arXiv subjects

Joerg Teschner

Publications and source records attributed to Joerg Teschner.

17 recordsLinked to original sources

arXiv2025

Higher-Rank Mathieu Opers, Toda Chain, and Analytic Langlands Correspondence

We study the Riemann-Hilbert problem associated to flat sections of oper connections of arbitrary rank on the twice-punctured Riemann sphere with irregular singularities of the mildest type. We construct the solutions in terms of the solutions to a single non-linear integral equation. It follows from this construction that the generating function of the submanifold of opers coincides with the Yang-Yang function of the quantum Toda chain, proving a conjecture by Nekrasov, Rosly and Shatashvili. In this way we may furthermore reformulate the quantization conditions of the Toda chain in terms of the connection problem, for which we also provide a solution. We finally interpret our results as a variant of the Analytic Langlands Correspondence for the real version of the Hitchin system corresponding to the Toda chain.

math-ph↗

arXiv2023

Classical limit of the geometric Langlands correspondence for $SL(2, \mathbb{C})$

The goal of this paper is to give an explicit description of the integrable structure of the Hitchin moduli spaces. This is done by introducing explicit parameterisations for the different strata of the Hitchin moduli spaces, and by adapting the Separation of Variables method from the theory of integrable models to the Hitchin moduli spaces. The resulting description exhibits a clear analogy with Drinfeld's first construction of the geometric Langlands correspondence. It can be seen as a classical limit of a version of Drinfeld's construction which is adapted to the complex number field.

math.DG↗

arXiv2021

Mathematical structures of non-perturbative topological string theory: from GW to DT invariants

We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore.

hep-th↗

arXiv2019

Trinion Conformal Blocks from Topological strings

In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory $T_2$. The partition functions exhibit jumps when passing from one chamber in the parameter space to another. Such jumps can be attributed to a change of the integration contour in the free field representation of Liouville conformal blocks. We compare the partition functions of the $T_2$ theories representing trifundamental half hypermultiplets in $N=2$, $d=4$ field theories to the partition functions associated to bifundamental hypermultiplets. We find that both are related to the same Liouville conformal blocks up to inessential factors. In order to establish this picture we combine and compare results obtained using topological vertex techniques, matrix models and topological recursion. We furthermore check that the partition functions obtained by gluing two $T_2$ vertices can be represented in terms of a four point Liouville conformal block. Our results indicate that the $T_2$ vertex offers a useful starting point for developing an analog of the instanton calculus for SUSY gauge theories with trifundamental hypermultiplets.

hep-th↗

arXiv2018

Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence

Single-valuedness of the eigenfunctions of the quantised Hitchin Hamiltonians is proposed as a natural quantisation condition. Separation of Variables can be used to relate the classification of eigenstates to the classification of projective structures with real holonomy. Using complex Fenchel-Nielsen coordinates one may reformulate the quantisation conditions in terms of the generating function for the variety of opers. These results are interpreted as a variant of the geometric Langlands correspondence.

math-ph↗

arXiv2017

A guide to two-dimensional conformal field theory

This is a review of two-dimensional conformal field theory including some of the relations to integrable models. An effort is made to develop the basic formalism in a way which is as elementary and flexible as possible at the same time. Some advanced topics like conformal field theory on higher genus surfaces and relations to the isomonodromic deformation problem are discussed, for other topics we offer a first guide to the literature.

hep-th↗

arXiv2017

Classical conformal blocks and isomonodromic deformations

The leading classical asymptotics of Virasoro conformal blocks on the Riemann sphere with n generic and n-3 "heavy" degenerate field insertions can be described in terms of the geometry of Garnier system describing the monodromy preserving deformations of second order Fuchsian differential equations on an n-punctured sphere. This allows us to characterise the leading classical asymptotics of Virasoro conformal blocks completely, and to clarify in which sense conformal field theory represents a quantisation of the isomonodromic deformation problem.

hep-th↗

arXiv2017

Supersymmetric field theories and geometric Langlands: The other side of the coin

This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on $N=4$ SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin - Witten and Nekrasov - Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional sigma models having Hitchin's moduli spaces as target-manifold.

hep-th↗

arXiv2015

Quantisation of super Teichmueller theory

We construct a quantisation of the Teichmueller spaces of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential.

hep-th↗

arXiv2015

Surface Operators and Separation of Variables

Alday, Gaiotto, and Tachikawa conjectured relations between certain 4d N=2 supersymmetric field theories and 2d Liouville conformal field theory. We study generalizations of these relations to 4d theories with surface operators. For one type of surface operators the corresponding 2d theory is the WZW model, and for another type - the Liouville theory with insertions of extra degenerate fields. We show that these two 4d theories with surface operators exhibit an IR duality, which reflects the known relation (the so-called separation of variables) between the conformal blocks of the WZW model and the Liouville theory. Furthermore, we trace this IR duality to a brane creation construction relating systems of M5 and M2 branes in M-theory. Finally, we show that this duality may be expressed as an explicit relation between the generating functions for the changes of variables between natural sets of Darboux coordinates on the Hitchin moduli space.

