6d SCFT Research Articles

Isomorphisms of 4D N=2 SCFTs from 6D

There exist 4d $\mathcal{N}=2$ SCFTs in class $\mathcal{S}$ which have different constructions as punctured Riemann surfaces, but which nevertheless appear to describe the same physics. Some of these class $\mathcal{S}$ theories have an alternative construction as torus-compactifications of 6d $(1,0)$ SCFTs. We demonstrate that the 6d SCFTs are isomorphic. Each 6d SCFT in question can be obtained from a parent 6d SCFT by Higgs branch renormalization group flow, and the parent theory possesses a discrete symmetry under which the relevant Higgs branch flows are exchanged. The existence of this discrete symmetry, which may be embedded in an enhanced continuous symmetry, proves that the original pair of class $\mathcal{S}$ theories are, in fact, isomorphic.

The geometry of decoupling fields

We consider 4d field theories obtained by reducing the 6d (1,0) SCFT of N M5-branes probing a ℂ2/ℤk 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 ℂ2/ℤk 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.

Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

We consider supersymmetric surface defects in compactifications of the 6d6d minimal (D_{N+3},D_{N+3})(DN+3,DN+3) conformal matter theories on a punctured Riemann surface. For the case of N=1N=1 such defects are introduced into the supersymmetric index computations by an action of the BC_1\,(\sim A_1\sim C_1)BC1(∼A1∼C1) van Diejen model. We (re)derive this fact using three different field theoretic descriptions of the four dimensional models. The three field theoretic descriptions are naturally associated with algebras A_{N=1}AN=1, C_{N=1}CN=1, and (A_1)^{N=1}(A1)N=1. The indices of these 4d4d theories give rise to three different Kernel functions for the BC_1BC1 van Diejen model. We then consider the generalizations with N>1N>1. The operators introducing defects into the index computations are certain A_{N}AN, C_NCN, and (A_1)^{N}(A1)N generalizations of the van Diejen model. The three different generalizations are directly related to three different effective gauge theory descriptions one can obtain by compactifying the minimal (D_{N+3},D_{N+3})(DN+3,DN+3) conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the A_NAN case, and derive various properties these operators have to satisfy as a consequence of 4d4d dualities following from the geometric setup. In some cases we are able to verify these properties which in turn serve as checks of said dualities. As a by-product of our constructions we also discuss a simple Lagrangian description of a theory corresponding to compactification on a sphere with three maximal punctures of the minimal (D_5,D_5)(D5,D5) conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.

Flavor symmetry of 5d SCFTs. Part II. Applications

In Part I of this series of papers, we described a general method for determining the flavor symmetry of any 5d SCFT which can be constructed by integrating out BPS particles from some 6d SCFT compactified on a circle. In this part, we apply the method to explicitly determine the flavor symmetry of those 5d SCFTs which reduce, upon a mass deformation, to some 5d mathcal{N} = 1 gauge theory carrying a simple gauge algebra. In these cases, the flavor symmetry of the 5d gauge theory is often enhanced at the conformal point. We use our method to determine this enhancement.

Higgsing towards E-strings

We explore 6d (1, 0) superconformal field theories with SU(3) and SU(2) gauge symmetries which cascade after Higgsing to the E-string theory on a single M5 near an E8 wall. Specifically, we study the 2d mathcal{N} = (0, 4) gauge theories which describe self-dual strings of these 6d theories. The self-dual strings can be also viewed as instanton string solitons of 6d Yang-Mills theories. We find the 2d anomaly-free gauge theories for self-dual strings, amending the naive ADHM gauge theories which are anomalous, and calculate their elliptic genera. While these 2d theories respect the flavor symmetry of each 6d SCFT only partially, their elliptic genera manifest the symmetry fully as these functions as BPS index are invariant in strongly coupled IR limit. Our consistent 2d (0, 4) gauge theories also provide new insights on the non-linear sigma models for the instanton strings, providing novel UV completions of the small instanton singularities. Finally, we construct new 2d quiver gauge theories for the self-dual strings in 6d E-string theory for multiple M5-branes probing the E8 wall, and find their fully refined elliptic genera.

Classification of 5d mathcal{N} = 1 gauge theories

We classify 5d mathcal{N} = 1 gauge theories carrying a simple gauge group that can arise by mass-deforming 5d SCFTs and 6d SCFTs (compactified on a circle, possibly with a twist). For theories having a 6d UV completion, we determine the tensor branch data of the 6d SCFT and capture the twist in terms of the tensor branch data. We also determine the dualities between these 5d gauge theories, thus determining the sets of gauge theories having a common UV completion.

