arXiv subjects

Andrea E. V. Ferrari

Publications and source records attributed to Andrea E. V. Ferrari.

At least 19 recordsLinked to original sources

arXiv2026

Vertex Superalgebras for Hypertoric Varieties and 3d Abelian Gauge Theories

Hypertoric (or toric hyperkähler) varieties are a class of symplectic singularities and their resolutions, obtained as Hamiltonian reductions of a symplectic vector space acted on by a torus. In physics, they appear as Higgs (and Coulomb) branches of 3d $\mathcal{N}=4$ supersymmetric quantum field theories with abelian gauge group. In this work, we construct an $\hbar$-adic (in the sense of microlocalisation) sheaf of vertex operator superalgebras over a given smooth hypertoric variety. Its global sections give the $A$-twisted boundary of the corresponding 3d gauge theory. We use this to prove that the associated affine variety of this hypertoric vertex operator superalgebra recovers the singular hypertoric variety. This proves the 3d Higgs branch conjecture for a large class of boundary vertex operator superalgebras. In particular, these vertex operator superalgebras are quasi-lisse. This is in contrast to the (purely even) hypertoric vertex operator superalgebras (and their $\hbar$-adic localisations) constructed previously by Kuwabara as global sections of sheaves on families of universal Poisson deformations of the hypertoric varieties. These are generally not quasi-lisse. We show that the vertex operator superalgebras defined in this paper are (fermionic) simple-current extensions of those defined by Kuwabara, and investigate the consequences for symplectic duality and characters. We observe that the latter are upgraded from partial (or false) theta functions to quasimodular forms.

math.QA↗

arXiv2025

Remarks on Associated Varieties and Minimal Tension Holography

We comment on certain analogies that have recently emerged between path integral localisation phenomena in minimal tension string theory on $\mathrm{AdS}_3\times S^3\times T^4$ and associated varieties of boundary vertex algebras of 3d $\mathcal{N}=4$ theories. We give evidence for the fact that the path integral localisation relies on an intriguing relation between the conformal boundary of $\mathrm{AdS}_3$ and the associated variety of $V_1(\mathfrak{psl}(2|2))$, (the complexification of) the affine algebra that enters the world-sheet description of the string theory. We explain the origin of certain operators (related to ``secret representations") that have appeared in this description. Based on the expectation that similar statements will hold for $\mathrm{AdS}_5\times S^5$, we explicitly write down free field realisations of $V_1(\mathfrak{psl}(4|4))$ that may hopefully be useful to study the world-sheet dual of free $\mathcal{N}=4$ SYM.

hep-th↗

arXiv2025

Difference Equations: from Berry Connections to the Coulomb Branch

In recent work, we demonstrated that a spectral variety for the Berry connection of a 2d $\mathcal{N}=(2,2)$ GLSM with Kähler vacuum moduli space $X$ and abelian flavour symmetry is the support of a sheaf induced by a certain action on the equivariant quantum cohomology of $X$. This action could be quantised to first-order matrix difference equations obeyed by brane amplitudes, and by taking the conformal limit, vortex partition functions. In this article, we elucidate how some of these results may be recovered from a 3d perspective, by placing the 2d theory at a boundary and gauging the flavour symmetry via a bulk A-twisted 3d $\mathcal{N}=4$ gauge theory (a sandwich construction). We interpret the above action as that of the bulk Coulomb branch algebra on boundary twisted chiral operators. This relates our work to recent constructions of actions of Coulomb branch algebras on quantum equivariant cohomology, providing a novel correspondence between these actions and spectral data of generalised periodic monopoles. The effective IR description of the 2d theory in terms of a twisted superpotential allows for explicit computations of these actions, which we demonstrate for abelian GLSMs.

hep-th↗

arXiv2025

Berry Connections for 2d $(2,2)$ Theories, Monopole Spectral Data & (Generalised) Cohomology Theories

We study Berry connections for supersymmetric ground states of 2d $\mathcal{N}=(2,2)$ GLSMs quantised on a circle, which are generalised periodic monopoles. Periodic monopole solutions may be encoded into difference modules, as shown by Mochizuki, or into an alternative algebraic construction given in terms of vector bundles endowed with filtrations. By studying the ground states in terms of a one-parameter family of supercharges, we relate these two different kinds of spectral data to the physics of the GLSMs. From the difference modules we derive novel difference equations for brane amplitudes, which in the conformal limit yield novel difference equations for hemisphere or vortex partition functions. When the GLSM flows to a nonlinear sigma model with Kähler target $X$, we show that the two kinds of spectral data are related to different (generalised) cohomology theories: the difference modules are related to the equivariant quantum cohomology of $X$, whereas the vector bundles with filtrations are related to its equivariant K-theory.

