8d gauge anomalies and the topological Green-Schwarz mechanism

Journal of High Energy Physics, Nov 2017

String theory provides us with 8d supersymmetric gauge theory with gauge algebras \( \mathfrak{s}\mathfrak{u}(N),\mathfrak{s}\mathfrak{o}(2N),\mathfrak{s}\mathfrak{p}(N),{\mathfrak{e}}_6,{\mathfrak{e}}_7\kern0.5em \mathrm{and}\kern0.5em {\mathfrak{e}}_8 \), but no construction for \( \mathfrak{so}\left(2N+1\right) \), \( {\mathfrak{f}}_4 \) and \( {\mathfrak{g}}_2 \) is known. In this paper, we show that the theories for \( {\mathfrak{f}}_4 \) and \( \mathfrak{so}\left(2N+1\right) \) have a global gauge anomaly associated to π d=8, while \( {\mathfrak{g}}_2 \) does not have it. We argue that the anomaly associated to π d in d-dimensional gauge theories cannot be canceled by topological degrees of freedom in general. We also show that the theories for \( \mathfrak{s}\mathfrak{p}(N) \) have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.

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8d gauge anomalies and the topological Green-Schwarz mechanism

HJE 8d gauge anomalies and the topological Green-Schwarz mechanism Kazuya Yonekura 0 1 2 6 7 8 0 4-1-1 Kitakaname , Hiratsuka-shi, Kanagawa 259-1292 , Japan 1 Fohringer Ring 6 , 80805 Munich , Germany 2 In~aki Garc a-Etxebarria 3 Department of Physics, School of Science, Tokai University 4 School of Natural Sciences, Institute for Advanced Study 5 Max Planck Institute for Physics 6 University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 7 1 Einstein Drive Princeton , NJ 08540 , U.S.A 8 boundary @(B String theory provides us with 8d supersymmetric gauge theory with gauge algebras su(N ), so(2N ), sp(N ), e6, e7 and e8, but no construction for so(2N +1), f4 and g2 is known. In this paper, we show that the theories for f4 and so(2N +1) have a global gauge anomaly associated to d=8, while g2 does not have it. We argue that the anomaly associated to d in d-dimensional gauge theories cannot be canceled by topological degrees We also show that the theories for sp(N ) have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green- Anomalies in Field and String Theories; Extended Supersymmetry; Field 1 Introduction and summary Brief review of traditional global anomaly Traditional anomaly associated to d Global anomaly: the traditional method A brief review of the strategy of Elitzur and Nair Exercise in four dimensions Computations in eight dimensions 4 Global anomaly: a new method using instantons The basic idea Relation to traditional anomaly Exercise in four dimensions Computations in eight dimensions 2 3 5 6 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 4.4 6.1 6.2 6.3 Uncancellability of traditional global anomaly Topological Green-Schwarz mechanism Topological Green-Schwarz mechanism for cohomological classes Topological Green-Schwarz mechanism in action: a 3d example The need for a subtler version in 8d A List of homotopy groups 1 Introduction and summary In ten dimensions with N =1 supersymmetry, no supersymmetric gauge theory is anomaly free, whereas with the supergravity multiplet only the gauge algebras so(32) and e8 e8 are allowed. Both possibilities are realized by string theory [1{3].1 Furthermore, in these models, the anomaly of the fermions is canceled by a special coupling of the gauge eld and the metric to the two-form eld, now known as the Green-Schwarz mechanism [1]. We study a question of a similar nature in eight dimensions.2 A vector multiplet in eight dimensional N =1 supersymmetry contains a gaugino which is a chiral fermion, and is 1An argument based purely on anomalies of the relevant supersymmetry multiplets would allow for the additional possibilities e8 u(1)248 and u(1)496, but more careful consideration of the Green-Schwarz couplings required by supersymmetry rules these theories out [3]. 2There are a series of works in six dimensions initiated by Kumar and Taylor [4]. An elegant review of case is actually realized in string theory. As is well-known, the su(N ) theory is realized on N D7-branes, and the so(2N ) theory with N 4 and sp(N ) theory are realized on N D7-branes on top of an O7 -plane. In addition, the F-theory 7-branes provide the e6;7;8 theories [6{9]. What then is the status for the other gauge algebras so(2N +1), f4 and g2? Is it simply that we do not know the construction yet, or are they inconsistent because of an anomaly?3 In this paper, we will rst show that the cases so(2N +1) with 2N +1 7 and f4 have global anomalies in the sense of Witten [12, 13] associated to 8 of respective groups. This will be done in two ways, one by using the traditional method of Elitzur and Nair [14], another by introducing a BPST instanton. The second method reveals a surprise: although the case sp(N ) does not have a global anomaly associated to 8, it has in fact a subtler anomaly detected by the invariant as discussed e.g. in [15, 16]. This begs the question how this is consistent with the fact that the D7-branes on an O7+-plane give rise to an eight-dimensional (8d) supersymmetric sp(N ) gauge theory. We suggest that this is via a coupling to topological degrees of freedom which cancel the anomaly, in a way analogous to the Green-Schwarz mechanism, although we have been unable to actually describe the topological quantum eld theory (TQFT) which does the job. Morally speaking, it should be given by the topological part of the Ramond-Ramond (RR) elds in the spacetime with an O7+-plane, but the authors have been unable to write it down. Instead we provide a simple three-dimensional (3d) model where the gauge anomaly is canceled by a TQFT. This also begs an independent question whether the traditional global anomaly associated to d in d-dimensions might be canceled by coupling to a TQFT. Although the answer to this question is already implicit in the original arguments of global anomalies [12, 13], we will make it more explicit that this is impossible, settling the non-existence of the 8d theories with gauge algebras so(2N + 1) and f4. Intriguingly, we do not nd any anomaly for g2 in this paper. It is not known whether this algebra can be realized in some compacti cation of string theory down to eight dimensions. The rest of the paper is organized as follows: in section 2, we start by reviewing the global anomaly associated to d and its relation to the mod-2 index and the invariant. In section 3, we then compute the anomalies of 8d N =1 gauge theories associated to d=8 using the traditional method. In section 4, we re-compute the same anomalies using a new method using the instanton background. We reproduce the results of section 3 and also nd a subtler anomaly for sp(N ). Sections 3 and 4 can be read independently. We then 3In perturbative type IIB string theory it might appear that it is possible to engineer the so(2N +1) theory by putting an odd number of D7 branes on top of an O7 plane. There are various arguments that this con guration is inconsistent: there is no homology group or K-theory class that could support the discrete RR torsion associated with the stuck D7 brane [10, 11], the would-be monodromy of the axiodilaton around the stack does not correspond to any element in the Kodaira classi cation, and a D3 probe of the con guration would have a 4d SU(2) anomaly in its worldvolume [10]. The analysis in this paper show in section 5 that the traditional global anomaly associated to d cannot be canceled by coupling it to a TQFT. Finally in section 6 we discuss how a subtler global anomaly can sometimes be canceled by a TQFT in a way analogous to the standard Green-Schwarz mechanism. We have an appendix listing d 11 of various Lie groups. 