Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves

Abstract

Abstract. We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on P 2 . The top nonvanishing equivariant Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators. Mathematics Subject Classification (2010).Primary: 14C05,17B69.

Supersymmetric gauge theories, intersecting branes and free fermions

“…In other words, we consider making the SU(N) into a global symmetry instead of a gauge symmetry at the boundary. This suggestion has recently been confirmed in [19]. So, depending on how we deal with the theory at infinity, there are reasons to expect both affine symmetry structures to appear and have a combined action of the Lie algebra…”

Geometric realizations of the basic representation of $$\widehat{\mathfrak {gl}}_r$$ gl ^ r

“…Proof The proof of (1) is completely analogous to [14,Lemma 3.2]. Part (2), is the dual of [17, Corollary 3.12].…”

“…However, the Clifford algebra is sometimes defined by letting generators of different colours commute (see for example [14,Section 1.2]). In this case it is necessary to modify the action of Cl on F to…”

“…, r s ) of r and consider the moduli space of framed torsion-free sheaves of rank s and second Chern class n, M(s, n). In [14], Licata and Savage showed that, under a suitable torus action, the localized equivariant cohomology of (a disjoint union of infinitely-many copies) of M(s, n) provides a suitable geometric analogue of bosonic/fermionic Fock space. This is accomplished by defining an action of the Heisenberg and Clifford algebras on this cohomology in terms of the top Chern classes of certain equivariant vector bundles on M(s, n).…”

“…This is accomplished by defining an action of the Heisenberg and Clifford algebras on this cohomology in terms of the top Chern classes of certain equivariant vector bundles on M(s, n). The construction given in [14] naturally corresponds to the homogeneous realization in [22], and thus our first step is to generalize their construction to an arbitrary partition. With this framework in place, we define a new set of operators using vector bundles on certain subvarieties of M(s, n).…”