arXiv subjects

Christopher Beem

Publications and source records attributed to Christopher Beem.

At least 19 recordsLinked to original sources

arXiv2026

Towards a classification of graded unitary ${\mathcal W}_3$ algebras

We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible ${\mathcal W}_3$ vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $\mathfrak{R}$-filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the $(3,q+4)$ minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible $\mathfrak{sl}_3$ affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the $(A_2,A_q)$ Argyres--Douglas theories.

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arXiv2025

On the semi-infinite cohomology of graded-unitary vertex algebras

Recently, the first author with A. Ardehali, M. Lemos, and L. Rastelli introduced the notion of graded unitarity for vertex algebras. This generalization of unitarity is motivated by the SCFT/VOA correspondence and introduces a novel Hilbert space structure on the state space of a large class of vertex algebras that are not unitary in the conventional sense. In this paper, we study the relative semi-infinite cohomology of graded-unitary vertex algebras that admit a chiral quantum moment map for an affine current algebra at twice the critical level. We show that the relative semi-infinite chain complex for such a graded-unitary vertex algebra has a structure analogous to that of differential forms on a compact K\"ahler manifold, generalizing a strong form of the classic construction of Banks--Peskin and Frenkel--Garland--Zuckerman. We deduce that the relative semi-infinite cohomology is itself graded-unitary, which establishes graded unitarity for a large class of vertex operator algebras arising from three- and four-dimensional supersymmetric quantum field theories. We further establish an outer USp$(2)$ action on the semi-infinite cohomology (which does not respect cohomological grading), analogous to the Lefschetz $\mathfrak{sl}(2)$ in K\"ahler geometry. We also show that the semi-infinite chain complex is quasi-isomorphic as a differential graded vertex algebra to its cohomology, in analogy to the formality result of Deligne--Griffiths--Morgan--Sullivan for the de Rham cohomology of compact K\"ahler manifolds. We conclude by observing consequences of these results to the associated Poisson vertex algebras and related finite-type derived Poisson reductions.

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arXiv2025

Graded Unitarity in the SCFT/VOA Correspondence

Vertex algebras that arise from four-dimensional, $\mathcal{N}=2$ superconformal field theories inherit a collection of novel structural properties from their four-dimensional ancestors. Crucially, when the parent SCFT is unitary, the corresponding vertex algebra is not unitary in the conventional sense. In this paper, we motivate and define a generalized notion of unitarity for vertex algebras that we call \emph{graded unitarity}, and which captures the consequences of four-dimensional unitarity under this correspondence. We also take the first steps towards a classification program for graded-unitary vertex algebras whose underlying vertex algebras are Virasoro or affine Kac--Moody vertex algebras. Remarkably, under certain natural assumptions about the $\mathfrak{R}$-filtration for these vertex algebras, we show that only the $(2,p)$ central charges for Virasoro VOAs and boundary admissible levels for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ Kac--Moody vertex algebras can possibly be compatible with graded unitarity. These are precisely the cases of these vertex algebras that are known to arise from four dimensions.

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arXiv2024

Free field realizations for rank-one SCFTs

In this paper, we construct the associated vertex operator algebras for all $\mathcal{N}=2$ superconformal field theories of rank one. We give a uniform presentation through free-field realizations, which turns out to be a particularly suitable framework for this task. The elementary building blocks of the construction are dictated by the low energy degrees of freedom on the Higgs branch, which are well understood for rank-one theories. We further analyze the interplay between Higgs and Coulomb data on the moduli space of vacua, which tightly constrain the overall structure of the free field realizations. Our results suggest a plausible bottom-up classification scheme for low-rank SCFTs incorporating vertex algebra techniques.

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arXiv2023

Simplifying the Type $A$ Argyres-Douglas Landscape

A well-established organisational principle for Argyres--Douglas-type $\mathcal{N}=2$ superconformal field theories in four dimensions is to characterise such theories by the data defining a(n irregular) Hitchin system on $\mathbb{CP}^1$. The dictionary between Hitchin system data and various features of the corresponding SCFT has been studied extensively, but the overall structure of the resulting space of SCFTs still appears quite complicated. In this work, we systematically delineate a variety of simplifications that arise within this class of constructions due to several large classes of isomorphisms between SCFTs associated with inequivalent Hitchin system data (and their exactly marginal gaugings). We restrict to the most studied class of theories, namely the type $A$ theories without outer automorphism twists.

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arXiv2023

Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions

We study the boundary vertex algebras of $A$-twisted $\mathcal{N}=4$ Abelian gauge theories in three dimensions. These are identified with the BRST quotient (semi-infinite cohomology) of collections of symplectic bosons and free fermions that reflect the matter content of the corresponding gauge theory. We develop various free field realisations for these vertex algebras which we propose to interpret in terms of their localisation on their associated varieties. We derive the free field realisations by bosonising the elementary symplectic bosons and free fermions and then calculating the relevant semi-infinite cohomology, which can be done systematically. An interesting feature of our construction is that for certain preferred free field realisations, the outer automorphism symmetry of the vertex algebras in question (which are identified with the symmetries of the Coulomb branch in the infrared) are made manifest.