2 Brief review of traditional global anomaly In this section, we review the global anomaly associated to d and its relation to the mod-2 index and the invariant. This section is completely standard and can be skipped by an experienced reader. 2.1 Traditional anomaly associated to d HJEP1(207) First let us brie y review how the traditional anomaly arises. Gauge transformations g on R d which go to 1 at in nity are classi ed topologically by d(G), so we will loosely write it as g 2 d(G). (More precise statement is that the equivalence class [g] of such gauge transformations under continuous deformation is classi ed by we consider a family of gauge elds a(t) parametrized by t 2 R given by d(G), [g] 2 d(G).) Then a(t) = f (t)g 1dg; where f (t) is a smooth function of t such that f (t! 1) = 0 and f (t!+1) = 1. Now, consider the fermion partition function det PLD= [a(t)] in the background eld a(t), where PL is the projector to left-handed fermions. The gauge eld con gurations at t ! 1 and t ! +1 are gauge equivalent. However, it can happen that det PLD= [a(t = +1)] = eiA det PLD= [a(t = The phase eiA represents the anomaly. In a situation in which det PLD= [a(t)] is naturally real, as in Witten's four-dimensional (4d) Sp(N ) anomalies [12] as well as 8d anomalies of strictly real representations, the anomaly takes values in eiA = 1 and it is simply related to the number of eigenvalues of the Dirac operator which cross zero when t is changed from 1 to +1. Moreover, by spectral ow considerations, this number can be represented by the number of zero modes in d + 1 dimensions. We take a gauge eld A on Rd+1 which is just a(t) by regarding t as one of the directions of Rd+1. Then we can compute the number of zero modes J of the Dirac operator on Rd+1 in the presence of the gauge eld A. The anomaly is simply given by eiA = ( 1)J . In summary, the traditional anomaly for real det PLD= [a(t)] is detected by considering a gauge eld con guration in one-higher dimension d + 1. We take a gauge eld A on Rd+1 which goes to a pure gauge at in nity A ! g 1dg, where g corresponds to a nontrivial element of d(G) at the sphere Sd at in nity. Then, if the number of fermion zero modes are odd, we have the traditional global anomaly. 2.2 Mod-2 index and the invariant Before moving on, we also review the so-called mod-2 index of Dirac operators to see the relation between the traditional anomaly [12] reviewed above and a more modern proposal (2.1) (2.2) in [15]. The mod-2 index can be de ned when the Dirac operator (times the imaginary number i) can be taken to be real-antisymmetric. This means the following. The Dirac operator is D = i D . Here D = @ + TaAa is a covariant derivative, where Ta is the representation of generators of the gauge algebra g, and for simplicity we suppress the spin connection. Then, the reality of the Dirac operator means that 1 and 1 Ta are the strict-real representation of Clif g, where Clif is the Cli ord algebra associated to the tangent bundle of a space-time manifold. This is possible if the representations of both of Clif and g are strict-real, or both of them are pseudo-real.4 Witten's Sp(N ) anomaly in 4d uses the case where they are both pseudo-real. In our application in this paper, the gauginos are in the adjoint representation which is strict-real, and also the 9-dimensional Cli ord algebra is strictly real (in the convention f ; g = 2 ). Now we can de ne the mod-2 index. First, recall the case of nite dimensional matrices. A nite dimensional real anti-symmetric matrix A can be transformed by an orthogonal matrix to a block diagonal form as A ! (0) (0) 0 6=0 1 ( 1)J = ( 1)Ind(D). is de ned by where each ( ) is one of the blocks of the block-diagonal matrix. Then, the number of zero eigenvalues modulo 2 does not change under smooth continuous deformations. On a closed spin manifold, the operator D can be regarded as an in nitedimensional anti-symmetric real matrix. This is because 1 are real-symmetric matrices, 1 Ta are real-antisymmetric, and the derivatives @ are real-antisymmetric (by integration by parts on a closed manifold). Therefore, the number of zero eigenvalues mod 2 is well de ned. This is the mod-2 index Ind(D) 2 Z2 of the Dirac operator D = i D . By using this notation, the global anomaly discussed in the previous subsection is given by Another way to describe anomalies is by the Atiyah-Patodi-Singer invariant, which where the sum is taken over nonzero eigenvalues of D. However, on a closed manifold, the nonzero eigenvalues appear in pairs as as can be seen in (2.3), and hence their contributions are canceled out. Therefore, we get exp( 2 i ) = ( 1)Ind(D); where we have used ( 1)dim ker D = ( 1)Ind(D). The left-hand side is the anomaly formula of [15], while the right-hand side is the anomaly formula of [12]. 4The strict and pseudo real conditions are rephrased as the existence of a charge conjugation matrix C such that CT = C for strict real and CT = C for pseudo real, respectively. The charge conjugation matrix of a representation of the algebra Clif g is given by CClif Cg in an obvious notation. Hence it is strictly real if both of Clif and g are strict/pseudo real. { 4 { (2.3) (2.4) (2.5) where A is the connection on Md, Af = f Af 1 +f df 1 its gauge transform, and we denote by A the anomalous phase, which we aim to determine. This can be done using descent, as usual, with the result that Z [A] = eiAZ [Af ] A = Z Bd+1 CS(Af ) CS(A) In this section, we compute the traditional global anomaly of the 8d adjoint fermion using the methods of Elitzur and Nair [14], elaborating on an observation by Witten [17]. A brief review of the strategy of Elitzur and Nair We will now brie y review the approach in [14, 17], to make the paper more self-contained. (See also [18] for a very clear and detailed exposition of the procedure.) The basic idea is to relate the computation of the global anomaly of a representation RG of G (which we assume to be free of local anomalies) to a local anomaly under gauge F , has known anomalies. The simplest case is that RG0 is a sum of copies of the trivial one-dimensional representation, which clearly has no local or global anomaly. Our aim is to relate the global anomaly of RG under G transformations to the local anomaly of RF under F transformations. We do so as follows. The local anomaly under f 2 F (which means that f is a gauge transformation in F , by abusing the notation) is the variation of the phase of the fermionic path integral Z on a manifold Md, which for simplicity we take to be Sd: (3.1) (3.2) (3.3) (3.4) where CS(A) is the Chern-Simons functional on d + 1 dimensions for A, and Bd+1 is some d + 1-dimensional manifold with boundary Md = Sd, which we take to be a ball. We have chosen some arbitrary extension of f to the interior of Bd+1. For convenience we will introduce (f; A) CS(Af ) CS(A) : Consider now the speci c case f 2 G when restricted to the boundary of Bd+1, while allowing f 2= G in the interior. We denote this boundary value of f by g. We also restrict A to belong to G. The anomalous phase of Z under f (viewed as an element of G), i.e. the global anomaly that we are after, can then be computed by (3.2). Using elementary properties of the Chern-Simons functional, one can verify that CS(Ag) = CS(A) for g 2 G, since we are assuming that G is free of local anomalies. This implies that (f; A) depends on F only up to equivalence under multiplication by elements of G, or equivalently it depends on F=G only. Since we are setting f = g 2 G at the boundary, we can collapse the boundary to a point for the purposes of computing (3.2), and write A = Z Sd+1 (f; A) ; { 5 { with the understanding that f is valued in F=G here. One can easily check that this expression gives a homomorphism from d+1(F=G) to R. The normalization can be determined as follows. Consider the long exact sequence in homotopy in the exact sequence, and we know the normalization of A for some of the generators, we can work out (by linearity) the normalization for the rest of the generators. 