Class $\mathcal{S}$ Anomalies from M-theory Inflow

We present a first principles derivation of the anomaly polynomials of 4d $\mathcal{N} = 2$ class $\mathcal{S}$ theories of type $A_{N-1}$ with arbitrary regular punctures, using anomaly inflow in the corresponding M-theory setup with $N$ M5-branes wrapping a punctured Riemann surface. The labeling of punctures in our approach follows entirely from the analysis of the 11d geometry and $G_4$ flux. We highlight the applications of the inflow method to the AdS/CFT correspondence.


INTRODUCTION
't Hooft anomalies are measures of degrees of freedom of quantum systems that are preserved under renormalization group flow. Thus, anomalies provide powerful tools for exploring phases and non-perturbative regimes of quantum theories.
In the last ten years, a new approach to studying quantum field theories (QFTs) has emerged with the discovery of N = 2 class S superconformal field theories (SCFTs) [1,2], where a large class of 4d N = 2 SCFTs are geometrically defined from reductions of 6d (2, 0) SCFTs on punctured Riemann surfaces. A choice of 6d SCFT and boundary data at the punctures completely specify a 4d SCFT and its various protected sectors. A typical theory in this class is non-Lagrangian and strongly coupled, and yet it can be analyzed from the geometric construction. The approach of the class S program has been generalized and adopted for studying SCFTs in different dimensions with varying amount of supersymmetry. The geometrization program has become a standard tool in the study of QFTs.
A key feature of the class S program is the richness of the variety of punctures on the Riemann surface. The anomalies of N = 2 class S SCFTs in the presence of regular punctures have been indirectly obtained from field theoretic arguments [3][4][5]. However, a direct derivation of the anomalies from the geometric definition of class S SCFTs is lacking. In this letter we use anomaly inflow in M-theory to provide a first principles derivation, building on [6]. Our procedure can be generalized to obtain the anomalies of other classes of SCFTs with geometric descriptions. Further, our prescription suggests a method for extracting the exact anomalies of a holographic SCFT from its gravity dual.
The 't Hooft anomalies of a d-dimensional QFT are neatly encoded in the (d + 2)-form anomaly polynomial. In this letter we derive the anomaly polynomials of 4d N = 2 class S SCFTs with regular punctures engineered from the 6d (2, 0) A N −1 SCFTs. First, we describe the relevant geometric setup from a stack of N M5-branes in M-theory, and the inflow procedure. Then we provide a novel description of the boundary data at punctures in terms of the four-form flux of M-theory. Finally, we compute the anomaly polynomial and discuss its implications for holography. A companion paper [7] to this letter contains more complete derivations and a broader study of the results and their implications.

SETUP AND INFLOW
A 4d N = 2 class S theory of type A N −1 is engineered in M-theory by taking the low-energy limit of a configuration with N coincident M5-branes wrapping a punctured Riemann surface. Let W 6 denote the 6d worldvolume of the M5-brane stack inside the ambient 11d space M 11 . The normal bundle to W 6 , denoted N W 6 , encodes the five transverse directions to the stack and generically has structure group SO (5). We study the case W 6 = M 4 × Σ g,n , where M 4 is external spacetime and Σ g,n is a Riemann surface of genus g with n punctures.
