arXiv subjects

Leonardo Santilli

Publications and source records attributed to Leonardo Santilli.

At least 19 recordsLinked to original sources

arXiv2026

Superconformal index of large N and long quivers

We develop a universal framework for computing generating functions in quiver gauge theories in the large $N$ limit. We apply this method to the superconformal index of 4d $\mathcal{N}=2$ gauge theories, as well as to the 0-instanton index of 5d $\mathcal{N}=1$ theories, both for finite quivers of arbitrary shape and in the long quiver limit, including both special unitary and orthosymplectic gauge groups, demonstrating its versatility. In the Hall--Littlewood limit, we show that the series expansion of the resulting index captures the expected Higgs branch chiral ring generators. Our exact expressions in the long quiver limit provide new predictions for the supergravity index of the holographically dual warped AdS$_5$ and AdS$_6$ backgrounds.

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arXiv2026

Twisted traces and quantization of moduli stacks of 3d $\mathcal{N}=4$ Chern-Simons-matter theories

We conjecture, and show in a plethora of examples, that the sphere partition function of 3d $\mathcal{N}=4$ Chern-Simons-matter theories equals a sum of twisted traces on tensor products of Verma modules over the quantization of the moduli spaces of vacua. This extends a conjecture of Gaiotto-Okazaki to Chern-Simons-matter theories. We also show that the partition function of every Abelian gauge theory with higher charges has such twisted trace decomposition, and uncover new Abelian dualities between theories with and without Chern-Simons couplings.

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arXiv2025

Non-commutative resolutions and pre-quotients of Calabi-Yau double covers

Following an earlier proposal arXiv:2307.02038 to apply the GLSM formalism to understand the so-called non-commutative resolution, this paper takes one important step further to extend this formalism to a much larger class of non-commutative resolutions. The proposal was initially motivated by the discovery of a new class of mirror pairs singular Calabi-Yau varieties arXiv:2003.07148, given by certain branched double covers over toric varieties of MPCP type. The overarching problem was to understand these mirror pairs from the viewpoint of homological mirror symmetry arXiv:alg-geom/9411018. In the present paper, we propose two main results along this line. First, one new insight is that the `gauge-fixing' condition on the branching locus of the double cover used in arXiv:2003.07148 can be relaxed in an interesting way. This turns out to produce GLSMs that describe a much larger class of non-commutative resolutions, leading to $A$-periods for a larger class of non-commutative resolutions, as well as the GKZ systems for their $A$-periods. Second, we show that the $A$-periods can also be realized as $A$-periods of a certain smooth CICY family in a toric variety of MPCP type, such that a suitable finite quotient of this family recovers the double cover CY we have started with. We call this CICY family the `pre-quotient' of the double cover CY. This realization strongly suggests that pre-quotient may provide an important approach for understanding homological mirror symmetry for singular double cover CY varieties and non-commutative resolutions.

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arXiv2025

Large $N$ limits of supersymmetric quantum field theories: A pedagogical overview

The different large $N$ limits of supersymmetric quantum field theories in three, four, and five dimensions are reviewed. We distinguish between the planar limit of SQCD theories, the M-theory limit suited in three and five dimensions, and the long quiver limit. The method to solve exactly the sphere partition functions in each type of limit is spelled out in a pedagogical way. After a comprehensive general treatment of the saddle point approximation in the large $N$ limit, we present an extensive list of examples and detail the calculations. The scope of this overview is to provide an entry-level, computation-oriented understanding of the techniques featured in the field theory side of the AdS/CFT correspondence.

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arXiv2024

(-1)-form symmetries from M-theory and SymTFTs

We explore $(-1)$-form symmetries within the framework of geometric engineering in M-theory. By constructing the Symmetry Topological Field Theory (SymTFT) for selected 5d $\mathcal{N}=1$, 4d $\mathcal{N}=2$ and 4d $\mathcal{N}=1$ theories, we formalize the geometric origin of these symmetries and compute the mixed anomaly polynomials involving $(-1)$-form and higher-form symmetries. Our findings consistently reveal both discrete and continuous $(-1)$-form symmetries, aligning with established field theory results, while also uncovering new $(-1)$-form symmetry factors and structural insights. In particular, we study the SymTFT of 4d $\mathcal{N}=1$ theories from M-theory on a class of spaces with $G_2$ holonomy, and obtain properties such as modified instanton sums and 4-group structures observed in other 4d gauge theories. Additionally, we systematically construct symmetry operators for continuous abelian symmetries, refining existing proposals, and providing an M-theory origin for them.

