The Geometry of Decoupling Fields

We consider 4d field theories obtained by reducing the 6d (1,0) SCFT of $N$ M5-branes probing a $\mathbb C^2/\mathbb Z_k$ singularity on a Riemann surface with fluxes. We follow two different routes. On the one hand, we consider the integration of the anomaly polynomial of the parent 6d SCFT on the Riemann surface. On the other hand, we perform an anomaly inflow analysis directly from eleven dimensions, from a setup with M5-branes probing a resolved $\mathbb C^2/\mathbb Z_k$ singularity fibered over the Riemann surface. By comparing the 4d anomaly polynomials, we provide a characterization of a class of modes that decouple along the RG flow from six to four dimensions, for generic $N$, $k$, and genus. These modes are identified with the flip fields encountered in the Lagrangian descriptions of these 4d models, when they are available. We show that such fields couple to operators originating from M2-branes wrapping the resolution cycles. This provides a geometric origin of flip fields. They interpolate between the 6d theory in the UV, where the M2-brane operators are projected out, and the 4d theory in the IR, where these M2-brane operators are part of the spectrum.

4 Wrapped M2-branes and flip fields 15 5 The case of genus one 18 5.1 The Y p,q quiver theories from inflow 19 5.2More quiver theories and flip fields at genus one 20 6 Central charges 23 6.1 Computational setup 24 6.2 Salient properties of the central charge 25 6.3 Exact results for uniform flux configurations 29 6.4 Perturbative analysis 31 6.5 Genus-one cases 32

Conclusion and outlook 32
A Review of the E 3 4 contribution to anomaly inflow 34 1 Introduction and summary The reduction of higher-dimensional superconformal field theories (SCFTs) to lower dimensions has proven to be a powerful framework to construct non-trivial field theories, study their properties, and, more broadly, organize the space of quantum field theories (QFTs) using topological and geometric tools.One of the most prominent realizations of this paradigm is provided by class S constructions [1,2] and their generalizations [3][4][5][6][7][8][9][10][11][12][13].
The central idea is to start with a 6d SCFT and reduce it on a Riemann surface, triggering a renormalization group (RG) flow that can yield a non-trivial 4d SCFT in the IR.
In most instances, the parent 6d SCFT admits a realization in string/M/F-theory.Indeed, an atomic classification of 6d (1,0) SCFTs has been proposed in F-theory [14].When an explicit string theoretic construction of the parent SCFT is available, we have two ways of thinking about the resulting 4d SCFT: it is the IR fixed point of a purely fieldtheoretic RG flow from six dimensions, and it is also the QFT capturing the low-energy dynamics of a string theory setup with four non-compact dimensions of spacetime.As a simple example, we may consider the 6d (2,0) SCFT of type A N −1 , which can be realized by a stack of N M5-branes.This 6d SCFT can be reduced on a smooth Riemann surface preserving 4d N = 2 or N = 1 supersymmetry, depending on the choice of topological twist [1,7].In M-theory, the M5-brane stack is wrapped on the Riemann surface, and the choice of topological twist is mapped to the topology of the normal bundle to the M5-branes.
In this work, we explore a generalization of this circle of ideas to 4d N = 1 SCFTs of class S k [8].In the class S k program, the parent 6d SCFT is the 6d (1,0) theory realized by a stack of N M5-branes probing a C2 /Z k singularity.The global symmetries of this SCFT for generic N , k include the SU(2) R R-symmetry and a U(1) × SU(k) × SU(k) flavor symmetry. 1This 6d (1,0) SCFT is reduced on a Riemann surface with a topological twist that preserves 4d N = 1 supersymmetry.The data that specify the construction include the topology of the Riemann surface, possible defects (punctures) that can decorate the Riemann surface, and a choice of fluxes for global symmetries of the 6d (1,0) SCFT.
For simplicity, in this paper we restrict our attention to the case of a smooth Riemann surface without punctures.Even in this simpler class of setups, the RG flow from six to four dimensions can exhibit subtle features, such as a non-trivial pattern of modes that decouple along the flow, as demonstrated in [12] for reductions on tori with flux.
In the construction of Lagrangian models for 4d SCFTs originating from reductions of a 6d SCFT, it is not unusual to encounter flip fields, i.e. gauge singlets φ that participate in a superpotential coupling W flip = φ O, where O is a gauge invariant operator (for example, a baryonic operator constructed from a bifundamental field in a quiver gauge theory).For a gauge theory of sufficiently large rank, the superpotential coupling W flip = φ O is irrelevant: in the deep IR, the flip field φ is expected to behave as a free field, and therefore decouple from the interacting SCFT.Flip fields are ubiquitous in the literature on class S and its generalizations [8,11,13,[15][16][17][18][19][20][21][22][23][24], and appear in particular in many models studied in [12].One of the aims of this work is to revisit this class of models, with the aim of shedding light on the origin of flip fields from a geometric M-theory perspective.
The main goal of this paper is to contrast the field-theoretic point of view on S k constructions with a point of view based on a direct construction from M-theory, as illustrated in figure 1.Our objective is two-fold: x H n M / H a + f 2 X l 3 d 7 r p X 9 K q V t z D i n N Z L d a d y Q p L a A f t o j J y 0 T G q o z N 0 g Z q I o g f 0 j N 7 R h 3 V v P V k v 1 u v 3 6 Y w 1 y W y j X 7 A + v w C l v K J d < / l a t e x i t > 6d (1, 0) SCFT (I SCFT   On the left: a stack of M5-branes with flat 6d worldvolume probes a C 2 /Z k singularity, yielding a 6d (1,0) SCFT, plus 6d modes associated to free tensor, vector multiplets.The interacting 6d (1,0) SCFT is reduced on a Riemann surface with fluxes.The outcome is organized into an interacting 4d SCFT, and a collection of 4d free fields, interpreted as flip fields.On the right: a stack of M5-branes probes a resolved C 2 /Z k singularity and is wrapped on a Riemann surface.At low energies, this M-theory setup gives the same 4d SCFT, plus other 4d modes coming from free tensor, vector multiplets on the Riemann surface.The blue, solid arrows denote anomaly inflow from the 11d bulk onto the M5branes.The red, hollow arrows denote the purely field-theoretical reduction of the 6d (1,0) SCFT on the Riemann surface.
(i) Identify the M-theory setups that correspond to class S k reductions on a smooth Riemann surface with non-zero fluxes for the global SU(k) 2 flavor symmetry of the parent 6d SCFT.
(ii) Exploit these M-theory setups to gain insights on the field theory flow from 6d to 4d.
Let us now proceed to summarize the main results of this paper.
As far as objective (i) is concerned, a natural proposal is as follows: in M-theory, we should consider a stack of M5-branes that probes a resolved C 2 /Z k singularity, further wrapped on a Riemann surface.The 11d background probed by the M5-branes is expected to be a flux background: a non-zero G 4 -flux threads non-trivial 4-cycles, obtained combining the Riemann surface with the 2-cycles originating from the resolution of the C 2 /Z k singularity.
The above discussion can be made more precise for k = 2. Indeed, in this case we can establish a connection to a class of AdS 5 solutions in 11d supergravity, first discussed by Gauntlett, Martelli, Sparks, and Waldram (GMSW) [25].These solutions take the form of a warped product AdS 5 × w M 6 , in which the internal space M 6 is a fibration of a 4manifold M 4 over a Riemann surface Σ g .The space M 4 can be regarded as a resolution of the orbifold S 4 /Z 2 : the north and south pole of the S 4 are fixed points of the Z 2 action, yielding locally R 4 /Z 2 singularities; the singularity at each pole is resolved introducing a 2cycle.The internal G 4 -flux configuration is specified by three positive integer flux quanta, which we denote by N , N N 1 , N S 1 .The flux quantum N measures G 4 -flux on the 4-cycle given by the fiber M 4 at a generic point of the base Σ g .The integer N N 1 quantifies instead the G 4 -flux on the 4-cycle obtained by combining the Riemann surface with the resolution 2-cycle at the north pole of S 4 .Similar remarks apply to N S 1 .We propose the following interpretation of this class of solutions: they describe the near-horizon geometry of a stack of N wrapped M5-branes probing a resolved C 2 /Z 2 singularity [26].
As discussed in [27], the topology and G 4 -flux configuration of the internal space M 6 for k = 2 admit a natural generalization for higher k.In this case, M 6 is still taken to be a fibration of a 4-manifold M 4 over a Riemann surface Σ, but M 4 is identified with the resolution of the orbifold S 4 /Z k .The fixed points of the orbifold action are locally R 4 /Z k and can be resolved by blow-up, introducing a collection of k − 1 resolution 2-cycles near each pole of the S 4 .The G 4 -flux configuration is described by a total of 2k − 1 flux quanta, , in direct analogy to the GMSW solution.Explicit AdS 5 solutions in which the internal space M 6 has the topology described in the previous paragraph are not known for k > 2. While the relevant BPS system is well-studied [25,28], the search for such solutions proves to be a challenging task.This is certainly an important problem, but one which we choose to set aside for the purposes of this paper.Our working assumption is that the topology and flux configuration for k > 2 can be realized in the near horizon limit of a well-defined M-theory setup.
Crucially, in order to extract the physical consequences of our working assumption we do not need an explicit holographic solution.Building on [29,30], systematic methods have been developed [31] (see also [32]) to compute the inflow anomaly polynomial I inflow 6 for the wrapped M5-branes probing the resolved singularity, using as input the topology and flux configuration of M 6 .The quantity I inflow 6 is a 6-form characteristic class that captures the anomalous variation of the bulk 11d supergravity action in the presence of the wrapped M5-branes.According to the standard anomaly inflow paradigm, I inflow 6 is expected to be canceled by the 't Hooft anomalies of the 4d degrees of freedom living along the noncompact directions of the M5-brane stack.The computation of I inflow 6 was performed in [26] for k = 2 and in [27] for general k, at cubic order in the flux quanta (which originate from the 2-derivative C 3 G 4 G 4 coupling in the M-theory effective action).In this work we complete the computation of I inflow 6 by deriving the terms linear in the flux quanta (which originate from the higher-derivative coupling C 3 X 8 ).
Comparison with the integrated anomaly polynomial in field theory.
In order to contrast the field-theoretical and M-theory perspectives on class S k theories, it is natural to compare the quantity I inflow 6 on the M-theory side with the quantity Σg I SCFT 8 on the field theory side.Here, I SCFT 8 denotes the anomaly polynomial of the parent interacting 6d (1,0) SCFT [33], and Σg I SCFT on the Riemann surface, taking into account both R-symmetry and flavor fluxes [12].In particular, the quantity Σg I SCFT depends not only on N , k, and the Euler characteristic χ of the Riemann surface, but also on the flavor fluxes for the Cartan subgroups of the SU (k)2 flavor symmetry, denoted by ).One of the main results of this paper is a precise match between Σg I SCFT . (1.1) The 6-form is the anomaly polynomial of the interacting 4d SCFT of class S k that we want to study. 2 The quantity I v,t 6 is the anomaly polynomial of a collection of free 4d fields, which we interpret as the reduction on Σ g of a free 6d tensor multiplet and free 6d vector multiplets.The quantity I flip 6 is also the anomaly polynomial of a collection of free 4d fields, but with a different interpretation: they are flip chiral multiplets, i.e. gauge singlets coupled to 4d operators of the interacting 4d SCFT via irrelevant interactions.They are free fields, but due to their superpotential couplings to non-trivial operators in the interacting SCFT, they give large contributions to the integrated anomaly polynomial Σ I SCFT 8 (here "large" refers to the fact that these contributions scale with N and the flux quanta, and are not order one fixed numbers).These contributions have to be suitably accounted for in order to extract the anomaly polynomial I SCFT 6 of the interacting 4d SCFT from the integrated polynomial Σ I SCFT