We are interested in setups that preserve 4d N = 2 supersymmetry (for M 4 = R 1,3 ). In this case, the structure group of N W 6 reduces from SO(5) to SO(2)×SO(3), and correspondingly N W 6 decomposes as N W 6 = N SO(3) ⊕ N SO (2) . The (universal cover) of SO(2)×SO(3) is identified with the U (1) r × SU (2) R R-symmetry of the 4d field theory. In summary, the tangent bundle to 11d spacetime restricted on W 6 decomposes as The total space of the N SO(2) fibration over Σ g,n is the cotangent bundle T * Σ g,n , and is hyper-Kähler. The twisting of N SO(2) over Σ g,n implements a partial topological twist of the 6d (2, 0) A N −1 theory living on the stack. Ifn denotes the Chern root of N SO(2) , then where c r 1 is the first Chern class of U (1) r ,t is the Chern root of T Σ g,n , and χ(Σ g,n ) = 2(1 − g) − n is the Euler arXiv:1812.04016v1 [hep-th] 10 Dec 2018 characteristic of the punctured Riemann surface. In order to specify the 4d theory, we must supplement each puncture with appropriate data, encoding the boundary conditions for the 6d theory. The puncture data is determined by the branching pattern of the M5-branes which governs the flavor symmetry of the 4d theory.
From the point of view of M-theory, the combined system of the M5-brane stack and the 11d bulk enjoys a nonanomalous diffeomorphism invariance. The total system is free from local anomalies in 11d due to a cancellation between the anomaly generated by the chiral massless degrees of freedom localized on W 6 , and anomaly inflow from the bulk.
The anomaly inflow from the bulk amounts to a classical anomalous variation of the M-theory effective action under 11d diffeomorphisms, due to the presence of the M5-brane stack. The latter acts as a magnetic source for the M-theory four-form G 4 with delta-function support on W 6 , dG 4 = 2π N δ W6 . In order to analyze anomaly inflow in the supergravity approximation we must smooth out the delta-function singularity [8,9]. This is achieved by cutting out a small tubular neighborhood of the M5brane stack. As a result, we are now considering Mtheory on a manifold with a boundary M 10 = ∂M 11 , which is diffeomorphic to an S 4 bundle over W 6 . The information about the original delta-function source is translated into a smoothed-out G 4 flux, The quantity f is a bump function that depends only on the radial distance away from the M5-brane stack, smoothly interpolating between −1 at the boundary M 10 and 0 away from it. The four-form E 4 is globally-defined, closed, invariant under the action of the structure group of N W 6 , and can be written locally as 3 . The integral of E 4 over the S 4 surrounding the stack measures the total magnetic charge N of the M5-branes.
The anomalous variation of the M-theory effective action is expressed as an integral over M 10 and is conveniently formulated in the framework of descent, The formal quantity I 12 is a twelve-form characteristic class constructed from E 4 and given by On the right-hand-side we suppressed wedge products for brevity, and we introduced the eight-form class I 8 , which is defined in terms of the Pontryagin classes of T M 11 as The inflow contribution to the anomaly polynomial of the 4d CFT is extracted by integrating I 12 over the total space of the S 4 bundle over Σ g,n , denoted M 6 , Anomaly cancellation requires I inf 6 to cancel against the CFT anomaly, up to decoupling modes, To compute the integral in (7), we excise small disks around each puncture on Σ g,n , together with the S 4 fibers on top of them. We thus obtain a space M 6 , which is an S 4 fibration over a smooth Riemann surface with n boundaries. We replace the excised portions of M 6 with suitable local geometries X α 6 , with α = 1, . . . , n, glued smoothly to M 6 . This decomposition of M 6 translates to where P α denotes the α th puncture on Σ g,n . We refer to I inf 6 (Σ g,n ) as the bulk contribution to I inf 6 . Each geometry X α 6 is locally S 2 × X α 4 , where the S 2 encodes the angular directions of N SO (3) , while X α 4 comprises the directions of the excised disk, together with the fibers of N SO(2) on top of it. More precisely, X α 4 is the local space that models T * Σ g,n in the vicinity of the puncture P α . Thus, the possible choices of X α 4 in Mtheory encode the puncture data. The space X α 4 admits a U (1) isometry, which is identified with the U (1) action on N SO (2) in the bulk of T * Σ g,n .