3.2 Exercise in four dimensions Let us illustrate all this discussion with a simple example [12, 14, 17], where we choose d = 4, G = SU(2), F = SU(3), RF = 3 and thus RG = 2, RG0 = 1. We have that RG RG0 is pseudo-real, so there is no local anomaly for transformations in G. We want to determine the global anomaly. The relevant homotopy groups are 5(SU(3)) = Z, 4(SU(3)) = 0, 4(SU(2)) = Z2, and 5(SU(3)=SU(2)) = 5(S5) = Z.5 We then read from (3.5) that the following sequence is exact6 HJEP1(207) : : : ! 5(SU(3)) ! | {z Z } | 5 SU(3) SU(2) {z Z } ! 4(SU(2)) ! 0 : | {z Z2 } By exactness, it must be that is multiplication by 2, and is reduction modulo 2. Consider now some choice of extension g of g, the non-trivial generator of 4(SU(2)), into the bulk of B5. As before, we allow for the extension to be given by arbitrary elements of SU(3) in the bulk. When projected down to SU(3)=SU(2), and after compactifying B5 ! S5, g gives us an odd multiple of the fundamental generator q of 5(SU(3)=SU(2)). Denote the fundamental generator of 5(SU(3)) by f. For the case of interest, where we have a 3 of SU(3), A(f) = (f) Z S5 is simply the winding number,7 so A(f) = 2 . Since (f) = 2q, and A induces a homomorphism, we learn that A(q) = A(g) = , and there is a global anomaly for the SU(2) transformation g, as rst found in [12]. More generally, due to the relation of to the SU(3) anomaly polynomial in six dimensions, we have that A(g) = 2 I3(RF ), where we de ne for SU(N ), N > 2 TrRF (F k) = Ik(RF ) Tr (F k) : with d = 2k 2. We will use below the familiar fact that Ik(RF ) = 0 if RF is real or pseudo-real and k 2 2Z + 1. 5The coset space SU(n)=SU(n 1) is topologically S2n 1. 6By \0" in homotopy exact sequences we mean the trivial group of one element. All the groups involved in our computations are abelian, which justi es the notation. 7Here we are abusing notation somewhat: A is not the anomalous phase for the SU(3) theory, which would be given instead by the integral of over a disk, not a sphere. (3.6) (3.7) (3.8) { 6 { Let us now move on to the actual cases of interest: N =1 theories in eight dimensions. We have a fermion in the adjoint representation, which has no local anomaly, but can potentially have a global anomaly of the type just described whenever 8(G) 6= 0. This is the case for G 2 fSU(2); SU(3); SU( 4 ); SO(7) : : : SO(10); SO(N ); G2; F4g, where we take N > 10. See the list of homotopy groups in the appendix A. We have separated the SO(7) to SO(10) cases from the rest since from SO(11) on the homotopy groups relevant for our computation become stable. The group SU( 4 ): let us consider rst the SU( 4 ) case, which we will embed in SU(5). Using the fact that SU(5)=SU( 4 ) ' S9, and the known homotopy groups of spheres and SU(n), we nd that the following portion of the homotopy long exact sequence is exact: is reduction modulo 24. The adjoint of SU(5) is free of local anomalies in eight dimensions, so (f ) = 0 in SU(5). The adjoint of SU(5) decomposes as AdjSU( 4 ) 4 4 1. Since 1 has no global anomaly, we learn that the adjoint of SU(4) has no global anomaly The group SU(3): using the embedding into SU( 4 ), the relevant portion of the long exact sequence in homotopy in this case is : : : 9(SU(3)) 8(SU(3)) 9(SU( 4 )) 8(SU( 4 )) 9(S7) 8(S7) 0 (3.10) where we used SU( 4 )=SU(3) ' S7. We have that 9(SU(3)) = Z3, while 9(SU( 4 )) = 9(S7) = Z2. Exactness of the sequence then implies that vanishes, and thus we end up with the short exact sequence 0 ! 8(SU(3)) ! 8(SU( 4 )) ! | {z Z12 } | {z Z24 } that the anomaly of the adjoint of SU(3) is given by twice the anomaly of the adjoint of SU( 4 ), which as explained above vanishes. { 7 { Remark on SU(N ): physically, we can explain the above results for SU(3) and SU( 4 ) as follows. Let us consider SU(N ) theory with an adjoint fermion. Let us also add a scalar in the adjoint representation. The scalar eld does not contribute to the anomaly. Now, suppose that this SU(N ) theory is anomaly free. Then, by giving an appropriate expectation value to the adjoint scalar, we can break SU(N ) to SU(N 1) U(1). The fermion is now in the representation AdjSU(N 1) N 1 N 1 1 of SU(N 1). Since the original SU(N ) theory was assumed to be anomaly free, the new SU(N 1) gauge group must also be anomaly free under the RG ow from SU(N ) to SU(N 1). The fermions 1 do no contribute to the anomaly because we can add mass terms to them. Thus we conclude that SU(N 1) with an adjoint fermion is anomaly free. For large enough N , 8(SU(N )) is zero. Thus, we expect that SU(N ) does not have anomaly associated to 8(SU(N )) for any small N , such as N = 3 and N = 4. SO(N ) for N in the stable range: this case has been discussed in [19], also using the general approach of [14]. We proceed by embedding SO(N ) into U(N ). We choose N large enough such that we are in the stable range for all the homotopy groups entering our computation. More concretely, we are assuming N 11. The relevant homotopy groups were computed in [20]. See also [21] for a concise summary of the results used here, and in following sections. From the long exact sequence in homotopy (3.5) we have that 0 ! 10(U=SO) ! 9(SO) ! 9(U) ! 9(U=SO) ! 8(SO) ! 0 | {z Z2 } we end up with the short exact sequence in the right. We thus nd that A = I5(RU ). Since the adjoint of SO(N ) embeds as the antisymmetric of U(N ), we nd that there is an anomaly whenever I5( ) = N 16 is odd, i.e. whenever N is odd. The group F4: we analyze F4 by embedding into E6. The quotient space E6=F4 (known as \IV" in Cartan's classi cation of symmetric spaces) has well understood homotopy groups at low enough ranks [22]. In particular, for i Proceeding as above, we end up with the short exact sequence 15 we have that i(E6=F4) = i(S9). 0 ! 9(E6) ! 9 | {Zz } E6 F4 | {z } Z ! 8(F4) ! 0 : Consider rst the branching 27 ! 26 = 1 for the generator of 9(E6),8 we conclude that the 26 representation of F4 is anomalous. In turn, the adjoint of E6 decomposes as 78 ! 26 52, where the 52 is the adjoint representation of F4. Since the adjoint of E6 gives rise to neither local or global anomalies, it must be the case that the contribution of the 26 cancels against the 1. Since the 27 of E6 has a local anomaly, and 8One way to see this may be to use SO(10) E6 under which 27 ! 