arXiv subjects

Andrea Brini

Publications and source records attributed to Andrea Brini.

At least 19 recordsLinked to original sources

arXiv2026

The Saito determinant for extended affine Weyl discriminant strata

We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.

math.RT↗

arXiv2025

Conifold gap and all-genus mirror symmetry for local $\mathbb{P}^2$

The Conifold Gap Conjecture asserts that the polar part of the Gromov-Witten potential of a Calabi-Yau threefold near its conifold locus has a universal expression described by the logarithm of the Barnes $G$-function. In this paper, I prove the Conifold Gap Conjecture for the local projective plane. The proof relates the higher genus conifold Gromov-Witten generating series of local $\mathbb{P}^2$ to the thermodynamics of a certain statistical mechanical ensemble of repulsive particles on the positive half-line. As a corollary, this establishes the all-genus mirror principle for local $\mathbb{P}^2$ through the direct integration of the BCOV holomorphic anomaly equations.

math.AG↗

arXiv2025

Dubrovin duality and mirror symmetry for ADE resolutions

We show that, under Dubrovin's notion of ''almost'' duality, the Frobenius manifold structure on the orbit spaces of the extended affine Weyl groups of type $\mathrm{ADE}$ is dual, for suitable choices of weight markings, to the equivariant quantum cohomology of the minimal resolution of the du Val singularity of the same Dynkin type. We also provide a uniform Lie-theoretic construction of Landau-Ginzburg mirrors for the quantum cohomology of $\mathrm{ADE}$ resolutions. The mirror B-model is described by a one-dimensional LG superpotential associated to the spectral curve of the $\widehat{\mathrm{ADE}}$ affine relativistic Toda chain.

math.AG↗

arXiv2024

Refined Gromov-Witten invariants

We study the enumerative geometry of stable maps to Calabi-Yau 5-folds $Z$ with a group action preserving the Calabi-Yau form. In the central case $Z=X \times \mathbb{C}^2$, where $X$ is a Calabi-Yau 3-fold with a group action scaling the holomorphic volume form non-trivially, we conjecture that the disconnected equivariant Gromov-Witten generating series of $Z$ returns the Nekrasov-Okounkov equivariant K-theoretic PT partition function of $X$ and, under suitable rigidity conditions, its refined BPS index. We show that in the unrefined limit the conjecture reproduces known statements about the higher genus Gromov-Witten theory of $X$; we prove it for $X$ the resolved conifold; and we establish a refined cycle-level local/relative correspondence for local del Pezzo surfaces, implying the Nekrasov-Shatashvili limit of the conjecture when $X$ is the local projective plane. We further establish B-model physics predictions of Huang-Klemm for refined higher genus mirror symmetry for local $\mathbb{P}^2$. In particular, we prove that our refined Gromov-Witten generating series obey extended holomorphic anomaly equations, are quasi-modular functions of $Γ_1(3)$, have leading asymptotics at the conifold point given by the logarithm of the Barnes double-Gamma function, and satisfy a version of the higher genus Crepant Resolution Correspondence with the refined orbifold Gromov-Witten theory of $[\mathbb{C}^3/μ_3]$. This refines results, and partially proves conjectures, of Lho-Pandharipande, Coates-Iritani, and Bousseau-Fan-Guo-Wu.

math.AG↗

arXiv2023

Mirror symmetry for extended affine Weyl groups

We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin-Zhang in arXiv:hep-th/9611200 on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model mirror is given by a one-dimensional Landau-Ginzburg superpotential constructed from a suitable degeneration of the family of spectral curves of the affine relativistic Toda chain for the corresponding affine Poisson--Lie group. As applications of our mirror theorem we give closed-form expressions for the flat coordinates of the Saito metric and the Frobenius prepotentials in all Dynkin types, compute the topological degree of the Lyashko-Looijenga mapping for certain higher genus Hurwitz space strata, and construct hydrodynamic bihamiltonian hierarchies (in both Lax-Sato and Hamiltonian form) that are root-theoretic generalisations of the long-wave limit of the extended Toda hierarchy.