BULK CONTRIBUTION TO INFLOW
To write down the class E 4 on M 6 it is convenient to recall that S 4 can be realized as an S 1 φ × S 2 Ω fibration over an interval. The subscript φ is a reminder that we use the coordinate φ (with period 2π) to parametrize S 1 φ . The label Ω is inserted for convenience, to distinguish S 2 Ω from other two-spheres discussed below. The interval is parametrized with a coordinate µ ∈ [0, 1]. At µ = 0 the radius of S 2 Ω goes to zero, while at µ = 1 S 1 φ shrinks to zero. The non-triviality of the N SO(2) bundle is captured by Dφ = dφ − A, where A is a connection with field strength dA = 2πn, see (2). Using this notation, the general E 4 reads The function γ depends on µ only, satisfies γ(0) = 0, γ(1) = 1, and has no zeros within the interval (0, 1), but is otherwise arbitrary. The two-form e Ω 2 is the closed, SO(3)-invariant completion of the volume form on S 2 Ω , normalized to integrate to 1. The overall normalization in (10) is fixed by (3).
The class I 8 on M 6 is obtained via the decomposition of p 1 (T M 11 ), p 2 (T M 11 ) under (1), using standard formulae for Pontryagin classes of direct sums of vector bundles. Notice that p 1 (T Σ g,n ) =t 2 , p 1 (N SO(2) ) =n 2 , while where c R 2 is the second Chern class of SU (2) R . The only terms in I 8 that can contribute to the integral over M 6 are those linear int, We are now in a position to compute the integral of I 12 over M 6 . To this end, it is useful to recall the Bott-Cattaneo formula [10] The result reads The quantity I inf 6 (Σ g,n ) coincides with the dimensional reduction along Σ g,n of the inflow eight-form anomaly polynomial for a stack of M5-branes [6].

PUNCTURE GEOMETRY AND FLUX
To understand X α 6 , first consider a small disk around a generic point on Σ g,n with polar coordinates (r Σ , β). The local geometry is an S 1 β × S 1 φ × S 2 Ω fibration over the half-strip spanned by r Σ and the µ interval depicted in Figure 1.  We now map the (r Σ , µ) half-strip to a quadrant of R 2 with coordinates (ρ, η). The qualitative features of this map are highlighted in Figure 1. The region 0 < η < η max on the η axis corresponds to the µ interval at r Σ = 0, while the region η > η max corresponds to r Σ > 0 at µ = 1.
We define a new angle χ = φ + β, and we regard the whole X α 4 as an S 1 β fibration over the 3d base space spanned by (ρ, η, χ). We demand that S 1 χ shrinks along the η axis in the base space, so that we identify the base space with R 3 with cylindrical coordinates (ρ, η, χ). The non-triviality of the S 1 β fibration is captured by for L a function of ρ, η.
The function L is smooth in the interior of the (ρ, η) quadrant, but it approaches a discontinuous, piecewise constant function of η for ρ → 0. More precisely, we need L = 1 for 0 < η < η max , and L = 0 for η > η max . This ensures that we reproduce the features of the previous description-that S 1 φ shrinks at η > η max and S 1 β shrinks on [0, η max ]. The discontinuity in L implies that the S 1 β fibration has a monopole source of charge +1 on the η axis located at η = η max .
So far we have done a rewriting of the local geometry near a generic point on the Riemann surface, i.e. a nonpuncture. Although in this case the local geometry is trivial, the formulation in terms of the S 1 β fibration (13) features a monopole source where the S 1 β fiber shrinks. This setup lends itself to a natural generalization. Consider a fibration as in (13) with p monopoles labeled by a = 1, . . . , p, located at η = η a and with η p = η max . This configuration is depicted in Figure 2. Denote the piecewise constant values of L by L = a for η a−1 < η < η a ; p+1 = 0 .
