady invariant <cit.> (see <cit.> for an accessible introduction in English) of the C^* algebra bundle on the manifold X. We refer the interested reader to <cit.>, a concise paper with definition and computation of K_N^*(S^3) as an example.



 §.§ 6D (2,0) DE-type Theories
 



  
D-type



Here, the IIB side is still a pure geometry ^2/Γ_D_N with a non-Abelian quotient by the dihedral group Γ_D_N⊂ SU(2) of 4N-8 elements, such that the “2-form" charge lattice of non-dynamical defects in the 6D theory is the D_N weight lattice. Therefore, it is straightforward to determine:

    Ab(Γ_D_N) = {[   _2 ⊕_2 N  even,;       _4  N  odd. ].


However, the K-theoretic symmetry from IIA side needs another type of twist specified by an element of H^1(X, _2). According to section 4.2 of <cit.>, 6D (2,0) SCFT of D-type can be engineered by putting a stack of k NS5 brane on top of a ON5_A brane, an object that implements the combination of I_4, an inversion of the transverse directions, and (-1)^F_L, the worldsheet left-moving fermion parity. The K-theory charge of this configuration was computed in <cit.>. Concretely, they computed ^-K_(k+1) ξ_3^*(S^3/_2), the K-theory on S^3/_2 ≅^3 with a ξ_3 the generator of H^3(X, ) and a non-trivial 1-form twist - ∈ H^1(S^3/_2, _2) ≅_2 := {±} as:

    ^-K_(k+1)ξ_3^0(S^3/_2)    = 0, 
    
        ^-K_(k+1)ξ_3^1(S^3/_2)    = {[  _2 ⊕_2 k even,;      _4  k odd. ].

Here k + 1 = κ describes the D-type theory of type SO(2k), and the “+1" accounts for the fact that the orbifold IIA/I_4 (-1)^F_L := ON5_A^0 itself carries one unit of H_3 flux, which could be viewed as the combination of ON5_A (without H_3 flux) and an NS5 brane.



  
E-type

For decades, it was believed that 6D (2,0) SCFT can only be engineered from type IIB string theory via its compactification on a ^2/Γ_E_N singularity, where N = 6, 7, 8. So we learn that their even-form symmetry is _m for Γ_E_n with m = 9-n, which coincides with the center of the ADE Lie group.

On the other hand, the IIA construction of such theories is not available. One could only construct them from M-theory using the exotic U-fold construction <cit.> which is inherently strongly coupled and thus non-geometric. In M-theory, the five-dimensional internal manifold can be given as (× T^3)/_p where _p is a non-geometric quotient given by:

    _p = _p^·_p^ρ,

where _p^ is a 2π/p rotation on the  complex plane, _p^ρ is an order-p element in the non-geometric part SL(2, )_ρ of the U-duality group SL(2, )_ρ× SL(3, ) ≅ E_3, 3() acting on T^3, whereas the geometric factor SL(3, ) is not involved in the quotient. Such quotients are only possible when the complexified Kähler moduli is set fixed, so these configurations are inherently strongly coupled and beyond the reach of perturbative IIA. As stated, we leave the study of charges in M-theory to future work.





 §.§ Worldsheet Analysis
 

Here we summarize the worldsheet analysis of D-brane charges in various literature in the 6D (2,0) cases in <cit.> for the convenience of the readers. As an illustration, we focus on the analysis of D-type cases. Here we are considering (2)_k WZW models for even k with D-type non-diagonal modular invariants, which is also known as (3)_k WZW model. We remark that this only covers the worldsheet of IIA construction of the D-type (2,0) theories. There, the spectrum of this theory in terms of holomorphic and anti-holomorphic states is given by:

    =    ⊕_j = 0^k/2_2j⊗_2j⊕⊕_j = 1^k/2_2j-1⊗_k+1-2j   (k/2 odd), 
     =    ⊕_j = 0^k/4-1(_2j⊕_k-2j) ⊗( _2j-1⊕_k+1-2j) 
       ⊕ 2 _k/2⊗_k/2   (k/2 even).


The exponent μ, ν, …∈ε are indices for the bulk fields that control the bulk modular S-matrix S_μν and fusion coefficients _μν^   ρ.  For concrete Hilbert spaces as given above, the number of exponents is given as the number of _a ⊗_a (“diagonal entries") after expanding out the tensor product, which then equals to the number of untwisted boundary states. One can count the exponents to get the number n = k/2 + 2, i.e., there are n (untwisted) boundary states. See <cit.> for a pedagogical treatment of the boundary CFT formalism, including definitions and explicit expressions of these states. 

Here we consider the standard situation that the boundary corresponds to the regular representation, so the set of index of such boundary states (usually denoted as a, b, …) coincides with the set of exponents. Then the boundary state charges are labeled by nodes on D_n-type Dynkin diagram, where the i-th node is associated with a charge q_i. These charges need to satisfy the charge equations modulo 2 <cit.>:

    2q_i = q_i-1 + q_i+1 (1 < i < n-2), 
        2q_1 = q_2, 2q_n-1 = 2q_n = q_n-2,
        2q_n-2 = q_n-3 + q_n-1 + q_n

Here i = 1, …, n label nodes on a D_n Dynkin diagram, where i = n-2 labels the trivalent node whose three adjacent nodes are labeled by i = n-3, n-1, n. Solving these relations would give us the charge group K = _2 ⊕_2 for k even and K = _4 for k odd.

The analysis of A-type and E-type proceeds in a completely analogous fashion, where in the E-type we need to use the su(2)_k E-type modular invariant partition functions for k = 10, 16, 28. The outcomes all coincide with the center of ADE simply-laced Lie group via McKay correspondence.

unsrt


