Exts and vertex operators
TL;DR: 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 as mentioned in this paper.
Abstract: 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.
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TL;DR: In this paper, the relation between vortex counting in two-dimensional supersymmetric field theories and the refined BPS invariants of the dual geometries was studied, which can also be mapped to the computation of degenerate conformal blocks in 2-dimensional CFTs.
Abstract: 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.
402 citations
TL;DR: Alday et al. as discussed by the authors studied the origin of the conformal block expansion from a CFT point of view and found a special orthogonal basis of states in the highest weight representations of the algebra of mutually commuting Virasoro and Heisenberg algebras.
Abstract: 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.
313 citations
TL;DR: In this paper, the universal part of the effective twisted superpotential of the quiver gauge theory was derived for a d-dimensional torus and a two-dimensional cigar with the same deformation.
Abstract: We study macroscopically two dimensional ${\mathcal{N}=(2,2)}$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product ${\mathbb{T}^{d} \times \mathbb{R}^{2\epsilon}}$ of a d-dimensional torus and a two dimensional cigar with ${\Omega}$ -deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories and the Yangian ${\mathbf{Y}_{\epsilon}(\mathfrak{g}_{\Gamma})}$ , quantum affine algebra ${\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})}$ , or the quantum elliptic algebra ${\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})}$ associated to Kac–Moody algebra ${\mathfrak{g}_{\Gamma}}$ for quiver ${\Gamma}$ .
191 citations
TL;DR: In this article, the supersymmetric partition function of the M-theory was computed on a two-torus, with arbitrary supersymmetry preserving twists, using the topological vertex formalism.
Abstract: We consider M-theory in the presence of M parallel M5-branes prob- ing a transverse AN−1 singularity. This leads to a superconformal theory with (1,0) supersymmetry in six dimensions. We compute the supersymmetric partition func- tion of this theory on a two-torus, with arbitrary supersymmetry preserving twists, using the topological vertex formalism. Alternatively, we show that this can also be obtained by computing the elliptic genus of an orbifold of recently studied M-strings. The resulting 2d theory is a (4,0) supersymmetric quiver gauge theory whose Higgs branch corresponds to strings propagating on the moduli space of SU(N) M−1 instan- tons on R 4 where the right-moving fermions are coupled to a particular bundle.
190 citations
TL;DR: In this article, the convolution algebra in the equivariant K-theory of the Hilbert scheme of A2 was shown to be isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of GL∞.
Abstract: In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A2. We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of GL∞. We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K-theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K-theory of the moduli space of framed torsion free sheaves over P2, virtual fundamental classes, shuffle algebras, …).
103 citations
References
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26 Jun 2003
TL;DR: In this paper, the authors investigated various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.
Abstract: We study \( \mathcal{N} = 2 \) supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in the Ω-background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.
1,350 citations
Book•
01 Jan 1997
TL;DR: This book discusses K-Theory, Symplectic Geometry, Flag Varieties, K- theory, and Harmonic Polynomials, and Representations of Convolution Algebras.
Abstract: Preface.- Chapter 0. Introduction.- Chapter 1. Symplectic Geometry.- Chapter 2. Mosaic.- Chapter 3. Complex Semisimple Groups.- Chapter 4. Springer Theory.- Chapter 5. Equivariant K-Theory.- Chapter 6. Flag Varieties, K-Theory, and Harmonic Polynomials.- Chapter 7. Hecke Algebras and K-Theory.- Chapter 8. Representations of Convolution Algebras.- Bibliography.
1,144 citations
Book•
17 Sep 1999942 citations
587 citations
Book•
01 Jan 1999
TL;DR: The Weil Model and the Cartan Model were proposed by Cartan as discussed by the authors, who considered the Weil model as an extension of Cartan's formula and showed that it can be used in the context of equivariant cohomology in topology.
Abstract: 1 Equivariant Cohomology in Topology.- 3 The Weil Algebra.- 4 The Weil Model and the Cartan Model.- 5 Cartan's Formula.- 6 Spectral Sequences.- 7 Fermionic Integration.- 8 Characteristic Classes.- 9 Equivariant Symplectic Forms.- 10 The Thom Class and Localization.- 11 The Abstract Localization Theorem.- Notions d'algebre differentielle application aux groupes de Lie et aux varietes ou opere un groupe de Lie: Henri Cartan.- La transgression dans un groupe de Lie et dans un espace fibre principal: Henri Cartan.
513 citations