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Matthew Buican

Publications and source records attributed to Matthew Buican.

At least 37 records · Page 2Linked to original sources
arXiv2021

$a\times b=c$ in $2+1$D TQFT

We study the implications of the anyon fusion equation $a\times b=c$ on global properties of $2+1$D topological quantum field theories (TQFTs). Here $a$ and $b$ are anyons that fuse together to give a unique anyon, $c$. As is well known, when at least one of $a$ and $b$ is abelian, such equations describe aspects of the one-form symmetry of the theory. When $a$ and $b$ are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and $a$ and $b$ lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian $a$ and $b$ can also be an indication of zero-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, $M_{24}$), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.

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arXiv2020

Non-Abelian Anyons and Some Cousins of the Arad-Herzog Conjecture

Long ago, Arad and Herzog (AH) conjectured that, in finite simple groups, the product of two conjugacy classes of length greater than one is never a single conjugacy class. We discuss implications of this conjecture for non-abelian anyons in 2+1-dimensional discrete gauge theories. Thinking in this way also suggests closely related statements about finite simple groups and their associated discrete gauge theories. We prove these statements and provide some physical intuition for their validity. Finally, we explain that the lack of certain dualities in theories with non-abelian finite simple gauge groups provides a non-trivial check of the AH conjecture.

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arXiv2020

$\mathcal{N}=4$ SYM, Argyres-Douglas Theories, and an Exact Graded Vector Space Isomorphism

In this first of two papers, we explain in detail the simplest example of a broader set of relations between apparently very different theories. Our example relates $\mathfrak{su}(2)$ $\mathcal{N}=4$ super Yang-Mills (SYM) to a theory we call "$(3,2)$." This latter theory is an exactly marginal diagonal $SU(2)$ gauging of three $D_3(SU(2))$ Argyres-Douglas (AD) theories. We begin by observing that the Schur indices of these two theories are related by an algebraic transformation that is surprisingly reminiscent of index transformations describing spontaneous symmetry breaking on the Higgs branch. However, this transformation breaks half the supersymmetry of the SYM theory as well as its full $\mathcal{N}=2$ $SU(2)_F$ flavor symmetry. Moreover, it does so in an interesting way when viewed through the lens of the corresponding 2D vertex operator algebras (VOAs): affine currents of the small $\mathcal{N}=4$ super-Virasoro algebra at $c=-9$ get mapped to the $\mathcal{A}(6)$ stress tensor and some of its conformal descendants, while the extra supersymmetry currents on the $\mathcal{N}=4$ side get mapped to higher-dimensional fermionic currents and their descendants on the $\mathcal{A}(6)$ side. We prove these relations are facets of an exact graded vector space isomorphism (GVSI) between these two VOAs. This GVSI respects the $U(1)_r$ charge of the parent 4D theories. We briefly sketch how more general $\mathfrak{su}(N)$ $\mathcal{N}=4$ SYM theories are related to an infinite class of AD theories via generalizations of our example. We conclude by showing that, in this class of theories, the $\mathcal{A}(6)$ VOA saturates a new inequality on the number of strong generators.

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arXiv2019

Galois Conjugation and Multiboundary Entanglement Entropy

We revisit certain natural algebraic transformations on the space of 3D topological quantum field theories (TQFTs) called "Galois conjugations." Using a notion of multiboundary entanglement entropy (MEE) defined for TQFTs on compact 3-manifolds with disjoint boundaries, we give these abstract transformations additional physical meaning. In the process, we prove a theorem on the invariance of MEE along orbits of the Galois action in the case of arbitrary Abelian theories defined on any link complement in $S^3$. We then give a generalization to non-Abelian TQFTs living on certain infinite classes of torus link complements. Along the way, we find an interplay between the modular data of non-Abelian TQFTs, the topology of the ambient spacetime, and the Galois action. These results are suggestive of a deeper connection between entanglement and fusion.

