Abstract
Abstract We study conformal field theory with the symmetry algebra $ \mathcal{A}\left( {2,\ p} \right)={{{\widehat{\mathfrak{gl}}{(n)_2}}} \left/ {{\widehat{\mathfrak{gl}}{{{\left( {n-p} \right)}}_2}}} \right.} $ . In order to support the conjecture that this algebra acts on the moduli space of instantons on ℂ2/ℤ p , we calculate the characters of its representations and check their coincidence with the generating functions of the fixed points of the moduli space of instantons. We show that the algebra $ \mathcal{A}\left( {2,\ p} \right) $ can be realized in two ways. The first realization is connected with the cross-product of p Virasoro and p Heisenberg algebras: $ {{\mathcal{H}}^p} $ × Vir p . The second realization is connected with: $ {{\mathcal{H}}^p} \times \widehat{\mathfrak{sl}}{(p)_2}\times \left( {\widehat{\mathfrak{sl}}{(2)_p}\times {{{\widehat{\mathfrak{sl}}{(2)_{n-p }}}} \left/ {{\widehat{\mathfrak{sl}}{(2)_n}}} \right.}} \right) $ . The equivalence of these two realizations provides the non-trivial identity for the characters of $ \mathcal{A}\left( {2,\ p} \right) $ . The moduli space of instantons on ℂ2/ℤ p admits two different compactifications. This leads to two different bases for the representations of $ \mathcal{A}\left( {2,\ p} \right) $ . We use this fact to explain the existence of two forms of the instanton pure partition functions.
Highlights
The moduli space of instantons on C2/Zp admits two different compactifications
The nontrivial fact about the moduli space of instantons M is that one can construct the action of some symmetry algebra A on the equivariant cohomologies of the moduli space M
In [17] it was shown that the basis in the space of the equivariant cohomologies can be labelled by the fixed points of the torus acting on the moduli space
Summary
We concentrate on the counting of the fixed points of the moduli space. We confine ourselves to the case of r = 2, which means that we consider the U(2) instantons on C2/Zp, whose moduli space is N M(2, N)Zp. It is convenient to numerate the torus fixed points in this case by the pairs of Young diagrams colored in p colors. It should be noted here that the coloring parameters of the Young diagrams are connected with the topological characteristics of the instantons corresponding to the fixed points of torus action on the moduli space. 2.2 Counting of the non equivalent generating functions of the colored Young diagrams. One can check that the sum of the cardinalities of the equivalence classes equals to the number of generating functions (2.16) with given s p s l=1 p s − 2l [(p−s)/2]. In order to perform this we add the set of p − 1 holomorphic currents of spin 1/2 to the p stress-energy tensors
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