arXiv subjects

Charles A. S. Young

Publications and source records attributed to Charles A. S. Young.

At least 19 recordsLinked to original sources

arXiv2025

Higher Chiral Algebras in a Polysimplicial Model

Vertex algebras are equivalent to translation-equivariant chiral algebras on $\mathbb{A}^1$, in the sense of Beilinson and Drinfeld. In this paper we give an algebraic construction of a chiral algebra on $\mathbb{A}^n$; this can be seen as an algebraic construction of a higher-dimensional vertex algebra. We introduce a model, in dg commutative algebras, of the derived algebra of functions on the configuration space of $k$ distinct labelled marked points in $\mathbb{A}^n$. Working in this model -- which we call the polysimplicial model -- we obtain a dg operad of chiral operations on a degree-shifted copy of the canonical sheaf. We prove that there is a quasi-isomorphism, to this dg operad, from the Lie-infinity operad. This result makes the shifted canonical sheaf into a first example of a homotopy polysimplicial chiral algebra on $\mathbb{A}^n$, in a sense which generalizes to higher dimensions Malikov and Schechtman's notion of a homotopy chiral algebra.

math.QA↗

arXiv2024

Raviolo vertex algebras, cochains and conformal blocks

Raviolo vertex algebras were introduced recently by Garner and Williams in arXiv:2308.04414. Working at the level of cochain complexes, in the present paper we construct spaces of conformal blocks, or more precisely their duals, coinvariants, in the raviolo setting. We prove that the raviolo state-field map correctly captures the limiting behaviour of coinvariants as marked points collide.

math.QA↗

arXiv2023

Towards non-perturbative BV-theory via derived differential geometry

We propose a global geometric framework which allows one to encode a natural non-perturbative generalisation of usual Batalin-Vilkovisky (BV-)theory. Namely, we construct a concrete model of derived differential geometry, whose geometric objects are formal derived smooth stacks, i.e. stacks on formal derived smooth manifolds, together with a notion of differential geometry on them. This provides a working language to study generalised geometric spaces that are smooth, infinite-dimensional, higher and derived at the same time. Such a formalism is obtained by combining Schreiber's differential cohesion with the machinery of Töen-Vezzosi's homotopical algebraic geometry applied to the theory of derived manifolds of Spivak and Carchedi-Steffens. We investigate two classes of examples of non-perturbative classical BV-theories in the context of derived differential cohesion: scalar field theory and Yang-Mills theory.

hep-th↗

arXiv2023

Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine $\mathfrak{sl}_2$ Gaudin model

In this work we will use a general procedure to construct higher local Hamiltonians for the affine $\mathfrak{sl}_2$ Gaudin model. We focus on the first non-trivial example, the quartic Hamiltonians. We show by direct calculation that the quartic Hamiltonians commute amongst themselves and with the quadratic Hamiltonians which define the model. We go on to introduce a certain next-to-leading-order semi-classical limit of the model. In this limit, we are able to write down the full hierarchy of higher local Hamiltonians and prove that they commute.

math.QA↗

arXiv2022

Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex

The notion of a Manin triple of Lie algebras admits a generalization, to dg Lie algebras, in which various properties are required to hold only up to homotopy. This paper introduces two classes of examples of such homotopy Manin triples. These examples are associated to analogs in complex dimension two of, respectively, the punctured formal 1-disc, and the complex plane with multiple punctures. The dg Lie algebras which appear include certain higher current algebras in the sense of Faonte, Hennion and Kapranov arXiv:1701.01368. We work in a ringed space we call rectilinear space, and one of the tools we introduce is a model of the derived sections of its structure sheaf, whose construction is in the spirit of the adelic complexes for schemes due to Parshin and Beilinson.

