Schedule for: 24w5222 - Generalized Geometry Meets String Theory

Abstract: Generalized geometry, introduced by Hitchin in the early 2000s, has not only grown into a formalism for understanding previously mysterious constructions in physics (especially in supersymmetric sigma models) but has also led to the discovery of new mathematical structures such as generalized complex manifolds, generalized Riemannian and Kähler structures, and many other extensions of classical geometric structures such as G2 structures. One of the themes underlying these developments, which originates in the ideas of the Weinstein school of Poisson geometry, is the appearance of differentiable stacks and shifted symplectic structures on them. I will give an overview of these developments aimed at a general mathematical audience with an interest in geometry.

Abstract: I will start by reviewing the definition for the Ricci tensor of a pair formed by a generalized metric and a divergence operator on an arbitrary Courant algebroid. A generalized metric $E_{-}$ is called generalized Einstein with divergence $\delta$ if the Ricci tensor of the pair $(E_{-}, \delta)$ vanishes identically. After recalling what an odd exact Courant algebroid is, I will come to main result of my talk, which is a description of all left-invariant generalized Einstein metrics with left-invariant divergence on odd exact Courant algebroids over 3-dimensional Lie groups. This work is in collaboration with Vicente Cortés. For details, see arxiv: 2311.00380.

Abstract: A Kähler metric on a smooth manifold is a compatible pair of a complex structure and a symplectic form that combine to a Riemannian metric. The space of Kähler metrics in a given Kähler class is an open affine space over smooth functions. Alternatively, one can describe Kähler metrics as an open affine space over Lagrangian sections of the smooth cotangent bundle with the canonical symplectic structure. Generalized Kähler metrics of symplectic type share a similar description. They form a torsor over Lagrangian bisections of a symplectic groupoid over the manifold. The formal space of Lagrangian branes in a symplectic manifold carries a natural integrable complex structure. We introduce the concept of holomorphically varying Lagrangian branes and describe a procedure to equip the complex parameter space of holomorphic families of Lagrangian branes with a symplectic structure. We then conclude that there is a Kähler metric on the moduli of prequantized generalized Kähler metrics of symplectic type. This lays the foundation for Hamiltonian geometry on the space of generalized Kähler metrics.

Abstract: After a brief summary of the main feature of EGG and its structures that are relevant in supergravity, I will discuss a work in progress on non-supersymmetric backgrounds.

Abstract: Generalized complex structures encompass complex and symplectic ones and go beyond them: there are generalized complex manifolds that are neither complex nor symplectic. However, they are only possible on almost complex manifolds, and hence in even dimensions. In this talk, I will discuss how Bn-generalized geometry, a simple and natural variation of generalized geometry, provides a similar framework for manifolds of any dimension and I will focus on the analogue of generalized complex structures for 3-manifolds. This talk is based on recent and ongoing joint work with J. Porti.

Abstract: The Hull-Strominger system of equations arose in mathematical physics and has been proposed as a tool for uniformization problems in complex geometry, in particular Reid’s fantasy. In this talk I will describe a new approach to solving these equations using generalized Ricci flow/pluriclosed flow. I will furthermore describe a conceptual modification of these equations which leads to a different approach to Reid’s fantasy.

Abstract: Let $M$ be a smooth manifold with tangent bundle $T$ and cotangent bundle $T^*$. By a generalized structure on $M$, we mean an endomorphism of $T \oplus T^*$ that squares to $\pm Id_{T \oplus T^*}$. In this talk, we consider pairs of generalized structures on $M$ whose product is a generalized metric. An example of such commuting pairs is given by generalized K\"ahler structures. There are three other types of such commuting pairs: generalized para-K\"ahler, generalized chiral and generalized anti-K\"ahler structures. We discuss the integrability of these structures and explain how para-hyper-Hermitian and Born geometry fit into this generalized context.

