arXiv subjects

Niklas Garner

Publications and source records attributed to Niklas Garner.

At least 19 recordsLinked to original sources

arXiv2026

W-algebras of the Deligne-Cvitanovi\'{c} Exceptional series and the minimal 3d ${\mathcal N}=4$ SCFT

We propose a three-dimensional field theory construction that realizes the vertex algebras associated with the intermediate Lie algebras and the related $C_2$-cofinite minimal $W$-algebras of the Deligne-Cvitanovi\'c (DC) series as boundary algebras. The construction is based on the minimal three-dimensional ${\mathcal N}=4$ superconformal field theory coupled to a topological field theory. For a Neumann-type boundary condition compatible with the topological $A$-twist, the algebra of boundary local operators realizes the minimal $W$-algebra $W_{-h^\vee/6}(\mathfrak{g},f_{\text{min}})$. While this boundary condition is not deformable to the $B$-twist, we argue that a holomorphic-topological ($HT^B$) twist instead realizes the level-one affine algebras of the intermediate Lie algebras, providing a uniform three-dimensional origin for these vertex algebra structures.

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arXiv2025

Unravelling the Holomorphic Twist II: Anomalies and Extended Supersymmetry

Twists of four-dimensional supersymmetric quantum field theories (SQFTs) isolate protected sectors with rich algebraic structures. We develop a unified framework for analyzing symmetries and anomalies in four-dimensional holomorphically twisted SQFTs, combiningcomplex-geometric and algebraic perspectives. This approach clarifies the connections between existing formulations in the literature and resolves several open questions left unanswered in the first installment of this series. We place particular emphasis on theories with extended supersymmetry, where the holomorphic twist gives rise to enhanced algebraic and geometric structures. We explain how these features emerge and govern the organization of the twisted theory. Furthermore, we demonstrate how a superconformal deformation of the twisted theory naturally leads to the associated vertex operator algebra, clarifying how the higher algebraic structures of the holomorphically twisted theory give rise to vertex algebras structures.

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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

A Conjecture of Warnaar-Zudilin from Deformations of Lie Superalgebras

We prove a collection of $q$-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra $L_k(\mathfrak{osp}_{1|2n})$ into the principal subsuperspace of $L_k(\mathfrak{sl}_{1|2n+1})$ in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers $N$, $k$ the character of the principal subspace of type $A_N$ at level $k$ can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.

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arXiv2024

Scattering off of Twistorial Line Defects

The recently devised chiral algebra bootstrap computes the form factors of a special class of ``twistorial'' 4d QFTs as correlation functions of the theory's 2d celestial chiral algebra. Examples of twistorial theories include self-dual Yang-Mills theory coupled to special massless matter content, and certain form factors in these theories are equivalent to a subset of MHV amplitudes in massless QCD, coupled to the same matter. In this paper, we extend the chiral algebra bootstrap to include scattering in the presence of charged sources, using a self-dual dyon in a twistorial theory as our main example. Self-dual theories in the presence of such sources lift to holomorphic gauge theories on non-Hausdorff twistor space, and we generalize the Koszul duality construction of Costello and Paquette to this setting. With this approach, we easily reproduce a recent formula of Adamo, Bogna, Mason, and Sharma for $n$-point MHV scattering of gluons off the self-dual dyon.

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arXiv2024

Mirror Symmetry and Level-rank Duality for 3d $\mathcal{N} = 4$ Rank 0 SCFTs

We introduce a family of 3d $\mathcal{N} = 4$ superconformal field theories that have zero-dimensional Coulomb and Higgs branches and propose that the rational vertex operator algebras $W^{\text{min}}_{k - \scriptstyle{\frac{1}{2}}}(\mathfrak{sp}_{2N})$ and $L_{k}(\mathfrak{osp}_{1|2N})$ model the modular tensor categories of line operators in their topological $A$ and $B$ twists, respectively. Our analysis indicates that the action of 3d mirror symmetry on this family of theories is related to a novel level-rank duality and leads to several conjectural $q$-series identities of independent interest.

