# Geometry, Gravity and Supersymmetry

Europe/Berlin
02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

### 02.430

#### Mainz Institute for Theoretical Physics, Johannes Gutenberg University

Staudingerweg 9 / 2nd floor, 55128 Mainz
Description
The goal of the workshop is to bring together geometers and mathematical physicists working on problems related to two related areas: the classification of supergravity backgrounds and the study of moduli spaces and special geometric structures, in order to first gauge the progress that has been made so far in these research areas, and then to identify directions in which these areas can develop further.
• Monday, 24 April
• 09:30 10:30
Arrival/Registration 1h
• 10:30 10:40
Introduction 10m
• 10:40 11:25
Progress on supersymmetric solutions of gauged supergravities 45m
The construction of supersymmetric solutions of gauged supergravities in N=2, d=4 and 5 dimensions is a very complicated, highly non-linear problem both in the Abelian and non-Abelian cases. In the N=2,d=5 Abelian case we have recently proposed a new Ansatz for the Kaehler base space that simplifies the problem and establishes a connection with the N=2,d=4 case, for cubic models [arXiv:1611.09383]. In the non-Abelian case we have characterized all the timelike supersymmetric solutions (arXiv:0806.1477, arXiv:1512.07131) and we have constructed, analytically, (multi) black-hole, black-ring and black-string solutions and also globally-regular monopoles and instanton solutions with non-trivial SU(2) fields. The fact that they are given in a fully analytic solution (unlike all the previous non-Abelian solutions known so far) allows us to study them in detail. We find that they exhibit glaring violations of the no-hair conjecture. We embed some of these solutions in 10-dimensional Heterotic string theory to identify their elementary brane constituents and compute their entropy (when there is a horizon) from a microscopic point of view. A very important role in our constructions is played by the relation found by Kronheimer between monopoles in $R^3$ and instantons in 4-dimensional hyper-Kaehler spaces admitting the action of a U(1) group. We use it to construct multi-dyonic instanton solutions. We also find that singular monopoles, usually discarded, can be used in black-hole solutions since the singularities are covered or smoothed by the non-trivial geometry.
Speaker: Prof. Tomas Ortin (IFT-UAM/CSIC)
• 11:25 11:45
Break 20m
• 11:45 12:15
Killing superalgebras and high supersymmetry 30m
I will talk about joint work with José Figueroa-O'Farrill on the algebraic structure of the Lie superalgebra generated by the Killing spinors of an 11-dimensional supergravity background. I will explain that any such Killing superalgebra can be regarded as an appropriate deformation of a subalgebra of the Poincaré superalgebra and discuss applications to the classification of highly supersymmetric backgrounds. In particular, we will see that preserving more than half the supersymmetry implies the supergravity field equations. If time permits, I will also mention ongoing work on the existence of half-BPS backgrounds whose Killing superalgebras are deformations of the Killing superalgebra of the elementary M2-brane.
Speaker: Dr Andrea Santi (University of Bologna)
• 12:15 15:45
Lunch/Discussion 3h 30m
• 15:45 16:30
Quaternionic Heisenberg group and Heterotic String Solutions with non-constant dilaton in dimensions 7 and 5 45m
Smooth solutions of the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton based on the quaternionic Heisenberg group are constructed. It is shown that through appropriate contractions the solutions found in the $G_2$-heterotic case converge to the heterotic solutions on 6-dimensional inner non-K\"ahler spaces previously found and, moreover, to new heterotic solutions with non-constant dilaton in dimension 5. All the solutions satisfy the heterotic equations of motion up to the first order of \alpha^{\prime}.