math.AG↗

arXiv2023

Discrete Mathematics

The purpose of the present work is to provide short and supple teaching notes for a $30$ hours introductory course on elementary \textit{Enumerative Algebraic Combinatorics}. We fully adopt the \textit{Rota way}. The themes are organized into a suitable sequence that allows us to derive any result from the preceding ones by elementary processes. Definitions of \textit{combinatorial coefficients} are just by their \textit{combinatorial meaning}. The derivation techniques of formulae/results are founded upon constructions and two general and elementary principles/methods: - The \textit{bad element} method (for \textit{recursive} formulae). As the reader should recognize, the bad element method might be regarded as a combinatorial companion of the idea of \textit{conditional probability}. - The \textit{overcounting} principle (for \textit{close form} formulae). Therefore, \textit{no computation} is required in \textit{proofs}: \textit{computation formulae are byproducts of combinatorial constructions}. We tried to provide a self-contained presentation: the only prerequisite is standard high school mathematics. We limited ourselves to the \textit{combinatorial point of view}: we invite the reader to draw the (obvious) \textit{probabilistic interpretations}.

math.HO↗

arXiv2023

Capelli-Deruyts bitableaux and the classical Capelli generators of the center of the enveloping algebra $U(gl(n))$

In this paper, we consider a special class of Capelli bitableaux, namely the Capelli-Deruyts bitableaux. The main results we prove are the hook coefficient lemma and the expansion theorem. Capelli-Deruyts bitableaux of rectangular shape are of particular interest since they are central elements in the enveloping algebra. The expansion theorem implies that these central element is explicitely described as a polynomial in the classical Capelli central elements. The hook coefficient lemma implies that the Capelli-Deruyts bitableaux are (canonically) expressed as the products of column determinants.

math.CO↗

arXiv2023

On quasi-tame Looijenga pairs

We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact log Gromov-Witten invariants of Looijenga pairs to other curve counting invariants of Gromov-Witten/Gopakumar-Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the log invariants in presence of infinite scattering. The resulting identity of $q$-series appears to be new and of independent combinatorial interest.

math.AG↗

arXiv2022

Enumerative geometry of surfaces and topological strings

This survey covers recent developments on the geometry and physics of Looijenga pairs, namely pairs $(X,D)$ with $X$ a complex algebraic surface and $D$ a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to $(X,D)$, including the log Gromov--Witten invariants of the pair, the Gromov--Witten invariants of an associated higher dimensional Calabi--Yau variety, the open Gromov--Witten invariants of certain special Lagrangians in toric Calabi--Yau threefolds, the Donaldson--Thomas theory of a class of symmetric quivers, and certain open and closed BPS-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.

math-ph↗

arXiv2022

Quantum immanants, double Young-Capelli bitableaux and Schur shifted symmetric functions

In this paper are introduced two classes of elements in the enveloping algebra $\mathbf{U}(gl(n))$: the \emph{double Young-Capelli bitableaux} $[\ \fbox{$S \ | \ T$}\ ]$ and the \emph{central} \emph{Schur elements} $\mathbf{S}_λ(n)$, that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element $\mathbf{S}_λ(n)$ is the sum of all double Young-Capelli bitableaux $[\ \fbox{$S \ | \ S$}\ ]$, $S$ row (strictly) increasing Young tableaux of shape $\widetildeλ$. The Schur elements $\mathbf{S}_λ(n)$ are proved to be the preimages - with respect to the Harish-Chandra isomorphism - of the \emph{shifted Schur polynomials} $s_{λ|n}^* \in Λ^*(n)$. Hence, the Schur elements are the same as the Okounkov \textit{quantum immanants}, recently described by the present authors as linear combinations of \emph{Capelli immanants}. This new presentation of Schur elements/quantum immanants doesn't involve the irreducible characters of symmetric groups. The Capelli elements $\mathbf{H}_k(n)$ are column Schur elements and the Nazarov-Umeda elements $\mathbf{I}_k(n)$ are row Schur elements. The duality in $\boldsymbolζ(n)$ follows from a combinatorial description of the eigenvalues of the $\mathbf{H}_k(n)$ on irreducible modules that is {\it{dual}} (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the $\mathbf{I}_k(n)$. The passage $n \rightarrow \infty$ for the algebras $\boldsymbolζ(n)$ is obtained both as direct and inverse limit in the category of filtered algebras, via the \emph{Olshanski decomposition/projection}.