hep-th↗

arXiv2025

On Spectral Data for $(2,2)$ Berry Connections, Difference Equations & Equivariant Quantum Cohomology

We study supersymmetric Berry connections of 2d $\mathcal{N}=(2,2)$ gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin-Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arise naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a quantization of the Cherkis-Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with Kähler target $X$, we show that the difference modules are related to the equivariant quantum cohomology of $X$.

hep-th↗

arXiv2025

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.

hep-th↗

arXiv2024

3d $\mathcal{N}=4$ Mirror Symmetry, TQFTs, and 't Hooft Anomaly Matching

Any local unitary 3d $\mathcal{N}=4$ superconformal field theory (SCFT) has a corresponding "universal" relevant deformation that takes it to a gapped phase. This deformation preserves all continuous internal symmetries, $\mathcal{S}$, and therefore also preserves any 't Hooft anomalies supported purely in $\mathcal{S}$. We describe the resulting phase diagram in the case of SCFTs that arise as the endpoints of renormalization group flows from 3d $\mathcal{N}=4$ Abelian gauge theories with any number of $U(1)$ gauge group factors and arbitrary integer charges for the matter fields. We argue that the universal deformations take these QFTs to Abelian fractional quantum Hall states in the infrared (IR), and we explain how to match 't Hooft anomalies between the non-topological ultraviolet theories and the IR topological quantum field theories (TQFTs). Along the way, we give a proof that 3d $\mathcal{N}=4$ mirror symmetry of our Abelian gauge theories descends to a duality of these TQFTs. Finally, using our anomaly matching discussion, we describe how to connect, via the renormalization group, abstract local unitary 3d $\mathcal{N}=4$ SCFTs with certain 't Hooft anomalies for their internal symmetries to IR phases (partially) described by Abelian spin Chern-Simons theories.

hep-th↗

arXiv2024

Machine learning and optimization-based approaches to duality in statistical physics

The notion of duality -- that a given physical system can have two different mathematical descriptions -- is a key idea in modern theoretical physics. Establishing a duality in lattice statistical mechanics models requires the construction of a dual Hamiltonian and a map from the original to the dual observables. By using simple neural networks to parameterize these maps and introducing a loss function that penalises the difference between correlation functions in original and dual models, we formulate the process of duality discovery as an optimization problem. We numerically solve this problem and show that our framework can rediscover the celebrated Kramers-Wannier duality for the 2d Ising model, reconstructing the known mapping of temperatures. We also discuss an alternative approach which uses known features of the mapping of topological lines to reduce the problem to optimizing the couplings in a dual Hamiltonian, and explore next-to-nearest neighbour deformations of the 2d Ising duality. We discuss future directions and prospects for discovering new dualities within this framework.

cond-mat.stat-mech↗

arXiv2024

$L_1(\mathfrak{psl}_{n|n})$ from BRST reductions, associated varieties and nilpotent orbits

We verify a conjecture of Beem and the first author stating that a certain family of physically motivated BRST reductions of beta-gamma systems and free fermions is isomorphic to $L_1(\mathfrak{psl}_{n|n})$, and that its associated variety is isomorphic as a Poisson variety to the minimal nilpotent orbit closure $\overline{\mathbb{O}_{\mathrm{min}}(\mathfrak{sl}_n)}$. This shows in particular that $L_1(\mathfrak{psl}_{n|n})$ is quasi-lisse. Combining this with other results in the literature (in particular work of Ballin et al.), this paper provides a concrete and important example of how one can extract two symplectic dual varieties from a rather well-known vertex operator algebra.

math.QA↗

arXiv2024

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.

hep-th↗

arXiv2024

Anomalies of Generalized Symmetries from Solitonic Defects

We propose the general idea that 't Hooft anomalies of generalized global symmetries can be understood in terms of the properties of solitonic defects, which generically are non-topological defects. The defining property of such defects is that they act as sources for background fields of generalized symmetries. 't Hooft anomalies arise when solitonic defects are charged under these generalized symmetries. We illustrate this idea for several kinds of anomalies in various spacetime dimensions. A systematic exploration is performed in 3d for 0-form, 1-form, and 2-group symmetries, whose 't Hooft anomalies are related to two special types of solitonic defects, namely vortex line defects and monopole operators. This analysis is supplemented with detailed computations of such anomalies in a large class of 3d gauge theories. Central to this computation is the determination of the gauge and 0-form charges of a variety of monopole operators: these involve standard gauge monopole operators, but also fractional gauge monopole operators, as well as monopole operators for 0-form symmetries. The charges of these monopole operators mainly receive contributions from Chern-Simons terms and fermions in the matter content. Along the way, we interpret the vanishing of the global gauge and ABJ anomalies, which are anomalies not captured by local anomaly polynomials, as the requirement that gauge monopole operators and mixed monopole operators for 0-form and gauge symmetries have non-fractional integer charges.

