New Geometrical Approach to Superstrings
Alexander Belopolsky
TL;DR
This work develops a geometrical framework for superstring perturbation theory based on differential forms on supermanifolds and integration over moduli, enabling amplitudes to be computed from BRST cohomology classes with arbitrary representatives. It builds a full chiral/antichiral BRST structure using Baranov-Schwarz projections, Plücker forms, and picture-changing operators within a topological superconformal field theory setting to handle arbitrary backgrounds and non-discrete states. The approach yields a geometrical proof of the dilaton theorem, clarifies how background symmetries arise from BRST cohomology, and demonstrates spacetime supersymmetry without reliance on picture changing, with explicit treatment of type II strings. Overall, the framework provides a background-agnostic, mathematically robust route to multi-loop superstring amplitudes and symmetry analyses, including both even and odd background symmetries.
Abstract
We present a new geometrical approach to superstrings based on the geometrical theory of integration on supermanifolds. This approach provides an effective way to calculate multi-loop superstring amplitudes for arbitrary backgrounds. It makes possible to calculate amplitudes for the physical states defined as BRST cohomology classes using arbitrary representatives. Since the new formalism does not rely on the presence of primary representatives for the physical states it is particulary valuable for analyzing the discrete states for which no primary representatives are available. We show that the discrete states provide information about symmetries of the background including odd symmetries which mix Bose and Fermi states. The dilaton is an example of a non-discrete state which cannot be covariantly represented by a primary vertex. The new formalism allows to prove the dilaton theorem by a direct calculation.