math.QA↗

arXiv2021

Affine opers and conformal affine Toda

For $\mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $\text{Aut}\, \mathcal O$-equivariant identification between $\text{Fun}\,\text{Op}_{\mathfrak g}(D)$, the algebra of functions on the space of ${\mathfrak g}$-opers on the disc, and $W\subset π_0$, the intersection of kernels of screenings inside a vacuum Fock module $π_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $δ_{-1}\left|0\right>$. We show that the latter endows $π_0$ with a canonical notion of translation $T^{\text{(aff)}}$, and use it to define the densities in $π_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $\text{Aut}\,\mathcal O$-action defines a bundle $Π$ over $\mathbb P^1$ with fibre $π_0$. We show that the product bundles $Π\otimes Ω^j$, where $Ω^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $\nabla^{\text{(aff)}} - αT^{\text{(aff)}}$, $α\in \mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[\mathbf v_j dt^{j+1} ] \in H^1(\mathbb P^1, Π\otimes Ω^j,\nabla^{\text{(aff)}})$ of the de Rham cohomology of $\nabla^{\mathrm{(aff)}}$. Any choice of ${\mathfrak g}$-Miura oper $χ$ gives a connection $\nabla^{\mathrm{(aff)}}_χ$ on $Ω^j$. Using coinvariants, we define a map $\mathsf F_χ$ from sections of $Π\otimes Ω^j$ to sections of $Ω^j$. We show that $\mathsf F_χ\nabla^{\text{(aff)}} = \nabla^{\text{(aff)}}_χ\mathsf F_χ$, so that $\mathsf F_χ$ descends to a well-defined map of cohomologies. Under this map, the classes $[\mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(\mathbb P^1, Ω^j,\nabla^{\text{(aff)}}_χ)$ defined by the ${\mathfrak g}$-oper underlying $χ$.

math.QA↗

arXiv2020

An analog of the Feigin-Frenkel homomorphism for double loop algebras

We prove the existence of a homomorphism of vertex algebras, from the vacuum Verma module over the loop algebra of an untwisted affine algebra, whose construction is analogous to that of the Feigin-Frenkel homomorphism from the vacuum Verma module at critical level over an affine algebra.

math.QA↗

arXiv2020

Affine Gaudin models and hypergeometric functions on affine opers

We conjecture that quantum Gaudin models in affine types admit families of local higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals. We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle $Ω$. Each fibre is isomorphic to the direct product of the space of sections of the square of $Ω$ with the direct product, over the exponents $j$ not equal to 1, of the twisted cohomology of the $j^{\rm th}$ tensor power of $Ω$.

math.QA↗

arXiv2020

Cubic hypergeometric integrals of motion in affine Gaudin models

We construct cubic Hamiltonians for quantum Gaudin models of affine types $\hat{\mathfrak{sl}}_M$. They are given by hypergeometric integrals of a form we recently conjectured in arXiv:1804.01480. We prove that they commute amongst themselves and with the quadratic Hamiltonians. We prove that their vacuum eigenvalues, and their eigenvalues for one Bethe root, are given by certain hypergeometric functions on a space of affine opers.

math.QA↗

arXiv2016

Cyclotomic Gaudin models with irregular singularities

Generalizing the construction of the cyclotomic Gaudin algebra from arXiv:1409.6937, we define the universal cyclotomic Gaudin algebra. It is a cyclotomic generalization of the Gaudin models with irregular singularities defined in arXiv:math/0612798. We go on to solve, by Bethe ansatz, the special case in which the Lax matrix has simple poles at the origin and arbitrarily many finite points, and a double pole at infinity.

math.QA↗

arXiv2016

Cyclotomic discriminantal arrangements and diagram automorphisms of Lie algebras

Recently a new class of quantum integrable models, the cyclotomic Gaudin models, were described in arXiv:1409.6937, arXiv:1410.7664. Motivated by these, we identify a class of affine hyperplane arrangements that we call cyclotomic discriminantal arrangements. We establish correspondences between the flag and Aomoto complexes of such arrangements and chain complexes for nilpotent subalgebras of Kac-Moody type Lie algebras with diagram automorphisms. As a byproduct, we show that the Bethe vectors of cyclotomic Gaudin models associated to diagram automorphisms are nonzero.