8
. The role of flip fields in the field-theory flow from 6d to 4d is studied in detail in [12] with several examples where the Riemann surface is a torus.
Our analysis furnishes a general expression for the contribution I flip 6 of flip fields, for a Riemann surface of arbitrary genus and for arbitrary values of k and the flux parameters.Our findings match the results of [12] in the case of genus one.
The case k = 2 at genus one: D3-branes at the tip of Cone(Y p,q ).In the special case k = 2 and genus one, the GMSW solution is related by a chain of string theory dualities to the AdS 5 × Y p,q solutions in Type IIB string theory [25], which are holographically dual to the SCFTs engineered by a stack of D3-branes at the tip of the Calabi-Yau cone over Y p,q .By analyzing this special case, we find further evidence in favor of (1.1), by verifying the relation k = 2, genus g = 1: The above equality holds exactly in N , and not only at large N .We confirm the map between the p, q integers of Y p,q and the flux quanta on the M-theory side, established in [34].Notice that the term I v,t 6 in the general relation (1.1) is absent in (1.2), because it is proportional to the Euler characteristic of the Riemann surface.

Flip fields and M2-brane operators.
The anomaly polynomial I flip 6 encodes the anomalies of the flip fields we encounter upon reducing the 6d (1,0) SCFT on the Riemann surface.From the data in I flip 6 it is straightforward to extract the charges and multiplicities of the operators that get flipped.Such charges and multiplicities can be matched precisely with those of wrapped M2-brane states in the M-theory setup.
In the case k = 2, one can resort to the explicit AdS 5 GMSW solution to study supersymmetric M2-brane probes, by identifying the calibrated 2-cycles in the internal space M 6 [34].Moreover, the charges of the operators originating from these wrapped M2-brane probes can be extracted systematically from the terms in the uplift ansatz for G 4 that are linear in the external gauge fields.These are in turn conveniently extracted from the same 4-form E 4 that we utilize in the anomaly inflow computation.
For k > 2 we do not have an explicit holographic solution, nor do we have a solution describing the flux background probed by the M5-brane stack.For these reasons, a direct analysis of the calibration conditions for wrapped M2-brane probes is challenging.Nonetheless, we can identify non-trivial 2-cycles in the internal space M 6 .Motivated by the analogy with the k = 2 case, we make the working assumption that the relevant nontrivial 2-homology classes in M 6 admit a calibrated representative, so that the associated wrapped M2-brane operators are BPS.We can then proceed to compute their charges from the 4-form E 4 utilized in the anomaly inflow analysis.We obtain a perfect match with the charges of the operators that are flipped by the fields in I flip 6 .Moreover, we can also reproduce their multiplicities: they are simply given by the units of G 4 flux threading the relevant 2-cycle (combined with the Riemann surface), by virtue of a standard Landau-level degeneracy argument [35], which we review in section 4.
The identification of charges and multiplicities of flipped operators, and wrapped M2brane operators, suggests the following physical picture.The wrapped M2-branes operators are associated to blow-up modes for the C 2 /Z k singularity.In the 6d (1,0) SCFT, however, such modes are not present [36][37][38][39][40][41][42][43][44].In contrast, we expect the M2-brane operators to be part of the spectrum of the 4d theory obtained by reduction on the Riemann surface.Indeed, in Lagrangian models, they are baryonic operators.As a result, a mechanism is needed to interpolate between six and four dimensions; this mechanism is precisely given by the flip fields from the term I flip 6 .They act as Lagrange multipliers that project away the wrapped M2-brane operators in the integration of the anomaly polynomial on Σ g .In the 4d theory, they are expected to be free fields and decouple, thus effectively reintroducing the M2-brane operators.

Central charges.
We study a-maximization [45] on the combination −I inflow maximum of the trial a central charge.The resulting a and c are compatible with the Hofman-Maldacena bounds [46].

Organization of the paper.
The rest of this paper is organized as follows.In section 2 we review the M-theory flux setups studied in [27], giving a brief account of the isometries and topology of the relevant internal space M 6 .In section 3 we discuss the anomaly inflow computation (including the E 4 X 8 contribution) and we present in detail the main relation (1.1), giving the explicit expressions for I v,t 6 and I flip 6 .Section 4 is devoted to the match between the charges and multiplicities of the flip fields entering I flip 6 , and those of operators originating from M2branes wrapping resolution 2-cycles in M 6 .In section 5 we focus on the case in which the Riemann surface is a torus, establishing a precise correspondence with D3-brane theories dual to AdS 5 × Y p,q , for k = 2.We also consider some explicit Lagrangian examples with k = 2, 3.In section 6 we use a-maximization to compute conformal and flavor central charges from −I inflow 6 − I v,t 6 and establish various properties of these quantities.We conclude with an outlook in section 7. Several appendices collect useful material and detailed derivations.

Review of the eleven dimensional flux setups
In this section we summarize the basic features of the internal space M 6 in the putative 11d flux backgrounds of relevance for the 4d field theories of S k .For a more detailed account of the geometry and homology of M 6 , we refer the reader to [27].M 6 is characterized by the fibration where Σ g is a Riemann surface of genus g, and M 4 is the manifold obtained by resolving the fixed points of the orbifold S 4 /Z k via a blow-up procedure, This M 4 is locally a multi-center Gibbons-Hawking space, with k − 1 2-cycles separated by k (unit-charge) Kaluza-Klein monopoles aligned along a common axis.It admits two U(1) isometries, and can be expressed in turn as a fibration where M 2 is a compact 2d space.Schematically, the metric on M 4 can be cast in the form where the angular coordinates η and θ span the 2d space M 2 .The S 1 ψ circle shrinks everywhere on the boundary ∂M 2 , at η, θ = 0, π.Before the blow-up, the orbifold fixed points are labeled by η = 0, π, which we refer to here as the north and south poles, Figure 2: Illustration of the topology of the unresolved S 4 /Z k space (left) and resolved M 4 space (right), with the ψ and ϕ angles suppressed, taken from [27].The circle S 1 ψ vanishes along the entire boundary ∂M 2 , while S 1 ϕ vanishes only at the 2k monopoles, labeled by the index i.The blue bubbles represent the resolution 2-cycles connected by unit-charge Kaluza-Klein monopoles.
respectively.After the blow-up, each pole is replaced by a chain of k −1 resolution 2-cycles.The k monopoles in the north carry Kaluza-Klein charge +1, while the k monopoles in the south carry charge −1, with the relative sign accounting for the opposite orientations relative to M 2 at η = 0, π.There is a U(1) gauge symmetry associated with each resolution 2-cycle, so there is an overall U(1) k−1 symmetry in both the north and the south.The topology of the S 4 /Z k and its resolved counterpart M 4 are illustrated in figure 2.
The function L(η, θ) is piecewise constant on ∂M 2 , with its difference across a given monopole measuring that monopole's Kaluza-Klein charge.Labeling the resolution 2-cycles in the north by i = 1, . . ., k − 1 and those in the south by i = k + 1, . . ., 2k − 1, we have explicitly where t is a periodic coordinate parameterizing the boundary ∂M 2 .
In the full internal space M 6 , twisting of M 4 over the Riemann surface introduces a U(1) connection over Σ g to the form dψ. N = 1 supersymmetry is preserved specifically by a topological twist [6,7] dψ → Dψ = dψ − 2πχA Σ . (2.6) The quantity χ = 2(1 − g) is the Euler characteristic of the genus-g Riemann surface, with volume form V Σ normalized as Σg V Σ = 1, and A Σ is the local antiderivative of V Σ .This topological twist leads to nontrivial relations in the homology of M 6 .Consider first the 2-cycles in M 6 .There are two types: the boundary 2-cycles in M 4 , and the Riemann surface Σ g itself, at the position of each of the monopoles.We thus have 4k total 2-cycles, one of each type for every i = 1, . . ., 2k.However, as described in [27], only 2k − 1 of these 2-cycles are independent.The situation is analogous for the 4-cycles.The region M 4 constitutes one 4-cycle in the full M 6 space, as do the 2k pairings of the boundary 2-cycles with Σ g , The topological twist (2.6) trivializes certain linear combinations of these 4-cycles, however, The corresponding basis of 2k − 1 2-cycles C α 2 can be taken to be Poincaré-dual to these 4-cycles.Quantization of the M-theory 4-form flux G 4 associates each 4-cycle with an integer, subject to linear constraints inherited from (2.9), namely, In the basis (2.10), we have 2k − 1 independent flux quanta, All told, the space M 6 is characterized by the integer parameters k, χ, N , N N i , N S i .