hep-th↗

arXiv2015

Line operators in theories of class $\mathcal{S}$, quantized moduli space of flat connections, and Toda field theory

Non-perturbative aspects of $\mathcal{N}=2$ supersymmetric gauge theories of class $\mathcal{S}$ are deeply encoded in the algebra of functions on the moduli space $\mathcal{M}_\text{flat}$ of flat $SL(N)$-connections on Riemann surfaces. Expectation values of Wilson and 't Hooft line operators are related to holonomies of flat connections, and expectation values of line operators in the low-energy effective theory are related to Fock-Goncharov coordinates on $\mathcal{M}_\text{flat}$. Via the decomposition of UV line operators into IR line operators, we determine their noncommutative algebra from the quantization of Fock-Goncharov Laurent polynomials, and find that it coincides with the skein algebra studied in the context of Chern-Simons theory. Another realization of the skein algebra is generated by Verlinde network operators in Toda field theory. Comparing the spectra of these two realizations provides non-trivial support for their equivalence. Our results can be viewed as evidence for the generalization of the AGT correspondence to higher-rank class $\mathcal{S}$ theories.

hep-th↗

arXiv2012

Irregular singularities in Liouville theory

Motivated by problems arising in the study of N=2 supersymmetric gauge theories we introduce and study irregular singularities in two-dimensional conformal field theory, here Liouville theory. Irregular singularities are associated to representations of the Virasoro algebra in which a subset of the annihilation part of the algebra act diagonally. In this paper we define natural bases for the space of conformal blocks in the presence of irregular singularities, describe how to calculate their series expansions, and how such conformal blocks can be constructed by some delicate limiting procedure from ordinary conformal blocks. This leads us to a proposal for the structure functions appearing in the decomposition of physical correlation functions with irregular singularities into conformal blocks. Taken together, we get a precise prediction for the partition functions of some Argyres-Douglas type theories on the four-sphere.

hep-th↗

arXiv2009

Gauge Theory Loop Operators and Liouville Theory

We propose a correspondence between loop operators in a family of four dimensional N=2 gauge theories on S^4 -- including Wilson, 't Hooft and dyonic operators -- and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these N=2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun's formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of 't Hooft and dyonic loop operators in these N=2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmuller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.

hep-th↗

arXiv2006

On the time-dependent description for the decay of unstable D-branes

We discuss how to describe time-dependent phenomena in string theory like the decay of unstable D-branes with the help of the world-sheet formulation. It is shown in a nontrivial well-controlled example that the coupling of the tachyons to propagating on-shell modes which escape to infinity can lead to time-dependent relaxation into a stationary final state. The final state corresponds to a fixed point of the RG flow generated by the relevant field from which the tachyon vertex operator is constructed. On the way we set up a fairly general formalism for the description of slow time-dependent phenomena with the help of conformal perturbation theory on the world-sheet.

hep-th↗

arXiv2005

H(3)+ correlators from Liouville theory

We prove that arbitrary correlation functions of the H(3)+ model on a sphere have a simple expression in terms of Liouville theory correlation functions. This is based on the correspondence between the KZ and BPZ equations, and on relations between the structure constants of Liouville theory and the H(3)+ model. In the critical level limit, these results imply a direct link between eigenvectors of the Gaudin Hamiltonians and the problem of uniformization of Riemann surfaces. We also present an expression for correlation functions of the SL(2)/U(1) coset model in terms of correlation functions in Liouville theory.

hep-th↗

arXiv2004

On Boundary Perturbations in Liouville Theory and Brane Dynamics in Noncritical String Theories

We study certain relevant boundary perturbations of Liouville theory and discuss implications of our results for the brane dynamics in noncritical string theories. Our results include (i) There exist monodromies in the parameter $μ_{\rm B}$ of the Neumann-type boundary condition that create an admixture represented by the Dirichlet type boundary condition. (ii) Certain renormalization group flows can be studied perturbatively, which allows one to determine the results of the corresponding brane decays. (iii) There exists a simple renormalization group flow that can be calculated exactly. In all the cases that we have studied, the RG flow acts like a covering transformation for the mondromies mentioned under (i).

hep-th↗

arXiv2003

Classical and Quantum D-branes in 2D String Theory

We investigate two classes of D-branes in 2-d string theory, corresponding to extended and localized branes, respectively. We compute the string emission during tachyon condensation and reinterpret the results within the $c=1$ matrix model. As in hep-th/0304224, we find that the extended branes describe classical eigenvalue trajectories, while, as found in hep-th/0305159, the localized branes correspond to the quantum field that creates and destroys eigenvalues. The matrix model relation between the classical probe and the local collective field precisely matches with the descent relation between the boundary states of D-strings and D-particles.

hep-th↗