arXiv subjects

Luca Cassia

Publications and source records attributed to Luca Cassia.

17 recordsLinked to original sources

arXiv2026

M2-brane partition functions and HD supergravity from equivariant volumes

We use the results of our companion paper arXiv:2502.20444, which explores the equivariant generalization of the constant-map contribution in topological string theory on toric Calabi--Yau manifolds $X$, to establish a holographic correspondence with M2-brane partition functions. We rigorously test this conjecture within the perturbative regime, incorporating all finite $N$ corrections on the field theory side. Our approach involves additional key details, such as incorporating effective 4d higher-derivative supergravity corrections to introduce refinement. A central result is the derivation of the Airy function representation for the squashed $S^3$ partition function of the field theory, for an arbitrary squashing parameter. We demonstrate that this Airy function structure is universal across all M2-brane models and provide a general expression in terms of the equivariant volume of $X$, incorporating the mesonic deformations corresponding to complexified masses. This expression is then evaluated explicitly for several examples, including ABJM theory, its flavored generalizations, circular quivers, and beyond, demonstrating agreement with the available field theory localization results. We extend the analysis to the superconformal and twisted indices of M2-brane models, and their spindle generalizations, leaving their full perturbative completion for future work. Finally, we explore avenues for generalizing these results to other brane systems, explicitly applying the idea to D3-branes.

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arXiv2025

Constant maps in equivariant topological strings and geometric modeling of fluxes

We study the equivariant generalization of topological strings on toric manifolds, focusing in particular on defining the contributions of constant maps in the genus expansion of the partition function. This approach regularizes the integration over non-compact Calabi-Yau spaces, producing finite results at each order in the expansion, as illustrated by a broad set of explicit examples. Our investigation highlights the geometric modeling of flux compactifications and clarifies the link between the effective supergravity framework and the equivariant topological string formalism, building on recent developments by Martelli and Zaffaroni. We conclude that the connection between topological string theory and supergravity/field theory involves switching between geometric moduli and fluxes, shedding light on the role of ensemble averages in string theory. We propose an exact non-perturbative holographic match with the corresponding M2-brane partition functions, which we test perturbatively at all orders in the gauge group rank $N$ in a companion paper. A special case of our proposal for vanishing flux reformulates the Ooguri-Strominger-Vafa conjecture within the equivariant topological string framework.

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arXiv2025

Superintegrability of $q,t$-matrix models and quantum toroidal algebra recursions

$q,t$-deformed matrix models give rise to representations of the deformed Virasoro algebra and more generally of the quantum toroidal $\mathfrak{gl}_1$ algebra. These representations are described in terms of finite difference equations that induce recursion relations for correlation functions. Under suitable assumptions, these recursions admit unique solutions expressible through "superintegrability" formulas, i.e. explicit closed formulas for averages of Macdonald polynomials. In this paper, we discuss examples arising from localization of 3d $\mathcal{N}=2$ theories, which include $q,t$-deformation of well known classical ensembles: Gaussian, Laguerre and Jacobi. We explain how relations in the quantum toroidal algebra can be used to give a new and universal proof of the known superintegrability formulas, as well as to derive new formulas for models that have not been previously studied in the literature. Finally, we make some remarks regarding the relation between superintegrability and orthogonal polynomials.

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arXiv2025

Generalized Macdonald functions and quantum toroidal gl(1) algebra

The Macdonald operator is known to coincide with a certain element of the quantum toroidal $\mathfrak{gl}(1)$ algebra in the Fock representation of levels $(1,0)$. A generalization of this operator to higher levels $(r,0)$ can be built using the coproduct structure, it is diagonalized by the generalized Macdonald symmetric functions, indexed by $r$-tuple partitions and depending on $r$ alphabets. In this paper, we extend to the generalized case some of the known formulas obeyed by ordinary Macdonald symmetric functions, such as the $e_1$-Pieri rule or the identity relating them to Whittaker vectors obtained by Garsia, Haiman, and Tesler. We also propose a generalization of the five-term relation, and the Fourier/Hopf pairing. In addition, we prove the factorized expression of the generalized Macdonald kernel conjectured previously by Zenkevich.