Anomaly polynomials in class S k from inflow
In this section we argue that the inflow anomaly polynomial for wrapped M5-branes probing a resolved C 2 /Z k singularity is to be identified with the anomaly polynomial of a class S k theory, obtained from reduction of the parent 6d (1,0) SCFT on a smooth Riemann surface with non-trivial SU(k) 2 flavor fluxes.The identification holds up to the contribution of a suitable collection of free fields, which we discuss in detail.

Integrated anomaly polynomial from six dimensions
Here we review briefly the integration of the 6d 8-form anomaly polynomial I SCFT 8 on a smooth genus-g Riemann surface, with a non-trivial topological twist and flavor fluxes [12].Let us stress that I SCFT 8 denotes the anomaly polynonomial of the interacting 6d (1,0) SCFT realized by a stack of M5-branes probing a C 2 /Z k singularity [33], without the inclusion of decoupled sectors, such as the center-of-mass mode of the stack.
The R-symmetry of the parent 6d (1,0) theory is SU(2) R .In the reduction to four dimensions, the Chern root of the SU(2) R bundle is shifted to implement the topological twist that preserves 4d N = 1 supersymmetry, The label R on the 4d background Chern class c 1 (R ) is a reminder that this is a reference R-symmetry, which does not generically coincide with the 4d For generic N , k, the 6d SCFT admits a U(1) s ×SU(k) b ×SU(k) c flavor symmetry.The Chern roots of the SU(k) b , SU(k) c bundles are denoted by b i , c i (i = 1, . . ., k), respectively, and they are subject to the constraints k i=1 b i = 0 = k i=1 c i .The reduction to four dimensions is performed with the following Chern root shifts, Notice in particular that, for simplicity, in this work we do not turn on a flavor flux for the U(1) s symmetry.The quantities c 1 (t), c 1 (β i ), c 1 (γ i ) are the first Chern classes of background fields for 4d U(1) global symmetries.We observe that c 1 (β i ), c 1 (γ i ), as well as the flavor fluxes We find it convenient to adopt the following parametrizations of the above constraints, with the conventions We are now in a position to quote the result of the integration of the anomaly polynomial I SCFT 8 of the parent 6d (1,0) SCFT on the Riemann surface Σ g , Σg

Anomaly inflow from eleven dimensions
The inflow anomaly polynomial for a stack of M5-branes probing a background associated with the internal geometry M 6 and background flux configuration G 4 is given by [31] The closed and gauge-invariant 4-form E 4 is constructed from G 4 by including the external gauge fields associated with the isometries and the non-trivial cohomology classes of M 6 .The 8-form characteristic class X 8 is built with the Pontryagin classes p i (T M 11 ) of the tangent bundle of the 11d spacetime.
The computation of the inflow anomaly polynomial for the flux setups reviewed in section 2 is discussed in detail in [27] for general k, where the contribution of the E 3  4 term was studied.The contribution of the E 4 X 8 term for general k is derived in appendix B. We have a completely explicit expression for −I inflow 6 in terms of k, N , χ, the flux parameters , and the first Pontryagin class p 1 (T W 4 ) of the external spacetime.Since this expression is quite complicated, however, we refrain from reproducing it in the main text.The E 3 4 contribution to −I inflow region near resolved southern orbifold point region near resolved northern orbifold point Figure 3: Pictorial depiction of the strategy used to evaluate the E 4 X 8 contribution to the inflow anomaly polynomial.In order to obtain the resolved orbifold M 4 , we consider the central "cylindrical" region of S 4 away from the poles, and we glue in the resolved orbifolds at each pole.These include resolution 2-cycles, depicted as small green circles in the figure.The central contribution is obtained by taking a full S 4 and removing small polar caps.The contribution of the latter is computed by taking the limit k = 1 in the contribution of the resolved orbifolds.
subtract these polar caps for k = 1, we get the contribution of the central "cylindrical" region.Finally, we glue back in the resolved orbifolds with the appropriate value k > 1, in order to get the final desired result.
In concluding, we observe that the E 3 4 contribution was computed in [27] using a different strategy, but is nonetheless compatible with this cut-and-glue picture.Both for E 4 X 8 , and for E 3  4 , the contribution associated to the central region of M 4 is equal to the inflow anomaly polynomial for a 4d N = 1 SCFT originating from the 6d (2, 0) SCFT of type A N −1 with twist parameters p, q satisfying p = q [7].

Matching the two sides: decoupled modes and flip fields
Let us now discuss the relation between Σg I SCFT 8 and −I inflow 6 .To this end, it is convenient to introduce the following notation, This quantity is the anomaly polynomial of a free, positive-chirality Weyl fermion in four dimensions, with prescribed charges q R , q t , q β i , q γ i under the 4d U(1) symmetries U(1) R , U(1) t , U(1) β i , U(1) γ i , in the notation of section 3.1.The quantities c 1 ( β i ), c 1 ( γ i ) are the unconstrained Chern roots defined by (3.4).
-12 - We can write the difference between Σg I SCFT 8 and −I inflow 6 in terms of a collection of free fermions, with anomaly given by (3.7) for appropriate charges.More precisely, we find where I v,t 6 and I flip 6 are given by Some remarks on our notation are in order.The quantities L i a,b can be identified with the components of the positive roots of the Lie algebra su(k), The multiplicity factors m a,b are given in terms of the unconstrained flavor fluxes N β i , N γ i introduced in (3.4) by the following expressions, (3.12) The relation (3.8) holds provided that we make the following identifications among quantities related to anomaly inflow from eleven dimensions, and quantities related to integration from six dimensions, The quantities (A k−1 ) ij are the entries of the Cartan matrix of su(k), Notice the k−i label on southern quantities, as opposed to the i label on northern quantities.
The free-field contributions in I v,t 6 are interpreted as originating from the reduction on Σ g of free fields in six dimensions, as suggested by the fact that they are proportional to the Euler characteristic χ.The 8-form anomaly polynomials of a free 6d (1,0) tensor multiplet, and a free 6d (1,0) vector multiplet, are readily computed and integrated on the Riemann surface, with the result This observation suggests us to interpret the first line of I v,t 6 in (3.9) as a contribution of one tensor and k − 1 vectors.The former is associated with the center of mass of the M5-brane stack, while the latter is thought of as the Cartan generators of SU (k).By a similar token, we interpret the other terms in (3.9) as coming from the reduction of 6d W-bosons of SU(k), whose charges are indeed given by the roots of su(k).
Interpretation of I flip 6 .While the free-field contributions in I v,t 6 have a 6d interpretation, those in I flip 6 are interpreted in four-dimensional terms.More precisely, we identify I flip 6 with the anomaly polynomial of a collection of flip fields.Here, by flip field we mean a 4d N = 1 chiral multiplet φ that couples to an operator O of the interacting 4d SCFT with a superpotential coupling of the form The field φ has canonical kinetic terms and, if it were not for (3.16), would be completely decoupled from the 4d SCFT.Since the superpotential has R-charge 2 and is neutral under other global symmetries, the coupling (3.16) implies where Ψ φ denotes the fermion in the chiral multiplet φ, and q X is a shorthand notation for q t , q β i , q γ i .The charges q R [Ψ φ ], q X [Ψ φ ] are the charges given I flip 6 in (3.10).They are readily translated into charges of the operators O that get flipped, Notice how all flipped operators have zero charge under U(1) R .
In section 4 we match the charges of the flipped operators (3.18) with charges of wrapped M2-brane states, and we comment further on the physical mechanism underlying the flipping of these operators.
Flavor fluxes versus resolution fluxes: a geometric picture.Let us motivate the identification (3.13) by considering the geometric interpretation of the flux quanta N βi , N γi in the 6d setup.Prior to compactification over the Riemann surface, the internal space of the 6d theories is S 4 /Z k , and has two orbifold fixed points which can be resolved by the set of 2-cycles C i 2 defined in (2.7).Associated with each 2-cycle is a U(1) flavor symmetry.The only 4-cycle in such a setup is the (unresolved) S 4 /Z k that is analogous to there is no analogue of the 4-cycles C i 4 defined in (2.8).As a consequence, the flux quanta N βi , N γi assigned to the U(1) βi , U(1) γi flavor symmetries in the compactification are naturally associated with C i 2 but not C i 4 .From the perspective of the expansion flux quanta are intrinsically paired to 4-cycles [27].So we conjecture that N βi , N γi are really flux quanta with respect to the 4-cycles Poincaré-dual to the resolution 2-cycles C i 2 (after reduction on the Riemann surface).In contrast, the flux quanta N N i , N S i were defined with respect to the 4-cycles C 4,N i , C 4,S i described in (2.10).Direct comparison between Σg I SCFT 8 and −I inflow 6 therefore requires that we find the transformation matrices relating these two distinct bases of homology classes.As worked out in appendix D, these transformation matrices turn out to be block diagonal, with each of the two nontrivial blocks proportional to A k−1 , the Cartan matrix of su(k).One may heuristically interpret this as a remnant of the enhanced SU(k) symmetry present at each orbifold fixed point before being resolved into k − 1 2-cycles with U(1) symmetries.Indeed, the identification in (3.13) is precisely the change of basis we have described, with an additional sign change for the southern flux quanta to preserve their positivity.
Next, we argue that the factor of 1/k in the identification (3.13) of the field strength F ϕ 2 with the Chern root c 1 (t) can be attributed to the different periodicities of t and ϕ.More specifically, the periodicity of the angular variable t in the 6d theories is reduced from 2π to 2π/k by the Z k quotient, but the periodicity of ϕ in the inflow computation directly from 11d is simply 2π by construction.In the former case, we gauge the U(1) t isometry as (Dt) g = Dt − A t 1 , while in the latter case we gauge the U(1) ϕ isometry as (Dϕ) g = Dϕ + A ϕ 1 .This motivates the identifications, Lastly, the factor of 2 appearing in the identification (3.13) between F ψ 2 /2π and c 1 (R ) is needed to ensure an appropriate R-charge normalization [34].