10 16. We consider an E6 bundle on S10 such that SO(10) E6 is identi ed with the tangent bundle of S10. From the index theorem, it is possible to show that 16 has one net zero mode while 10 has no net zero mode, by using the fact that the Euler number of S10 is 2. { 8 { (3.12) (3.13) contribution of the 52. We thus learn that the adjoint of F4 has a global anomaly in The group G2: we analyze G2 by embedding into F4, via the chain G2 where the elided representations are either singlets or appear an even number of times, which cannot give rise to a Z2 valued global anomaly. From the results in the previous section, we conclude that the adjoint (14) of G2 is free of global anomalies in eight dimensions. Remaining cases: SU(2), SO(N ) with 7 N 10: these are somewhat more technical, but have been computed in [19] and papers cited therein. The result is that SU(2) has no anomaly, and the SO(N ) cases have anomaly as in the stable case described above, i.e. whenever N is odd. 4 Global anomaly: a new method using instantons In this section, we discuss a way to nd global anomalies of Weyl fermions in general representation RG of the gauge group G in d spacetime dimensions, by introducing a codimension-4 gauge instanton. Rather than attempting a completely general classi cation of anomalies, here we just consider a speci c setup to see the anomaly of RG and later discuss what kind of anomaly the speci c setup is detecting. 4.1 The basic idea The basic idea is to consider a gauge theory soliton and see the anomaly of the zero modes living on it. For concreteness we focus our attention to the case where the gauge soliton is the familiar codimension-4 instanton. However, the idea here is applicable to more general solitons. Let us take a subgroup SU(2) G, and the maximal subgroup H G which commutes with SU(2). Namely, we have [SU(2) H]=C G, where C is some subgroup of the center of SU(2) H. An example is that [SU(2) subgroup C does not play any role, we will often omit C and loosely write SU(2) H Suppose that the representation RG is decomposed under SU(2) H as RG ! M(nSU(2) n 1 RHn ); (4.1) where nSU(2) is the n-dimensional (i.e., spin (n 1)=2) irreducible representation of SU(2), and RHn is some representation of H which is not necessarily irreducible. If we consider an instanton of SU(2) with unit instanton charge, there are fermion zero modes living on it. The representation nSU(2) produces Nn = 16 (n3 n) zero modes, where { 9 { Nn is twice the Dynkin index of nSU(2). Then, we have a gauge group H which is unbroken by the instanton, and the zero modes produce localized Weyl fermions in the representation rH := M NnRHn : (4.2) d 4.2 More concretely, let us suppose that the spacetime is X = Rd 4 S4, and the instanton is put on S4. Then, at low energies, we get a (d unbroken gauge group H and Weyl fermions in d 4)-dimensional gauge theory with the 4 dimensions in the representation rH . If this theory is anomalous, that means that the original theory is also anomalous. If RG does not have a perturbative anomaly in d dimensions, then neither does rH in 4 dimensions. However, there can be global anomalies as we will see explicitly later. HJEP1(207) Relation to traditional anomaly Now we study the relation of our anomaly associated to rH in d 4 dimensions and the traditional anomaly reviewed in section 2. In the traditional anomaly, the important point is that the gauge con guration approaches to a pure gauge at the in nity of Rd+1. However, we have considered the S4 compacti cation in our discussion above, so the relation of our anomaly to the traditional one is not obvious. In a little more detail, the traditional anomaly can be detected by the mod-2 index (or more generally the invariant) on Sd+1 as reviewed in section 2, where Sd+1 is the one point compacti cation of Rd+1. On the other hand, the anomaly discussed above is detected by the mod-2 index (or the invariant) in S4 Sd 3. We put an SU(2) instanton on S4, and also put a nontrivial gauge con guration of the gauge eld of H associated to d 4(H) on Sd 3. To connect the two anomalies, we need to do a nontrivial manipulation which we now explain. The main point is that if the condition d 4(G) = 0: : d 4(H) ! d(G): is satis ed, then we can de ne a homomorphism Mathematically, this homomorphism is described as follows: on Sd+1, consider an embedding Sd 3 Sd+1 and take a tubular neighborhood B4 Sd 3 Sd+1 of Sd 3. The standard instanton bundle on B4 and an H-bundle on Sd 3 de nes a G-bundle on B4 Sd 3. The instanton bundle on B4 is assumed to be pure trivial (i.e., pure gauge) on @B4 = S3. Since we assumed that d 4(G) is trivial, this G-bundle is trivial on the Sd 3) = S3 Sd 3 because any element of d 4(H) becomes zero in d 4(G). Therefore the bundle can be extended to the whole of Sd+1. This de nes an element in d(G). More physically, we can state the construction in the following way. Our anomaly discussed above can be seen by considering a manifold Y = Sd 3 S4 as explained above. Let us decompactify it to R R4. We put an instanton of SU(2) on R 4 which is (4.3) (4.4) Away from f0g d 3 such that it approaches to a pure gauge B ! h 1dh at the in nity of R h corresponds to a nontrivial element of d 4(H). The relevant zero modes are localized near the intersection of these two con gurations. By taking the instanton size to be very small, we get a codimension-4 object which we call \instanton bran".9 On Rd+1 = R R4, the instanton brane is localized near f0g, while the gauge eld B of the gauge subgroup H is localized near f0g R4. R4, the B is trivial up to gauge transformations. Now, suppose that the original gauge group G satis es the condition (4.3). In this case, the h 2 d 4(H) becomes trivial in d 4(G) and hence we can almost deform the gauge eld B to the trivial con guration B = 0 as a gauge eld of G. However, the important point is that the deformation B ! 0 is possible only away from the instanton brane. Near the instanton brane it is not guaranteed that we can make B ! 0 because the SU(2) instanton con guration may obstruct such deformation. In this way, we get a gauge eld con guration on Rd+1 which is localized near the submanifold Z = R f0g and is trivial away from Z up to gauge transformations. The gauge eld B is now localized near f0g 2 Rd+1. This situation may be described as \an instanton brane with a soliton inside it associated to d 4(H)". The soliton inside the instanton brane supports the fermion zero modes relevant to our anomaly. In the above argument, the world volume Z of the instanton brane is extending to in nity. However, the gauge eld B at in nity is of the form h 1dh, and hence in the coordinate patch R n f0g we can make a gauge transformation such that B = 0 near in nity. Then, the total gauge eld con guration near in nity of Z is simply that of the instanton brane without B. Then we can compactify the world volume of the instanton brane to e.g., Z = Sd 3 0 f g R d 2 R3. In summary, we have obtained a gauge con guration which is localized on a compact (h) 2 submanifold Z Rd+1. Because Z is compact, the gauge eld must approach to a pure gauge at the in nity of Rd+1. Therefore, this con guration is characterized topologically by an element of d(G). Namely, corresponding to each h 2 d 4(H), we get an element d(G) if the condition (4.3) is satis ed. It may also be checked by a topological argument that this map : d 4(H) 3 h 7! (h) 2 d(G) (4.5) is a homomorphism from d 4(H) to d(G). The map gives the relation between our anomaly and the traditional anomaly when the condition (4.3) is satis ed. If rH has a global anomaly under h 2 d 4(H), the instanton brane with the corresponding soliton has odd number of fermion zero modes. This means that the original representation RG of the gauge group G has an anomaly under (h) 2 d(G) and hence the theory su ers from the traditional global anomaly. How about the inverse direction? If the gauge con guration related to (h) does not give odd number of fermion zero modes for every h, can we conclude that there is no 9This is motivated by the fact that small instantons on 7-branes give D3-branes in string and F theory. However, our discussion does not rely on string theory at