SQCD and pairs of pants

We show that the 4d mathcal{N} = 1 SU(3) Nf = 6 SQCD is the model obtained when compactifying the rank one E-string theory on a three punctured sphere (a trinion) with a particular value of flux. The SU(6) × SU(6) × U(1) global symmetry of the theory, when decomposed into the SU(2)3× U(1)3× SU(6) subgroup, corresponds to the three SU(2) symmetries associated to the three punctures and the U(1)3× SU(6) subgroup of the E8 symmetry of the E-string theory. All the puncture symmetries are manifest in the UV and thus we can construct ordinary Lagrangians flowing in the IR to any compactification of the E-string theory. We generalize this claim and argue that the mathcal{N} = 1 SU(N + 2) SQCD in the middle of the conformal window, Nf = 2N + 4, is the theory obtained by compactifying the 6d minimal (DN +3, DN +3) conformal matter SCFT on a sphere with two maximal SU(N + 1) punctures, one minimal SU(2) puncture, and with a particular value of flux. The SU(2N + 4) × SU(2N + 4) × U(1) symmetry of the UV Lagrangian decomposes into SU(N + 1)2× SU(2) puncture symmetries and the U(1)3× SU(2N + 4) subgroup of the SO(12 + 4N ) symmetry group of the 6d SCFT. The models constructed from the trinions exhibit a variety of interesting strong coupling effects. For example, one of the dualities arising geometrically from different pair-of-pants decompositions of a four punctured sphere is an SU(N + 2) generalization of the Intriligator-Pouliot duality of SU(2) SQCD with Nf = 4, which is a degenerate, N = 0, instance of our discussion.

Anomaly inflow, accidental symmetry, and spontaneous symmetry breaking

We consider the 6d (1,0) SCFT on a stack of N M5-branes probing a ℂ2/ℤ2 singularity. In particular, we study its compactifications to four dimensions on a smooth genus-g Riemann surface with non-trivial flavor flux, yielding a family of 4d CFTs. By tracking the M-theory origin of the global symmetries of the 4d CFTs, we detect the emergence of an accidental symmetry and the spontaneous symmetry breaking of a U(1) generator. These effects are visible from geometric considerations and not apparent from the point of view of the compactification of the 6d field theory. These phenomena leave an imprint on the ’t Hooft anomaly polynomial of the 4d CFTs, which is obtained from recently developed anomaly inflow methods in M-theory [1]. In the large-N limit, we identify the gravity dual of the 4d setups to be a class of smooth AdS5 solutions first discussed by Gauntlett-Martelli-Sparks-Waldram. Using our anomaly polynomial, we compute the conformal central charge and a non-Abelian flavor central charge at large N , finding agreement with the holographic predictions.

4d mathcal{N} = 1 from 6d D-type mathcal{N} = (1, 0)

Compactifications of 6d mathcal{N} = (1, 0) SCFTs give rise to new 4d mathcal{N} = 1 SCFTs and shed light on interesting dualities between such theories. In this paper we continue exploring this line of research by extending the class of compactified 6d theories to the D- type case. The simplest such 6d theory arises from D5 branes probing D-type singularities. Equivalently, this theory can be obtained from an F-theory compactification using −2- curves intersecting according to a D-type quiver. Our approach is two-fold. We start by compactifying the 6d SCFT on a Riemann surface and compute the central charges of the resulting 4d theory by integrating the 6d anomaly polynomial over the Riemann surface. As a second step, in order to find candidate 4d UV Lagrangians, there is an intermediate 5d theory that serves to construct 4d domain walls. These can be used as building blocks to obtain torus compactifications. In contrast to the A-type case, the vanishing of anomalies in the 4d theory turns out to be very restrictive and constraints the choices of gauge nodes and matter content severely. As a consequence, in this paper one has to resort to non- maximal boundary conditions for the 4d domain walls. However, the comparison to the 6d theory compactified on the Riemann surface becomes less tractable.

Classifying 5d SCFTs via 6d SCFTs: arbitrary rank

According to a conjecture, all 5d SCFTs should be obtainable by rank- preserving RG flows of 6d SCFTs compactified on a circle possibly twisted by a background for the discrete global symmetries around the circle. For a 6d SCFT admitting an F-theory construction, its untwisted compactification admits a dual M-theory description in terms of a “parent” Calabi-Yau threefold which captures the Coulomb branch of the compacti- fied 6d SCFT. The RG flows to 5d SCFTs can then be identified with a sequence of flop transitions and blowdowns of the parent Calabi-Yau leading to “descendant” Calabi-Yau threefolds which describe the Coulomb branches of the resulting 5d SCFTs. An explicit description of parent Calabi-Yaus is known for untwisted compactifications of rank one 6d SCFTs. In this paper, we provide a description of parent Calabi-Yaus for untwisted compactifications of arbitrary rank 6d SCFTs. Since 6d SCFTs of arbitrary rank can be viewed as being constructed out of rank one SCFTs, we accomplish the extension to arbi- trary rank by identifying a prescription for gluing together Calabi-Yaus associated to rank one 6d SCFTs.