The charge k a of each monopole is measured by for S 2 a the 2-sphere surrounding the monopole in base space R 3 . The S 1 β circle shrinks at each monopole. Since the space X α 4 is a local model for T * Σ g,n in the neighborhood of the puncture P α , its geometry is constrained. In particular, k a > 0 for all a, so that the a are a sequence of decreasing integers. Furthermore, the local geometry near each monopole is an ALF hyper-Kähler space, modeled by a single-center Taub-NUT space with charge k a , denoted TN ka . This space has an R 4 /Z ka orbifold singularity which can be resolved to yield a smooth hyper-Kähler space TN ka . Now we discuss E 4 in the geometry X α 6 . The most general form of E 4 compatible with the symmetries is where the gauging of χ with the connection A is inherited from φ, and Dβ denotes Dβ as in (13)  Y , W are functions of ρ, η and are constrained by flux quantization of E 4 . Both Y and W must vanish on the ρ axis at η = 0, because S 2 Ω shrinks there. We start by defining the relevant cycles. For a = 1, . . . , p there is a four-cycle B a consisting of the interval [η a−1 , η a ] at ρ = 0, S 1 β , and S 2 Ω . For a ≥ 2, S 1 β shrinks at the endpoints of [η a−1 , η a ] and thus we also have a two-cycle S a , depicted in Figure 2.
Next, consider the arc C a connecting a point on the ρ axis to a point within the (η a , η a+1 ) interval, with a = 1, . . . , p−1, as depicted in Figure 2. The arc C a , together with S 2 Ω and the combination of S 1 χ and S 1 β that shrinks along (η a , η a+1 ), gives the four-cycle C a . The arc C p in Figure 2, combined with S 1 φ and S 2 Ω , gives a four-cycle C p that is equivalent to the bulk S 4 .
Supersymmetry requires the flux of E 4 through the C a and B a cycles to respectively carry the same sign. We choose the orientations such that Ba E 4 and Ca E 4 are positive to be consistent with the conditions C p ∼ = S 4 and, for the non-puncture, C 1 ∼ = B 1 ∼ = S 4 . One finds such that w 0 = 0 and {w a } p a=1 is an increasing sequence of positive integers.
The flux Ca E 4 equals Y evaluated at the endpoint of the C a arc on the η axis. Since the endpoint can be freely moved within (η a , η a+1 ), Y is piecewise constant along the η axis, and takes non-negative integer values, Although Y is discontinuous along the η axis, E 4 must be continuous. This condition gives y a − y a−1 = w a k a , where y 0 = 0 and we used C p ∼ = S 4 . Continuity of E 4 thus implies the partition of N labeling a regular puncture.
For each non-trivial two-cycle in X α 6 , we can turn on an additional contribution to E 4 of the form ω ∧F , for ω the Poincaré dual of the two-cycle and F the field strength of a background U (1) connection on M 4 . One such twocycle is S a depicted in Figure 2, with Poincaré dual denoted ω a . Additional two-cycles are introduced upon resolving the orbifold singularities at the monopoles. The resolved space TN ka admits k a − 1 two-cycles, with Poincaré duals { ω a,I } ka−1 I=1 . Their intersection pairings give the Cartan matrix C su(ka) of su(k a ), (20) Including these additional terms, E 4 reads where F a and F a,I are 4d field strengths. (21) only captures the Cartan subgroup of the full 4d flavor group G F , Let us now discuss I 8 in the puncture geometry. It is computed using the local decomposition The Pontryagin classes of T X α 4 are given in terms of the Chern roots λ 1 , λ 2 as p 1 (T X α 4 ) = λ 2 1 + λ 2 2 , p 2 (T X α 4 ) = λ 2 1 λ 2 2 . To account for the gauging of the angle χ in (16), the Chern roots are shifted by c r 1 , The relevant terms of I 8 are where p 1 (T X α 4 ) is taken as the class before the shift (24). The total p 1 (T X α 4 ) decomposes into a sum of p 1 ( TN ka ) terms, which satisfy TN ka p 1 ( TN ka ) = 2 k a [11].

