6d, N=(1,0) Coulomb Branch Anomaly Matching

6d QFTs are constrained by the analog of 't Hooft anomaly matching: all anomalies for global symmetries and metric backgrounds are constants of RG flows, and for all vacua in moduli spaces. We discuss an anomaly matching mechanism for 6d N=(1,0) theories on their Coulomb branch. It is a global symmetry analog of Green-Schwarz-West-Sagnotti anomaly cancellation, and requires the apparent anomaly mismatch to be a perfect square, $\Delta I_8={1\over 2}X_4^2$. Then $\Delta I_8$ is cancelled by making $X_4$ an electric / magnetic source for the tensor multiplet, so background gauge field instantons yield charged strings. This requires the coefficients in $X_4$ to be integrally quantized. We illustrate this for N=(2,0) theories. We also consider the N=(1,0) SCFTs from N small $E_8$ instantons, verifying that the recent result for its anomaly polynomial fits with the anomaly matching mechanism.

6d QFTs have chiral matter, so anomalies provide a useful handle. Gauge anomaly cancellation highly constrains the matter content [2,9,12,[16][17][18][19]. The analog of 't Hooft anomalies, for global symmetries, usefully constrains the low-energy theory: these anomalies must be constant along RG flows, and on the vacuum manifold, even if the symmetry is spontaneously broken. In the broken case, as in 4d [20], anomaly matching can require certain WZW-type low-energy interactions, to cancel apparent anomaly mismatches. This was discussed for 6d theories in [21], and applied to the case of N = (2, 0) theories on the Coulomb branch. We here apply analogous considerations to N = (1, 0) theories. Consider a 6d, N = (1, 0) theory with a Coulomb branch moduli space of vacua, associated with φ for the real scalar(s) of tensor multiplets. Let S origin denote the lowenergy theory at φ = 0. Moving to φ = 0, the theory reduces at low-energy as  [3,4,5]. 2 The notation is because it reduces, on an S 1 , to a 5d N = 1, U (1) vector multiplet.
remaining interactions in the low-energy theory. We here discuss an anomaly matching mechanism, which cancels ∆I 8 provided that it is a perfect square: More generally, with multiple tensors, we need where the I index runs over the tensor multiplets, and Ω IJ is a positive definite, symmetric metric on the space of tensor multiplets, which is implicit in the ∧· product in (1.3).
The mechanism is analogous to that of [22,23] for canceling anomalies of local symmetries. A reducible gauge anomaly I 8 can be cancelled via an additional tensor multiplet contribution ∆I 8 of the form 3 (1.3). This is achieved by making X I 4 into electric / magnetic sources for the tensor multiplet field strengths H I . Our sign conventions 4 are such that Ω IJ is positive definite. The full theory is then gauge anomaly free if I 8 + ∆I 8 = 0.
We apply a similar mechanism to global symmetries; rather than canceling an unwanted I 8 of opposite sign, here the tensor multiplet's ∆I 8 provides the 't Hooft anomaly matching deficit. This is achieved by making X 4 (the · is shorthand for multiple tensors, i.e. the I index in (1.3)) act as electric / magnetic sources for the tensor multiplets, so in (1.6) is not invariant under global symmetry background gauge transformations, B 2 must also correspondingly transform, such that H is invariant, δH = 0: 2 . (1.7) 3 In [22,23], the H I also includes the tensor from the gravity multiplet, which has opposite chirality from those of the matter multiplets, and correspondingly enters into Ω IJ with opposite signature [23]. Here we decouple gravity, so Ω IJ has a definite signature. We take it to be positive. 4 We take matter fermions to contribute positively to I 8 , while gauginos contribute negatively.
Then the positive ∆I 8 (1.3) from tensor multiplets can e.g. cancel a negative I 8 gauge anomaly.
Then variation of (1.4) will compensate for the apparent discrepancy from (1.2).
Because B 2 has quantized charges, the coefficients in X 4 must be correspondingly appropriately quantized. The general X 4 can be expanded in characteristic classes is the Pontryagin class for the rigid, background spacetime curvature, p 1 (T ) ≡ 1 2 tr(R/2π) 2 , c 2 (R) and c 2 (F i ) are Chern classes of the SU (2) R and F i flavor symmetry background field strengths. The Chern classes c 2 (R) and c 2 (F i ) will here always be normalized to integrate to one for the minimal associated instanton configuration in the background gauge fields; as we will discuss, the corresponding statement for p 1 (T )/4 is less clear. Such background gauge field instanton configurations are codimension 4 strings 5 , with H charge given by n SU(2) R or n i (the i index runs over all global symmetries). These charges must reside in an integral lattice, so there is a quantization condition We expect that n grav in (1.8) is also quantized, but are uncertain about the normalization.
Note also that the susy completion of (1.4) will give terms L ef f ∼ −φF µν F µν , as in [2], now coupling the real scalar φ of the tensor multiplets to the background field strengths.
The outline is as follows. In section 2, we elaborate on the above anomaly matching mechanism. In section 3, we discuss the N = (2, 0) theories, from a N = (1, 0) perspective.
In section 4, we review the 6d N = (1, 0) theories associated with small E 8 instantons, and their recently-obtained anomaly polynomial [27]. In section 5, apply the anomaly matching mechanism to the small E 8 instanton theory on its Coulomb branch.
Note added: Just prior to posting this paper, the outstanding paper [28] appeared. It uses essentially the same kind of anomaly matching mechanism as discussed here, to derive new results for anomaly polynomials for many classes of N = (1, 0) theories.

