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Notes on instantons in topological field theory and beyond
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ITEP-TH-04/07 IHES-P/07/08
NOTES ON INSTANTONS IN TOPOLOGICAL FIELD THEORY
AND BEYOND
E. FRENKEL, A. LOSEV, AND N. NEKRASOV
Abstract. This is a brief summary of our studies of quantum field theories in a
special limit in which the instantons are present, the anti-instantons are absent, and
the perturbative corrections are reduced to one-loop. We analyze the corresponding
models as full-fledged quantum field theories, beyond their topological sector. We
show that the correlation functions of all, not only topological (or BPS), observables
may be studied explicitly in these models, and the spectrummay be computed exactly.
An interesting feature is that the Hamiltonian is not always diagonalizable, but may
have Jordan blocks, which leads to the appearance of logarithms in the correlation
functions. We also find that in the models defined on Ka?hler manifolds the space of
states exhibits holomorphic factorization. In particular, in dimensions two and four
our theories are logarithmic conformal field theories.
1. Introduction
Most two- and four-dimensional quantum field theories have two kinds of coupling
constants: the actual coupling g, which in particular counts the loops in the perturba-
tive calculations, and the topological coupling, ?, the theta-angle, which is the chemical
potential for the topological sectors in the path integral. These couplings can be com-
bined into the complex coupling τ and its complex conjugate τ?. The idea is to study
the dependence of the theory on τ , τ? as if they were two separate couplings, not
necessarily complex conjugate to each other.
For example, in the four-dimensional gauge theory one combines the Yang-Mills
coupling g and the theta-angle ? as follows:
(1.1) τ =
?
2π
+
4πi
g2
.
For the two dimensional sigma model with the complex target space X, endowed with
a Hermitian metric gij and a (1, 1) type two-form Bij one defines
(1.2) τij = Bij + igij .
If dB = 0, th
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