Wrapped M2-branes and flip fields
Operators from M2-branes wrapping calibrated 2-cycles.The calibration conditions for probe M2-branes wrapping 2-cycles in the internal space M 6 were derived in [34], where they were also analyzed for the k = 2 GMSW solutions.The homology classes of the resolution 2-cycles of M 4 (at a generic point on the Riemann surface) admit a calibrated representative.In the notation of section 2, these homology classes are the C i 2 in (2.7) with labels i = 1 and i = 3. Wrapping M2-brane probes on such 2-cycles yields BPS particle states in the external spacetime.
A direct analysis of the calibration conditions for k ≥ 3 is challenging.Indeed, we do not have an explicit AdS 5 solution in which the internal space has the topology and flux quanta of M 6 .Solutions describing the flux background probed by the M5-brane stack are not known, either.For these reasons, we refrain from studying the calibration conditions for M2-brane probes for k ≥ 3, and we make the following working assumption: the homology classes of the resolution 2-cycles of M 4 admit calibrated representative 2-cycles.We wrap probe M2-branes on such cycles, getting BPS states in the external spacetime.
In what follows, we study for generic k ≥ 2 the charges and multiplicities of such states.Crucially, these data do not depend on the (putative, for k > 2) explicit calibrated representative, but only on its homology class.
Charges of M2-branes operators.The method for the computation of the charges of wrapped M2-brane operators is explained in [34].The key point is to use the standard coupling of the 11d 3-form C 3 to the worldvolume of the M2-brane probe.The desired charges are extracted by integrating this 3d coupling along the compact directions of the 2-cycle wrapped by the M2-brane, thus extracting terms linear in the external gauge fields.
In order to determine how the external gauge fields enter the 11d 3-form C 3 , we resort to the 4-form E 4 used in the anomaly inflow computation.More precisely, we proceed as follows: extract the terms in E 4 that are linear in the external 2-form field strengths , and cast these terms as a total derivative of a 3-form δC 3 that is linear in the corresponding external gauge fields All the relevant charges of interest are then obtained by integrating δC 3 on the 2-cycle wrapped by the M2-brane.
After these general preliminaries, we are in a position to outline the results we obtain for the setups of interest in this work.The 3-form δC 3 derived from E 4 takes the form where the label β runs over all independent 2-cycles in M 6 , while I is a collective label for the isometry directions ψ, ϕ.The 2-forms (ω 2,β ) g are obtained from the harmonic 2-forms on M 6 by means of the replacements dψ → dψ+A ψ 1 , dϕ → dϕ+A ϕ 1 .The 2-forms (Ω 2,I ) g are derived in the process of constructing the equivariant completion of the harmonic 4-forms on M 6 .We refer the interested reader to [27], where the explicit expressions of (ω 2,β ) g and (Ω 2,I ) g can be found.
The charges under the U(1) field strengths for flavor symmetries and isometries, respectively F β 2 and F I 2 , can then by computed from the integrals The results can be summarized in the following table, up to an overall orientation choice, where we have written the flavor charges in terms of the entries of the Cartan matrix of su(k).Notice that the charges are given in a basis that corresponds to the external gauge -16 -fields in the anomaly inflow computation.Making use of (3.13), they are readily translated to the basis used in the integration of the anomaly polynomial of the 6d SCFT, In the above tables we have also reported the multiplicity of the M2-brane operators, which is derived as explained below.
Multiplicities of M2-branes operators.The multiplicities, or degeneracies, of the states originating from an M2-brane probe wrapping a 2-cycle can be derived using a Landau-level argument [35].The probe M2-branes of interest in this work wrap a resolution 2-cycle C i 2 in M 4 , and sit at a point on Σ g .Moreover, a non-trivial G 4 -flux is turned on along the 4-cycle that results from combining the resolution 2-cycle C i 2 and the Riemann surface Σ g .As a result, the M2-brane behaves like a point particle on Σ g , in the presence of a non-zero magnetic field.This quantum-mechanical system exhibits a well-known Landau degeneracy of states, which is simply given by the total magnetic flux, measured in units of the minimal magnetic flux that can be turned on.This quantized magnetic flux is indeed identified with the quantized G 4 -flux through the 4-cycle obtained combining Σ g and C i 2 .In conclusion, the expected degeneracies of the wrapped M2-brane operators of interest are simply given by the values of the corresponding G 4 -flux.
Comparison to flip fields.The anomaly polynomial I flip 6 is interpreted as a sum over flip fields.The charges of the corresponding flipped operators are collected in (3.18), for a generic pair of labels a < b, a, b = 1, . . ., k.These pairs correspond to all positive roots of su(k).Those pairs a < b with b = a + 1 correspond to the simple roots of su(k).Based on standard intuition regarding M-theory on a Z k singularity, we expect the M2-branes wrapping the (k − 1) resolution 2-cycles at the north and south poles to correspond to the simple roots of su(k) N , su(k) S .(The full set of positive roots corresponds to BPS bound states of the M2-brane states corresponding to the simple roots.)Now, the identity shows a match between the charges of the flipped operators in (3.18), and the charges of M2-branes wrapping resolution 2-cycles in (4.4).
The multiplicities of the flip fields for generic a < b are reported in (3.12).Let us specialize to b = a + 1, and combine (3.12) with the relations (3.13) between the flavor fluxes and the resolution fluxes.We obtain We thus verify that, for pairs with b = a+1, corresponding to simple roots, the multiplicities that enter I flip 6 coincide with the degeneracies given by the Landau-level argument discussed above.
M-theory origin of the flipping mechanism.Recall that the flip fields enter the 4d theory via the schematic superpotential coupling Based on the previous analysis, we identify the flipped operators O with the operators originating from M2-branes wrapping the resolution 2-cycles in the internal space.For generic N , the coupling (4.7) is irrelevant.It is therefore important in the UV, where its effect is to project out the operators O.This fits with the fact that the 6d parent SCFT admits no blow-up modes for the C 2 /Z k singularity [36][37][38][39][40][41][42][43][44].In contrast, (4.7) is irrelevant in the deep IR, where the flip fields φ become free fields and decouple.The operators O are thus effectively reintroduced in the 4d theory.

The case of genus one
In this section, we consider several explicit examples at genus one in order to gather evidence for a series of connected claims: • Since I v,t 6 drops out of (3.8) when χ = 0, in this case the quantity −I inflow 6 provides direct access to the anomaly polynomial of the corresponding 4d SCFT, (5.1) • The quantity we have called (5.2) Note that both the E 3 contribution to the inflow anomaly polynomial recorded in appendix A and the E 4 X 8 contribution derived in appendix B are expressed in a form which only applies for higher-genus Riemann surfaces, i.e. χ < 0. The genus-one result can be obtained either by first using (3.13) and subsequently fixing χ = 0, or by repeating an analogous anomaly inflow computation as in [27] with cohomology class representatives chosen consistently with χ = 0 from the beginning (which we describe in appendix C).We have verified that both paths produce the same result, Table 1: Field content for the infinite family of Y p,q quiver gauge theories.See, e.g.[47].
a,b I free 6 q R = 1; q t = 0; a,b I free 6 q R = 1; q t = 0; q β i = 0; q γ i = N L i a,b . (5.3) We explore how the expression (5.3) can be used to verify the claims highlighted above in various examples.
5.1 The Y p,q quiver theories from inflow Consider the case k = 2 for genus one.Here we find that the equation (5.3) reproduces the anomaly polynomial of the Y p,q quiver gauge theories, which can be engineered on a stack of D3-branes at the tip of the Calabi-Yau cone over Y p,q [47].The Y p,q are an infinite family of Sasaki-Einstein manifolds labeled by positive integers p and q with 0 ≤ q ≤ p [25].The holographic duals of the corresponding AdS 5 × Y p,q solutions in Type IIB string theory were constructed in [47], using an iterative procedure on the quiver for Y p,p .The field content of this family of quiver gauge theories is summarized in table 1.The quiver associated with Y p,q has 2p gauge groups, represented diagrammatically by 2p nodes.All fields are in either a spin-0 or spin-1/2 representation of a global SU(2) symmetry.There are two additional global U(1)'s, labeled here as U(1) B and U(1) F .
-19 - The anomaly polynomial for general Y p,q quiver gauge theories can be computed directly from the field content and associated degeneracies, and is given by In virtue of a chain of dualities connecting the GMSW solution in 11d supergravity to the AdS 5 × Y p,q solutions in Type IIB, the internal manifold M 6 in the corresponding inflow setup is defined by k = 2 and χ = 0.The quantity −I inflow 6 can be obtained for example from (5.3), Under the field strength redefinitions, and the identifications between the integers p, q and the resolution flux quanta N β 1 , N γ 1 , we verify an exact match, . This match supports our claim (5.1) that the topological and geometric data of M 6 fully characterize the anomaly polynomial of the corresponding (genus-one) 4d SCFT.