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arXiv2024

B-twisted Gaiotto-Witten theory and topological quantum field theory

We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto-Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern-Simons-Rozansky-Witten theories and supergroup Chern-Simons theories and include as particular examples global forms of $\mathfrak{gl}(1 \vert 1)$-Chern-Simons theory and toral Chern-Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory.

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arXiv2023

Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs

We initiate the study of boundary Vertex Operator Algebras (VOAs) of topologically twisted 3d $\mathcal{N}=4$ rank-0 SCFTs. This is a recently introduced class of $\mathcal{N}=4$ SCFTs that by definition have zero-dimensional Higgs and Coulomb branches. We briefly explain why it is reasonable to obtain rational VOAs at the boundary of their topological twists. When a rank-0 SCFT is realized as the IR fixed point of a $\mathcal{N}=2$ Lagrangian theory, we propose a technique for the explicit construction of its topological twists and boundary VOAs based on deformations of the holomorphic-topological twist of the $\mathcal{N}=2$ microscopic description. We apply this technique to the $B$ twist of a newly discovered family of 3d $\mathcal{N}=4$ rank-0 SCFTs ${\mathcal T}_r$ and argue that they admit the simple affine VOAs $L_r(\mathfrak{osp}(1|2))$ at their boundary. In the simplest case, this leads to a novel level-rank duality between $L_1(\mathfrak{osp}(1|2))$ and the minimal model $M(2,5)$. As an aside, we present a TQFT obtained by twisting a 3d $\mathcal{N}=2$ QFT that admits the $M(3,4)$ minimal model as a boundary VOA and briefly comment on the classical freeness of VOAs at the boundary of 3d TQFTs.

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arXiv2023

Enhanced symmetries in minimally-twisted three-dimensional supersymmetric theories

We show that the action of residual supersymmetries in holomorphic-topological twists of $N = 2$ theories in three dimensions naturally extends to the action of certain infinite dimensional Lie superalgebras. We demonstrate this in a range of examples, including $N = 4$ Yang-Mills theories and superconformal Chern-Simons theories, describing how the symmetries are implemented at the level of local operators.

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arXiv2023

Higgs and coulomb branches from superconformal raviolo vertex algebras

We propose a method for extracting the Higgs and Coulomb branches of a three-dimensional N = 4 quantum field theory from the algebra of local operators in its holomorphic-topological twist using the formalism of raviolo vertex algebras. Our construction parallels that of the chiral ring and twisted chiral ring of an N = 2 superconformal vertex operator algebra.

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arXiv2023

Raviolo vertex algebras

We develop an algebraic structure modeling local operators in a three-dimensional quantum field theory which is partially holomorphic and partially topological. The geometric space organizing our algebraic structure is called the raviolo (or bubble) and replaces the punctured disk underlying vertex algebras; we refer to this structure as a raviolo vertex algebra. The raviolo has appeared in many contexts related to three-dimensional supersymmetric gauge theory, especially in work on the affine Grassmannian. We prove a number of structure theorems for raviolo vertex algebras and provide simple examples that share many similarities with their vertex algebra counterparts.

math.QA↗

arXiv2023

Twistorial monopoles & chiral algebras

We initiate the study of how the insertion of magnetically charged states in 4d self-dual gauge theories impacts the 2d chiral algebras supported on the celestial sphere at asymptotic null infinity, from the point of view of the 4d/2d twistorial correspondence introduced by Costello and the second author. By reducing the 6d twistorial theory to a 3d holomorphic-topological theory with suitable boundary conditions, we can motivate certain non-perturbative enhancements of the celestial chiral algebra corresponding to extensions by modules arising from 3d boundary monopole operators. We also identify the insertion of 4d (non-abelian) monopoles with families of spectral flow automorphisms of the celestial chiral algebra.

