Ricci flow, Courant algebroids, and renormalization of Poisson-Lie T-duality
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. Ševera}, Lett. Math. Phys. 105, No. 12, 1689--1701 (2015; Zbl 1344.53064)]. Physically, Poisson-Lie T-duality gives an isomorphism of two-dimensional \(\sigma\)-models (after a suitable reduction of the target spaces) as Hamiltonian systems. Let \((M,g)\) be a Riemannian manifold. Then, recall that, following \textit{D. Friedan} [``Nonlinear models in \(2+\varepsilon\)-dimensions'', Phys. Rev. Lett. 45, No. 13, 1057--1060 (1980; \url{doi:10.1103/physrevlett.45.1057})], the 1-loop renormalization group of the standard two-dimensional \(\sigma\)-model with the standard action functional is the Ricci flow \(dg/dt=-2\text{Ric}_g+L_Xg\), where \(L_X\) denotes the Lie derivative of an arbitrary vector field \(X\in\Gamma(TM)\). As it is clear from T-duality, it is convenient to introduce a closed 3-form \(H\) into the picture, and consequently, we shall consider \(\text{Ric}_{(g,H)}\), the Ricci tensor of the \(g\)-preserving connection on \(M\) with the torsion \(T\) given by \(g(T(-,-),-)=-H(-,-,-)\). In fact, we write \(H=db+H_0\), where \(b\) is a two-form on \(M\), and \(H_0\) is a closed 3-form. In this setting, it is not difficult to write an explicit action functional for the corresponding \(\sigma\)-model, and the 1-loop renormalization group flow is now \[ \frac{de}{dt}=-2\text{Ric}_{(g,H)}+L_X e+i_XH_0-d\alpha,\quad (1) \] where \(e=g+b\), and \(\alpha\in\Gamma(T^*M)\). Insightfully in [J. Geom. Phys. 114, 506--522 (2017; Zbl 1358.53071)], \textit{J. Streets} reinterpreted this expression in terms of Courant algebroids; the pair \((X,\alpha)\) is seen as a section of the standard Courant algebroid \(TM\oplus T^*M\), and the tensor field \(e\) is replaced by its graph (which is a subbundle of \(TM\oplus T^*M\)). In fact, it can be proved that this graph is a generalized metric. Under this light, (1) becomes an evolutionary partial differential equation of this subbundle. It is worthwhile to note that a generalized metric in an exact Courant algebroid is equivalent to a Riemannian metric and a closed 3-form.\newline \newline The compatibility of Poisson-Lie T-duality with the 1-loop renormalization group flow was showed in [\textit{K. Sfetsos} et al., ``Renormalization of Lorentz non-invariant actions and manifest T-duality'', Nucl. Phys. B 827, No. 3, 545--564 (2010; \url{doi:10.1016/j.nuclphysb.2009.11.001})] in the special case of \(M=\{pt\}\) under certain mild conditions, and an expression for a generalized Ricci tensor of a generalized metric was obtained. Then two natural questions arose. First, prove that the obtained expression really gives the generalized Ricci flow (1), and secondly, if it can be extended to Poisson-Lie T-duality when \(M\) is non-trivial. These two questions are the original motivations of the paper under review. Let \(V^+\) be a generalized metric in a Courant algebroid \(E\), with a compatible connection \(\nabla\) on the vector bundle that underlies \(E\). The main conceptual notion introduced in this article is the definition of a generalized Ricci flow \(\text{GRic}^{(\nabla)}_{V^+}\) for an arbitrary Courant algebroid (not only for exact ones). This object can be regarded either as a bilinear form or as an infinitesimal deformation of \(V_+\) (in the space of generalized metrics). Although the definition may be tricky in a first look, the authors introduced Feynmann-like diagrams to deal with anchors, connections or torsion of connections, and then the definition of \(\text{GRic}^{(\nabla)}_{V^+}\) is quite natural. These pictorial representations are used throughout the article to prove technical results and it is expected that they become very useful features in the field. This suitable definition of a generalized Ricci flow for arbitrary Courant algebroids implies (see Theorem 3) the compatibility between Poisson-Lie T-duality and the 1-loop renormalization group flow, generalizing the case of \(M=\{pt\}\) and simplifying the proof. As final remarks, note that in this work the authors use an usual connection on the vector bundle underlying the Courant algebroid to obtain \(\text{GRic}^{(\nabla)}_{V_+}\). In the literature (see for instance, [\textit{M. Gualtieri}, in: The many facets of geometry. A tribute to Nigel Hitchin. Oxford: Oxford University Press. 368--394 (2010; Zbl 1239.53104); \textit{M. Garcia-Fernandez}, ``Torsion-free generalized connections and heterotic supergravity'', Commun. Math. Phys. 332, No. 1, 89--115 (2014; \url{doi:10.1007/s00220-014-2143-5})]), there exists a definition of Ricci tensor of a generalized connection. The authors show (Remark 2) that the two definitions of Ricci tensor of a generalized metric coincide up to an inner derivation of the Courant algebroid. Finally, since \(\text{GRic}^{(\nabla)}_{V_+}\) is defined as an infinitesimal deformation of \(V_+\) the authors wonder if it defines (in the case of a compact manifold) a well-defined flow for some finite time.
generalized metric, Severa 3-form, Poisson-Lie T-duality
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