More quiver theories and flip fields at genus one
Next we revisit some explicit examples at genus one reported in [12] in order to provide further evidence for the interpretation of I flip 6 in (3.8) and the equality (5.2).In these examples, we consider N to be generic, but make the implicit assumption that N is large enough to ensure that all the couplings between flip fields and baryons are irrelevant.It would be interesting to consider in greater detail low values of N , but we refrain from such analysis in this work.Next to the fields are their charged summarized using fugacities.This theory has a combination of cubic and quartic superpotential terms.Again these are most conveniently generated by taking all terms consistent with the symmetries.Additionally there are superpotential terms coming from the flipping.The theory also has an R-symmetry, a convenient choice for which is to give all the bifundamentals R-charge 1  2 , and R-charge 1 for the adjoints and their singlets.combinations of u(1) t , u(1) 1 and u(1) 2 while the last is 2u(1) 2 u(1) 1 .
Next we preform a-maximization.We take the R-symmetry to be: where we take u(1) R to rotate the bifundamentals with charge 1 2 and the adjoints and their associated singlets with charge 1.The flipping fields attached to the bifundamental then also have R charge 1 while those attached to the adjoits have R charge 0. Performing the a-maximization we find that: ↵ = 3 p 5 6 .With this value we find that a = 5 p 5 4 .However with this R-charge the adjoint flipping fields are below the unitary bound.Therefore the natural conjecture is that these fields become free at some point along the flow leading to an accidental u(1) that mixes with the R-symmetry.We can now repeat the a-maximization, but taking these to be free fields where we find: ↵ = p 13 3 6 . We find that all fields have dimensions above the unitary bound and that the superpotential coupling the flipping fields to the adjoints is irrelevant.All of these are consistent with our claim.We can also preform a-maximization considering all the flipping fields as free, where we indeed find that, compared to that point, the superpotential coupling the flipping fields to the adjoints is irrelevant while the one coupling the flipping fields to the bifundamentals is relevant.
So to conclude we expect the theory in figure 7 to flow to an interacting fixed point consisting of the quiver theory, without the adjoint flipping, plus two free chiral fields.We next want to evaluate the index of this fixed point.Again for simplicity we shall first ignore the two free chiral fields.From 6d we expect an su(4) ⇥ su(2) ⇥ u(1) global symmetry.We -23 -  12 , 1 2 , 0).Next to the fields are their charges summarized through fugacities.The theory has a quartic superpotential involving the four bifundamentals as well as the superpotential coming from the flipping.There is also an R-symmetry where it is convenient to give R-charge 1  2 to the four bifundamentals Qi and R-charge 2 N 2 to the flipping fields , .
the resulting theory depends on the value of N .For N > 2, this superpotental is irrelevant and the theory should flow to the same fixed point, but with free singlets.However, for N = 2, this superpotental is relevant and the theory should flow to a new fixed point.We shall now discuss the latter case in more detail.Note that for this special case of N = 2, Q1 is a 2 ⇥ 2 matrix and the notation Q 2 1 stands for det Q1, and similarly for Q2.Each of these terms is invariant under a corresponding su(2) global symmetry and the global su(4) symmetry which is present in the absence of these terms [30] is broken to su(2) ⇥ su(2) ⇥ u(1) in the presence of these terms, where u(1) is the baryonic symmetry which acts as +1 on Q1 and Q2 and as 1 on Q3 and Q4.⇡ 0.027, so the R-charges change only slightly compared to their naive value.One can check that all gauge invariant fields are above the unitary bound so this is consistent with the theory flowing to an interacting fixed point.
We can next evaluate the anomalies for this theory.Particularly, for the conformal -17 -Figure 4: The quiver diagram for the 4d class S 2 theory corresponding to a torus with flux (1, 0, 0).Next to the fields are their charges summarized through fugacities.We use mostly standard notation except for two points: lines from a group to itself represent N 2 hypermultiplets forming the adjoint plus singlet representations of the group; we write an X over a field to represent the fact that the baryon of that field is flipped.The theory has a cubic superpotential for every triangle which can also be derived by considering the most general cubic superpotential that is gauge invariant and consistent with the symmetry allocation.There is also the superpotential term which is not generally cubic coming from the flipping.All fields, save the flipping fields, have the free R-charge 2  3 .
The Higgs branch of this N = 2 A ne A 1 theory was evaluated in [22], see section 5.2.1 and in particular equation (5.19).Furthermore it was found to be the closure of the next to minimal orbit of usp(4) as in Table 3 of [23] and Tables 10 and 12 of [24], where another description sets it as the Z 2 orbifold of the closure of the minimal nilpotent orbit of SL(4) (alternatively known as the reduced moduli space of 1 SU(4) instanton on C 2 ).This emphasizes that the global symmetry on this part is indeed usp(4).The unrefined Hilbert Series takes the form and it admits the highest weight generating function [24,25] HW G A ne Quiver with µ 1 and µ 2 the fugacities for the highest weights of usp( 4).From this one deduces the -15 - Figure 6: On the left is the quiver diagram for the 4d class S 3 theory corresponding to a torus with only 1 flux, while on the right is a table summarizing the charges of the various fields.Note that several di↵erent fields have the same charges and so are represented with the same letter.This theory has a rather large cubic superpotential involving the 12 triangles in the diagram.Again these are most conveniently generated by taking all cubic terms consistent with the symmetries.Additionally there are the superpotential terms coming from the flipping, which in general are not cubic.It is again convenient to choose the R-symmetry so that all non-flipping fields have R-charge 2  3 .
marginal operators in the [2] su(2) , as well as the marginal operators in the adjoint of the full global symmetry G which must be present to cancel the contribution of the conserved currents.These contribute 10 exactly marginal operators.So we now have more than those expected from (4.1).
We can also consider closing the last puncture with a baryon charged under 2 .The resulting flux leads to the symmetry breaking pattern su(6) !su(4) ⇥ su(2) ⇥ u(1) which we expect to be the 4d global symmetry.In the field theory the vev leads to a quiver with an su(N ) group with N flavors.This group confines in the IR leading to the identification of the groups it's connected to and making the flipping fields massive.After the dust settles we end with the so called L 222 [20] quiver theory in figure 7.This theory can also be derived from 4 NS branes on the circle, with two types of orientation.
We next proceed to analyze this theory in detail.First we need to evaluate the conformal R-symmetry.By inspection one can see that there is only one u(1) that can mix with the R-symmetry, which in our notation is u(1) 1 .The remaining u(1)'s can be grouped into 4 baryonic u(1)'s, each rotating one of the four pairs of bifundamentals with opposite charges while the adjoints and their associated singlets being neutral.Three of these u(1)'s are -22 - Figure 6: On the left is the quiver diagram for the 4d class S 3 theory corresponding to a torus with only 1 flux, while on the right is a table summarizing the charges of the various fields.Note that several di↵erent fields have the same charges and so are represented with the same letter.This theory has a rather large cubic superpotential involving the 12 triangles in the diagram.Again these are most conveniently generated by taking all cubic terms consistent with the symmetries.Additionally there are the superpotential terms coming from the flipping, which in general are not cubic.It is again convenient to choose the R-symmetry so that all non-flipping fields have R-charge 2  3 .
marginal operators in the [2] su(2) , as well as the marginal operators in the adjoint of the full global symmetry G which must be present to cancel the contribution of the conserved currents.These contribute 10 exactly marginal operators.So we now have more than those expected from (4.1).
We can also consider closing the last puncture with a baryon charged under 2 .The resulting flux leads to the symmetry breaking pattern su(6) !su(4) ⇥ su(2) ⇥ u(1) which we expect to be the 4d global symmetry.In the field theory the vev leads to a quiver with an su(N ) group with N flavors.This group confines in the IR leading to the identification of the groups it's connected to and making the flipping fields massive.After the dust settles we end with the so called L 222 [20] quiver theory in figure 7.This theory can also be derived from 4 NS branes on the circle, with two types of orientation.
We next proceed to analyze this theory in detail.First we need to evaluate the conformal R-symmetry.By inspection one can see that there is only one u(1) that can mix with the R-symmetry, which in our notation is u(1) 1 .The remaining u(1)'s can be grouped into 4 baryonic u(1)'s, each rotating one of the four pairs of bifundamentals with opposite charges while the adjoints and their associated singlets being neutral.Three of these u(1)'s are -22 -Figure 4: Explicit examples of quiver gauge theories of class S k realized on a torus with fluxes, taken from [12].The gauge nodes are SU(N ) groups.An arrow connecting two distinct nodes is a bifundamental chiral multiplet.An arrow connecting a node to itself represents an adjoint-plus-singlet chiral multiplet.An "X" on an arrow signals that the baryon associated with that field is flipped.The charges under the global, non-Rsymmetries are given in terms of fugacities.The reference R-symmetry charges for the various quivers are as follows.In quiver (a), all fields (except the flip fields) have R-charge 2/3.In quiver (b), all fields (except the flip fields) have R-charge 1/2.In quiver (c), all fields (except the flip fields) have R-charge 2/3.In quiver (d), the bifundamentals have R-charge 1/2 and the adjoint-plus-singlet's have R-charge 1.For a description of the superpotential of these models, we refer the reader to [12].