INFLOW ANSWER AND CFT COMPARISON
We now have the necessary components to compute I inf 6 (P α ) = X α 6 I 12 in (9). We use the standard parametrization of I 6 for 4d N = 2 SCFTs where n v and n h are the effective numbers of vector multiplets and hypermultiplets respectively; k G is the flavor central charge of a factor G of the 4d flavor group.
A direct computation of the integrals yields Note that there is an enhancement of the k a − 1 Cartan components to the second Chern class of the full non-Abelian SU (k a ) factor in (22).
The partition of N in (19) defines a Young diagram with rows We define k i = i − i+1 and N i = i j=1 j . It follows that (27)-(28) are equivalently written as (12), n inf h (Σ g,n ) = 1 6 (4 N 3 − N ) χ(Σ g,n ) . (33) According to (9), the total n inf v , n inf h are These quantities can now be compared to the known CFT answers [4], as presented in [6]. We find The inflow and CFT contributions cancel, up to minus the anomaly of a free 6d (2, 0) tensor multiplet reduced on a genus-g Riemann surface Σ g,0 with no punctures. We identify this free tensor multiplet with the center-ofmass mode of the M5-brane stack. Our results show that this mode is insensitive to the presence of punctures.

CONCLUSION AND APPLICATIONS TO HOLOGRAPHY
In this letter we provided a first principles derivation of the anomaly polynomials of 4d N = 2 A N −1 class S theories with arbitrary regular punctures, using anomaly inflow in the corresponding M-theory setup with N M5branes wrapping a punctured Riemann surface.
In our approach, the puncture data are entirely specified by the topological properties of the 11d geometry and G 4 flux in the vicinity of the puncture. Remarkably, the anomaly inflow cancels exactly the known anomalies of the 4d SCFTs, up to the contribution of the center-ofmass free tensor multiplet on the M5-brane stack.
Our method for analyzing N = 2 regular punctures is generalizable to irregular punctures and setups with less supersymmetry. Many interesting QFTs can be realized via branes probing geometries in string theory and Mtheory. In such cases, inflow can be a robust tool to compute anomalies, and therefore provides a handle on non-perturbative aspects of these QFTs.
We conclude with a discussion of applications to holography. An important motivation for our analysis of the local puncture geometry and E 4 flux comes from the holographic M-theory duals of N = 2 and N = 1 class S theories with punctures [3,12]. In particular, the fibration in (13) is related to and inspired by the Bäcklund transform of [3]. The solutions are warped products of AdS 5 with an internal space M hol 6 with four-form flux G hol 4 . We observe that the topological properties of M hol 6 in [3] are the same as those of M 6 in (7). Furthermore, where E 4 is E 4 with all 4d connections turned off and G hol 4 is the four-form flux of [3]. In the bulk of Σ g,n E 4 = S 4 , but E 4 is non-trivial in the puncture geometry and encodes the puncture labelling.
Kaluza-Klein reduction of 11d supergravity on M hol 6 yields a 5d gauged supergravity model with an AdS 5 vacuum. The full reduction ansatz requires a G hol 4 that captures the fluctuations of the AdS 5 gauge fields beyond the linearized level. E 4 is a natural candidate for constructing such an ansatz [9].
In the solutions of [3] the classical objects M hol 6 , G hol 4 provide the exact topological data of M 6 , E 4 to all orders in N . This data determines the E 4 and I 8 needed to carry out the inflow procedure, which (subtracting the O(1) contribution of decoupling modes) yields the exact anomaly coefficients of the dual SCFT. This route to the exact a and c central charges bypasses a computation with the AdS 5 effective action, which would require a detailed knowledge of higher-derivative corrections. An interesting question is whether (37) extends to more general AdS 5 solutions in M-theory, with varying amount of supersymmetry. If so, we may use inflow and classical data of the supergravity solution to access exact anomaly coefficients, providing a systematic way to compute quantum corrections in AdS 5 .