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arXiv2025

Symplectic cuts and open/closed strings II

In arXiv:2306.07329 we established a connection between symplectic cuts of Calabi-Yau threefolds and open topological strings, and used this to introduce an equivariant deformation of the disk potential of toric branes. In this paper we establish a connection to higher-dimensional Calabi-Yau geometries by showing that the equivariant disk potential arises as an equivariant period of certain Calabi-Yau fourfolds and fivefolds, which encode moduli spaces of one and two symplectic cuts (the maximal case) by a construction of Braverman arXiv:alg-geom/9712024. Extended Picard-Fuchs equations for toric branes, capturing dependence on both open and closed string moduli, are derived from a suitable limit of the equivariant quantum cohomology rings of the higher Calabi-Yau geometries.

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arXiv2025

Symplectic cuts and open/closed strings I

This paper introduces a concrete relation between genus zero closed Gromov-Witten invariants of Calabi-Yau threefolds and genus zero open Gromov-Witten invariants of a Lagrangian $A$-brane in the same threefold. Symplectic cutting is a natural operation that decomposes a symplectic manifold $(X,ω)$ with a Hamiltonian $U(1)$ action into two pieces glued along an invariant divisor. In this paper we study a quantum uplift of the cut construction defined in terms of equivariant gauged linear sigma models. The nexus between closed and open Gromov-Witten invariants is a quantum Lebesgue measure associated to a choice of cut, that we introduce and study. Integration of this measure recovers the equivariant quantum volume of the whole CY3, thereby encoding closed Gromov-Witten invariants. Conversely, the monodromies of the quantum measure around cycles in Kähler moduli space encode open Gromov-Witten invariants of a Lagrangian $A$-brane associated to the cut. Both in the closed and the open string sector we find a remarkable interplay between worldsheet instantons and semiclassical volumes regularized by equivariance. This leads to equivariant generating functions of GW invariants that extend smoothly across the entire moduli space, and which provide a unifying description of standard GW potentials. The latter are recovered in the non-equivariant limit in each of the different phases of the geometry.

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arXiv2024

$β$-ensembles and higher genera Catalan numbers

We propose formulas for the large $N$ expansion of the generating function of connected correlators of the $β$-deformed Gaussian and Wishart-Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of $β$ with $1/β$ and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit $β=1$, these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic $β$, we define the higher genus Catalan polynomials $C_{g,ν}(β)$ whose coefficients are integer numbers.

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arXiv2023

From equivariant volumes to equivariant periods

We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on $S^1$, $D^2$ and $D^2 \times S^1$, respectively. We define these objects and study their dependence on equivariant parameters for non-compact toric Kähler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the non-equivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov-Witten invariants of the target.

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arXiv2022

On refined Chern-Simons and refined ABJ matrix models

We consider the matrix model of $U(N)$ refined Chern-Simons theory on $S^3$ for the unknot. We derive a $q$-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as $q$-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent $q$-difference operator for a similar refinement of ABJ theory and show that the corresponding $q$-Virasoro constraints are equal to those of refined Chern-Simons for a gauge super-group $U(N|M)$. Our equations and solutions are manifestly symmetric under Langlands duality $q\leftrightarrow t^{-1}$ which correctly reproduces 3d Seiberg duality when $q$ is a specific root of unity.

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arXiv2021

Virasoro constraints revisited

We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction on the rank of the matrix model.

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arXiv2020

On matrix models and their $q$-deformations

Motivated by the BPS/CFT correspondence, we explore the similarities between the classical $β$-deformed Hermitean matrix model and the $q$-deformed matrix models associated to 3d $\mathcal{N}=2$ supersymmetric gauge theories on $D^2\times_{q}S^1$ and $S_b^3$ by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the $W$-algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results.