Speaker: Prof. Stefan Ivanov (Sofia University)
• 16:30 17:00
Completeness in Projective Special Geometry 30m
One way to construct quaternionic Kähler manifolds (target spaces for sigma models in $N=2$ supersymmetric theories of gravity) is via the supergravity c-map (Ferrara-Sabharwal'90) from projective special Kähler manifolds and its combination with the supergravity r-map (deWit-Van Proeyen'92) from projective special real manifolds. After briefly introducing the structures involved I will discuss conditions for the geodesic completeness of projective special real and projective special Kähler manifolds. It is known that geodesic completeness is preserved under both the r-map and the c-map (Cortés-Han-Mohaupt'12). I will argue that projective special Kähler manifolds are complete if the condition of "regular boundary behavior" is satisfied (Cortés-Dyckmanns-Suhr'17). Projective real manifolds will be shown to be complete if they are closed as hypersurfaces in Euclidian space (Cortés-Nardmann-Suhr).
Speaker: Dr Stefan Suhr (Department of Mathematics, University of Hamburg)
• Tuesday, 25 April
• 09:30 10:15
Conformal symmetry and supergravity 45m
Conformal symmetry is the maximal spacetime symmetry. Supergravity plays an important role in the developments of quantum theories including gravity, as in superstring theory. Considering supergravity as a broken superconformal theory lead to many insights in the structure of supergravity. We review these developments and report on recent investigations on constrained multiplets, the Einstein equation and conserved currents in a supergravity theory from the conformal point of view.
Speaker: Prof. Antoine Van Proeyen (KU Leuven)
• 10:15 10:45
Hidden symmetries of black holes in maximal supergravities 30m
Hidden symmetries corresponding to Killing tensors are a notable feature of known rotating black hole solutions in supergravity. I will discuss the presence of these symmetries in general, non-extremal black holes of maximal supergravities in diverse dimensions, their relation to commuting symmetry operators, and other physical consequences.
Speaker: David Chow (Max Planck Institute for Gravitational Physics)
• 10:45 11:05
Break 20m
• 11:05 11:50
A Riemann-Hilbert approach to attractor geometries and extremal black holes. 45m
We review in which sense the stationary-axisymmetric sector of four- and five-dimensional gravity is integrable, and how the non-linear equations of motion can be viewed as the integrability conditions of a linear system. Solving this linear system can in turn be formulated as a matrix Riemann-Hilbert factorisation problem. While this has been known (at least) since the seminal work of Breitenlohner and Maison in the late 1980's, explicit solutions of Riemann-Hilbert problems are hard to obtain, and have so far been restricted to cases where the space-time is asymptotically flat, and where the monodromy matrix has first order poles in the spectral parameter. We present an efficient method for obtaining explicit matrix factorisations which avoids these assumptions, and show that the factorisation of monodromy matrices with second order poles leads to attractor geometries (near horizon geometries of extremal black holes) and extremal black hole solutions. Examples include both over-rotating and under-rotating extremal black holes and their near-horizon geometries. We also study the action of the Geroch group on monodromy matrices and on solutions of the linear system. Group elements which realise Harrison-type transformations relate attractor geometries to the corresponding extremal black hole solutions. A family of consistent non-trivial deformations of the monodromy matrix is shown to correspond to a family of Geroch group elements depending explicitly on the spectral parameter.
Speaker: Dr Thomas Mohaupt (Department of Mathematical Sciences, University of Liverpool)
• 11:50 15:45
Lunch/Discussion 3h 55m
• 15:45 16:30
New AdS/CFT pairs from non-Abelian T-duality 45m
I will discuss recent applications of non-Abelian T-duality in the context of Holography. I will discuss the usefulness of the method as a solution generating technique of AdS solutions as well as its interpretation at the level of the CFTs dual to the newly generated backgrounds.
Speaker: Prof. Yolanda Lozano (University of Oviedo)
• 16:30 17:00
Duality and the Generalised Geometry of (2,1) Sigma Models in Superspace 30m
The geometry, isometry symmetries and superspace formulation of two-dimensional non-linear sigma models with manifest (2,1) supersymmetry is revisited. Following Rocek and Verlinde, we then dualise the models by gauging the isometries using suitable (2,1) gauge multiplets and introducing Lagrange multiplier superfields that constrain the field strengths of the gauge multiplets to vanish. Integrating out the Lagrange multipliers leads to the original action, whereas integrating out the vector multiplets yields the dual action. The geometry of the dual model is defined implcitly in terms of the appropriate vector superfield potentials. Superspace reductions and the relation to the usual Buscher rules are investigated.