all. HJEP1(207) global anomaly associated to d(G)? At the level of the above argument, it is not possible because we have not yet shown that the map : d 4(H) ! d(G) is surjective. However, the surjectivity may be shown case by case. In practice, we encounter this problem in the case of the G2 gauge group in d = 8. In this case, 8(G2) = Z2. Now, the point is that is de ned purely topologically, so it is independent of the representation RG. Thus, if we can nd some representation RG which has the anomaly under (h), then that (h) must be a nontrivial element of 8(G2) = Z2. This establishes the surjectivity of : d 4(H) ! d(G) in this particular case. We will show that the 7-dimensional representation of G2 is really anomalous under (h). For d = 8, the only class of simple Lie groups for which (4.3) is not satis ed is Sp(N ). For this case, our anomaly detected by d 4(H) is actually di erent from the traditional anomaly associated to d(G). Indeed, we will see that 8d G = Sp(N ) Super-Yang-Mills for N 2 has the global anomaly related to 4(H), even though we have 8(Sp(N )) = 0 . We will also argue that the anomaly of Sp(N ) may be canceled by a TQFT, while the anomaly associated to d(G) cannot be canceled by a TQFT. 4.3 Exercise in four dimensions Let us see again Witten's original anomaly for d = 4 and G = Sp(N ) by using the argument in section 4.1. We present the argument for G = Sp(1) = SU(2), but the generalization to Sp(N ) is obvious. nontrivial one. The maximal subgroup H SU(2) which commutes with SU(2) is given by its center H = Z2. The condition (4.3) is satis ed because 0(SU(2)) = 0. Also, there is a possibility of an anomaly because H = Z2 is discrete and 0(H) has two elements, trivial one and Let us consider the representation of G = SU(2), RG = M cnnSU(2); n (4.6) (4.7) where nSU(2) is the n dimensional representation of SU(2) and cn are some non-negative integers. Then the rH de ned by (4.2) is given by rH = X n=even ! cnNn Z2 + X n=odd ! cnNn 1Z2 : where 1Z2 and Z2 are the trivial and nontrivial representations of Z2, respectively. Then we get a d 4 = 0 dimensional theory with P n=even cnNn zero modes which transform nontrivially under the gauge subgroup H = Z2. There is anomaly of the path integral if P n=even cnNn is odd. This can be seen, for example, from the fact that the path integral in d 4 = 0 is an ordinary integral, and the integral measure is changed by ( 1)Pn=even cnNn under the H = Z2 gauge transformation. Therefore, the gauge invariance is violated if P n=even cnNn is odd. For example, we have N2 = 1, N4 = 10, N6 = 35 and so on. The fact that 4SU(2) does not have global anomaly (because N4 = even) will be used later. As discussed in general above, there is a homomorphism : 0(Z2) = Z2 ! 4(SU(2)). This map is constructed by considering a small pointlike SU(2) instanton whose world line sweeps S1, and then including a nontrivial H = Z2 holonomy around this S1. This construction was essentially discussed in gure 1 of [23] in a slightly di erent context. We study global anomalies of 8d Super-Yang-Mills (SYM) which automatically have maximal supersymmetry. The matter content can be easily seen from dimensional reduction of ten-dimensional (10d) SYM, which contains the gauge eld and a Majorana-Weyl gaugino. After reducing two dimensions, the Majorana-Weyl in 10d can be considered as Majorana (but not Weyl) in 8d which is equivalent to Weyl (but not Majorana) in 8d. This is analogous to the fact that 4d gauginos can be considered as either Majorana or Weyl. For our purposes, it is convenient to consider it as Weyl. Before continuing, we note that the anomaly matching of 4d N =2 theories was studied in [24] on its Higgs branch which is assumed to have the form of the one-instanton moduli space of a Lie group G. Such a 4d N =2 theory would be realized on the worldvolume of the core of a BPST instanton in the 8d N =1 supersymmetric G gauge theory. Therefore the analysis and the actual computation in that paper is very much related to those in this paper, and the results are consistent. (h) 2 Simply laced groups: we do not discuss simply laced groups because of the following reasons. Simply laced groups other than SU(2) satis es the condition (4.3) given by 4(G) = 0, so if there is an anomaly for some h 2 4(H), that anomaly comes from 8(G) which cannot be canceled by TQFT, as already implicitly seen in the argument of [12, 13] and as will be more explicitly explained in section 5. However, we know that the SYM theories for all the simply laced groups appear in F-theory. Therefore, a priori it is expected that there is no anomaly. One can check it explicitly by considering some examples of subgroups SU(2) H G. For G = SU(2), we can only have H = Z2 and hence the anomaly associated to 4(H) is trivial. The group SO(2N + 1) with N > 2: let us consider SO (or more precisely Spin) groups. Let us take a subgroup SU(2) SU(2)0 SO(2N 3); (4.8) and take H = SU(2)0 SO(2N 3). In this case, the adjoint representation Adj(SO(2N +1)) decomposes as of SO(2N 3). 2SU(2) 2SU(2)0 (2N 3)SO(2N 3) Adj(SU(2) SU(2)0 SO(2N 3)) (4.9) where 2SU(2) and 2SU(2)0 are the 2-dimensional (doublet) representations of SU(2) and SU(2)0, respectively, and (2N 3)SO(2N 3) is the 2N 3 dimensional vector representation One can compute rH de ned in (4.2) as rH = 2SU(2)0 (2N 3)SO(2N 3) + (H singlets): (4.10) This representation contains an odd number (i.e., 2N 3) of SU(2)0 doublets, and hence su ers from the global anomaly in 4d. Therefore, we conclude that SO(2N +1) has a global anomaly in 8d. This anomaly is associated to 8(SO(2N + 1)) = Z2 as discussed above. The group G2: the group G2 contains a subgroup SU(2)1 SU(2)2 which can be seen from the a ne Dynkin diagram. One can compute the decomposition of the adjoint representation and get the result HJEP1(207) for H = SU(2)2, or This gives for H = SU(2)1. In either case, there is no global anomaly in 4d. One can also check that the homomorphism : 4(H) ! 8(G) = Z2 is surjective. To see this, take the 7-dimensional representation of G2 which decomposes as 2SU(2)1 4SU(2)2 Adj(SU(2)1 SU(2)2) Depending on which SU(2) to be taken as H, we get either (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) rH = 4SU(2)2 + (H singlets) rH = 10 2SU(2)1 + (H singlets) 7G2 ! 2SU(2)1 2SU(2)2 + 3SU(2)2 : rH = 2SU(2)0 + (H singlets) and hence the representation 7G2 su ers from the global anomaly associated to (h) 2 8(G2) for a nontrivial element h 2 4(SU(2)). Because 8(G2) = Z2, the must be surjective. Therefore, we conclude that the G2 SYM theory does not have a traditional anomaly associated to 8(G2). It would be interesting to study whether this theory is completely anomaly free beyond the level of the traditional anomaly. The group F4: the case F4 can be treated easily by using the fact that it contains a subgroup SO(9) F4 under which the adjoint representation decomposes as Adj(F4) ! Adj(SO(9)) 24SO(9); where 24SO(9) is the 24-dimensional spinor representation of SO(9). We further take the subgroup SU(2) SU(2)0 SO(5) as we did above for SO(2N + 1). One can check from the decomposition 24SO(9) = 2SU(2) 22SO(5) + 2SU(2)0 22SO(5) that 24SO(9) does not contribute to the 4d anomaly of SU(2)0. Therefore, F4 su ers from the global anomaly as in the case of SO(9). The group Sp(N ) with N > 1: this class does not satisfy the condition (4.3), so the anomaly associated to 4(H) is di erent from 8(G). Indeed, for N > 1, we have 8(Sp(N )) = 0 and hence there is no traditional anomaly. We take the subgroup Sp(1) Sp(N 1) Sp(N ); under which the adjoint representation decomposes as Adj(Sp(N )) ! 