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arXiv2022

Free field realisation and the chiral universal centraliser

In the TQFT formalism of Moore-Tachikawa for describing Higgs branches of theories of class $\mathcal{S}$, the space associated to the unpunctured sphere in type $\mathfrak{g}$ is the universal centraliser $\mathfrak{Z}_G$, where $\mathfrak{g}=Lie(G)$. In more physical terms, this space arises as the Coulomb branch of pure $\mathcal{N}=4$ gauge theory in three dimensions with gauge group $\check{G}$, the Langlands dual. In the analogous formalism for describing chiral algebras of class $\mathcal{S}$, the vertex algebra associated to the sphere has been dubbed the \emph{chiral universal centraliser}. In this paper, we construct an open, symplectic embedding from a cover of the Kostant-Toda lattice of type $\mathfrak{g}$ to the universal centraliser of $G$, extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of $\mathfrak{Z}_G$, we propose a free field realisation of the chiral universal centraliser for any simple group $G$. We exploit this realisation to develop free field realisations of chiral algebras of class $\mathcal{S}$ of type $\mathfrak{a}_1$ for theories of genus zero with up to six punctures. These realisations make generalised $S$-duality completely manifest, and the generalisation to more than six punctures is conceptually clear, though technically burdensome.

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arXiv2022

Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing

The correspondence between four-dimensional $\mathcal{N}=2$ superconformal field theories and vertex operator algebras, when applied to theories of class $\mathcal{S}$, leads to a rich family of VOAs that have been given the monicker chiral algebras of class $\mathcal{S}$. A remarkably uniform construction of these vertex operator algebras has been put forward by Tomoyuki Arakawa in arXiv:1811.01577. The construction of arXiv:1811.01577 takes as input a choice of simple Lie algebra $\mathfrak{g}$, and applies equally well regardless of whether $\mathfrak{g}$ is simply laced or not. In the non-simply laced case, however, the resulting VOAs do not correspond in any clear way to known four-dimensional theories. On the other hand, the standard realisation of class $\mathcal S$ theories involving non-simply laced symmetry algebras requires the inclusion of outer automorphism twist lines, and this requires a further development of the approach of arXiv:1811.01577. In this paper, we give an account of those further developments and propose definitions of most chiral algebras of class $\mathcal S$ with outer automorphism twist lines. We show that our definition passes some consistency checks and point out some important open problems.

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arXiv2021

Schur Indices of Class $\mathcal{S}$ and Quasimodular Forms

We investigate the unflavoured Schur indices of class $\mathcal S$ theories of modest rank, and in the case of $\mathcal{N}=4$ super Yang-Mills theory with special unitary gauge group of somewhat more general rank, with an eye towards their modular properties. We find closed form expressions for many of these theories in terms of quasimodular forms of level one or two, with the curious feature that in general they are sums of quasimodular forms of different weights. For type $\mathfrak{a}_1$ theories, the index can be fixed by taking a simple Ansatz for the family of quasimodular forms appearing in the expansion of this type and demanding that the result be sufficiently regular as $q\to0$. For higher rank cases, an equally simple construction is lacking, but we nevertheless find that these indices can be expressed in terms of mixed-weight quasimodular forms.

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arXiv2021

A geometric free field realisation for the genus-two class $\mathcal{S}$ theory of type $\mathfrak{a}_1$

We present a free field realisation for the vertex operator algebra associated to the genus-two, class $\mathcal{S}$ superconformal field theory of type $\mathfrak{a}_1$. The free field realisation is in the style of recent work by the authors, and is formulated in terms of a one-dimensional isotropic lattice vertex algebra along with two pairs of symplectic fermions. Our realisation makes manifest an enhanced ${\rm USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an ${\rm SU(2)}$ outer automorphism that has been observed in recent work of Kiyoshige and Nishinaka and significantly simplifies the structure of the algebra. Along the way, we also produce a realisation of the generic subregular Drinfel'd-Sokolov $\mathcal{W}$ algebra of type $\mathcal{c}_2$ in terms of the generic principle $\mathcal{W}$ algebra of type $\mathfrak{c}_2$ and a one-dimensional isotropic lattice vertex algebra.

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arXiv2020

Argyres-Douglas Theories in Class S Without Irregularity

We make a preliminary investigation into twisted $A_{2n}$ theories of class S. Contrary to a common piece of folklore, we establish that theories of this type realise a variety of models of Argyres-Douglas type while utilising only regular punctures. We present an in-depth analysis of all twisted $A_2$ trinion theories, analyse their interrelations via partial Higgsing, and discuss some of their generalised S-dualities.