Speaker: Dr Mohab Abou Zeid (Georg-August-Universität Göttingen)
• Wednesday, 26 April
• 09:30 10:15
Dynamical symmetry enhancement near black hole horizons 45m
In this talk I will sketch a general proof of the (super)symmetry enhancement occurring near black hole horizons, a phenomenon previously observed only on a case by case basis. I will also show that the symmetry algebra for all supersymmetric black hole horizons with non-trivial fluxes includes an sl(2,R) subalgebra.
Speaker: Prof. Ulf Gran (Chalmers University of Technology)
• 10:15 10:45
The ASK/PSK-correspondence and the r-map 30m
We formulate a correspondence between affine and projective special Kähler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special Kähler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special Kähler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative $\alpha'$-corrections in heterotic and type-II string compactifications with $N=2$ supersymmetry. We show that the completeness of the deformed supergravity r-map metric depends solely on the (well-understood) completeness of the undeformed metric and the sign of the deformation parameter.
Speaker: Mr Peter-Simon Dieterich (Universität Hamburg)
• 10:45 11:05
Break 20m
• 11:05 11:50
Rigidity of 2-step Carnot algebras 45m
We study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions for each of the choices. Explicit criteria for rigidity of pseudo $H$- and $J$-type algebras are given. In particular, we establish the relation of the so-called $J^2$-condition to rigidity, and we explore these conditions in relation to pseudo $H$-type algebras. Based on work [(arXiv:1603.00373 [math.RT])][1] with Mauricio Godoy-Molina, Irina Markina and Alexander Vasil'ev. [1]: https://arxiv.org/abs/1603.00373
Speaker: Prof. Boris Kruglikov (University of Tromsø)
• 11:50 15:45
Lunch/Discussion 3h 55m
• 15:45 16:30
Left-invariant Einstein metrics on $S^3 \times S^3$ 45m
The classification of homogeneous compact Einstein manifolds in dimension six is an open problem. We consider the remaining open case, namely left-invariant Einstein metrics $g$ on $G = \mathrm{SU}(2) \times \mathrm{SU}(2) = S^3 \times S^3$. Einstein metrics are critical points of the total scalar curvature functional for fixed volume. The scalar curvature $S$ of a left-invariant metric $g$ is constant and can be expressed as a rational function in the parameters determining the metric. The critical points of $S$, subject to the volume constraint, are given by the zero locus of a system of polynomials in the parameters. In general, however, the determination of the zero locus is out of reach with current technology. Instead, we consider the case where $g$ is, in addition to being left-invariant, assumed to be invariant under a non-trivial finite group $\Gamma$ of isometries. When $\Gamma\not\cong \mathbb{Z}_2$ we prove that the Einstein metrics on $G$ are given by (up to homothety) either the standard metric or the nearly K\"ahler metric, based on representation-theoretic arguments and computer algebra. For the remaining case $\Gamma\cong \mathbb{Z}_2$ we present partial results.
Speaker: Dr Alexander Haupt (University of Hamburg)
• 16:30 17:00
Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimension 30m
Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. We study such spaces $M$ in three important cases. First, we assume $\langle\cdot,\cdot\rangle$ is invariant, in which case the Einstein property requires that $G$ is either solvable or semisimple. Next, we investigate the case where $G$ is solvable. Here, $M$ is compact and $M=G/\Gamma$ for a lattice $\Gamma$ in $G$. We show that in dimensions less or equal to $7$, compact quotients $M=G/\Gamma$ exist only for nilpotent groups $G$. We conjecture that this is true for any dimension. In fact, this holds if Schanuel's Conjecture on transcendental numbers is true. Finally, we consider semisimple Lie groups $G$, and find that $M$ splits as a pseudo-Riemannian product of Einstein quotients for the compact and the non-compact factors of $G$. This is joint work with Yuri Nikolayevsky (La Trobe University).