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arXiv2019

Peculiar Index Relations, 2D TQFT, and Universality of SUSY Enhancement

We study certain exactly marginal gaugings involving arbitrary numbers of Argyres-Douglas (AD) theories and show that the resulting Schur indices are related to those of certain Lagrangian theories of class $\mathcal{S}$ via simple transformations. By writing these quantities in the language of 2D topological quantum field theory (TQFT), we easily read off the $S$-duality action on the flavor symmetries of the AD quivers and also find expressions for the Schur indices of various classes of exotic AD theories appearing in different decoupling limits. The TQFT expressions for these latter theories are related by simple transformations to the corresponding quantities for certain well-known isolated theories with regular punctures (e.g., the Minahan-Nemeschansky $E_6$ theory and various generalizations). We then reinterpret the TQFT expressions for the indices of our AD theories in terms of the topology of the corresponding 3D mirror quivers, and we show that our isolated AD theories generically admit renormalization group (RG) flows to interacting superconformal field theories (SCFTs) with thirty-two (Poincaré plus special) supercharges. Motivated by these examples, we argue that, in a sense we make precise, the existence of RG flows to interacting SCFTs with thirty-two supercharges is generic in a far larger class of 4D $\mathcal{N}=2$ SCFTs arising from compactifications of the 6D $(2,0)$ theory on surfaces with irregular singularities.

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arXiv2019

Rationalizing CFTs and Anyonic Imprints on Higgs Branches

We continue our program of mapping data of 4D $\mathcal{N}=2$ superconformal field theories (SCFTs) onto observables of 2D chiral rational conformal field theories (RCFTs) by revisiting an infinite set of strongly coupled Argyres-Douglas (AD) SCFTs and their associated logarithmic 2D chiral algebras. First, we turn on discrete flavor fugacities (for continuous flavor symmetries) in a known correspondence between certain unrefined characters of these logarithmic theories and unrefined characters of a set of unitary 2D chiral RCFTs. Motivated by this discussion, we then study 4D Higgs branch renormalization group flows (i.e., flows activated by vevs for which only $su(2)_R\subset su(2)_R\times u(1)_R$ is spontaneously broken) emanating from our AD theories from the point of view of the unitary 2D theories and find some surprises. In particular, we argue that certain universal pieces of the topological data underlying the 2D chiral algebra representations associated with the 4D infrared (IR) theory can be computed, via Galois conjugation, in the topological quantum field theory (TQFT) underlying the unitary ultraviolet (UV) chiral RCFT. The mapping of this topological data from UV to IR agrees with the fact that, in our theories, the moduli spaces we study consist of free hypermultiplets at generic points if and only if the UV TQFT is a theory of abelian anyons.

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arXiv2018

Flowing from 16 to 32 Supercharges

We initiate a study of an infinite set of renormalization group flows with accidental supersymmetry enhancement. The ultraviolet fixed points are strongly interacting four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) with no known Lagrangian descriptions, and the infrared fixed points are SCFTs with thirty-two (Poincaré plus special) supercharges.

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arXiv2018

Nonunitary Lagrangians and unitary non-Lagrangian conformal field theories

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this note, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) by performing simple calculations involving sets of non-unitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full $\mathcal{N}=2$ superconformal algebra. As a byproduct, we find relations between characters in unitary and non-unitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.

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arXiv2017

N=2 S-duality Revisited

Using the chiral algebra bootstrap, we revisit the simplest Argyres-Douglas (AD) generalization of Argyres-Seiberg S-duality. We argue that the exotic AD superconformal field theory (SCFT), $T_{3,{3\over2}}$, emerging in this duality splits into a free piece and an interacting piece, T_X, even though this factorization seems invisible in the Seiberg-Witten (SW) curve derived from the corresponding M5-brane construction. Without a Lagrangian, an associated topological field theory, a BPS spectrum, or even an SW curve, we nonetheless obtain exact information about T_X by bootstrapping its chiral algebra, chi(T_X), and finding the corresponding vacuum character in terms of Affine Kac-Moody characters. By a standard 4D/2D correspondence, this result gives us the Schur index for T_X and, by studying this quantity in the limit of small S^1, we make contact with a proposed S^1 reduction. Along the way, we discuss various properties of T_X: as an N=1 theory, it has flavor symmetry SU(3)XSU(2)XU(1), the central charge of chi(T_X) matches the central charge of the bc ghosts in bosonic string theory, and its global SU(2) symmetry has a Witten anomaly. This anomaly does not prevent us from building conformal manifolds out of arbitrary numbers of T_X theories (giving us a surprisingly close AD relative of Gaiotto's T_N theories), but it does lead to some open questions in the context of the chiral algebra / 4D N=2 SCFT correspondence.

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arXiv2017

On Irregular Singularity Wave Functions and Superconformal Indices

We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU(N). As an application, we give closed-form expressions for the Schur indices of all (A_{N-1}, A_{N(n-1)-1}) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the (A_N, A_M) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of "Type IV" AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple "monopole vev RG" formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems.