6d 't Hooft anomalies, and a new mechanism for their matching
By the descent procedure [29][30][31][32], the anomalous variation of the effective action of a 6d theory is given in terms of the anomaly polynomial 6 8-form I 8 : where δ denotes the variation, M 6 is 6d spacetime 7 , the subscript on X 6 is the form number, and the superscript the order in the gauge or global symmetry variation parameter. Now suppose that the theory has a moduli space of vacua, and the theory at the origin has anomaly polynomial I origin 8 , while the theory away from the origin has a naively different anomaly polynomial I away,naive 8 . The naive difference leads to an apparent mismatch (2. 2) The variation of the low-energy effective action must make up for this difference: As an example, consider N = (2, 0) theories on their Coulomb branch: with dΩ 3 = φ * (ω 4 ) the volume form on the S 4 Nambu-Goldstone manifold, and ∂M 7 = M 6 .
It was conjectured in [21] that c(G) = |G|h G , which fits with the G = SU (N ) cases [34], and also SO(2N ) [35], as derived via M-theory M5 branes and bulk anomaly inflow.
The interaction (2.5) remains even when the global symmetry background is turned off, F Sp(2) → 0. This is related to the fact that the 't Hooft anomaly difference, ∆I 8 ∝ p 2 (F Sp (2) ), is irreducible (i.e. it includes tr F 4 Sp (2) , not just (tr F 2 Sp(2) ) 2 ). This is similar to the 4d Wess-Zumino-Witten interaction [20] for matching the irreducible 't Hooft anomaly differences of non-Abelian SU (N ≥ 3) global symmetries. Reducible t Hooft anomaly differences, on the other hand, lead to WZW-type interactions that become trivial when the background symmetry gauge fields are set to zero. That will be the case for the reducible differences (1.2) to be discussed here.
For 't Hooft anomaly discrepancies of the form (1.2) on the Coulomb branch (1.1), the needed compensating variation (2.3) is where we define X (0) 3 and X (1) 2 via the usual descent notation, as in (2.1): 2 . (2.7) The variation (2.6) arises from the term (1.4) in the low-energy effective action. Unlike Note that a self-dual string's charge Q is quantized as 8 which expresses the compactness of the gauge invariance of B. More generally, the lattice of allowed dyonic string charges must be self-dual [37]. The general 4-form X 4 in (1.2) can be expanded as in (1.8), in terms of properly normalized characteristic classes. So 8 The 1 2 here is from the 6d string's Dirac quantization, eg = 1 2 2π n, see e.g. . boundary, Σ 4 p 1 ∈ 24Z if Σ 4 is spin (this follows from the spin 1/2 index theorem, since A = 1 + p 1 /24 + . . .); for compact Σ 4 that is not necessarily spin, Σ 4 p 1 ∈ 3Z. But here we are interested non-compact Σ 4 , or Σ 4 with boundary, where the index theorems include boundary contributions, η, and the quantization conditions are weaker, see e.g. [40]. The Q contribution from n grav could likewise have boundary contributions. We will not consider the n grav quantization issue further here. We will see that the E 8 instanton example gives n grav = 1 with the normalization in (1.8).  where it was noted that (2.5) can be obtained by taking dΩ 3 to source H 3 with coefficient α m and ⋆H 3 with coefficient α e , see also [41]. This seemed to require ∆c/12 = α e α m , with α e = α m , apparently in conflict with self-duality of H 3 , and unclear quantization of α e,m . 9 I. e. c 2 (F G ) = λ(G) −1 1 2 tr(F G /2π) 2 , where λ(G) can be computed as in e.g. [38,39].

Review: the small E 8 instanton theory, E 8 [N ], and its anomaly polynomial
We will illustrate the anomaly matching mechanism for the case S origin = The Coulomb branch corresponds to moving the M5 branes to φ ∼ x 11 = 0 (the Higgs branch corresponds to dissolving the M5s into E 8 instantons, necessarily at x 11 = 0). The added free-hypermultiplet corresponds to the CM location of the M5 branes in the x 6,7,8,9 directions. By considering anomaly inflow, as in [34] but including the effect of the M9 brane, the anomaly polynomial of this theory was obtained in [27]  Here +f.h. denotes "free-hyper:" The notation in (4.1) is much as in [27] 3) Our normalization is such that all Σ 4 c 2 (F ) = 1 for the minimal instanton configuration.
In this notation, the anomaly polynomial of the N = (2, 0) theory of N M5 branes, keeping only SO(4) ⊂ SO (5)  in anomalies between the LHS and RHS of (5.1) is It's indeed a perfect square, as required. Moreover, writing this X 4 as in (1.2), the coefficients are indeed integrally quantized (the 1 2 's in (5.2) all cancel or combine to 1) The N = 1 case of (5.4) and (5.5) coincides with the N = 1 case of (5.1) and (5.2). More generally, all N tensor multiplets on the RHS of (5.4) participate in the anomaly matching mechanism, hence the overall N in (5.5), with an associated lattice of integral charges.