In the school year 2011/2012 I gave online lectures on quantum integrability.

The last meeting in 2011 is on Tuesday 20 December at 15h30 CET. It was devoted to solution of exercises.


Below are the contents of the course. Number of the list item corresponds to the number of the lecture.

Fall term

  1. Before integrability
    • Definition of su(2) XXX spin chain and numerical study of the spectrum
    • Perron-Frobenius theorem
    • Uniqueness of the antiferromagnetic vacuum
  2. Coordintae Bethe Ansatz for SU(n) spin chain
    • Construction of wave function from plane waves
    • Nested transfer matrix, periodicity condition
    • RTT=TTR relation and possibility to diagonalize
    • su(2) - mapping solutions to states in irreps
    • Analytic derivation of energy of antiferromagnetic vacuum.
  3. Nested/Algebraic Bethe Ansatz
    • Nested Bethe Ansatz for su(3)
    • Discovery of Algebraic Bethe Ansatz
  4. Fusion procedure
    • Particle in symmetric and antisymmetric irreps
    • R-matrix for scattering of arbitrary representation with fundamental
    • Transfer matrices in arbitrary representations, basic examples of bilinear identities.
  5. Character identities and Hirota dynamics
    • Character formulas for su(n)
    • Hirota relations
    • Solution of Hirota equations in the infinite strip
    • Baxter equation for su(2) spin chain
    • su(n) generalization and various TQ-type relations as Plucker identities (Backlund flows, higher Baxter equation etc).
  6. Numerical study of solutions for su(2) spin chain
  7. Baxter operators

Spring term

  1. Integrable QFT, factorized scattering matrix
  2. Continuous limit of spin chains
  3. (generalized) Wiener-Hopf method
  4. Solution of IQFT spectral problem in finite volume (from TBA, from spin chains, from Hirota postulated or derived)
  5. Review and more abstract reformulation of learned material (in terms of Yangians etc.)