Ivan K. Kostov

A bstract: We investigate the universal behaviour of a one-parameter generalisation of the six-vertex model on planar graphs, which we refer to as the seven-vertex model, or 7vM for quick reference. The 7vM is characterised by a temperature coupling and its continuum limit exhibits massive, dilute and dense phases similarly to the O ( n ) loop model. However, there is an important distinction: the loop weights are no longer topological and the dynamics of the loops is now entangled with the local geometry of the lattice. From the dual matrix model we derive explicit expressions for the sphere and disk partition functions in the continuum limit. The disk partition function for fixed length is a deformation of the Bessel integral known as the Krätzel function. We argue that the 7v matrix model (7vMM) and Matrix Quantum Mechanics (MQM) provide two complementary non-perturbative realisations of sine-Liouville gravity. Specifically, we find that the continuum limits of 7vMM and MQM share the same classical spectral curve but describe two different types of branes in sine-Liouville gravity. The 7vMM precisely covers the range of parameters where the Minkowskian MQM lacks a simple interpretation in terms of multiple tachyon scattering. We investigate the flow relating the dilute and dense phases and argue that this flow is the gravitational analogue of the massless flow in the sine-Gordon model with imaginary mass coupling. The two endpoints of the flow are described by a free boson coupled to Liouville gravity and compactified on circles with two different radii.

The finite-volume thermodynamics of a massive integrable QFT is described in terms of a grand canonical ensemble of loops immersed in a torus and interacting through scattering factors associated with their intersections. The path integral of the loops is evaluated explicitly after decoupling the pairwise interactions by a Hubbard-Stratonovich transformation. The HS fields are holomorphic fields depending on the rapidity and can be expanded in elementary oscillators. The torus partition function is expressed as certain expectation value in the Fock space of these oscillators. In the limit where one of the periods of the torus becomes asymptotically large, the effective field theory becomes mean field type. The mean field describes the infinite-volume thermodynamics which is solved by the Thermodynamical Bethe Ansatz.

The computation of a certain class of four-point functions of heavily charged BPS operators boils down to the computation of a special form factor — the octagon. In this paper, which is an extended version of the short note [1], we derive a non-perturbative formula for the square of the octagon as the determinant of a semi-infinite skew-symmetric matrix. We show that perturbatively in the weak coupling limit the octagon is given by a determinant constructed from the polylogarithms evaluating ladder Feynman graphs. We also give a simple operator representation of the octagon in terms of a vacuum expectation value of massless free bosons or fermions living in the rapidity plane.