Notes on Certain (0,2) Correlation Functions

Abstract

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.

Summing the instantons in half-twisted linear sigma models

“…For a suitable choice of ρ a the low energy well approximated by a non-linear sigma model (NLSM) with target-space the classical moduli space of the gauge theory, 10) and complexified Kähler class B + iJ linear in τ a . A useful notion for the study of the V-model is the cone K c ⊂ R r defined as the set of ρ a ∈ R r for which the D-terms have a solution.…”

“…A look at the symmetry charges shows that the E i must remain linear in the Σ a to maintain the classical U(1) L × U(1) R symmetry. Thus, the most general form of E-parameters takes the form 10) where the E ai µ are complex parameters. Since the monomials in the S i correspond to generators of the component of Aut(V ) connected to the identity, there is a direct relation between the E-parameters and the elements of the group Aut(V ) discussed in section 2.1.…”

“…As in [10,29] a Hermitian metric on O(d i ) → P 1 has been used in some of the definitions, so that the sections of the bundles in braces count zero modes of the kinetic operator in a fixed instanton background. Under this twist the supercharge Q + becomes the nilpotent world-sheet scalar operator Q T .…”

An introduction to quantum sheaf cohomology

“…However, this has been shown not to be the case. Historically, [1] first conjectured examples of (0,2) analogues of quantum cohomology, which were verified in the work [17] which gave a mathematical definition of (0,2) quantum cohomology and computed some basic examples. Later, [2] found a general physical argument explaining why (0,2) quantum cohomology can exist physically.…”

“…An analogue of quantum cohomology has been known since 2004, see e.g. [2,6,7,12,17,24,20,21,22,27,28]. Very recently, for deformations of the tangent bundle, there now exists a (0,2) monomial-divisor mirror map [23].…”