2Sp(1) (2N 2)Sp(N 1) Adj(Sp(1) Sp(N 1)): HJEP1(207) with obvious notations. Taking H = Sp(N 1) and introducing an instanton of SU(2) = Sp(1), we get to 8(G). rH = (2N 2)Sp(N 1) + (H singlets): This is anomalous in 4d. Therefore, the SYM with the Sp(N ) gauge group for N > 1 su ers from some new global anomaly which is di erent from the traditional one associated However, the Sp(N ) SYM is realized by an O7+-plane in string theory. Therefore, it must be possible to cancel the anomaly somehow. Also, because of the tight constraints from supersymmetry, there is no freedom to add local propagating degrees of freedom. Therefore, the anomaly must be canceled by coupling to a TQFT. (4.18) (4.19) (4.20) 5 Uncancellability of traditional global anomaly In view of the recent developments on anomalies, one can ask whether the traditional anomaly associated to d(G) can be canceled by coupling to a TQFT. We here argue that this is not possible. In this section, the gauge eld is treated as a background eld because its path integral plays no role. Essentially we follow the original arguments of [12, 13], with the possible existence of a TQFT in mind. Let a be some gauge eld con guration (possibly trivial) which has compact support on Rd. Let g be a gauge transformation representing a nontrivial element of d(G) which also has compact support on Rd. Let ag = g 1ag + g 1dg be a gauge transform of a by g, which again has compact support. As reviewed in section 2.1, the traditional anomaly can be seen by going from a to ag by a path like (1 f (t))a + f (t)ag where f (t ! 1) = 0 and f (t ! +1) = 1. The anomaly is represented by the change of the phase of the fermion partition function det PLD= [a(t)] under this continuous local deformation of the gauge eld. The question is whether such an anomaly can also be produced by a theory without massless propagating degrees of freedom. We denote the partition function of such a gapped theory as ZTQFT[a]. This notation implies that the low energy limit is described by a TQFT. The locality principle in quantum eld theory suggests the following. If a theory is gapped, the change of the partition function under any local continuous deformation of background elds can be captured by a local e ective action in the low energy limit. This can be seen by the following argument. Under a small (i.e., topologically trivial, but not necessarily in nitesimal) deformation a of the eld a, the change of the partition function is given by ZTQFT[a + a] ZTQFT[a] = exp i J a Z where J is a local operator coupled to a (or more explicitly the current operator to which the gauge eld is coupled). There may be other terms of the form ( a)2K + in the action, but the argument below is the same even if we include them. By expanding a, it is reduced to the computation of correlation functions of J . In a gapped theory, the correlation functions decays exponentially fast, and in the limit of very large mass gap, they are given just by contact terms. This means that the result of (5.1) is given by a local polynomial (possibly with derivatives) of a. Notice that there is no room for TQFT to change this conclusion, because we are just considering topologically trivial deformation a which has compact support on Rd. As a result of the above argument, the e ective action de ned by S[a] = log ZTQFT[a] log ZTQFT[0] is a local polynomial action of a (neglecting irrelevant higher dimensional operators) for a topologically trivial a.10 Moreover, the absence of perturbative anomaly implies that S[a] is invariant under in nitesimal gauge transformations. Remember that both a and ag are topologically trivial on R d with compact support. Namely, they are not just pure gauge, but are literally zero outside a compact region. Then the di erence of the logarithm of the partition functions between a and ag is given by log ZTQFT[ag] log ZTQFT[a] = S[ag] S[a]; However, in even dimensions, there is no such local polynomial action S[a] which is invariant under in nitesimal gauge transformations, and which produces the anomaly S[ag] S[a] 6= 0 for nontrivial element g 2 d(G). Instead, we have S[ag] = S[a]. We conclude that the anomaly associated to d(G) in d = even dimensions cannot be produced by a gapped system without massless propagating degrees of freedom. This excludes the possibility that the traditional anomaly can be canceled by a TQFT. In odd dimensions, a Chern-Simons action can have S[ag] for the parity anomaly if we regard it as a kind of global anomaly. However, the ChernSimons action is just a local action, and a TQFT does not play any role. Therefore, the 10We remark that there is a di erence between a local action and a local polynomial action. To explain this point and also to illustrate the argument below (5.1), let us consider a gapped theory in odd dimensions in which the ZTQFT[a] is given by e 2 i , where is Atiyah-Patodi-Singer invariant of some Dirac operator coupled to a. See e.g., [15, 25] for discussions on such a theory. Under a local continuous deformation, the change of is captured by the change of Chern-Simons action. Notice that itself is not represented by a polynomial of a. Only the di erence S[a] = log ZTQFT[a] log ZTQFT[0] for topologically trivial a can be represented by a Chern-Simons action which is a polynomial of a. However, the invariant is local in some appropriate sense because it satis es the gluing law [26] (see [25] for physics explanation). Our action S[a] is a polynomial of a as the argument below (5.1) clearly shows. We can enumerate such local polynomial actions. S[a] 6= 0. This is relevant (5.1) (5.2) (5.3) global anomaly which is not cancelled by a Chern-Simons counterterm cannot be cancelled by a TQFT in odd dimensions either. 6 Topological Green-Schwarz mechanism We have seen that the 8d Sp(N ) SYM su ers from the new global anomaly which is di erent from the traditional one associated to 8(G). Since the 8d Sp(N ) SYM can be realized in string theory, the anomaly must be canceled, and the cancellation must be carried out by coupling to a TQFT. In this section, we discuss how this kind of subtle anomaly might be sometimes canceled by a TQFT in a way analogous to the standard Green-Schwarz mechanism. In section 6.1, we rst explain an analogue of the Green-Schwarz mechanism which uses topological degrees of freedom, using a TQFT which couples to ordinary homology cycles. Then, in section 6.2, we discuss an example where this method actually works in 3d: an SU(N )=ZN gauge theory coupled to an adjoint. But unfortunately this does not work in 8d Sp(N ) theory, as we explain in section 6.3. We then discuss some of the expected properties of the 8d TQFT. Most of the contents of section 6.1 and section 6.2 are not new, but we put emphasis on some of the points relevant to the discussion of the 8d Sp(N ) anomaly. 