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arXiv2024

Hawking-Page transition on a spin chain

The accessibility of the Hawking-Page transition in AdS$_5$ through a 1d Heisenberg spin chain is demonstrated. We use the random matrix formulation of the Loschmidt echo for a set of spin chains, and randomize the ferromagnetic spin interaction. It is shown that the thermal Loschmidt echo, when averaged, detects the predicted increase in entropy across the Hawking-Page transition. This suggests that a 1d spin chain exhibits characteristics of black hole physics in 4+1 dimensions. We show that this approach is equally applicable to free fermion systems with a general dispersion relation.

quant-ph↗

arXiv2024

Higher form symmetries and orbifolds of two-dimensional Yang-Mills theory

We undertake a detailed study of the gaugings of two-dimensional Yang-Mills theory by its intrinsic charge conjugation 0-form and centre 1-form global symmetries, elucidating their higher algebraic and geometric structures, as well as the meaning of dual lower form symmetries. Our derivations of orbifold gauge theories make use of a combination of standard continuum path integral methods, networks of topological defects, and techniques from higher gauge theory. We provide a unified description of higher and lower form gauge fields for a $p$-form symmetry in the geometric setting of $p$-gerbes, and derive reverse orbifolds by the dual $(-1)$-form symmetries. We identify those orbifolds in which charge conjugation symmetry is spontaneously broken, and relate the breaking to mixed anomalies involving $(-1)$-form symmetries. We extend these considerations to gaugings by the non-invertible 1-form symmetries of two-dimensional Yang-Mills theory by introducing a notion of generalized $θ$-angle.

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arXiv2024

Dynamical quantum phase transitions from random matrix theory

We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transition persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.

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arXiv2024

Explicit large $N$ von Neumann algebras from matrix models

We construct a large family of quantum mechanical systems that give rise to an emergent type III$_1$ von Neumann algebra in the large $N$ limit. Their partition functions are matrix integrals that appear in the study of various gauge theories. We calculate the real-time, finite temperature correlation functions in these systems and show that they are described by an emergent type III$_1$ von Neumann algebra at large $N$. The spectral density underlying this algebra is computed in closed form in terms of the eigenvalue density of a discrete matrix model. Furthermore, we explain how to systematically promote these theories to systems with a Hagedorn transition, and show that a type III$_1$ algebra only emerges above the Hagedorn temperature. Finally, we empirically observe in examples a correspondence between the space of states of the quantum mechanics and Calabi--Yau manifolds.

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arXiv2023

Linear Quivers at Large-$N$

Quiver theories constitute an important class of supersymmetric gauge theories with well-defined holographic duals. Motivated by holographic duality, we use localisation on $S^d$ to study long linear quivers at large-N. The large-N solution shows a remarkable degree of universality across dimensions, including $d = 4$ where quivers are genuinely superconformal. In that case we upgrade the solution of long quivers to quivers of any length.

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arXiv2023

Massive flows in AdS$_6$/CFT$_5$

We study five-dimensional ${\cal N}=1$ Superconformal Field Theories of the linear quiver type. These are deformed by a relevant operator, corresponding to a homogeneous mass term for certain matter fields. The free energy is calculated at arbitrary values of the mass parameter. After a careful regularisation procedure, the result can be put in correspondence with a calculation in the supergravity dual background. The F-theorem is verified for these flows, both in field theory and in supergravity. This letter presents some of the results in the companion paper arXiv:2211.13240.

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arXiv2023

Matrix Models and Holography: Mass Deformations of Long Quiver Theories in 5d and 3d

We enlarge the dictionary between matrix models for long linear quivers preserving eight supercharges in $d=5$ and $d=3$ and type IIB supergravity backgrounds with AdS$_{d+1}$ factors. We introduce mass deformations of the field theory that break the quiver into a collection of interacting linear quivers, which are decoupled at the end of the RG flow. We find and solve a Laplace problem in supergravity which realises these deformations holographically. The free energy and expectation values of antisymmetric Wilson loops are calculated on both sides of the proposed duality, finding agreement. Our matching procedure sheds light on the F-theorem in five dimensions.

