Obstruction Theory for Supermanifolds and Deformations of Superconformal Structures

Abstract

The interplay between geometry and supergeometry, from an algebraic point of view, sets the theme guiding the considerations in this thesis. In the smooth setting there is a sense in which these geometries can be identified and, in the complex (i.e., holomorphic) setting, such an identification no longer holds. As such, for at least this reason, complex algebraic supergeometry can find interest in its own right. It is the subject of this thesis and we study it here under two broad headings: obstruction theory and deformation theory.

Under the umbrella of obstruction theory, we focus largely on foundational aspects of supermanifolds and their description by means of supersymmetric thickenings. We start from the general principle that: any supermanifold will define a supersymmetric thickening but not necessarily conversely. One of the key objectives in this part of the thesis is in precisely formulating and proving this principle by elementary methods. We complement the proof given with examples of obstructed thickenings on the complex projective plane. To illustrate obstruction theory more generally for complex supermanifolds, we include and comment on a collection of examples from the literature, in addition to providing some new examples. Moreover, we will also consider the splitting problem for complex supermanifolds. Upon obtaining a characterisation of the obstruction classes to splitting via the grading vector field, we present a new proof of the Koszul splitting theorem for supermanifolds.

Regarding deformation theory, we concern ourselves with the construction of (odd) infinitesimal deformations of superconformal structures. These are structures on supermanifolds and, in the one dimensional case, arise under the guise of super Riemann surfaces. Explicit and elementary constructions of (odd) deformations are given for N = 1 and N = 2 super Riemann surfaces. One of the key objectives in this part of the thesis is on establishing precise relations between (1) the deformation theory of these super Riemann surfaces and (2) their obstruction theory as supermanifolds.

We conclude this thesis with a brief sketch on the state of supermoduli spaces, in both the N = 1 and N = 2 setting, as it presently stands in the literature. These discussions lead naturally toward directions for future research and paint a grander scheme in which this thesis sits. Show more