Classifying 5d SCFTs via 6d SCFTs: rank one

Following a recent proposal, we delineate a general procedure to classify 5d SCFTs via compactifications of 6d SCFTs on a circle (possibly with a twist by a discrete global symmetry). The path from 6d SCFTs to 5d SCFTs can be divided into two steps. The first step involves computing the Coulomb branch data of the 5d KK theory obtained by compactifying a 6d SCFT on a circle of finite radius. The second step involves computing the limit of the KK theory when the inverse radius along with some other mass parameters is sent to infinity. Under this RG flow, the KK theory reduces to a 5d SCFT. We illustrate these ideas in the case of untwisted compactifications of rank one 6d SCFTs that can be constructed in F-theory without frozen singularities. The data of the corresponding KK theory can be packaged in the geometry of a Calabi-Yau threefold that we explicitly compute for every case. The RG flows correspond to flopping a collection of curves in the threefold and we formulate a concrete set of criteria which can be used to determine which collection of curves can induce the relevant RG flows, and, in principle, to determine the Calabi-Yau geometries describing the endpoints of these flows. We also comment on how to generalize our results to arbitrary rank.

Phases of 5d SCFTs from M-/F-theory on non-flat fibrations

We initiate the systematic investigation of non-flat resolutions of non-minimal singularities in elliptically fibered Calabi-Yau threefolds. Compactification of M-theory on these geometries provides an alternative approach to studying phases of five-dimensional superconformal field theories (5d SCFTs). We argue that such resolutions capture non-trivial holonomies in the circle reduction of the 6d conformal matter theory that is the F-theory interpretation of the singular fibration. As these holonomies become mass deformations in the 5d theory, non-flat resolutions furnish a novel method in the attempt to classify 5d SCFTs through 6d SCFTs on a circle. A particularly pleasant aspect of this proposal is the explicit embedding of the 5d SCFT’s enhanced flavor group inside that of the parent 6d SCFT, which can be read off from the geometry. We demonstrate these features in toric examples which realize 5d theories up to rank four.

5d partition functions with a twist

We derive the partition function of 5d mathcal{N}=1 gauge theories on the manifold {S}_b^3 times {varSigma}_{mathfrak{g}} with a partial topological twist along the Riemann surface, {varSigma}_{mathfrak{g}} . This setup is a higher dimensional uplift of the two-dimensional A-twist, and the result can be expressed as a sum over solutions of Bethe-Ansatz-type equations, with the computation receiving nontrivial non-perturbative contributions. We study this partition function in the large N limit, where it is related to holographic RG flows between asymptotically locally AdS6 and AdS4 spacetimes, reproducing known holographic relations between the corresponding free energies on S5 and S3 and predicting new ones. We also consider cases where the 5d theory admits a UV completion as a 6d SCFT, such as the maximally supersymmetric mathcal{N}=2 Yang-Mills theory, in which case the partition function computes the 4d index of general class S theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to {mathrm{mathcal{M}}}_3 times {varSigma}_{mathfrak{g}} with more general three-manifolds ℳ3 and focus in particular on {mathrm{mathcal{M}}}_3 = {varSigma}_{mathfrak{g}}^{prime}times {S}^1 , in which case the partition function relates to the entropy of black holes in AdS6.

Classifying global symmetries of 6D SCFTs

We characterize the global symmetries for the conjecturally complete collection of all six dimensional superconformal field theories (6D SCFTs) which are realizable in F-theory and have no frozen singularities. We provide comprehensive checks of earlier 6D SCFT classification results via an alternative geometric approach yielding new restrictions which eliminate certain theories. We achieve this by directly constraining elliptically fibered Calabi-Yau (CY) threefold Weierstrass models and find this allows bypassing all anomaly cancellation machinery. This approach reduces the problem of classifying which 6D SCFT gauge and global symmetries are realizable in F-theory models before RG-flow to characterizing features of elliptic fibrations associated to these theories obtained by analysis of polynomials determining their local models. We supply an algorithm with implementation producing from a given SCFT base an explicit listing of all compatible gauge enhancements and their associated global symmetry maxima consistent with the geometric constraints we derive while making manifest the corresponding geometric ingredients for these symmetries including any possible Kodaira type realizations of each algebra summand. In mathematical terms, this amounts to determining all potentially viable non-compact CY threefold elliptic fibrations at finite distance in the moduli space with Weil-Petersson metric which meet certain requirements including the transverse pairwise intersection of singular locus components. We provide local analysis exhausting nearly all CY consistent transverse singular fiber collisions, global analysis concerning all viable gluings of these local models into larger configurations, and many novel constraints on singular locus component pair intersections and global fiber arrangements. We also investigate which transitions between 6D SCFTs can result from gauging of global symmetries and find that continuous degrees of freedom can be lost during such transitions.

4d mathcal{N}=1 from 6d (1, 0)

We study the geometry of 4d mathcal{N}=1 SCFT’s arising from compactification of 6d (1, 0) SCFT’s on a Riemann surface. We show that the conformal manifold of the resulting theory is characterized, in addition to moduli of complex structure of the Riemann surface, by the choice of a connection for a vector bundle on the surface arising from flavor symmetries in 6d. We exemplify this by considering the case of 4d mathcal{N}=1 SCFT’s arising from M5 branes probing ℤk singularity compactified on a Riemann surface. In particular, we study in detail the four dimensional theories arising in the case of two M5 branes on ℤ2 singularity. We compute the conformal anomalies and indices of such theories in 4d and find that they are consistent with expectations based on anomaly and the moduli structure derived from the 6 dimensional perspective.

Equivalence of several descriptions for 6d SCFT

We show that the three different looking BPS partition functions, namely the elliptic genus of the 6d mathcal{N}=left(1, 0right) Sp(1) gauge theory with 10 flavors and a tensor multiplet, the Nekrasov partition function of the 5d mathcal{N}=1 mathrm{S}mathrm{p}(2) gauge theory with 10 flavors, and the Nekrasov partition function of the 5d mathcal{N}=1 mathrm{S}mathrm{U}(3) gauge theory with 10 flavors, are all equal to each other under specific maps among gauge theory parameters. This result strongly suggests that the three gauge theories have an identical UV fixed point. Type IIB 5-brane web diagrams play an essential role to compute the SU(3) Nekrasov partition function as well as establishing the maps.

Enhanced gauge symmetry in 6D F‐theory models and tuned elliptic Calabi‐Yau threefolds

We systematically analyze the local combinations of gauge groups and matter that can arise in 6D F‐theory models over a fixed base. We compare the low‐energy constraints of anomaly cancellation to explicit F‐theory constructions using Weierstrass and Tate forms, and identify some new local structures in the “swampland” of 6D supergravity and SCFT models that appear consistent from low‐energy considerations but do not have known F‐theory realizations. In particular, we classify and carry out a local analysis of all enhancements of the irreducible gauge and matter contributions from “non‐Higgsable clusters,” and on isolated curves and pairs of intersecting rational curves of arbitrary self‐intersection. Such enhancements correspond physically to unHiggsings, and mathematically to tunings of the Weierstrass model of an elliptic CY threefold. We determine the shift in Hodge numbers of the elliptic threefold associated with each enhancement. We also consider local tunings on curves that have higher genus or intersect multiple other curves, codimension two tunings that give transitions in the F‐theory matter content, tunings of abelian factors in the gauge group, and generalizations of the “E8” rule to include tunings and curves of self‐intersection zero. These tools can be combined into an algorithm that in principle enables a finite and systematic classification of all elliptic CY threefolds and corresponding 6D F‐theory SUGRA models over a given compact base (modulo some technical caveats in various special circumstances), and are also relevant to the classification of 6D SCFT's. To illustrate the utility of these results, we identify some large example classes of known CY threefolds in the Kreuzer‐Skarke database as Weierstrass models over complex surface bases with specific simple tunings, and we survey the range of tunings possible over one specific base.

On the Defect Group of a 6D SCFT

On the Defect Group of a 6D SCFT

3d TQFT from 6d SCFT

We study the six-dimensional (2, 0) superconformal field theory on S 1 × S 2 × M via compactification to five dimensions, where M is a three-manifold. Twisted along M, the five-dimensional theory has a half of $ \mathcal{N}=\left( {2,2} \right) $ supersymmetry on S 2, the other half being broken by a superpotential. We show that in the limit where M is infinitely large, the twisted theory reduces to a three-dimensional topological quantum field theory which is closely related to Chern-Simons theory for the complexified gauge group.

Tinkertoys for the D N series

We describe a procedure for classifying 4D N=2 superconformal theories of the type introduced by Davide Gaiotto. Any punctured curve, C, on which the 6D (2,0) SCFT is compactified, may be decomposed into 3-punctured spheres, connected by cylinders. The 4D theories, which arise, can be characterized by listing the "matter" theories corresponding to 3-punctured spheres, the simple gauge group factors, corresponding to cylinders, and the rules for connecting these ingredients together. Different pants decompositions of $ correspond to different S-duality frames for the same underlying family of 4D N=2 SCFTs. In a previous work [1], we developed such a classification for the A_{N-1} series of 6D (2,0) theories. In the present paper, we extend this to the D_N series. We outline the procedure for general D_N, and construct, in detail, the classification through D_4. We discuss the implications for S-duality in Spin(8) and Spin(7) gauge theory, and recover many of the dualities conjectured by Argyres and Wittig [2].