arXiv subjects

Jörg Teschner

Publications and source records attributed to Jörg Teschner.

10 recordsLinked to original sources

arXiv2026

Analytic Langlands correspondence from SoV

The analytic Langlands correspondence proposed by Etingof, Frenkel and Kazhdan describes the solution to the spectral problems naturally arising in the quantisation of the Hitchin integrable systems in terms of real opers, certain second order differential operators on a Riemann surface having real monodromy. We prove this correspondence in the cases associated to the group $\mathrm{PSL}(2,\mathbb{C})$, and Riemann surfaces of genus zero with a number of punctures larger than three. A crucial ingredient is a unitary integral transformation mapping products of solutions to the ordinary differential equation associated to a real oper to eigenfunctions of the quantised Hitchin Hamiltonians. This allows us to construct joint eigenfunctions of Hecke operators and Hitchin Hamiltonians from real opers.

math.FA↗

arXiv2024

Schur Quantization and Complex Chern-Simons theory

Any four-dimensional Supersymmetric Quantum Field Theory with eight supercharges can be associated to a certain complex symplectic manifold called the "K-theoretic Coulomb branch" of the theory. The collection of K-theoretic Coulomb branches include many complex phase spaces of great interest, including in particular the "character varieties" of complex flat connections on a Riemann surface. The SQFT definition endows K-theoretic Coulomb branches with a variety of canonical structures, including a deformation quantization. In this paper we introduce a canonical "Schur" quantization of K-theoretic Coulomb branches. It is defined by a variant of the Gelfand-Naimark-Segal construction, applied to protected Schur correlation functions of half-BPS line defects. Schur quantization produces an actual quantization of the complex phase space. As a concrete application, we apply this construction to character varieties in order to quantize Chern-Simons gauge theory with a complex gauge group. Other applications include the definition of a new quantum deformation of the Lorentz group, and the solution of certain spectral problems via dualities.

hep-th↗

arXiv2024

From quantum curves to topological string partition functions II

We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of $d=4$, $\mathcal{N}=2$ supersymmetric field theories of class $\mathcal{S}$. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalisation of the tau-functions associated to these jumps define a natural line bundle playing a key role in the geometric characterisation of the topological string partition functions proposed here.

hep-th↗

arXiv2024

Quantum Analytic Langlands Correspondence

The analytic Langlands correspondence describes the solution to the spectral problem for the quantised Hitchin Hamiltonians. It is related to the S-duality of $\cal{N}=4$ super Yang-Mills theory. We propose a one-parameter deformation of the Analytic Langlands Correspondence, and discuss its relations to quantum field theory. The partition functions of the $H_3^+$ WZNW model are interpreted as the wave-functions of a spherical vector in the quantisation of complex Chern-Simons theory. Verlinde line operators generate a representation of two copies of the quantised skein algebra on generalised partition functions. We conjecture that this action generates a basis for the underlying Hilbert space, and explain in which sense the resulting quantum theory represents a deformation of the Analytic Langlands Correspondence.

hep-th↗

arXiv2022

Integrable sigma models at RG fixed points: quantisation as affine Gaudin models

The goal of this paper is to make first steps towards the quantisation of integrable non-linear sigma models using the formalism of affine Gaudin models, by approaching these theories through their conformal limits. We focus mostly on the example of the Klimčík model, which is a two-parameter deformation of the Principal Chiral Model on a Lie group $G$. We show that the UV fixed point of this theory is described classically by two decoupled chiral affine Gaudin models, encoding its left- and right-moving degrees of freedom, and give a detailed analysis of the chiral and integrable structures of these models. Their quantisation is then explored within the framework of Feigin and Frenkel. We study the quantum local integrals of motion using the formalism of quantised affine Gaudin models and show agreement of the first two integrals with known results in the literature for $G={\rm SU}(2)$. Evidence is given for the existence of a monodromy matrix satisfying the Yang-Baxter algebra for this model, thus paving the way for the quantisation of the non-local integrals of motion. We conclude with various perspectives, including on generalisations of this program to a larger class of integrable sigma models and applications of the ODE/IQFT correspondence to the description of their quantum spectrum.

hep-th↗

arXiv2020

From quantum curves to topological string partition functions

This paper describes the reconstruction of the topological string partition function for certain local Calabi-Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations. Quantum curves are characterised as solutions to a Riemann-Hilbert problem. The isomonodromic tau-functions associated to these Riemann-Hilbert problems admit a family of natural normalisations labelled by the chambers in the extended Kähler moduli space of the local CY under consideration. The corresponding isomonodromic tau-functions admit a series expansion of generalised theta series type from which one can extract the topological string partition functions for each chamber.

hep-th↗

arXiv2018

Toda conformal blocks, quantum groups, and flat connections

This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the $\mathcal{W}$-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra ${\mathfrak s}{\mathfrak l}_3$ on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of flat $\mathrm{SL}(3)$-connections. It is shown that the matrix elements of the monodromies can be expressed as Laurent polynomials of more elementary operators which have a simple definition in the free field representation. These operators are identified as quantised counterparts of natural higher rank analogs of the Fenchel-Nielsen coordinates from Teichmüller theory. Possible applications to the study of the non-Lagrangian SUSY field theories are briefly outlined.

hep-th↗

arXiv2015

Exact results on N=2 supersymmetric gauge theories

This is the introduction to the collection of review articles "Exact results on N=2 supersymmetric gauge theories". The first three sections are intended to give a general overview over the physical motivations behind this direction of research, and some of the developments that initiated this project. These sections are written for a broad audience of readers with interest in quantum field theory, assuming only very basic knowledge of supersymmetric gauge theories and string theory. This will be followed by a brief overview over the different chapters collected in this volume, while the last section indicates some related developments that we were unfortunately not able to cover.

hep-th↗

arXiv2014

Supersymmetric gauge theories, quantisation of moduli spaces of flat connections, and Liouville theory

This is the 11th article in the collection of reviews "Exact results in N=2 supersymmetric gauge theories", ed. J. Teschner. It describes an approach to understanding the 4d/2d relations discovered by Alday, Gaiotto and Tachikawa by establishing a triangle of relations between the zero mode quantum mechanics obtained by localisation of class $\cal S$ theories, the quantum theory obtained by quantisation of Hitchin moduli spaces, and conformal field theory.

hep-th↗

arXiv1996

Coupling supergravity to non-supersymmetric matter

By introducing a nonlinearly transforming goldstino field non-super\-sym\-metric matter can be coupled to supergravity. This implies the possibility of coupling a standard model with one Higgs to supergravity.

hep-th↗