Ricci flow, Courant algebroids, and renormalization of Poisson-Lie T-duality
type: Item
instance of: scholarly article
title: Ricci flow, Courant algebroids, and renormalization of Poisson-Lie T-duality
zbMATH Open document ID: 1378.53095
author: Fridrich Valach
author: Pavol Ševera
published in: Letters in Mathematical Physics
publication date: 2017-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1610.09004
review text: One of the central conjectures in mathematical physics is the SYZ conjecture (formulated in [A. Strominger, S.-T. Yau, E. Zaslow; Nuclear Phys. B, 479(1-2) : 243--259, 1996]), which may be stated by saying that mirror symmetry between complex and symplectic structures on Calabi-Yau manifolds can be understood as an application of T-duality. Physically, T-duality is an equivalence between two-dimensional \(\sigma\)-models with different target spaces. Mathematically, as pointed out by \textit{P. Bouwknegt} et al. [Commun. Math. Phys. 249, No. 2, 383--415 (2004; Zbl 1062.81119)], T-duality relates topologically distinct torus bundles, and provides a method to transport geometrical structures between them. An essential point in this work was the interplay between the topology of the \(\sigma\)-model target and the Neveu-Schwarz 3-form flux \(H\), a closed 3-form with integral periods. Notably, in [in: A celebration of the mathematical legacy of Raoul Bott. Based on the conference, CRM, Montreal, Canada, June 9--13, 2008. Providence, RI: American Mathematical Society (AMS). 341--365 (2010; Zbl 1200.53062)], \textit{G. R. Cavalcanti} and \textit{M. Gualtieri} proved that T-duality can be understood as a Courant algebroid isomorphism between the spaces in question. In fact, this description, allows us to transport Dirac structures, generalized Riemannian metrics or generalized complex and Kähler structures. In fact, historically, exact Courant algebroids (i.e., Courant algebroids \(E\) over a smooth manifold \(M\) that fits in the short exact sequence \(0\rightarrow T^*M\rightarrow E \rightarrow TM\rightarrow 0\)) and their reductions were introduced by \textit{P. Ševera} [``Letters to Alan Weinstein about Courant algebroids'', Preprint, \url{arXiv:1707.00265}] as the underlying geometric structure of Poisson-Lie T-duality; a kind of ``non-abelian'' T-duality, where the abelian Lie groups (tori) of original T-duality are replaced by non-abelian Lie groups [\textit{P. Ševe…
reviewed by: David Fernández
Mathematics Subject Classification ID: 53D18
Mathematics Subject Classification ID: 81T17
zbMATH DE Number: 6793830
zbMATH Keywords: Courant algebroids
zbMATH Keywords: generalized Ricci tensor
zbMATH Keywords: renormalization group flow
zbMATH Keywords: Ricci flow
zbMATH Keywords: T-duality
zbMATH Keywords: Dirac structures
zbMATH Keywords: generalized complex structures
zbMATH Keywords: generalized metric, Severa 3-form, Poisson-Lie T-duality
MaRDI profile type: Publication
OpenAlex ID: W2545492900
arXiv ID: 1610.09004
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DOI: 10.1007/S11005-017-0968-5
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description: scientific article