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arXiv2023

Line Operators in $U(1|1)$ Chern-Simons Theory

We analyze the non-semisimple category of line operators in Chern-Simons gauge theories based off the Lie superalgebra $\mathfrak{gl}(1|1)$. Our proposal is that the category of line operators $\mathcal{C}$ can be identified with the derived category of modules for a boundary vertex operator algebra $\mathcal{V}$ realized as a certain infinite-order simple current extension of the affine current algebra $V(\mathfrak{gl}(1|1))$ by boundary monopole operators. By translating this simple current extension of $V(\mathfrak{gl}(1|1))$ to the unrolled, restricted quantum group $\overline{U}^E(\fgl(1|1))$, we show that our category of line operators admits a second description in terms of a quasi-quantum group $\mathcal{A}$ realized by uprolling. We also compare our results across an expected physical duality with the cyclic orbifold of a free, $B$-twisted hypermultiplet and find a slight discrepancy at the level of braiding and associator. We end with a detailed analysis of coupling to background flat $GL(1, \C)$ connections and the resulting category of non-genuine line operators.

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arXiv2022

Vertex Operator Algebras and Topologically Twisted Chern-Simons-Matter Theories

We consider several topologically twisted Chern-Simons-matter theories and propose boundary VOAs whose module categories should model the category of line operators of the 3d bulk. Our main examples come from the topological $A$ and $B$ twists of the exotic $\mathcal{N}=4$ Chern-Simons-matter theories of Gaiotto-Witten, but we show that there is a topological "$A$-twist" for a much larger class of $\mathcal{N}\neq4$ theories. We illustrate a particular example of this new class of theories that admits the $p=2$ singlet VOA $\mathfrak{M}(2)$ on its boundary and comment on its relation to the $\psi \to \infty$ limit of the Gaiotto-Rap{\v c}{\'a}k corner VOA $Y_{1,1,0}[\psi]$.

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arXiv2022

Twisted Formalism for 3d $\mathcal{N}=4$ Theories

We describe the topological $A$ and $B$ twists of 3d $\mathcal{N}=4$ theories of hypermultiplets gauged by $\mathcal{N}=4$ vector multiplets as certain deformations of the holomorphic-topological ($HT$) twist of those theories, utilizing the twisted superfields of Aganagic-Costello-Vafa-McNamara describing $HT$-twisted 3d $\mathcal{N}=2$ theories. We rederive many known results from this perspective, including state spaces on Riemann surfaces, deformations induced by flavor symmetries, the boundary VOAs of Costello-Gaiotto, and the category of line operators as proposed by Costello-Dimofte-Gaiotto-Hilburn-Yoo. Along the way, we show how the secondary product of local operators in the holomorphic-topological twist is related to the secondary product in the fully topological twist.

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arXiv2022

TASI Lectures on the Mathematics of String Dualities

In these lecture proceedings, we describe some of the fundamental mathematical concepts that underlie supersymmetric string theory and field theory, and their role in describing and testing dualities. In particular, we provide a pedagogical introduction to topological and holomorphic twisting, descent, and higher algebraic structures. Our primary examples are worldsheet theories of topological strings, namely the A- and B-models, which we briefly review. These proceedings are based on lectures given by the second author at TASI 2021.

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arXiv2021

A QFT for non-semisimple TQFT

We construct a family of 3d quantum field theories $\mathcal T_{n,k}^A$ that conjecturally provide a physical realization -- and derived generalization -- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(i\pi/k)$. The theories $\mathcal T_{n,k}^A$ are defined as topological twists of certain 3d $\mathcal N=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted $\mathcal N=4$ matter sector (the source of non-semisimplicity). We show that $\mathcal T_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\mathfrak{sl}_n$-type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in $\mathcal T_{n,k}^A$ to the derived category of modules for a boundary Feigin-Tipunin algebra, and -- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ -- to the derived category of $U_q(\mathfrak{sl}_n)$ modules. We analyze many other key features of $\mathcal T_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PSL(n,\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.

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arXiv2020

Generalized affine Springer theory and Hilbert schemes on planar curves

We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp-Mellit, we also show the rational Cherednik algebra acts on the HOMFLY homologies of torus knots. This work was inspired in part by a construction in three-dimensional $\mathcal{N}=4$ gauge theory.

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