hep-th↗

arXiv2024

Generalized Symmetries and Anomalies of 3d N=4 SCFTs

We study generalized global symmetries and their 't Hooft anomalies in 3d N=4 superconformal field theories (SCFTs). Following some general considerations, we focus on good quiver gauge theories, comprised of balanced unitary nodes and unbalanced unitary and special unitary nodes. While the global form of the Higgs branch symmetry group may be determined from the UV Lagrangian, the global form of Coulomb branch symmetry groups and associated mixed 't Hooft anomalies are more subtle due to potential symmetry enhancement in the IR. We describe how Coulomb branch symmetry groups and their mixed 't Hooft anomalies can be deduced from the UV Lagrangian by studying center charges of various types of monopole operators, providing a concrete and unambiguous way to implement 't Hooft anomaly matching. The final expression for the symmetry group and 't Hooft anomalies has a concise form that can be easily read off from the quiver data, specifically from the positions of the unbalanced and flavor nodes with respect to the positions of the balanced nodes. We provide consistency checks by applying our method to compute symmetry groups of 3d N=4 theories corresponding to magnetic quivers of 4d Class S theories and 5d SCFTs. We are able to match these results against the flavor symmetry groups of the 4d and 5d theories computed using independent methods. Another strong consistency check is provided by comparing symmetry groups and anomalies of two theories related by 3d mirror symmetry.

hep-th↗

arXiv2023

Non-invertible Symmetries and Higher Representation Theory II

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra $\mathfrak{so}(N)$, and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed 't Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

hep-th↗

arXiv2023

The Twisted Index and Topological Saddles

The twisted index of 3d $\mathcal{N}=2$ gauge theories on $S^1 \times Σ$ has an algebro-geometric interpretation as the Witten index of an effective supersymmetric quantum mechanics. In this paper, we consider the contributions to the supersymmetric quantum mechanics from topological saddle points in supersymmetric localisation of abelian gauge theories. Topological saddles are configurations where the matter fields vanish and the gauge symmetry is unbroken, which exist for non-vanishing effective Chern-Simons levels. We compute the contributions to the twisted index from both topological and vortex-like saddles points and show that their combination recovers the Jeffrey-Kirwan residue prescription for the twisted index and its wall-crossing.

hep-th↗

arXiv2023

Non-invertible Symmetries and Higher Representation Theory I

The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.

hep-th↗

arXiv2023

Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg Algebras

We consider 3d $\mathcal{N} = 4$ theories on the geometry $Σ\times\mathbb{R}$, where $Σ$ is a closed and connected Riemann surface, from the point of view of a quantum mechanics on $\mathbb{R}$. Focussing on the elementary mirror pair in the presence of real deformation parameters, namely SQED with one hypermultiplet (SQED[1]) and the free hypermulitplet, we study the algebras of local operators in the respective quantum mechanics as well as their action on the vector space of supersymmetric ground states. We demonstrate that the algebras can be described in terms of Heisenberg algebras, and that they act in a way reminiscent of Segal-Bargmann (B-twist of the free hypermultiplet) and Nakajima (A-twist of SQED[1]) operators.

hep-th↗

arXiv2022

The 3d Twisted Index and Wall-Crossing

We study the twisted index of 3d $\mathcal{N}=2$ supersymmetric gauge theories on $S^1 \times Σ$ in the presence of a real FI parameter deformation. This parameter induces a 1d FI parameter for the effective supersymmetric quantum mechanics on $S^1$. Using supersymmetric localisation, the twisted index can be expressed as a contour integral. We show that the contour prescription is modified in the presence of the 1d FI parameter, leading to wall-crossing phenomena for the twisted index. In particular, we derive a general wall-crossing formula for abelian gauge theories. We also examine the origin of wall-crossing as change of stability condition in the algebro-geometric interpretation of the twisted index. These ideas are illustrated for abelian theories with $\mathcal{N}=4$ supersymmetry and in a non-abelian example that reproduces wall-crossing phenomena associated to moduli spaces of stable pairs.

hep-th↗

arXiv2021

Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface

This paper studies supersymmetric ground states of 3d $\mathcal{N}=4$ supersymmetric gauge theories on a Riemann surface of genus $g$. There are two distinct spaces of supersymmetric ground states arising from the $A$ and $B$ type twists on the Riemann surface, which lead to effective supersymmetric quantum mechanics with four supercharges and supermultiplets of type $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ respectively. We compute the space of supersymmetric ground states in each case, graded by flavour and R-symmetries and in different chambers for real mass and FI parameters, for a large class of supersymmetric gauge theories. The results are formulated geometrically in terms of the Higgs branch geometry. We perform extensive checks of compatibility with the twisted index and mirror symmetry.

hep-th↗