Equivariant Cartan-Eilenberg supergerbes for the Green-Schwarz superbranes III. The wrapping anomaly and the super-${\rm AdS}_5\times\mathbb{S}^5$ background

TL;DR

This paper advances the geometric understanding of superstring theory in curved backgrounds, specifically super-${ m AdS}_5 imes ext{ extonehalfmaths}^5$, by analyzing wrapping anomalies and supergerbes within a supersymmetry-equivariant framework.

Contribution

It develops enhanced geometric tools to study superstring propagation in super-${ m AdS}_5 imes ext{ extonehalfmaths}^5$, focusing on wrapping anomalies and their relation to supergerbes and algebraic contractions.

Findings

Identification of the wrapping anomaly in the super-σ-model

Construction of a trivial super-1-gerbe over super-${ m AdS}_5 imes ext{ extonehalfmaths}^5$

Analysis of the contraction from curved to flat superspace

Abstract

This is a continuation of a programme, initiated in the work arXiv:1706.05682 [hep-th], of supersymmetry-equivariant geometrisation of the Green-Schwarz super-$(p+2)$-cocycles coupling to the topological charges carried by super-$p$-branes of the superstring theory on reductive homogeneous spaces of supersymmetry Lie supergroups. In the present part, the ideas and geometro-algebraic tools developed previously are substantially enhanced, adapted to and applied in the physically significant curved backround of Metsaev and Tseytlin, determining the propagation of the critical superstring in the super-${\rm AdS}_5\times\mathbb{S}^5$ geometry. The analysis brings to the fore the r\^ole, in the geometrisation scheme proposed, of the wrapping anomaly of the Poisson algebra of the Noether charges of the rigid symmetries of the relevant super-{\sigma}-model that lift the geometric symmetries of the supertarget. In particular, the significance of the charges quantifying the monodromy of the Grassmann-odd coordinates in the Kosteleck\'y-Rabin-type quotient of the supertarget is emphasised. A trivial super-1-gerbe is associated with the Metsaev-Tseytlin super-3-cocycle over the super-${\rm AdS}_5\times\mathbb{S}^5$ target. The issue of compatibility of the geometrisation with the \.In\"on\"u-Wigner contraction of the supersymmetry Lie superalgebra to its flat-superspace counterpart is investigated at some length, revealing the rigidity of the relevant Cartan-Eilenberg cohomology and signalling an attractive potential alternative to the non-contractible trivial super-1-gerbe constructed.