Supergeometry, super Riemann surfaces and the superconformal action functional
type: Item
instance of: scholarly article
title: Supergeometry, super Riemann surfaces and the superconformal action functional
zbMATH Open document ID: 1423.81006
DOI: 10.1007/978-3-030-13758-8
author: Enno Keßler
published in: Lecture Notes in Mathematics
publication date: 2019-04-11T00:00:00Z
review text: This monograph is concerned with the question how the superconformal action functional, a supersymmetric extension of the harmonic action functional on Riemann surfaces, is related to super Riemann surfaces and their moduli, laying the groundwork of the moduli space of super Riemann surfaces via the superconformal action functional and showing fantastic similarities to the theory of Riemann surfaces and harmonic maps. The superconformal action functional appeared in the context of string theory already in the 1970s [\textit{S. Deser} and \textit{B. Zumino}, ``A complete action for the spinning string'' Phys. Lett. B 65, No. 4, 369--373 (1976; \url{doi:10.1016/0370-2693(76)90245-8})], when the mathematical theory of supergeometry developed. Super Riemann surfaces, a supergeometric analogue of Riemann surfaces, appeared only a little later, their significance for string theory being appreciated [\textit{D. Friedan}, in: Unified string theories, Workshop, Santa Barbara/Calif. 1985, 162--213 (1986; Zbl 0648.53057); \textit{E. D'Hoker} and \textit{D. H. Phong}, ``The geometry of string perturbation theory'', Rev. Modern Phys. 60, No. 4, 917--1065 (1988; \url{doi:10.1103/revmodphys.60.917})]. The principal objective in this monograph is to build a bridge between the mathematical theory of supergeometry and super Riemann surfaces with the superconformal action functional motivated by physics. The reader is strongly referred to \textit{E. Witten}'s [Pure Appl. Math. Q. 15, No. 1, 517--607 (2019; Zbl 1421.81102); ibid. 15, No. 1, 213--516 (2019; Zbl 1421.81101); ibid. 15, No. 1, 57--211 (2019; Zbl 1423.32012); ibid. 15, No. 1, 3--56 (2019; Zbl 1421.58001)]. The monograph consists of 13 chapters and two appendices. Chapter 2 is concerned with linear superalgebra. There are, roughly speaking, three approaches to supermanifolds, namely, the Rogers-DeWitt approach, the approach via ringed spaces (also called the Berezin-Kostant-Leites approach) and the approach via functors of p…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 53Z05
Mathematics Subject Classification ID: 14D21
Mathematics Subject Classification ID: 81T60
Mathematics Subject Classification ID: 14M30
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 58A50
zbMATH DE Number: 7045636
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-13758-8
OpenAlex ID: W2971178665
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description: scientific article; zbMATH DE number 7045636