6.1 Topological Green-Schwarz mechanism for cohomological classes Suppose X and Y are some characteristic classes in cohomology with Zk coe cient which are associated to gauge elds of group G. If we are given a manifold M equipped with gauge elds, then we get elements of cohomology X 2 H (M ; Zk) and Y 2 H (M ; Zk). Hp(M ; Zk) and Y 2 Hd p+1(M ; Zk) for an integer p. Then we want to consider a (d Let M be a d-dimensional manifold with G-bundle P . We also assume that X 2 form Zk gauge theory coupled to X and Y , which is roughly described by S = \ 2 i kb0da0 + b0X + ( 1)d p+1Y a0 00; p)(6.1) Z M where b0 and a0 are (d p)-form and (p 1)-form elds,11 and d is the exterior derivative. However, this action still does not make sense. This is because X and Y are elements of cohomology H (M ; Zk), but, at the level of classical Lagrangian, a0 and b0 are di erential forms which are not necessarily closed. Eventually this problem leads to the anomaly involving X and Y as discussed e.g. in [27]. We give an argument which is slightly di erent from [27]. For simplicity, we assume that the manifold M has no torsion in its cohomology so that H (M ; Zk) = H (M ; Z) Zk. First, we change the variables from a0 and b0 to a and b such that f := da = da0 + X; g := db = db0 + Y: (6.2) 1 k 11More precisely, they must be treated by using di erential cohomology theory, but we will be sloppy throughout the paper and just pretend as if they were the usual di erential forms. This does not a ect the nal conclusion. 1 k The uxes f = da and g = db are gauge invariant. The e ects of the background elds X and Y are now incorporated in quantization conditions of the uxes f and g as where [f ] and [g] are cohomology classes represented by f and g. Namely, the fractional part of the uxes of f and g are determined by X and Y , respectively. These shifts of the quantization conditions are analogous to the shift of 't Hooft magnetic ux in the presence of background elds of 1-form center symmetry in Yang-Mills theory [28, 29]. In the present case, the relevant symmetries are higher form (or more explicitly (d p)-form To obtain a gauge invariant action, we follow the standard procedure. Let N be a d + 1 dimensional manifold whose boundary is our d-manifold M . We assume that X and Y can also be extended to N . In this situation, we take the action as This action is gauge invariant. However, it depends on how we extend M and the cohomology classes X and Y on it to the manifold N . The standard way to see it is to consider another manifold N 0 and de ne the action SN0 . The di erence SN SN0 is given by considering a closed manifold L which is obtained by gluing N and N 0 along their common boundaries, and evaluating SL = 2 ik RL gf: Now, because of the quantization conditions (6.3), this integral on the closed manifold L is evaluated to be SN = 2 ik gf: Z N SL = 2 i Z k L Y X: (6.4) (6.5) This is the anomaly of this system. In terms of the topological defect operators, the (p 1) form symmetry whose background eld is X and the (d p)-form symmetry whose background is Y have associated `volume operators' supported on (d p)-dimensional submanifolds and on (p 1)dimensional submanifolds, respectively. This anomaly means that they have a braiding phase exp(2 i=k). Now, suppose that we consider a gauge theory of a gauge group G with some Weyl fermions. If the gauge theory has a global anomaly given by acteristic classes Y and X, then we can cancel the anomaly by adding the TQFT given by (6.4). This is what we mean by topological Green-Schwarz mechanism. There are also important topological constraints coming from the TQFT. The equations of motion of (6.4) are given by f = 0 and g = 0. However, because of the quantization conditions (6.3), we have to impose the conditions that X and Y are trivial as elements of H (M ; Zk). These constraints are analogous to the equation dH = tr RR tr F F in heterotic string theories, where H is the eld strength of the NS 2-form eld, R is the Riemann curvature, and F is the eld strength of the heterotic gauge group. This equation requires that tr RR tr F F is trivial in de Rham cohomology. 2k i RL Y X for some char In our 8d Sp(N ) SYM, the above topological constraints must work as follows, assuming the existence of some appropriate TQFT which cancels the anomaly of Sp(N ). In section 4, we found the anomaly by compactifying the theory on S4 and putting an instanton on it. The TQFT must forbid such a con guration. Namely, the necessary condition on the TQFT is that it must give the constraint that the instanton number on S4 is even. So the TQFT gives the constraints on the sum over topological sectors [30]. Topological Green-Schwarz mechanism in action: a 3d example Now let us discuss an example where the topological Green-Schwarz mechanism as described above is in action. Consider the 3d Majorana fermion in the adjoint of su(N ), where we assume N is even. It has an anomaly captured by the 4d topological term HJEP1(207) N Z 2 2 i c2(F ) where c2(F ) / tr F ^ F is the second Chern class in the normalization where it integrates to one for the SU(N ) one-instanton. (A complex fundamental fermion needs 1=2 ChernSimons. An adjoint Majorana fermion then needs level 1=2 2N 1=2 = N=2.) Now we try to gauge it with SU(N )=ZN . Let us impose time-reversal invariance so that we cannot add any Chern-Simons counterterms. Then this is anomalous, since a con guration of SU(N )=ZN can have instanton number 1=N . This is a version of the so-called parity anomaly, but we are imposing time-reversal invariance and hence the gauge symmetry is anomalous. But the anomaly (6.6) can also be written modulo 2 i as 2 i (6.6) (6.7) where w2 2 H2(M4; ZN ) is the (generalized) Stiefel-Whitney class of the bundle.12 Therefore, a 3d TQFT which couples to a background ZN one-form symmetry with this anomaly can cancel it. Note that in terms of the line operator coupled to w2, the anomaly (6.7) just means that the self braiding phase is i = exp(2 i=4). In other words, the topological spin of the operator is 1=4, which is detected by \twisting" the line operator [33]. If N is a multiple of 4, the anomaly can be canceled by using the Z4 theory with d = 3; p = 2 and taking X; Y to be the mod 4 reduction of w2. However, in the present case of the anomaly of the form X2 (with X = w2), there is more economical choice which is applicable to any even N . We use the U(1)2 U(1) 1 Chern-Simons theory which is time-reversal invariant [34{36]. It has a Z2 1-form symmetry which can be coupled to the mod 2 reduction of w2, and the coupling can be done as a shifted quantization condition of the U(1)2 gauge eld as in (6.3). The anomaly is computed exactly as in the previous subsection, and is given by (6.7). 12More precisely, for N 2 mod 4, we need to use the Pontryagin square operation to make sense of the factor 1=4, see e.g. section 6 of [31] or [32]. If we assume that the manifold has no torsion in cohomology, we can lift w2 to an element of integer cohomology and de ne w2 ^ w2 there. 6.3 The idea above does not exactly work in our 8d Sp(N ) theory if we insist on using ordinary (co)homology. To see this, recall how we found the anomaly in section 4. As reviewed in section 2, the anomaly is given by the number of zero modes modulo 2 in a 9-dimensional manifold N . To nd an anomaly, in section 4 we rst considered the instanton brane Z which is realized by codimension-4 instanton of Sp(N ) embedded in SU(2) Sp(N ). This has codimension 4 and hence dim Z = 5. There are localized zero modes on Z in the fundamental representation of Sp(N 1) Sp(N ). These modes can be regarded as fermions living on the brane Z coupled to Sp(N 1). Then, we further evaluate the mod 2 index of these fermions on Z. The location of the instanton brane Z at the homological level is determined by the Poincare dual of q1 which is the Pontryagin class or the instanton number of the Sp(N ) bundle (which is equivalent to the second Chern class c2 of the fundamental representation of Sp(N )). Thus, naively, the topological Green-Schwarz mechanism requires (6.8) (6.9) naively: X [mod 2 index in 5d]: Then the anomaly is described as naively: i Z XY i Z Z [mod 2 index in 5d] by using the