On TYPE II SUPERSTRINGS: THE Σ MODEL AND COMPACTIFICATION TO D=4

Abstract

We present a geometric construction of the σ model describing type II superstrings propagating on an arbitrary Mtarget. Specializing the covering space of the internal target manifold to be the nine-dimensional group manifold SU(2)3, we discuss the massless vertices both in the (4+9)-dimensional a model and in the D=4 superconformal theory, and show how they are related via dimensional reduction.

Gravitational Instantons in Heterotic String Theory: The H-Map and the Moduli Deformations of (4,4) Superconformal Theories

“…In the first part of this section we consider the bosonic σ-model, in the second part we extend the construction to locally supersymmetric σ-models of (1,1) type. Our results correspond to the generalization, with dilaton coupling, of the construction presented in [23]. Freezing the two-dimensional gravitinos one obtains the σ-model action with global (1,1) supersymmetry, that can be utilized to discuss the structure of the corresponding superconformal theory.…”

“…Let's briefly explain the somewhat unusual notations and the meaning of the quantities appearing in eq. (4.1) [13,23]. In particular e + and e − are the vielbein on the world-sheet ∂M, whose geometry is described by the structure equations…”

The Heterotic Supersymmetric Sigma Model in the Canonical Exterior Formalism

“…The extension to the geometric action of the (1, 1) σ model is straightforward [26], and it contains many more terms because of the presence of a second bi-dimensional spinor μ A .…”

“…Having the same structure they can be decomposed into four independent sectors corresponding to the inner-inner direction V + ∧ V − the inter-outer directions V + ∧ ζ and V − ∧ ζ and the outer-outer direction ζ ∧ ζ . The first step is to consider the Maurer-Cartan two-form equation (15) and the one-forms defined in (17), (26) and (27) decomposed along the supergravity background one-form fields (V + , V − , ζ ).…”

“…By straightforward calculation it can be shown: (i) Considering (78) it can be seen that the coefficients of the components V + ∧ ζ , V − ∧ ζ and ζ ∧ ζ cancel automatically when the rheonomic parametrization (see (17), (26) and (27)) is introduced. On the other hand, the cancellation of the component V + ∧ V − gives rise to the following condition…”