Speaker: Dr Wolfgang Globke (University of Adelaide)
• Thursday, 27 April
• 09:30 10:15
Six-dimensional conformal theories and nilpotent orbits 45m
I will discuss a class of CFTs obtained by Higgsing the (2,0) ADE theory by a pair of nilpotent elements in the ADE group. In the A and D case, the nilpotent elements are labeled by Young diagrams, and the holographic dual is known explicitly in IIA supergravity. In the E case and in some D cases, only an F-theory description is available, using sevenbranes with a pole for the Higgs field. The effect of these "T-branes" is conjectural, but it passes several numerical tests. Compactifying to four dimensions should produce new "massive class S" theories; for some easy cases, the AdS5 duals are now emerging, even in presence of punctures.
Speaker: Prof. Alessandro Tomasiello (Università di Milano-Bicocca)
• 10:15 10:45
Gravity duals to 6d (1,0) SCFTs on punctured Riemann surfaces 30m
We present AdS(5) solutions of type IIA supergravity, holographically dual to compactifications of six-dimensional (1,0) superconformal field theories on Riemann surfaces with punctures. The solutions have O8-plane, D4, D6 and D8-brane sources. The D4-branes are smeared over a Riemann surface, and are interpreted as a large number of punctures.
Speaker: Dr Achilleas Passias (Uppsala University)
• 10:45 11:05
Break 20m
• 11:05 11:50
Homogeneous solutions of decomposable 11D supergravity 45m
We will describe a class of solutions of 11 dimensional supergravity (M, g_M, F) under the assumption that the Lorentz manifold (M, g_M) is a direct product of a p-dimensional Lorentz manifold (M, g) and q = 11 − p - dimensional Riemann manifold (M ̄,g ̄). We made also some assumption about the structure of the 4-form F and write down and analyze the equa- tions of 11d supergravity in this case under assumption that the manifold (M,g_M,F) is homogeneous. In the case p = 4, we describe all such homo- geneous solutions of 11d decomposable supergravity under assumption that the 4-form is a direct sum of the volume form of a homogeneous Lorentz manifold (M,g) and an invariant 4-form of a compact homogeneous Riemannian manifold (M ̄ , g ̄). This is a joint work with I. Crysikos and A. Taghavi-Chabert.
Speaker: Prof. Dmitri Alekseevsky (Moscow)
• 11:50 15:30
Lunch/discussion 3h 40m
• 15:30 16:00
Geometry of BPS vortex-antivortex moduli spaces 30m
Gauged nonlinear sigma-models on Riemannian surfaces are self-dual field theories which admit supersymmetric extensions. In contrast with the more familiar gauged *linear* sigma-models, their classical vacua accommodate several types of BPS solitonic particles, described by vortex moduli spaces (supporting Kähler metrics) to which path integrals localise. This leads to some intricate phenomena -- in particular, the moduli spaces typically acquire a boundary corresponding to coalescence of particles of distinct types. In my talk, I will discuss geometric aspects of vortex moduli that relate to the presence of such a boundary, focusing on the simplest model of this sort. (This is joint work with Martin Speight.)
Speaker: Dr Nuno Romao (Universität Augsburg)
• 16:00 16:30
Killing spinor identities, spinorial geometry and off-shell supergravities 30m
Killing spinor identities relate (components of) the equations of motion of supergravity theories for supersymmetric solutions to one another, and tell us which of them are automatic for a given supergravity background. In the context of off-shell supergravities they are valid for any possible Lagrangian with the same matter content, including higher derivative and string theory corrected theories. Using spinorial geometry techniques we discuss in detail the example of the supersymmetric solutions of $\mathcal{N}=2, d=5$ supergravity with Weyl tensor squared and Ricci scalar squared corrections in a consistent truncation which is sufficient to capture the corrections at first order in the ungauged theory, and comment briefly on the general untrunacted case, focusing on the maximally supersymmetric solutions.
