2. GRAVITY 7. MICROSCOPIC PHYSICS 13. TOPOLOGY OF DEFECTS 18. ANOMALOUS NON-CONSERVATION OF FERMIONIC CHARGE 22. EDGE STATES AND FERMION ZERO MODES ON SOLITON 26. LANDAU CRITICAL VELOCITY 29. CASIMIR EFFECT AND VACUUM ENERGY

The Universe in a Helium Droplet

Operator Content of Two-Dimensional Conformally Invariant Theories

Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

https://www.physics.rutgers.edu/~friedan/papers/Nucl_Phys_B271_93_1986.pdf

Conformal invariance, supersymmetry and string theory

Conformal algebra and multipoint correlation functions in 2D statistical models

Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge C≤1

Conformal quantum field theory models in two dimensions having Z3 symmetry

Liouville conformal field theory is considered with conformal boundary. There is a family of conformal boundary conditions parameterized by the boundary cosmological constant, so that observables depend on the dimensional ratios of boundary and bulk cosmological constants. The disk geometry is considered. We present an explicit expression for the expectation value of a bulk operator inside the disk and for the two-point function of boundary operators. We comment also on the properties of the degenrate boundary operators. Possible applications and further developments are discussed. In particular, we present exact expectation values of the boundary operators in the boundary sin-Gordon model.

Boundary Liouville field theory. 1. Boundary state and boundary two point function

In their recent paper, Alday et al. (Lett Math Phys 91:167–197, 2010) proposed a relation between $${\mathcal{N}=2}$$
 four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra $${\mathcal{A}={\sf Vir} \otimes\mathcal{H}}$$
 which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of $${\mathcal{A}}$$
 . The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called $${Z_{{\sf bif}}}$$
 appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin–Ono integrable hierarchy.

Vladimir Fateev

Papers

Conformal algebra and multipoint correlation functions in 2D statistical models

Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge C≤1

Conformal quantum field theory models in two dimensions having Z3 symmetry

Boundary Liouville field theory. 1. Boundary state and boundary two point function

On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture