To every 3-manifold M one can associate a two-dimensional N=(2,2) supersymmetric field theory by compactifying five-dimensional N=2 super-Yang–Mills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional N=2 gauge theories on one hand, and in the geometric approach to knot homologies, on the other. We study the relation between vortex counting in such two-dimensional N=(2,2) supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can also be mapped to the computation of degenerate conformal blocks in two-dimensional CFT’s. Degenerate limits of vertex operators
in CFT receive a simple interpretation via geometric transitions in BPS counting.

/pdf/vortex-counting-and-lagrangian-3-manifolds-5drxws8ipc.pdf

Vortex Counting and Lagrangian 3-manifolds

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.

On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

Introduction Hilbert scheme of points Framed moduli space of torsion free sheaves on $\mathbb{P}^2$ Hyper-Kahler metric on $(\mathbb{C}^2)^{[n]}$ Resolution of simple singularities Poincare polynomials of the Hilbert schemes (1) Poincare polynomials of Hilbert schemes (2) Hilbert scheme on the cotangent bundle of a Riemann surface Homology group of the Hilbert schemes and the Heisenberg algebra Symmetric products of an embedded curve, symmetric functions and vertex operators Bibliography Index.

Lectures on Hilbert schemes of points on surfaces

We study an analog of the AGT (Alday-Gaiotto-Tachikawa) relation in five dimensions. We conjecture that the instanton partition function of 5D$ \mathcal{N} = 1 $ pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four-dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.

Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra

M2 branes suspended between adjacent parallel M5 branes lead to light strings, the `M-strings' In this paper we compute the elliptic genus of M-strings, twisted by maximally allowed symmetries that preserve 2d (2,0) supersymmetry In a codimension one subspace of parameters this reduces to the elliptic genus of the (4,4) supersymmetric A_{n-1} quiver theory in 2d We contrast the elliptic genus of N M-strings with the (4,4) sigma model on the N-fold symmetric product of R^4 For N=1 they are the same, but for N>1 they are close, but not identical Instead the elliptic genus of (4,4) N M-strings is the same as the elliptic genus of (4,0) sigma models on the N-fold symmetric product of R^4, but where the right-moving fermions couple to a modification of the tangent bundle This construction arises from a dual A_{n-1} quiver 6d gauge theory with U(1) gauge groups Moreover we compute the elliptic genus of domain walls which separate different numbers of M2 branes on the two sides of the wall

M-Strings

We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $\tildeA$ acting on this space, and interpret the right generalization of the $\nabla$ using an involution of the algebra which is antilinear with respect to the conjugation $(q,t)\mapsto (q^{-1},t^{-1})$.

A proof of the shuffle conjecture

The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.

Exts and Vertex Operators

The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext-groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.

Exts and vertex operators

/pdf/a-proof-of-the-shuffle-conjecture-4ka5rr4bt9.pdf

We study supersymmetric gauge theories in five dimensions, using their relation to the K-theory of the moduli spaces of torsion free sheaves. In the spirit of the BPS/CFT correspondence the partition function and the expectation values of the chiral, BPS protected observables are given by the matrix elements and more generally by the correlation functions in some q-deformed conformal field theory in two dimensions. We show that the coupling of the gauge theory to the bi-fundamental matter hypermultiplet inserts a particular vertex operator in this theory. In this way we get a generalization of the main result of [6] to K-theory. The theory of interpolating Macdonald polynomials is an important tool in our construction.

Erik Carlsson

Papers

A proof of the shuffle conjecture

Exts and Vertex Operators

Exts and vertex operators

A proof of the shuffle conjecture

Five dimensional gauge theories and vertex operators