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arXiv2017

Anyonic Chains, Topological Defects, and Conformal Field Theory

Motivated by the three-dimensional topological field theory / two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, that can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic "topological symmetries" that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry / topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones.

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arXiv2016

Aspects of Superconformal Multiplets in D>4

We explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and six-dimensional superconformal field theories. At the same time, we provide a detailed argument for the complete classification of unitary irreducible representations in five dimensions using a combination of physical and mathematical techniques.

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arXiv2016

Conformal Manifolds in Four Dimensions and Chiral Algebras

Any N=2 superconformal field theory (SCFT) in four dimensions has a sector of operators related to a two-dimensional chiral algebra containing a Virasoro sub-algebra. Moreover, there are well-known examples of isolated SCFTs whose chiral algebra is a Virasoro algebra. In this note, we consider the chiral algebras associated with interacting N=2 SCFTs possessing an exactly marginal deformation that can be interpreted as a gauge coupling (i.e., at special points on the resulting conformal manifolds, free gauge fields appear that decouple from isolated SCFT building blocks). At any point on these conformal manifolds, we argue that the associated chiral algebras possess at least three generators. In addition, we show that there are examples of SCFTs realizing such a minimal chiral algebra: they are certain points on the conformal manifold obtained by considering the low-energy limit of type IIB string theory on the three complex-dimensional hypersurface singularity x_1^3+x_2^3+x_3^3+A x_1x_2x_3+w^2=0. The associated chiral algebra is the A(6) theory of Feigin, Feigin, and Tipunin. As byproducts of our work, we argue that (i) a collection of isolated theories can be conformally gauged only if there is a SUSY moduli space associated with the corresponding symmetry current moment maps in each sector, and (ii) N=2 SCFTs with a>=c have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators).

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arXiv2016

A Small Deformation of a Simple Theory

We study an interesting relevant deformation of the simplest interacting N=2 SCFT---the original Argyres-Douglas (AD) theory. We argue that, although this deformation is not strictly speaking Banks-Zaks like (certain operator dimensions change macroscopically), there are senses in which it constitutes a mild deformation of the parent AD theory: the exact change in the "a" anomaly is small and is essentially saturated at one loop. Moreover, contributions from IR operators that have a simple description in the UV theory reproduce a particular limit of the IR index to a remarkably high order. These results lead us to conclude that the IR theory is an interacting N=1 SCFT with particularly small "a" and "c" central charges and that this theory sheds some interesting light on the spectrum of its AD parent.

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arXiv2015

Argyres-Douglas Theories, S^1 Reductions, and Topological Symmetries

In a recent paper, we proposed closed-form expressions for the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) superconformal field theories (SCFTs) in the Schur limit. Following up on our results, we turn our attention to the small S^1 regime of these indices. As expected on general grounds, our study reproduces the S^3 partition functions of the resulting dimensionally reduced theories. However, we show that in all cases---with the exception of the reduction of the (A_1, D_4) SCFT---certain imaginary partners of real mass terms are turned on in the corresponding mirror theories. We interpret these deformations as R symmetry mixing with the topological symmetries of the direct S^1 reductions. Moreover, we argue that these shifts occur in any of our theories whose four-dimensional N=2 superconformal U(1)_R symmetry does not obey an SU(2) quantization condition. We then use our R symmetry map to find the four-dimensional ancestors of certain three-dimensional operators. Somewhat surprisingly, this picture turns out to imply that the scaling dimensions of many of the chiral operators of the four-dimensional theory are encoded in accidental symmetries of the three-dimensional theory. We also comment on the implications of our work on the space of general N=2 SCFTs.

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arXiv2015

Argyres-Douglas Theories, the Macdonald Index, and an RG Inequality

We conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)_R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S^1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general N=2 superconformal field theories.

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arXiv2015

On the Superconformal Index of Argyres-Douglas Theories

We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres-Douglas (AD) superconformal field theories (SCFTs): the (A_1,A_{2n-3}) and the (A_1,D_{2n}) theories. While these SCFTs can be realized at special points on the Coulomb branch of certain N=2 gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the (A_1, A_{2n-3}) and (A_1, D_{2n}) indices by using a relation to two-dimensional q-deformed Yang-Mills theory and data from the class S construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the (A_1, A_3) and (A_1, D_4) SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S^1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S^1 reduction of our indices in a separate paper.

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