Poincare duality q1 $ Z. However, we need more information than ordinary (co)homology to compute the mod 2 index. In other words, there is no formula which gives the mod 2 index in terms of the cohomological characteristic classes of the gauge bundle (and metric). This can be seen from the fact that we can have a nontrivial mod 2 index on S5 as shown in the original paper of global anomaly [12], but there is no cohomological characteristic classes because the classifying space BSp(N ) of the Sp(N ) group has H5(BSp(N )) = 0.13 Moreover, to specify the instanton brane Z, it is important that the fermions on Z has a de nite spin structure because the mod 2 index depends on it. However, that information is missing in the homology class of Z determined by the Poincare dual of q1. The spin structure is determined as follows. The instanton has an anti-self-dual curvature, and in particular locks the SU(2) gauge bundle and the normal bundle to the locus of the instanton brane Z. This means that the holonomy of the normal bundle is reduced from SO( 4 ) to SU(2). This reduction of the structure group of the normal bundle, combined with the spin structure on the total space time, gives the spin structure to the normal bundle, and hence to the tangent bundle. The above considerations suggest that we need a concept of Poincare duality which gives us a de nite spin structure on the submanifold Z. Indeed mathematicians have 13The cohomology H (BSp(N ); Z) is freely generated as a ring by Pontryagin classes 1; q1; q2; ; qN where qi 2 H4i(BSp(N ); Z). Cohomology with more general coe cients can be obtained by universal coe cients theorem from H (BSp(N ); Z). Also, there is no nontrivial gravitational characteristic classes on S5 which can mix with q1 to give a nontrivial value. developed such a concept: KO-homology of Baum-Douglas, see e.g. string theory articles which use them, [37, 38]. R [Md] uv = R PD[u] v. The essential idea behind it is as follows. In the case of ordinary Z-coe cient homology u 2 Hi(Md; Z) on a d-manifold Md, an orientation of Md de nes the `volume form' [Md] 2 Hd(Md; Z), and then PD[u] 2 Hd i(Md; Z). Furthermore, for v 2 Hj (Md; Z), These concepts generalize to K and KO-(co)homologies. First, the KO-homology denes groups KOi(Md) for a given positive integer i and a d-dimensional manifold Md, as the ordinary homology Hi(Md) does. There is also KO-cohomology groups, denoted by KOi(Md). A KO-orientation of Md is a spin structure and de nes the KO-theoretic theory means that we take the index of the appropriate Dirac operator. `volume form' [Md] 2 KOd(Md). PD[u] 2 KOd i(Md), such that R[Md] uv = R Then, a class u 2 KOi(Md) has a Poincare dual PD[u] v. Here, the integration symbol in KO In the present context, an Sp(N ) bundle P on a 9-manifold M9 gives an element [P ] of the symplectic K-theory on M9, which is canonically isomorphic to KO4(M9) by Bott periodicity.14 Its Poincare dual PD[P ] 2 KO5(M9) in the KO sense is exactly our instanton brane Z equipped with the spin structure described above. An important formal di erence between the integration of KO theory and the integraneeds to be in Hd. This is no longer the case for the KO theory: in general, tion in ordinary (co)homology is that in the latter, to have non-vanishing R[Md] u, the u Z [Md] u 2 KO i d(pt) for u 2 KOi(Md) where KOi(pt) is a KO-cohomology on a single point pt. We have i KOi 0 Z 1 Z2 2 Z2 3 0 4 Z 5 0 6 0 7 0 and KOi = KOi+8. For example, the mod-2 index of an Sp bundle P over a spin 5-manifold M5 can be understood as the integration in KO theory. Indeed, R M5 2 KO 1(pt) = Z2. This can be evaluated in two steps: the Poincare dual of is an element PD[ ] 2 KO1(M5), which is a one-cycle with a spin structure. Then we have := [P ] 2 KO4(M5), and therefore Z M5 PD[ ] 1 2 KO 1(pt) = Z2: Note that the mod-2 index of a circle with a spin structure is 0 if the spin structure is NS (anti-periodic) and 1 if it is R (periodic). The class also gives instanton numbers when integrated over 4-manifolds. Therefore, it plays the role of both q1 and [mod 2 index in 5d]. So, very roughly speaking, we may identify X + Y . 14There is also a symplectic version of K-(co)homology called KSp-(co)homology, and Bott periodicity relates KSp and KO by KSpi=KO i 4=KOi+4. (6.10) (6.11) (6.12) Let us come back to the question of the 8d Sp theory. Given an Sp bundle P on a 8d spin manifold M , we consider a fermion in the adjoint representation. There is no perturbative anomaly and no global anomaly in the traditional sense associated to 8(Sp). There is however an anomaly, whose phase is characterized by Z N Sym2 2 Z2 (6.13) where N is the 9d manifold, is the class in KSp0(N ) = KO4(N ) associated to the vector bundle in the fundamental representation of an Sp bundle P over N , and Sym2 : KO4 ! KO8 = KO0 is the symmetric power sending the fundamental representation of Sp to the adjoint representation. Very roughly, we may think of it as Sym2 So, we need a 8d TQFT such that it can couple to elements 2 KO4(M ), and it has an anomaly characterized by (6.13). Note that if it can couple to an ordinary cohomology have said that this TQFT has a Zk 3-form symmetry. Since it can couple to an element in KO4(M ), it has some generalized notion of symmetry. The anomaly (6.13) suggests that the required TQFT is a KO-theoretic version of abelian Chern-Simons theory which is somewhat analogous to the theory U(1)2 U(1) 1 mentioned at the end of section 6.2. The authors hope to come back to this problem in 2 H4(M; Zk) instead, we would the future. Acknowledgments I.G.-E. thanks Miguel Montero and Diego Regalado for discussions. KO gratefully acknowledges support from the Institute for Advanced Study. YT is partially supported in part byJSPS KAKENHI Grant-in-Aid (Wakate-A), No.17H04837 and JSPS KAKENHI Grantin-Aid (Kiban-S), No.16H06335, and also supported in part by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. The work of KY is supported in part by the WPI Research Center Initiative (MEXT, Japan), and also supported by JSPS KAKENHI Grant-in-Aid (17K14265). A List of homotopy groups Table 1 lists low-degree homotopy groups of compact Lie groups. The table itself can be found in the appendix of [39]. For computations, see [40]. We only consider simply connected groups 1(G) = 0, while all compact simple Lie groups have 2(G) = 0 and 3(G) = Z. 4(G) is trivial except for G = Sp(N ), in which case 4(G) = Z2. 4(G) also has a derivation uniform to all G in terms of the root system, see [41].15 15The authors thank the discussions at https://mathoverflow.net/questions/259487/. SU(3) SU( 4 ) SU(5) Sp(N 3) SU(N 6) Spin(7) Spin(8) Spin(9) Spin(10) Spin(11) Spin(12) 13) G2 F4 E6 E7 E8 Z2 Z2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Z2 Z2 Z Z Z Z 0 0 0 0 0 0 0 0 0 0 0 0 Z12 Z6 Z3 Z2 Z 2 Z Z 0 Z Z Z Z Z Z Z Z Z 0 0 0 0 0 Z2 Z12 Z24 (Z2)2 (Z2)3 (Z2)2 Z2 Z2 Z2 Z2 Z2 Z2 0 0 0 (Z2)2 (Z2)3 (Z2)2 Z Z2 Z3 Z3 Z2 Z Z Z2 Z2 Z2 Z6 Z2 Z 0 0 Z120 Z2 Z24 Z8 Z15 Z120 0 Z30 Z120 0 Z8 Z8 Z4 Z2 0 0 0 0 0 0 0 Z Z Z Z Z Z Z2 Z Z4 Z4 0 Z Z Z Z Z Z 0 Z2 Z2 Z2 Z Z2 Z2 d For the in nite series SU(N ), the homotopy groups d(SU(N )) become stable for N > d=2. For the in nite series Spin(N ), the homotopy groups d(Spin(N )) become stable for N > d + 1. 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Iñaki García-Etxebarria, Hirotaka Hayashi, Kantaro Ohmori, Yuji Tachikawa, Kazuya Yonekura. 8d gauge anomalies and the topological Green-Schwarz mechanism, Journal of High Energy Physics, 2017, 177, DOI: 10.1007/JHEP11(2017)177