# Final PhD Seminar

## Summation of instantons in O(3) NLSM and quantum-classical duality

## Audience

**Members of RSPE welcome**

**Mr Boris Runov**

Instantons are special classical solutions of euclidean field equations encountered in gauge theories and sigma models. They describe tunnelling between topologically different vacua of quantum field theory. Instantons are essential for understanding non-perturbative low energy behaviour of QFT. In particular, it was conjectured by Polyakov that instantons are responsible for confinement of quarks in QCD. We consider instantons in one the O(3) non-linear sigma model in two dimensions. O(3) NLSM shares several non-trivial properties with Yang-Mills theory with gauge group SU(3) (bosonic part of QCD), in particular, degenerate classical vacua, asymptotic freedom and existence of confinement

phase at low energies.

We are interested mainly in the structure of the ground state of the theory. The properties of the ground state are encoded in the partition function, a weighted sum over all possible configurations of the system. Full partition function of O(3) NLSM including contributions of instantons can be well approximated by partition function of another 2 dimensional QFT known as Boukhvostov-Lipatov model. This model is integrable, which means that it enjoys Yangian symmetry and has an infinite number of conserved charges.

In this talk I will present the results of my work on solution of Boukhvostov-Lipatov model. I will describe its integrable structure in details and show how integrability was used to compute the ground state energy of this theory for arbitrary fermion mass and coupling constant. I will also present some insights into more general theory of integrable systems, in particular into the nature of special property of their integrals of motion known as the ODE/IM correspondence, which was essential for computation of the vacuum energy.