Computational setup
The anomaly polynomial of a 4d N = 1 SCFT contains where R is the generator of the superconformal R-symmetry and F R 2 its background field strength, and p 1 (T W 4 ) is the first Pontryagin class of the 4d worldvolume W 4 .Its central charges are [48] In the presence of additional flavor symmetries, one may also compute the associated flavor central charges [49], where T a G are generators of the flavor symmetry G, with the normalization tr(T a G T b G ) = δ ab /2 in the fundamental representation.
As described before, we can access I SCFT 6 through either of the equalities in (1.1), repeated here for convenience, For concreteness, suppose we restrict our attention to the first equality above, we can define a trial R-symmetry as The various T G are the generators of the U(1) G flavor symmetries, and the factor of 2 in front of T ψ is inserted to ensure it has the appropriately normalized R-charge [34].This is equivalent to carrying out the following replacements in the anomaly polynomial(s), from which we can construct a trial central charge, where A RRR and A R are respectively the coefficients of (F R 2 /2π) 3 and F R 2 /2π in the polynomial I SCFT 6 ({q G }).The central charge a corresponds to the local maximum of a trial ({q G }) with respect to the 2k − 1 real parameters q G .Such a solution is hereafter denoted by {q G max }.Up to an overall normalization, the flavor central charge b G for a given flavor symmetry G can be extracted by plugging {q G max } into I SCFT 6 ({q G }), and then reading off the coefficient We can also similarly find c.Analytically searching for a local maximum of a trial ({q G }) amounts to performing a two-step process: first solving the quadratic equations ∂a trial /∂q G = 0 simultaneously for all G, and then identifying a solution {q G max } that yields a negative-definite Hessian matrix [∂ 2 a trial /∂q G i ∂q G j ] {q G }={q G max } .As far as the former is concerned, Bezout's theorem asserts that a trial ({q G }) has at most 2 2k−1 (or infinitely many) critical points.Hence, the computational complexity of the problem scales roughly as 4 k at large k.This presents a formidable challenge even for state-of-the-art algebraic solvers, in which case we have to resort to numerical methods.In fact, the only scenario where we are able to analytically solve the a-maximization problem for generic combinations of χ, N , N N i , N S i for i = 1, . . ., k − 1 is k = 2, in which we are able to fully reproduce the result of [26].As we will discuss later in this section, there exists a family of theories with certain special flux configurations {N N i , N S i } that significantly reduce the number of independent extremization parameters q G , thus rendering the a-maximization problem analytically solvable for k > 2 as well.

Salient properties of the central charge
The central charge a thus obtained for the SCFTs via a-maximization exhibits a number of crucial properties which we collect in this subsection.
Uniqueness.For a given choice of k, χ, N , N N i , N S i , if the central charge a exists then it is necessarily unique.The proof of this statement proceeds as follows.Given the replacements made in (6.6), the trial central charge a trial ({q G }) defined in (6.7) is a cubic polynomial of the 2k − 1 variables q G where G = ϕ, N i , S i for i = 1, . . ., k − 1. Suppose a local maximum of a trial exists at some point {q G max } ∈ R 2k−1 .We can project the (2k − 1)-dimensional vector q = (q G 1 , . . ., q G 2k−1 ) onto an arbitrary line R p ⊂ R 2k−1 passing through {q G max }.Restricted to this line, the trial central charge becomes a univariate cubic polynomial a trial (q p ) = a trial ({q G })| Rp where q p = proj Rp q.
A univariate cubic polynomial admits at most one local maximum, so by construction {q G max } is the unique local maximum of a trial (q p ) along any R p .Since R p can be chosen to connect {q G max } to any other point in the parameter space, along which {q G max } is always the unique local maximum, we conclude that a trial has at most one local maximum at {q G } = {q G max }. and an O(N, 6 is of O(N3 ) according to (3.9), we have checked that it always contribute only at O(N ) to the central charge a. Hence in the large-N, N N i ,S i limit (or loosely, the large-N limit), a can be effectively determined by performing a-maximization directly on −I inflow,E 3 4
We observe that the large-N inflow anomaly polynomial , constructed in [27] and reviewed in appendix A, satisfies an interesting identity.Suppose we have two distinct setups with generally different Euler characteristics, χ A , χ B , of the (g ≥ 2) Riemann surfaces, and they are wrapped by different numbers, N A , N B , of M5-branes, then the corresponding large-N inflow anomaly polynomials follow Consequently, we have an analogous scaling relation for the trial central charge, This motivates the definitions of "reduced flux quanta," such that the central charge, which is the (unique) local maximum of a trial , obeys in the large-N limit.It follows from (6.8) that the same scaling relation applies to the other central charges c and b G as well.
Existence and flux positivity.As alluded to earlier, a central charge does not necessarily exist for an arbitrary combination of k, χ, N , N N i , N S i .This is possible because while Bezout's theorem states that there can be at most 2 2k−1 critical points of a trial ({q G }), they may be all saddle points (possibly with a local minimum swapped in).Moreover, given a choice of orientation for the internal space M 6 , we define the charge (or more precisely the number of M5-branes) N to be positive under this orientation.Similarly, we can define the rest of the flux quanta using the same choice of orientation for the relevant cycles. 4or a supersymmetric theory, we expect all of these flux quanta to have the same sign as N .It is therefore important for us to examine the range of parameters within which our construction admits a central charge.
In figure 5 we illustrate the range of existence of the central charge a for a variety of specific configurations of N N i and N S i , given fixed k, χ, N .For k = 2 there is only one trivial pair of axes, i.e.N N 1 and N S 1 , but for k = 3 there are four independent flux quanta, so we show here several representative 2d cross-sections in the (discrete) space of flux quanta.In all cases considered we observe a clear-cut boundary separating the regions of the flux lattice with or without a central charge.We also note that a always exists when the flux quanta are strictly nonnegative; the red "exclusion region" always lies in the Figure 5: 2d exclusion plots visualizing the existence/nonexistence of the central charge a for k = 2 and k = 3.The choices of k, χ, N are labeled on top of each plot, whereas the specific configuration of the (integral) flux quanta, N N i , N S i , can be read off from the coordinates of a given dot.A dot is green if a exists for the corresponding combination of flux quanta; it is red if a does not exist.Note that the upper four plots are symmetric with respect to the exchange of axes as a consequence of the D 2 symmetry of M 4 .
three quadrants with at least one negative flux quantum.In fact, we find that the same qualitative feature applies to general k: setups characterized by strictly nonnegative flux configurations have (unique) central charges.
Furthermore, we can deduce from the large-N scaling relation (6.11) that as we decrease |χ| or N , the boundary of the exclusion region retreats towards the two positive axes but never crosses them.This is because the ratio χ B N B /χ A N A cannot change sign as long as we keep g ≥ 2. In other words, the central charge is guaranteed to exist everywhere in the first quadrant (where all flux quanta are positive) once the signs of χ and N are fixed, thus conforming to our expectation that all fluxes in SCFTs should be of uniform sign.
For the case of k = 2 studied in [26], we can see that the relative signs of fluxes are fixed directly in the holographic dual, namely, the GMSW solution [25].If explicit holographic duals are found for k > 2 in the future, we anticipate the same uniform-flux-sign condition to hold.Dihedral symmetry.It was noted in [27] that the inflow anomaly polynomial is invariant (up to a sign) under the two parity transformations of M 4 .Unsurprisingly, this nice property carries over to the central charge.Let us first consider the action of a north-south involution, the trial central charge a trial ({N N i }, {N S i }, q ϕ , {q N i }, {q S i }) is invariant under q ϕ , q N i ↔ −q S i (6.12)

Exact results for uniform flux configurations
The a-maximization problem is much more tractable for uniform flux configurations, that is, N N i = N S i := N N for i = 1, 2, . . ., k − 1, than for arbitrary configurations.It is indeed exactly solvable for sufficiently small k.As described earlier, finding the central charges for k = 2 and k = 3 with uniform flux configurations is effectively a 1d maximization problem, and in both cases the various central charges can be written as reasonably compact closedform expressions.For k = 2, we get whereas for k = 3, we have The divergence of the expressions above when N = 1 and N N = 0 shall not worry us as long as we are working in the large-N limit.Moreover, it can be easily checked that a and c are the same at leading order.Interestingly, we observe that . This pattern has a natural generalization for higher k, as we will soon see.
While being fully analytic, the central charges we find for k ≥ 4 cannot be reduced to similarly compact forms.Nevertheless, figure 6 illustrates the functional dependence of a, a/c, A Rϕϕ , A Rii on N N for a range of k.Note that the scaling relation (6.8) implies up to O(N 3 , N 3 N ), changing χ and N amounts to rescaling the axes of the plots of a, A Rϕϕ , A Rii without altering their qualitative behavior.It can be seen that all the central charges are monotonic in k and N N .We also note that the ratio a/c is well within the Hofman-Maldacena bounds [46] on N = 1 SCFTs, Let us briefly comment on the flavor central charge b i ∝ A Rii .In general, because of the D 2 symmetry of M 4 , there are (k − 1)/2 independent A Rii when all the resolution flux quanta are equal, i.e.
for i = 1, 2, . . ., (k − 1)/2 , hence the notation Specifically, there is one independent A Rii for k = 2, 3, two for k = 4, 5, and three for k = 6, 7. It is evident from the separation between lines of like color in figure 6 that The inequalities are simultaneously saturated when N N = 0.