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arXiv2020

From Exact Results to Gauge Dynamics on $\mathbb{R}^3\times S^1$

We revisit the vacuum structure of the $\mathcal{N}=1$ Intriligator-Seiberg-Shenker model on $\mathbb{R}^3\times S^1$. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Ünsal, we argue that previously overlooked non-perturbative effects generate a Higgs-type potential on the classical Coulomb branch of the low-energy effective 3d $\mathcal{N}=2$ theory. In particular, on part of the Coulomb branch we encounter the first instance of a dynamically-generated quintic monopole superpotential.

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arXiv2019

Exact SUSY Wilson loops on $S^3$ from $q$-Virasoro constraints

Using the ideas from the BPS/CFT correspondence, we give an explicit recursive formula for computing supersymmetric Wilson loop averages in 3d $\mathcal{N}=2$ Yang-Mills-Chern-Simons $U(N)$ theory on the squashed sphere $S^3_b$ with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level $κ_2$ and Fayet-Illiopoulos parameter $κ_1$. For these values of $κ_1$ and $κ_2$ the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent $D^2 \times S^1$ theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

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arXiv2019

Branes, partition functions and quadratic monopole superpotentials

We obtain the brane setup describing 3d $\mathcal{N}=2$ dualities for $USp(2N_c)$ and $U(N_c)$ SQCD with monopole superpotentials. This classification follows from a complete analysis of affine and twisted affine compactifications from 4d. The analysis leads to a new duality for the unitary case that has been previously overlooked in the literature. We check this by matching of the three sphere partition function of the two sides of this new duality and find a perfect agreement. Furthermore we use the partition function to predict new 3d $\mathcal{N}=2$ dualities for SQCD with monopole superpotentials and tensorial matter.

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arXiv2018

$USp(2N_c)$ SQCD$_3$ with antisymmetric: dualities and symmetry enhancements

We study various aspects of the 4d/3d reduction of $\mathcal{N}=1$ dualities involving $USp(2N_c)$ gauge theories with $2N_f$ fundamentals and one antisymmetric. We discuss the non-trivial role played by the monopole superpotentials in the reduction and obtain new 3d dualities for models with both symplectic and unitary gauge groups. For $N_f=4$ we observe interesting webs of dualities and symmetry enhancements, recovering and extending some results already appeared in the literature.

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arXiv2017

c-extremization from toric geometry

We derive a geometric formulation of the 2d central charge $c_r$ from infinite families of 4d $\mathcal{N}=1$ superconformal field theories topologically twisted on constant curvature Riemann surfaces. They correspond to toric quiver gauge theories and are associated to D3 branes probing five dimensional Sasaki-Einstein geometries in the AdS/CFT correspondence. We show that $c_r$ can be expressed in terms of the areas of the toric diagram describing the moduli space of the 4d theory, both for toric geometries with smooth and singular horizons. We also study the relation between a-maximization in 4d and c-extremization in 2d, giving further evidences of the mixing of the baryonic symmetries with the exact R-current in two dimensions.

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arXiv2017

Surveying 4d SCFTs twisted on Riemann surfaces

Within the framework of four dimensional conformal supergravity we consider $\mathcal{N}=1,2,3,4$ supersymmetric theories generally twisted along the abelian subgroups of the R-symmetry and possibly other global symmetry groups. Upon compactification on constant curvature Riemann surfaces with arbitrary genus we provide an extensive classification of the resulting two dimensional theories according to the amount of supersymmetry that is preserved. Exploiting the c-extremization prescription introduced in arXiv:1211.4030 we develop a general procedure to obtain the central charge for 2d $\mathcal{N}=(0,2)$ theories and the expression of the corresponding R-current in terms of the original 4d one and its mixing with the other abelian global currents.

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