Research seminar on String Mathematics (SS21)

Quantum integrable systems and certain gauge theories

The quantization of (algebraic) integrable systems is a very intricate problem. One of the issues is that ‘quantum integrability’ does not (yet) have a mathematical and uniform definition because it is unclear what a quantum counterpart of the Arnold—Liouville theorem should be. One common feature of so-called quantum integrable systems in Physics is that they are essentially solvable by the so-called Bethe ansatz.


In this research seminar we want to take a look at quantum integrable systems through certain supersymmetric gauge theories. Since this topic is closer to Physics than Mathematics, we mainly focus on the big picture in each talk. The outline below is a suggestion but we could spend more time on certain aspects or skip others if we wish to.


Time: Tuesday 12:15 - 13:45 pm (mostly biweekly starting from April 13, see dates below).

Location: As usual these days 'in' Zoom. The link is either available through STiNE or upon request (see organizers at the bottom).

Note: There might be one or two additional talks by external speakers. These will be announced in the next weeks and added here.

Seminar plan

Overview of Nekrasov—Shatashvili / ‘Bethe/gauge correspondence'

(April 13) Speaker: Florian

  • Recap of classical (Poisson) integrable systems

  • Quantum integrable systems

  • Outlook: classical and quantum Gaudin models

  • Relation between certain 2d gauge theories and quantum integrable systems (e.g. Yang—Mills—Higgs and Nonlinear Schrödinger)

(‘Bethe/gauge correspondence’)

  • Discussion of next talks

  • Main reference: [GS1], [GS2], [STS].

  • Slides


Classical & Quantum Gaudin models

(April 27) Speaker: Duong

  • Lax form, r-matrix of integrable systems

  • Classical Gaudin model

  • Quantization via Bethe ansatz

  • References: [STS], ...

  • Slides


Bethe ansatz

(May 18) Speaker: Arpan

  • Broader introduction of the Bethe ansatz (possibly including the thermodynamic Bethe ansatz)

  • References: [E], [LM], [vT]. Also: [F], [M] (mainly §19).

  • Slides


Introduction to the non-linear Schrödinger equation (NLS)

(June 01) Speaker: Carl

  • N-particle periodic NLS

  • Bethe ansatz

  • Quantum integrable system

  • References: [GS1], [GS2] (compare below), ...

  • Slides


Review of Yang--Mills--Higgs (YMH)

(June 15) Speaker: Ivan

  • Basic setup of YMH equations (= Hitchin's equations...)

  • U(1)-action on solutions

  • Corresponding partition function


Note: The last talk has been moved from June 29 to July 06 because of the workshop BPS 2021.


TBA

(July 06) Speaker: Murad


Possible topics

(Quantum) Integrable systems

  • Algebraic integrable systems: Lax form, r-matrix

  • Quantum integrable systems: quantization, Bethe ansatz, classical & quantum Gaudin model as example

  • Main reference: [STS].


Nonlinear Schrödinger equation and the Bethe ansatz

Following [GS1, §4] (without §4.1, 4.2.):

  • Hamiltonian of the nonlinear Schrödinger equation etc. (also introduce ‘Yang system’ and ‘Yang(—Yang) function from the almost identical [GS2, §4]).

  • Wave functions.

  • Relation to quantization of T*H (H: maximal torus), see [GS2] around (5.43).

  • Main references: The two (overlapping) papers [GS1], [GS2].


Bethe/gauge correspondence I: Partition function of wave functions of Yang—Mills(—Higgs)

  • Partition functions of TFTs (mathematical…).

  • Yang—Mills(—Higgs), just a brief overview.

  • Torus partition functions (without details of computation but with relation to Bethe ansatz, see [GS1, (3.32)]).

  • Main reference: [GS1, §2, 3]


Bethe/gauge correspondence II: Relation between NLS and YMH

  • Nonlinear Schrödinger (NLS) and Yang--Mills(--Higgs) (YM(H)).

  • Match between the partition functions of NLS and YMH.

  • Main reference: [GS2, §5.3]


Nekrasov—Shatashvili: Omega-background

  • 4d N=2 gauge theories and relation to integrable systems.

  • Prepotential and twisted superpotential.

  • Relation to quantum integrable systems.

  • Example: either Toda or elliptic Calogero—Moser.

  • Main reference: [NS]


Main references

[E] Eckle, Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

[F] Fadeev, How the algebraic Bethe ansatz works for integrable models

[GS1] Gerasimov, Shatashvili: Higgs Bundles, Gauge Theories and Quantum Groups

[GS2] Gerasimov, Shatashvili: Two-dimensional gauge theories and quantum integrable systems

[LM] Levkovich-Maslyuk: Lectures on the Bethe Ansatz

[M] Musardo, Statistical Field Theory

[NS] Nekrasov, Shatashvili: Quantization of integrable systems and four dimensional gauge theories

[STS] Semenov-Tian-Shansky: Quantum integrable systems

[vT] van Tongeren: Introduction to the thermodynamic Bethe ansatz

Organizers

Murad Alim (murad.alim you-know uni-hamburg.de)

Arpan Saha (arpansaha2007 you-know gmail.com)

Florian Beck (florian.beck you-know uni-hamburg.de)