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arXiv2023

3d defects in 5d: RG flows and defect F-maximization

We use a combination of AdS/CFT and supersymmetric localization to study codimension-2 defects in 5d SCFTs and their gauge theory deformations. The 5d SCFTs are engineered by $(p,q)$ 5-brane webs, with defects realized by D3-branes ending on the 5-brane webs. We obtain the defect free energies and find that gauge theory descriptions of the combined 5d/3d systems can be connected to the UV defect SCFTs through a form of F-maximization which extremizes over different gauge theory defects. This leads to a match between the defect free energies obtained from supersymmetric localization in the gauge theories on the one hand and string theory results on the other. We extend this match to defect RG flows.

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arXiv2022

Crystal bases and three-dimensional $ \mathcal{N}=4 $ Coulomb branches

We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional $ \mathcal{N} =4 $ gauge theories. The result holds for simply-laced, non-simply laced and affine quivers. Two equivalent derivations are given in the non-simply laced case, either by application of the axiomatic rules or by folding a simply-laced quiver. We also study the effect of turning on real masses and the ensuing simplification of the crystal. We present a multitude of explicit examples of the equivalence. Finally, we put forward a correspondence between infinite crystals and Hilbert spaces of theories with isolated vacua.

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arXiv2022

Multiple phases and meromorphic deformations of unitary matrix models

We study a unitary matrix model with Gross-Witten-Wadia weight function and determinant insertions. After some exact evaluations, we characterize the intricate phase diagram. There are five possible phases: an ungapped phase, two different one-cut gapped phases and two other two-cut gapped phases. The transition from the ungapped phase to any gapped phase is third order, but the transition between any one-cut and any two-cut phase is second order. The physics of tunneling from a metastable vacuum to a stable one and of different releases of instantons is discussed. Wilson loops, $β$-functions and aspects of chiral symmetry breaking are investigated as well. Furthermore, we study in detail the meromorphic deformation of a general class of unitary matrix models, in which the integration contour is not anchored to the unit circle. The ensuing phase diagram is characterized by symplectic singularities and captured by a Hasse diagram.

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arXiv2021

Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for the case of the space of Hermitian matrices, where the distributions have already appeared in the physics literature. For the case when the symmetric space is the space of $m \times m$ symmetric positive definite matrices, we show how to efficiently compute by evaluating Pfaffians at specific values of $m$. Equivalently, we can obtain the same result by constructing specific skew orthogonal polynomials with regards to the log-normal weight function (skew Stieltjes-Wigert polynomials). Other symmetric spaces are studied and the same type of result is obtained for the quaternionic case. Moreover, we show how the probability density functions are a particular case of diffusion reproducing kernels of the Karlin-McGregor type, describing non-intersecting Brownian motions, which are also diffusion processes in the Weyl chamber of Lie groups.

math-ph↗

arXiv2021

Schur expansion of random-matrix reproducing kernels

We give expansions of reproducing kernels of the Christoffel-Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a $q$-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and $q$-deformed cases on the real line, we use extensions of Dotsenko-Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguerre polynomials, Painlevé tau functions and conformal block expansions is discussed in relationship to the Schur expansion obtained.

math-ph↗

arXiv2021

Phases of five-dimensional supersymmetric gauge theories

Five-dimensional $\mathcal{N}=1$ theories with gauge group $U(N)$, $SU(N)$, $USp(2N)$ and $SO(N)$ are studied at large rank through localization on a large sphere. The phase diagram of theories with fundamental hypermultiplets is universal and characterized by third order phase transitions, with the exception of $U(N)$, that shows both second and third order transitions. The phase diagram of theories with adjoint or (anti-)symmetric hypermultiplets is also determined and found to be universal. Moreover, Wilson loops in fundamental and antisymmetric representations of any rank are analyzed in this limit. Quiver theories are discussed as well. All the results substantiate the $\mathcal{F}$-theorem.

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