math.QA↗

arXiv2015

Populations of Solutions to Cyclotomic Bethe Equations

We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain "extended" master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an "extended" non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a ${\mathbb Z}_2$-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space.

math.QA↗

arXiv2014

Vertex Lie algebras and cyclotomic coinvariants

Given a vertex Lie algebra $\mathscr L$ equipped with an action by automorphisms of a cyclic group $Γ$, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over `local' Lie algebras $\mathsf L(\mathscr L)_{z_i}$ assigned to marked points $z_i$, by the action of a `global' Lie algebra ${\mathsf L}^Γ_{\{z_i \}}(\mathscr L)$ of $Γ$-equivariant functions. On the other hand, the universal enveloping vertex algebra $\mathbb V (\mathscr L)$ of $\mathscr L$ is itself a vertex Lie algebra with an induced action of $Γ$. This gives `big' analogs of the Lie algebras above. From these we construct the space of `big' cyclotomic coinvariants, i.e. coinvariants with respect to ${\mathsf L}^Γ_{\{z_i \}}(\mathbb V(\mathscr L))$. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in arXiv:1409.6937. At the origin, which is fixed by $Γ$, one must assign a module over the stable subalgebra $\mathsf L(\mathscr L)^Γ$ of $\mathsf L(\mathscr L)$. This module becomes a $\mathbb V(\mathscr L)$-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

math.QA↗

arXiv2014

Cyclotomic Gaudin models: construction and Bethe ansatz

To any simple Lie algebra $\mathfrak g$ and automorphism $σ:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case $σ= \text{id}$. We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.

math.QA↗

arXiv2014

Gaudin models for gl(m|n)

We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra gl(m|n). In particular, we prove the completeness of the Bethe ansatz in the case of tensor products of fundamental representations.

math.QA↗

arXiv2011

Integrable achiral D5-brane reflections and asymptotic Bethe equations

We study the reflection of magnons from a D5-brane in the framework of the AdS/CFT correspondence. We consider two possible orientations of the D5-brane with respect to the reference vacuum state, namely vacuum states aligned along "vertical" and "horizontal" directions. We show that the reflections are of the achiral type. We also show that the reflection matrices satisfy the boundary Yang-Baxter equations for both orientations. In the horizontal case the reflection matrix can be interpreted in terms of a bulk S-matrix, S(p, -p), and factorizability of boundary scattering therefore follows from that of bulk scattering. Finally, we solve the nested coordinate Bethe ansatz for the system in the vertical case to find the Bethe equations. In the horizontal case, the Bethe equations are of the same form as those for the closed string.

hep-th↗

arXiv2011

On q,t-characters and the l-weight Jordan filtration of standard quantum affine sl2 modules

The Cartan subalgebra of the sl2 quantum affine algebra is generated by a family of mutually commuting operators, responsible for the l-weight decomposition of finite dimensional modules. The natural Jordan filtration induced by these operators is generically non-trivial on l-weight spaces of dimension greater than one. We derive, for every standard module of quantum affine sl2, the dimensions of the Jordan grades and prove that they can be directly read off from the t-dependence of the q,t-characters introduced by Nakajima. To do so we construct explicit bases for the standard modules with respect to which the Cartan generators are upper-triangular. The basis vectors of each l-weight space are labelled by the elements of a ranked poset from the family L(m,n).

math.QA↗

arXiv2010

Multiplicity in Supersymmetric Spin Chains

We discuss a simple procedure for obtaining new integrable spin chains from old by replacing each single state of the original model by some collection of states. This works whenever the Lax matrix of the chain has a certain form. The simplest example is the su(n) XX model. We apply the techniques of the nested algebraic Bethe ansatz to solve such systems, in the bosonic and supersymmetric cases.

math-ph↗