math.RT↗

arXiv2021

Five-dimensional gauge theories and the local B-model

We propose an effective framework for computing the prepotential of the topological B-model on a class of local Calabi--Yau geometries related to the circle compactification of five-dimensional $\mathcal{N}=1$ super Yang--Mills theory with simple gauge group. In the simply-laced case, we construct Picard--Fuchs operators from the Dubrovin connection on the Frobenius manifolds associated to the extended affine Weyl groups of type $\mathrm{ADE}$. In general, we propose a purely algebraic construction of Picard--Fuchs ideals from a canonical subring of the space of regular functions on the ramification locus of the Seiberg--Witten curve, encompassing non-simply-laced cases as well. We offer several precision tests of our proposal. Whenever a candidate spectral curve is known from string theory/brane engineering, we perform non-perturbative comparisons with the gauge theory prepotentials obtained from the K-theoretic blow-up equations, finding perfect agreement. We also employ our formalism to rule out some proposals from the theory of integrable systems of Seiberg--Witten geometries for non-simply laced gauge groups.

hep-th↗

arXiv2021

On the log-local principle for the toric boundary

Let $X$ be a smooth projective complex variety and let $D=D_1+\cdots+D_l$ be a reduced normal crossing divisor on $X$ with each component $D_j$ smooth, irreducible, and nef. The log-local principle of van Garrel-Graber-Ruddat conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of $(X,D)$ is equivalent to the genus 0 local Gromov-Witten theory of $X$ twisted by $\bigoplus_{j=1}^l\mathcal{O}(-D_j)$. We prove that an extension of the log-local principle holds for $X$ a (not necessarily smooth) $\mathbb{Q}$-factorial projective toric variety, $D$ the toric boundary, and descendent point insertions.

math.AG↗

arXiv2021

Stable maps to Looijenga pairs: orbifold examples

In arXiv:2011.08830 we established a series of correspondences relating five enumerative theories of log Calabi-Yau surfaces, i.e. pairs $(Y,D)$ with $Y$ a smooth projective complex surface and $D=D_1+\dots +D_l$ an anticanonical divisor on $Y$ with each $D_i$ smooth and nef. In this paper we explore the generalisation to $Y$ being a smooth Deligne-Mumford stack with projective coarse moduli space of dimension 2, and $D_i$ nef $\mathbb{Q}$-Cartier divisors. We consider in particular three infinite families of orbifold log Calabi-Yau surfaces, and for each of them we provide closed form solutions of the maximal contact log Gromov-Witten theory of the pair $(Y,D)$, the local Gromov-Witten theory of the total space of $\bigoplus_i \mathcal{O}_Y(-D_i)$, and the open Gromov-Witten theory of toric orbi-branes in a Calabi-Yau 3-orbifold associated to $(Y,D)$. We also consider new examples of BPS integral structures underlying these invariants, and relate them to the Donaldson-Thomas theory of a symmetric quiver specified by $(Y,D)$, and to a class of open/closed BPS invariants.

math.AG↗

arXiv2021

Stable maps to Looijenga pairs

A log Calabi-Yau surface with maximal boundary, or Looijenga pair, is a pair $(Y,D)$ with $Y$ a smooth rational projective complex surface and $D=D_1+\dots + D_l \in |-K_Y|$ an anticanonical singular nodal curve. Under some positivity conditions on the pair, we propose a series of correspondences relating five different classes of enumerative invariants attached to $(Y,D)$: 1) the log Gromov-Witten theory of the pair $(Y,D)$, 2) the Gromov-Witten theory of the total space of $\bigoplus_i \mathcal{O}_Y(-D_i)$, 3) the open Gromov-Witten theory of special Lagrangians in a Calabi-Yau 3-fold determined by $(Y,D)$, 4) the Donaldson-Thomas theory of a symmetric quiver specified by $(Y,D)$, and 5) a class of BPS invariants considered in different contexts by Klemm-Pandharipande, Ionel-Parker, and Labastida-Marino-Ooguri-Vafa. We furthermore provide a complete closed-form solution to the calculation of all these invariants.