Perturbative analysis
Even though it is exceptionally challenging to analytically determine the central charge through a-maximization for arbitrary combinations of k, χ, N , N N i , N S i , we can use perturbation theory to solve the equations ∂a trial /∂q G = 0 order by order in the regime where The first and the last inequalities are required to ensure that the O(N ) contributions from I inflow,E4 X 8 and I v,t 6 are negligible. 5We find that the perturbative expansions of the central charges a = c and the anomaly coefficient A Rϕϕ can be written as For uniform flux configurations, these perturbative expansions have been checked to be consistent with the previously shown exact expressions (6.16), (6.18), (6.20), (6.22) derived for k = 2 and k = 3.The symmetry (6.25) between flavor central charges b N i ,S i ∝ A R(N i ,S i )(N i ,S i ) no longer holds for nonuniform flux configurations.We list below the perturbative expansions of various flavor central charges for k = 2 and k = 3, so the reader can compare them to their uniform-flux analogs (6.19) and (6.23).For k = 2, we obtain whereas for k = 3, we obtain (1.1), which is central to our analysis.The charges and multiplicities in I flip 6 are then interpreted in terms of M2-brane operators, associated to blow-up modes of the C 2 /Z k singularity.We thus get a physical picture of the role of flip fields: they are necessary to interpolate between six dimensions, where such blow-up modes are not present, and four dimensions, where they are part of the SCFT.
The results of this paper suggest several directions for future research.Firstly, it would be interesting to find explicit AdS 5 solutions in 11d supergravity, which generalize the GMSW solutions from k = 2 to higher values of k.This work shows that the topology and flux configuration of M 6 give rise, via inflow, to the anomaly polynomial of a 4d SCFT of class S k with fluxes for the SU(k) b , SU(k) c flavor symmetries.This observation is a strong hint that AdS 5 solutions should exist, whose internal space has the topology and G 4 -flux quanta of M 6 .
The special case in which the Riemann surface is a torus also deserves further investigation.For k = 2, we have obtained a precise match between the M-theory inflow anomaly polynomial, and the anomaly polynomial of the SCFT realized by N D3-branes at the tip of the cone over the Sasaki-Einstein space Y p,q (with p, q determined by the flavor fluxes in the M-theory construction).It is natural to study generalizations to higher values of k, for instance exploring possible connections to other families of explicit Sasaki-Einstein metrics, such as [51,52].
Another natural direction for further study is to consider 4d theories of class S Γ , i.e. theories obtained from reduction of the 6d (1,0) SCFT realized by N M5-branes probing the singularity C 2 /Γ, with Γ an ADE subgroup of SU (2).Based on our results, we conjecture that the pattern of charges of flip fields for these models should be given in terms of the roots and Cartan matrix of g Γ , the ADE Lie algebra associated to Γ.It would be interesting to perform explicit checks of this conjecture, for instance against the Lagrangian models of [53].
FB is supported by the European Union's Horizon 2020 Framework: ERC Consolidator Grant 682608.

A Review of the E 3 4 contribution to anomaly inflow
In this appendix, we record the E 3 4 contribution to −I inflow 6 , derived in [27] for χ < 0, where ) are values of functions parameterizing basis forms in cohomology at the positions of the monopoles, while i , w i α are constants on the intervals composing ∂M 2 .Explicitly, in the basis introduced in (2.10), we have where the charge n i is +1 for 1 ≤ i ≤ k (northern region) and −1 for k + 1 ≤ i ≤ 2k (sourthern region).We observe that the 4-form λ 2 has legs along M 4 only, and is supported at the locations of the monopoles.Upon twisting M 4 over the Riemann surface, and activating the external gauge fields, the Chern roots λ 1 , λ 2 get shifted: the sum λ 1 + λ 2 , which is associated with the angle ψ in the base of the Taub-NUT S 1 ϕ fibration, is shifted by the total connection for the angle ψ, consisting of a contribution along Riemann surface the topological twist), and a contribution along the external spacetime, The difference λ 1 − λ 2 , which is instead associated with the S 1 ϕ fiber, is shifted by the external gauge field for the ϕ isometry, Recalling (B.1), the Chern roots of (the polar regions of) M 4 after twisting and gauging take the form After these preliminaries, we can proceed with the computation of X 8 that captures the northern and southern caps of M 4 .To compute the Pontryagin classes of the total 11d spacetime, we can apply the splitting principle, with reference to the schematic decomposition T M 11 → T W 4 ⊕ T Σ g ⊕ T M 4 , (B.7) using (B.6) for the Chern roots of the last summand.We obtain and hence (neglecting terms with more than six legs in the external spacetime) As noted above, λ is supported at the locations of the monopoles.Upon expanding (B.9), we encounter terms without λ, terms linear in λ, and terms with λ 2 .Higher powers of λ vanish, because they have too many legs along M 4 .Next, we observe that the terms that are linear in λ do not contribute to E 4 X 8 .This is due to the fact that, in our construction of E 4 , we have imposed that E 4 be regular as we approach the locations of the monopoles.As a result, E 4 localized at a monopole cannot provide the additional M 4 legs that would be necessary (together with λ) to saturate the integral in the M 4 directions.Furthermore, we can drop all terms in X 8 that have purely external legs.Taking these considerations into account, we see that the relevant terms in X 8 are given by We may now consider the parametrization of E 4 given by equation (4.8) of [27], and reported here for convenience, For the explicit parameterization and properties of the various forms appearing in this expansion we refer the reader to appendix B of [27].Upon expanding E 4 X and collecting terms that can saturate the M 6 integration, we arrive at The terms with γ 4 are omitted, as it can be easily verified that they drop from the computation, given our prescription to compute integrals of λ 2 described below.For the first line of (B.12) we just need the integral The terms with λ 2 are handled recalling (B.2) and (B.3), which imply the prescription Here Z stands for an arbitrary quantity on M 4 , Z(t i ) denotes Z evaluated at the i-th monopole, and n i = +1 for i = 1, • • • , k and n i = −1 for i = k + 1, • • • , 2k are the monopole charges.We also need to recall that where the omitted terms are not relevant for the M 6 integration.We also need where the W 's are the 0-forms that enter the parametrization of the harmonic 2-forms ω 2,α .We find it convenient to introduce the shorthand notation Collecting the various contributions to the integral of (B.12), we obtain To evaluate the above sums, we need to recall some relations regarding the quantities U , Y , W for our choice of basis of 4-and 2-cycles in the parameterization for E 4 , The above relations imply in particular where W Σ 0,α is a constant such that ω 2,α is closed.Note especially that the expression for Ω α 4 is not the same as the naïve χ = 0 limit of (B.17) in [27], otherwise flux quanta N i = C 4,i N α Ω α 4 for i = 1, 2, . ., 2k are ill-defined.This is also necessary so as to recover the sum rules, that are consistent with their higher-genus cousins.As a reminder, we choose to parameterize a given basis of (co)homology classes using the expansion coefficients, where the various cycles are introduced in section 2. These coefficients can be expressed in terms of various auxiliary differential forms defined earlier, (C.6) Recall that the four-form flux E 4 (restricting only to continuous zero-form symmetries) can be expanded as follows, where we used the shorthand notation N i = N α a α i , Ũ0,ψ = N α U α 0,ψ , Ũ0,ϕ = N α U α 0,ϕ , Ỹ0 = N α Y α 0 .The resulting inflow anomaly polynomial can be written as  To conclude, this independent derivation for genus one from first principles confirms the validity of using (3.13) to acquire an inflow anomaly polynomial for arbitrary g ≥ 1, including the case χ = 0. -43 -as in (2.10).One can then proceed to define the "natural" basis 2-cycles C α 2 to be those that are Poincaré-dual to these basis 4-cycles.For simplicity, the basis of (co)homology classes described above will hereafter be referred to as the "resolution flux basis."On the other hand, as explained in section 3.3, the starting point for the basis implicitly used in the compactification of the 6d N = (1, 0) theories is instead an intuitive set of basis 2-cycles, where C C 2 is defined to be the Poincaré dual of C 4,C = M 4 .The rest of the basis 4-cycles (C 4,α ) can be similarly defined to be the Poincaré duals of (C α 2 ) .We will hereafter refer to this basis of (co)homology classes as the "flavor flux basis."