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arXiv2019

VOAs and rank-two instanton SCFTs

We analyze the N=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.

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arXiv2019

Free Field Realizations from the Higgs Branch

We present free field realizations for the associated vertex operator algebras of a number of four-dimensional $\mathcal{N}=2$ superconformal field theories. Our constructions utilize an exceptionally small set of chiral bosons whose number matches the complex dimensionality of the Higgs branch of the superconformal field theory. In the case of theories whose Higgs branches support additional degrees of freedom (free vector multiplets or decoupled interacting SCFTs), the corresponding "free field realizations" include additional ingredients: symplectic fermions in the case of vector multiplets and a $C_2$ co-finite VOA in the case of a residual interacting SCFT. The resulting picture is that the associated VOA can be constructed from the Higgs branch effective theory via free field realization. Our constructions also provide a natural realization of the $R$-filtration of the associated VOA.

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arXiv2018

Flavor symmetries and unitarity bounds in ${\mathcal N}=2$ SCFTs

In this letter I analyze the constraints imposed by unitarity on the flavor central charges of four-dimensional ${\mathcal N}=2$ SCFTs with general reductive global symmetry groups. I derive several general and far-reaching consequences of unitarity by computing the norms of flavor singlet Higgs branch operators appearing in the squares of "moment map" operators via the associated vertex operator algebra, and imposing the requirement that they be non-negative.

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arXiv2018

Secondary products in supersymmetric field theory

The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect to the primary product. We describe the mathematical structures involved and illustrate this general phenomenon in a range of physical examples arising from supersymmetric field theories in spacetime dimension two, three, and four. In the Rozansky-Witten twist of three-dimensional N=4 theories, this gives an intrinsic realization of the holomorphic symplectic structure of the moduli space of vacua. We further give a simple mathematical derivation of the assertion that introducing an Omega-background precisely deformation quantizes this structure. We then study the secondary product structure of extended operators, which subsumes that of local operators but is often much richer. We calculate interesting cases of secondary brackets of line operators in Rozansky-Witten theories and in four-dimensional N=4 super Yang-Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.

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arXiv2017

Vertex operator algebras, Higgs branches, and modular differential equations

Every four-dimensional ${\cal N}=2$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any ${\cal N}=2$ SCFT should obey a finite order modular differential equation. By way of the "high temperature" limit of the superconformal index, this allows the Weyl anomaly coefficient $a$ to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the "Deligne-Cvitanovi\'c exceptional series" of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class ${\cal S}$ theories, and ${\cal N}=4$ super Yang-Mills with $\mathfrak{su}(n)$ gauge group for small-to-moderate values of $n$.

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arXiv2016

More ${\mathcal N}=4$ superconformal bootstrap

In this long overdue second installment, we continue to develop the conformal bootstrap program for ${\mathcal N}=4$ superconformal field theories in four dimensions via an analysis of the correlation function of four stress-tensor supermultiplets. We review analytic results for this correlator and make contact with the SCFT/chiral algebra correspondence of arXiv:1312.5344. We demonstrate that the constraints of unitarity and crossing symmetry require the central charge $c$ to be greater than or equal to $3/4$ in any interacting ${\mathcal N}=4$ SCFT. We apply numerical bootstrap methods to derive upper bounds on scaling dimensions and OPE coefficients for several low-lying, unprotected operators as a function of the central charge. We interpret our bounds in the context of ${\mathcal N}=4$ super Yang-Mills (SYM) theories, formulating a series of conjectures regarding the embedding of the conformal manifold --- parametrized by the complexified gauge coupling --- into the space of scaling dimensions and OPE coefficients. Our conjectures assign a distinguished role to points on the conformal manifold that are self-dual under a subgroup of the S-duality group. This paper contains a more detailed exposition of a number of results previously reported in arXiv:1304.1803 in addition to new results.

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arXiv2016

Deformation quantization and superconformal symmetry in three dimensions

We investigate the structure of certain protected operator algebras that arise in three-dimensional N=4 superconformal field theories. We find that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. We identify several nontrivial conditions that the quantum algebra must satisfy in this basis. We consider examples of theories for which the moduli space of vacua is either the minimal nilpotent orbit of a simple Lie algebra or a Kleinian singularity. For minimal nilpotent orbits, the quantum algebras (and their preferred bases) can be uniquely determined. These algebras are related to higher spin algebras. For Kleinian singularities the algebras can be characterized abstractly - they are spherical subalgebras of symplectic reflection algebras - but the preferred basis is not easily determined. We find evidence in these examples that for a given choice of quantum algebra (defined up to a certain gauge equivalence), there is at most one choice of canonical basis. We conjecture that this is the case for general N=4 SCFTs.

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