math.AG↗

arXiv2020

The Koszul map K

The Bitableax correspondence isomorphism/Koszul map Theorem (BCK Theorem, for short, Theorem 6.5 below) describes a relevant pair of mutually inverse vector space isomorphisms, the Koszul map K : U(gl(n))-> Sym(gl(n)) and the bitableaux correspondence iWe describe a linear \emph{equivariant isomorphism} $\mathcal{K}$ from the enveloping algebra $\mathbf{U}(gl(n))$ to the algebra ${\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n))$ of polynomials in the entries of a ``generic'' square matrix of order $n$. The isomorphism $\mathcal{K}$ maps any {\textit{Capelli bitableau}} $[S|T]$ in $\mathbf{U}(gl(n))$ to the {\textit{(determinantal) bitableau}} $(S|T)$ in ${\mathbb C}[M_{n,n}]$ and any {\textit{Capelli *-bitableau}} $[S|T]^*$ in $\mathbf{U}(gl(n))$ to the {\textit{(permanental) *-bitableau}} $(S|T)^*$ in ${\mathbb C}[M_{n,n}]$. These results are far-reaching generalizations of the pioneering result of J.-L. Koszul [19] on the Capelli determinant in $\mathbf{U}(gl(n))$ (see, e.g. [24], [27]). We introduce {\textit{column}} Capelli bitableaux and *-bitableaux in Section 6; since they are mapped by the isomorphism $\mathcal{K}$ to {\textit{monomials}} in ${\mathbb C}[M_{n,n}]$, this isomorphism can be regarded as a sharpened version of the PBW isomorphism for the enveloping algebra $\mathbf{U}(gl(n))$. Since the center $\boldsymbolζ(n)$ of $\mathbf{U}(gl(n))$ equals the subalgebra of invariants $\mathbf{U}(gl(n))^{Ad_{gl(n)}}$, then $$ \mathcal{K} \big[ \boldsymbolζ(n) \big] = {\mathbb C}[M_{n,n}]^{ad_{gl(n)}}. $$somorphism B : Sym(gl(n)) -> U(gl(n)) that deeply link the enveloping algebra U(gl(n)) of the general linear Lie algebra gl(n) and the symmetric algebra Sym(gl(n)). The BCK Theorem can be regarded as a sharpened version of the PBW Theorem for the enveloping algebra U(gl(n)).

math.RA↗

arXiv2020

Young-Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants

We propose a new approach to a unified study of determinants, permanents, immanants, (determinantal) bitableaux and symmetrized bitableaux in the polynomial algebra $C[M_{n, n}]$ as well as of their Lie analogues in the enveloping algebra $U(gl(n))$. This leads to new relevant classes of elements in $U(gl(n))$: Capelli bitableaux, right Young-Capelli bitableaux and Capelli immanants. The set of standard Capelli bitableaux and the set of standard right Young-Capelli bitableaux are bases of $U(gl(n))$, whose action on the Gordan-Capelli basis of $C[M_{n, n}]$ have remarkable properties. Capelli immanants can be efficiently computed and provide a system of generators of $U(gl(n))$. The Okounkov quantum immanants are proved to be simple linear combinations of Capelli immanants. Several examples are provided throughout the paper.

math.RT↗

arXiv2020

Exterior powers of the adjoint representation and the Weyl ring of $E_8$

I derive explicitly all polynomial relations in the character ring of $E_8$ of the form $χ_{\wedge^k \mathfrak{e}_8} - \mathfrak{p}_{k} (χ_{1}, \dots, χ_{8})=0$, where $\wedge^k \mathfrak{e}_8$ is an arbitrary exterior power of the adjoint representation and $χ_{i}$ is the $i^{\rm th}$ fundamental character. This has simultaneous implications for the theory of relativistic integrable systems, Seiberg-Witten theory, quantum topology, orbifold Gromov-Witten theory, and the arithmetic of elliptic curves. The solution is obtained by reducing the problem to a (large, but finite) dimensional linear problem, which is amenable to an efficient solution via distributed computation.

math.RT↗

arXiv2019

$E_8$ spectral curves

I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ Lê-Murakami-Ohtsuki invariant of the Poincaré integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots.

hep-th↗