Abstract: Categorified principal bundles and connections on them can be used to model fields in string theory, and their relation with Courant algebroids has been a matter of study since the beginnings of generalized geometry. In this talk we will present this formalism and we will show in which way Courant algebroids of string/heterotic type encode infinitesimal symmetries of string-theoretic systems of equations such as the Hull-Strominger system. We will also provide a Chern correspondence for principal 2-bundles suggesting that algebraic methods might be applicable to the construction of moduli spaces for these theories, as in the classical case of Yang-Mills equations. This is based on arXiv:2310.12738 and ongoing work with L. Álvarez-Cónsul and M. Garcia-Fernandez.

Abstract: In the talk I will report some results on existence of special Hermitian structures, like balanced, pluriclosed, and generalized Kaehler, on mapping tori. In the first part of the talk I will show a construction via toric suspensions of hyperKaehler manifolds, obtained in collaboration with Gueo Grantcharov and Misha Verbitsky. Next, I will discuss joint results with Beatrice Brienza and Gueo Grantcharov on compact complex manifolds admitting a (not Bismut flat) pluriclosed metric with vanishing Bismut–Ricci form. The existence of generalized Kähler structures on mapping tori is also investigated.

Abstract: a-maximisation is a key property of 4d superconformal field theories that allows one to find the R-symmetry and central charge. Understanding the generic gravitational dual of this process is a central question in holography. I will describe some work in progress with Anthony Ashmore, Stephanie Baines and Michela Petrini, where we argue that it can be understood in terms of a notion of stability for AdS backgrounds in generalised geometry, in a way that physically connects closely to renormalisation group flow. For simplicity we will focus on the special case of Sasaki-Einstein geometries, where, using SL(d+1) generalised geometry, one reproduces (and extends) the volume minimisation procedure of Martelli-Sparks-Yau.

Abstract: I will review some old and new questions about the heterotic flux backgrounds with an emphasis on their global aspects.

Abstract: I will review some applications of generalized geometry to the study of supergravity solutions, focusing on type II. The Minkowski case leads to generalized K\"ahler or complex structures, depending on whether RR fluxes are present. The cosmological constant leads to a generalization of half-flat geometry with several notable examples. An open problem is the presence of universal bounds on the ratio of the internal diameter and the AdS curvature radius. I will also present similar results in different dimensions, with links to Sasaki--Einstein geometry and to generalized G2 structures. I will end with some comments on the case where supersymmetry is broken.

Carmen de la Victoria

Hotel Tent Granada

Abstract: The notion of a generalized Kähler (GK) structure was introduced in the early 2000’s by Hitchin and Gualtieri in order to provide a geometric framework of certain nonlinear sigma model theories that has been studied in physics. Since then, the subject developed rapidly. It is now realized, thanks to Hitchin, that GK structures are naturally attached to Kähler manifolds endowed with a holomorphic Poisson structure. Inspired by Calabi’s program in Kähler geometry, which aims at finding a ''canonical” Kähler metric in a fixed deRham class, and recent works by Goto and Gualtieri, I will present in this talk an approach towards a “generalized Kähler” version of Calabi’s problem motivated by an infinite dimensional moment map formalism. As an application, I will give an essentially complete resolution of this problem in the case of a toric complex Poisson variety. Based on a joint work with J. Streets and Y. Ustinovskiy.

Abstract: It is a long-standing problem to find the explicit supergravity backgrounds dual to generic N=1 deformations of N=4 super Yang-Mills. Using the formalism of generalised geometry, we show that the sought-for supergravity solutions naturally encode a kind of generalised holomorphic structure, dual to the superpotential of the field theory. With this perspective, we are able to solve almost all of the supersymmetry conditions for the deformed supergravity backgrounds, with the full solution following from a continuity argument. This formalism provides a powerful tool for studying the dual deformed field theories. For example, using only the holomorphic structure, we derive a new result for the Hilbert series of the deformed field theories.

