Notes On Supermanifolds and Integration

TL;DR

These notes survey the geometric framework of supermanifolds and two complementary integration formalisms (Berezinian sections and integral forms), with a focus on applications to superstring perturbation theory in the RNS formalism. They develop the structure of smooth and complex supermanifolds, their reduced spaces, submanifolds, and holomorphic splittings, and then build a comprehensive integration theory, including Stokes theorem, integration over fibers, and submanifold integration, via both Ber and integral-form perspectives. The work introduces differential and integral forms through Clifford and Weyl algebras, explains picture-changing operations, and clarifies how these notions underpin worldsheet and supermoduli-space constructions in string theory. It also discusses two viewpoints on relating complex manifolds to cs supermanifolds, the periods of holomorphic Berezinian sections, and the role of noncompact moduli and integration cycles in perturbative calculations. Collectively, the notes provide a practical, background-oriented toolkit for rigorously handling supergeometric aspects of superstring perturbation theory, including worldsheet integration and supermoduli-space considerations.

Abstract

These are notes on the theory of supermanifolds and integration on them, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.