Speaker: Dr Peter Sloane (Mesoamerican Centre for Theoretical Physics, Universidad Autonoma de Chiapas, Mexico)
• 16:30 17:00
Tamed symplectic vector bundles for four-dimensional Supergravity 30m
I will introduce a system of partial differential equations on a four-manifold M involving a Lorentzian metric coupled to a section of a Lorentzian submersion as well as to a positively-polarized two-form taking values on a tamed flat symplectic vector bundle. I will give a set of sufficient conditions for this system to reduce, when restricted to a contractible open set of M, to the standard local formulas used by physicists in the context of four-dimensional ungauged Supergravity, obtaining thus a mathematically rigorous and globally non-trivial formulation of the latter. We call the resulting theory a "Generalized Einstein-Section-Maxwell theory", or GESM for short. GESM provides a non-trivial extension of standard ungauged supergravity, and its supersymmetrization is expected to yield new Supergravity theories that escape the standard local classification of these theories. Furthermore, I will consider the global automorphism group of a generic GESM theory, giving then a precise definition and characterization of the global U-duality group of the theory. I will also discuss Dirac quantization, which is implemented in terms of a "Dirac system", namely a particular type smooth fiber subbundle of full integral lattices of the symplectic vector bundle of the theory. I will show that generic solutions of GESM theories correspond to the supergravity limit of four-dimensional locally geometric String Theory U-folds, providing some constrains on their global geometry and topology which follow from the global structure of GESM. Finally, I will comment on various open mathematical problems regarding the GESM system of partial differential equations and its moduli space of solutions on a globally hyperbolic Lorentzian four-manifolds. This is based on recent and ongoing work with Calin Lazaroiu.
Speaker: Dr Carlos Shahbazi (Institut für Theoretische Physik, Leibniz Universität Hannover)
• Friday, 28 April
• 09:30 10:15
New Developments in Heterotic Moduli 45m
I will begin by reviewing some recent developments in the heterotic moduli problem, both for six and seven dimensional compactifications to SU(3) and G2 structure manifolds respectively. After a brief overview of the infinitesimal (massless) moduli of six dimensional compactifications, I turn to the corresponding seven dimensional story. There are many similarities between the two situations, but also crucial differences that I will expand upon. Finally, I will report on some recent progress in understanding the finite deformations of six dimensional heterotic compactifications. We will see that the Maurer-Cartan equation of the corresponding differentially graded Lie algebra can be understood as the equation of motion of an effective theory derived from the heterotic superpotential. This effective theory has features which combine and generalise aspects of both Holomorphic Chern-Simons and Kodaira-Spencer theory.
Speaker: Dr Eirik Eik Svanes (LPTHE)
• 10:15 10:45
Non-linear anti-involutive symmetries of black hole entropy 30m
Freudenthal duality can be defined as an anti-involutive, non-linear map acting on symplectic spaces. After a general introduction on some aspects of extended (super)gravity theories in four dimensions and the structure of their U-orbits, I will consider their U-duality Lie groups "of type E7", and the corresponding notion of Freudenthal duality. I will elucidate and comment on the relation between the Hessian of the black hole entropy and the pseudo-Riemannian, rigid, para-special Kaehler metric of the pre-homogeneous vector spaces associated to the U-orbits. I will conclude with recent developments, including the extension to Abelian gaugings of supergravity (also in presence of hypermultiplets).
Speaker: Dr Alessio Marrani ("Enrico Fermi" Center, IT)
• 10:45 11:05
Break 20m
• 11:05 11:35
Special holonomy groups in supergeometry 30m
I will review my previous results about holonomy groups of superconnections on supermanifolds and will mention the joint work in progress with Andrea Santi about construction of superconnections with some classes of special holonomy groups.
Speaker: Dr Anton Galaev (University of Hradec Králové)