Abstract: Owing to insights from string theory, it has been understood for quite some time now that the Kähler cone of a Calabi–Yau threefold (i.e. a complex threefold with trivial canonical bundle) should not be considered in isolation but as part of an extended Kähler cone consisting of the individual Kähler cones of birationally equivalent Calabi–Yau threefolds. However, birational transformations do not preserve the Kähler property and it is a priori unclear how to make sense of the "Kähler cone" of a non-Kähler Calabi–Yau threefold. In light of this, it has been suggested by Yau that the Hull–Strominger system arising from the equations of motion of a heterotic superstring is a more appropriate setting for studying such geometric transitions than the Kähler Ricci-flat condition arising from the equations of motion of a Type II superstring. In cumulative work by García Fernández, Rubio, Tipler, and Shahbazi, it was observed that under certain technical assumptions, there is an infinitesimal Donaldson–Uhlenbeck–Yau-type correspondence between infinitesimal deformations of solutions of the Hull–Strominger system and certain objects in generalised geometry called holomorphic string algebroids, together with certain (1,1) Aeppli classes of the Calabi–Yau threefold. An implication of such a correspondence is that there is a natural affine structure on the moduli space of solutions of the Hull–Strominger system corresponding to a fixed holomorphic string algebroid. This affine structure is modelled on a vector space related to the (1,1) Aeppli cohomology of the Calabi–Yau threefold and may be regarded as the Hull–Strominger analogue of the Kähler cone. In this talk, which is based on upcoming joint work with Mario García Fernández and Raúl González Molina, I build on this observation by identifying an analogue of the hard Lefschetz property that implies the existence of a certain weaker version of a variation of polarised Hodge structure on the complexification of the Hull–Strominger analogue of the Kähler cone, and show that this property indeed holds in certain families of examples.

Abstract: I'll explain how the generalized Kähler class can be defined in terms of Morita equivalences of symplectic double groupoids and I’ll explain how this framework allows us to determine the fundamental degrees of freedom of a generalized Kähler metric in full generality. If time permits I’ll describe how these ideas can be explicitly illustrated in the case of compact Lie groups. This is joint work with Marco Gualtieri and Yucong Jiang.

Hotel Tent Granada

Abstract: Generalised Ricci flow plays an important role in string theory and the associated supergravity theories, while also linking various deep questions in geometry. I will discuss some recent results in the theory of this flow on a general class of Courant algebroids. I will show a fully generalised-geometric proof of various central properties, such as the short-time existence and uniqueness of solutions, scalar curvature monotonicity formula, and gradient property of the flow. This is joint work with Jeffrey Streets and Charles Strickland-Constable.

Abstract: I will discuss the old problem of determining the exact moduli of generic SU(3)-structure flux backgrounds of type II string theory. Using Exceptional Complex Structures (ECS), I will show that the infinitesimal deformations are counted by a spectral sequence in which the vertical maps are either de Rham or Dolbeault differentials (depending on the type of the ECS) and the horizontal maps are linear maps constructed from the flux and intrinsic torsion. I will argue that our results are exact to all orders in string perturbation theory and are valid for large flux. Nonetheless, the spectral sequences reproduce the naive results one would expect from the linearised-flux calculation around the supergravity background. I will also discuss obstructions from higher order deformations showing that a Tian-Todorov-like lemma implies the vanishing of obstructions. This has important implications for the Tadpole Conjecture, showing that higher order effects do not allow one to circumvent the conjecture.

Abstract: Lie algebroids have become a popular tool for describing geometric structures with singularities. This is especially prevalent in Poisson geometry where mildly degenerate Poisson structures can be understood as symplectic structures on Lie algebroids, but also certain generalized complex structures can be described in this manner. In this talk we will study a class of Lie algebroids, called of b^k-type, which are used to study geometric structures with poles of order k along a hypersurface. We will obtain a classification of such Lie algebroids in terms of distributions which satisfy a k-th order involutivity condition. If time permits we will discuss the implications of this classification for the deformation theory of symplectic structures with poles along a hypersurface. Joint work with Álvaro del Pino Gomez and Francis Bischoff.

Part III and IV of Carlos Shahbazi's course are scheduled after the end of the workshop to avoid overlap with the talks. Whoever is interested is welcome to join!