D Change of basis between flavor and resolution flux quanta
In this appendix we find the transformation matrix that relates the resolution flux basis and the flavor flux basis.Using the relations following from Poincaré duality and regularity of the basis two-forms in appendix C of [27], we can express the basis 2-cycles C α 2 in the resolution flux basis in terms of familiar 2-cycles.Let us start with the case of k = 2.One finds that which can be inverted to give The orthonormal pairing between cycles and cohomology representatives is preserved if (ω 2,β ) = (R 2 ) α β ω 2,α , such that

1 Introduction and summary 1 2 7 3 3 . 1
Review of the eleven dimensional flux setups Anomaly polynomials in class S k from inflow 9 Integrated anomaly polynomial from six dimensions 10 3.2 Anomaly inflow from eleven dimensions 11 3.3 Matching the two sides: decoupled modes and flip fields 12

decoupled modes in 4d (I decoupl 6 )SCFT 6 )
< l a t e x i t s h a 1 _ b a s e 6 4 = " S n 9 w I S a y C r u x e g l o n 9 X 2 p Z F 3 w x 4 = " > A A A C L 3 i c b V H L S s N A F J 3 4 N r 6 q L t 0 M t kI F K Y n 4 2 g i C I L r x g W 0 t N L F M J r d 1 c G Y S Z i Z C C f k o F 3 6 E X y A i i A t d + B e m t S 6 s n t X h n P u Y c y e I O d P G c Z 6 t k d G x 8 Y n J q W l 7 Z n Z u f q G w u F T X U a I o 1 G j E I 9 U I i A b O J N Q M M x w a s Q I i A g 5 X w e 1 h z 7 + 6 A 6 V Z J K u m G 4 M v S E e y N q P E 5 F K r c O Y F 0 G E y p S A N q M z e C n H J E 8 T c U M L x K d 7 H b g l f H h 5 V s e c 1 X S F 8 u 1 w 6 a e 1 c p 5 4 S f T 0 r r d s e y P B n Q K t Q d C p O H / g v c Q e k i A Y 4 b x U e v T C i i c j b K S d a N 1 0 n N n 5 K l G G U Q 2 Z 7 i Y a Y 0 F v S g Z Q I r b s i y P B a 7 4 V 6 2 O u J / 3 n N x L T 3 / J T J O D E g a V 6 S e + 2 E Y x P h 3 k 1 w y B R Q w 7 s 5 I V S x f D O m N 0 Q R m i c a m q R B E g F 6 A 4 d 3 L N Z 9 7 q f 9 b 8 j s P L s 7 n P Q v q W 9 W 3 O 2 K c 7 F Z P H A G V 5 h C K 2 g V l Z G L d t E B O k b n q I Y o e k A v 6 B 1 9 W P f W k / V q v X 2 X j l i D n m X 0 C 9 b n F w l p p i c = < / l a t e x i t > 4d N = 1 SCFT (I < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 8 P O T 1 c W a y e L Y R M X M K B Y K b 5 Z M r U = " > A A A C J n i c b V D N S s N A G N z 4 b / y r e v S y 2 A o V p C Q F t c e C I H p T b K v Q x L L Z f K 1 L d z d h d 1 M o I e / j w R f x 4 k E Q E T 3 4 K K a 1 H q z O a Z j5 Z p e Z I O Z M G 8 d 5 s 2 Z m 5 + Y X F p e W 7 Z X V t f W N w u Z W S 0 e J o t C k E Y / U T U A 0 c C a h a Z j h c B M r I C L g c B 3 0 T 0 b + 9 Q C U Z p F s m G E M v i A 9 y b q M E p N L n c K p F 0 C P y Z S C N K A y + y j E p b J 7 4 O y X 8 N X J a Q N 7 X t s V w r f L p f N O 7 T b 1 l B j r W W n f 9 k C G P 7 l O o e h U n D H w X + J O S B F N c N E p P H p h R B O R x y k n W r d d J z Z + S p R h l E N m e 4 m G m N A + 6 U F K h N Z D E W R 4 T x B z p 6 e 9 k f i f 1 0 5 M t + a n T M a J A U n z k 9 z r J h y b C I + m w C F T

8 )
< l a t e x i t s h a 1 _ b a s e 6 4 = " 0 4 E N P q D B u 8f g o J 4 b m L L w N b B S 2 i 4 = " > A A A C J X i c b V D L S s N A F J 3 U V 4 2 v q k s 3 g 1 W o I C U R X 8 u C C L p T s F p o Y p l M b t q h M 5 M w M x F K y P e 4 8 E t c u B B E U B f + i m m t C 6 t n d b j n n n s 5 J 0 g 4 0 8 Z x 3 q z S 1 P T M 7 F x 5 3 l 5 Y X F p e q a y u X e s 4 V R S a N O a x a g V E A 2 c S m o Y Z D q 1 E A R E B h 5 u g f z L U b + 5 A a R b L K z N I w B e k K 1 n E K D H F q F M 5 9 Q L o M p l R k A Z U b u + H O O I s w R E D H m r s e W 1 X C N + u b Z 1 3 D m 8 z T 4 m R n G / t 2 B 7 I 8 M f W q V S d u j M C / k v c M a m i M S 4 6 l U c v j G k q C j v l R O u 2 6 y T G z 4 g y j H L I b S / V k B D a J 1 3 I i N B 6 I I I c b w t i e n p S G w 7 / 0 9 q p i Y 7 9 j M k k N S B p s V J o U c q x i f G w C R w y B d T w Q U E I V a z 4 j G m P K E K L R B O X N E g i Q O / i 8 I 4 l e s T 9 b F R + b h f Z 3 c m k f 8 n 1 X t 0 9 q D u X e 9 W G M 2 6 h j D b Q J q o h F x 2 h B j p D F 6 i J K H p A z + g d f V j 3 1 p P 1 Y r 1 + r 5 a s s W c d / Y L 1 + Q U s I q S W < / l a t e x i t > 4d flip fields (I flip 6 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " E Q V Q s D G 2 M s 4 z 4 y + f r I C N 1 R + H j s A = " > A A A C m H i c b Z F b a 9 s w G I Z l 7 9 R 6 h 2 Y b u 9 l u x N J B B y X Y Z X S D 3 Q S 2 s v W u K 0 t b i L P w W f 5 s i + p g J D k Q T H 7 T / s v u + m 8 q J 2 Z 0 7 V 5 J 8 P J 8 O r 7 K a s G t i + O r I L x 3 / 8 H D R 1 v b 0 e M n T 5 / t D J 6 / O L O 6 M Q w n T A t t L j K w K L j C i e N O 4 E V t E G Q m 8 D y 7 / N L V z x d o L N f q p 1 v W O J N Q K l 5 w B s 6 j + e B 3 W f k W p R m W X L U M l U O z i g Ag T a P C I F K H y m p D Z S M c r w W 6 f V 9 Y I H M 3 m f V M K 3 r K U Y J S 1 D a m A I b U 0 7 3 d 4 / n h r z Y 1 k i 7 2 3 W r 3

Figure 1 :
Figure 1: Schematic depiction of the flows across dimensions considered in this work.On the left: a stack of M5-branes with flat 6d worldvolume probes a C 2 /Z k singularity, yielding a 6d (1,0) SCFT, plus 6d modes associated to free tensor, vector multiplets.The interacting 6d (1,0) SCFT is reduced on a Riemann surface with fluxes.The outcome is organized into an interacting 4d SCFT, and a collection of 4d free fields, interpreted as flip fields.On the right: a stack of M5-branes probes a resolved C 2 /Z k singularity and is wrapped on a Riemann surface.At low energies, this M-theory setup gives the same 4d SCFT, plus other 4d modes coming from free tensor, vector multiplets on the Riemann surface.The blue, solid arrows denote anomaly inflow from the 11d bulk onto the M5branes.The red, hollow arrows denote the purely field-theoretical reduction of the 6d (1,0) SCFT on the Riemann surface.

8 and I inflow 6 ,
including terms of order 1 in the flux parameters.The match takes the following form, ) leaving 2k − 1 independent 4-cycles.It is convenient to adopt a complete basis of 4-cycles C 4,α consisting of the M 4 bulk 4-cycle, the k − 1 4-cycles in the north, and the k − 1 4-cycles in the south,

I flip 6 in ( 3 . 8 )
is indeed the anomaly polynomial of the flip fields that appear in the field-theoretic construction on tori with fluxes.•The difference between Σg I SCFT 8 and I flip 6 as given by our formula (3.10) is indeed equal to the anomaly polynomial of the interacting SCFT in four dimensions,

Figure 7 :
Figure 7: The quiver diagram for the 4d class S 3 theory corresponding to a torus with two units of 1 flux and one of 2 flux.Next to the fields are their charged summarized using fugacities.This theory has a combination of cubic and quartic superpotential terms.Again these are most conveniently generated by taking all terms consistent with the symmetries.Additionally there are superpotential terms coming from the flipping.The theory also has an R-symmetry, a convenient choice for which is to give all the bifundamentals R-charge1  2 , and R-charge 1 for the adjoints and their singlets.

Figure 5 :
Figure 5: The quiver diagram for the 4d class S2 theory corresponding to a torus with flux (12 , 1 2 , 0).Next to the fields are their charges summarized through fugacities.The theory has a quartic superpotential involving the four bifundamentals as well as the superpotential coming from the flipping.There is also an R-symmetry where it is convenient to give R-charge1  2 to the four bifundamentals Qi and R-charge 2 N 2 to the flipping fields , .

First
we shall need to perform a-maximization to determine the superconformal Rsymmetry.It is straightforward to see that only the baryonic symmetry u(1) + u(1) can mix with the naive u(1) R of the KW model.Thus, we define:

3 6 = −I inflow 6 − I v,t 6 .
Large-N scaling relation.Recall that the central charges of the SCFT are encoded by the anomaly polynomial I SCFT The former can be further decomposed into an O N 3 , N 3 N i ,S i contribution I inflow,E 3 4 6

Figure 6 :
Figure 6: Plots of a, a/c, A Rϕϕ , A Rii for k = 2 to k = 7 with identical flux quantaN N i = N S i := N N for all i = 1, . . ., k − 1. N N istreated as a continuous real parameter for visualization purposes.All of the plots are evaluated with χ = −2 and N = 10.For a given k, there are (k − 1)/2 independent A Rii anomaly coefficients, which are proportional to the flavor central charges b i associated with the resolution cycles of M 4 .We use solid lines for i = 1, dashed lines for i = 2, and dotted lines for i = 3 where applicable.

1 2= C i=1 2 = 2
basis 2-cycles in the flavor basis are C N

Table 2 :
.33)5Appropriate powers of χ are inserted in the inequalities here based on the facts that NN i /|χ|N, NS i /|χ|N ∼ 1 are the "characteristic scales" and that I Central charge coefficients (with genus one) for various k.
subject to the condition that N α F α 2 = 0.The values of the various auxiliary functions evaluated at any t i are given by Ũ0,ψ (t i ) = Ũ0,ψ (t 1 ) − i − j )N j , (C.14)