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SUPERGRAVITY and GEOMETRY Theoretical Physics Has Always SUPERGRAVITY AND GEOMETRY GEORGE MOUTSOPOULOS Theoretical physics has always been a constant inspiration for the field of math- ematics. This remains true today, with string theory and mathematical physics blurring the lines between what constitutes physics and mathematics. From the physics point of view, string theory is a promising theory with which to unify con- cepts such as quantization and geometry, particles and spacetime. At the same time, string theory provides a wealth of new and elegant mathematical structures that beckon to have light shed on them. Here I will describe a few examples of the interplay between physics and mathematics that are directly relevant to my research. My research focuses on supergravity theory, a generalization of general relativ- ity that provides consistent spacetimes in which quantum strings can propagate. The geometry of supergravity provides a precise definition of a geometry that ex- tends (uniquely) lorentzian-signature riemannian geometry with Grassmann-odd spinorial degrees of freedom. As such, spinorial geometry is an integral part of my work. By adding these spinorial degrees of freedom, the geometry of supergravity may be seen to generalize manifolds to supermanifolds, diffeomorphism symmetry with supersymmetry, Killing vectors with Killing spinors, and Lie algebras to Lie superalgebras. In particular, a supergravity theory is a gravity theory such that its defining equations are invariant under supersymmetry. In order to introduce a mathematical problem in supergravity, let us consider first a classic problem in spinorial geometry, that of special spinorial holonomy. The holonomy is called special if their exist r-parallel spinors. One may further generalize from parallel spinors to spinors that satisfy the riemannian Killing spinor equation: (1) rX = λ X · : Here is a spinor field, λ is a function, r the Levi-Civita derivative, and X · is the action of the vector field X on in the Clifford module. Supergravity generalizes this equation to (2) rX = λ X · + ··· ; where the right-hand side contains other objects that are defined in supergravity theory. A spinor satisfying such an equation is called a supergravity Killing spinor. One may view supergravity Killing spinors as the Grassmann-odd spinorial analogue of Killing vectors. In particular, we may extend the isometry Lie algebra, which by definition consists of Killing vectors that leave a metric invariant, with supergravity Killing spinors. This defines the so-called Killing Lie superalgebra of a supergravity geometry: the Z2-graded vector space of Killing vectors and Killing spinors with a suitably defined Lie superbracket. The Killing Lie superalgebras may then be used to classify algebraically the geometries admitting Killing spinors. The implications of the existence of Killing spinors on a supergravity geometry is a major aspect of my research [1, 2, 3, 4]. Indeed, their existence severely restricts the possible geometries. For example, in 10 and 11 spacetime dimensions more than 16 Killing spinors imply homogeneity [5]. Recently, we studied the implications in Date: February 2016. 1 SUPERGRAVITY AND GEOMETRY 2 a particular three-dimensional supergravity theory [6]. By assuming one Killing spinor we found the necessary and sufficient conditions on the geometry and upon using a simplifying Ansatz, we found a large class of solutions, some of which are new and unexpected. Another example of my research is my current work on the Killing Lie superalge- bra of warped anti-de Sitter. Warped anti-de Sitter is a homogeneous deformation of anti-de Sitter geometry [7]. Anti-de Sitter geometry is a maximally symmetric lorentzian-signature space. A more familiar analogue of warped anti-de Sitter in euclidean signature would be the squashed three-sphere. In supergravity theories, anti-de Sitter is also a maximally supersymmetric space: it allows the maximum number of independent Killing spinors. Whether warped anti-de Sitter also allows Killing spinors and if its Killing Lie superalgebra characterizes the geometry are interesting questions. Another topic in my research is that of string theory dualities, broadly defined as isomorphisms of quantum theories. The case of T-duality is interesting because it hints towards a generalization of diffeomorphisms. More precisely, T-duality relates two strings propagating on cylinders of reciprocal radii. This is so because the momentum of a quantum string moving along the circle direction is integer valued as n=R, where n 2 Z, and the energy E of a string that winds m times along the circle is of the schematic form n 2 (3) E = + (m R)2 : R Exchanging momentum modes with winding modes gives the same energy provided one inverts R 7! 1=R. More generally for n-torus fibrations, the large diffeomor- phism symmetry group SO(n; Z) is enhanced to SO(n; n; Z), which includes such radii inversions. The notion of T-duality allows for exotic constructions called T-folds. They are an example of how string theory may generalize geometry where smoothness is lost. As a rough example, imagine we bend a cylinder whose ends have radii R and 1/R and then bind its ends. Impossible you say? The physics intuition is that particles may not jump through the discontinuity but strings may do so provided they exchange momentum modes with winding modes. This construction makes mathematical sense as a conformal field theory orbifold, certain aspects of which are studied in supergravity theory. T-duality is in fact a hidden symmetry of supergravity theory. Ideally, one would want a generalization of geometry in order to describe T-folds on an equal footing with smooth manifolds and where T-duality is manifest [8]. Yet another direction of my research concerns the dynamics of the exotic objects that \source" such non-smooth geometries [9]. The study of supersymmetry and the geometry in supergravity, but also the study of how geometry may be generalized in string theory, are the main ingre- dients that motivate my research. I have also worked on some other topics in mathematical physics, mostly related to spacetime geometry [10, 11, 12]. These works are published in high-energy physics or classical and quantum gravity jour- nals. Nevertheless, the nature of my work is on the very border where new physics and geometry ideas interact in a fascinating way. SUPERGRAVITY AND GEOMETRY 3 References [1] G. Moutsopoulos, Geometric and Non-Geometric Backgrounds of String Theory. PhD thesis, School of Mathematics, Edinburgh University, UK, 2008. http://hdl.handle.net/1842/3167. [2] N. S. Deger, G. Moutsopoulos, H. Samtleben, and O. Sarioglu, \All timelike supersymmetric solutions of three-dimensional half-maximal supergravity," JHEP 06 (2015) 147, arXiv:1503.09146 [hep-th]. [3] G. Moutsopoulos, \The NUT in the N=2 Superalgebra," Class.Quant.Grav. 27 (2010) 035008, arXiv:0908.0121 [hep-th]. [4] J. Figueroa-O'Farrill, E. Hackett-Jones, G. Moutsopoulos, and J. Simon, \On the maximal superalgebras of supersymmetric backgrounds," Class.Quant.Grav. 26 (2009) 035016, arXiv:0809.5034 [hep-th]. [5] J. M. Figueroa-O'Farrill, E. Hackett-Jones, and G. Moutsopoulos, \The Killing superalgebra of ten-dimensional supergravity backgrounds," Class.Quant.Grav. 24 (2007) 3291{3308, arXiv:hep-th/0703192 [hep-th]. [6] N. S. Deger and G. Moutsopoulos, \Supersymmetric solutions of N = (2; 0) Topologically Massive Supergravity," arXiv:1602.07263 [hep-th]. [7] G. Moutsopoulos, \Homogeneous anisotropic solutions of topologically massive gravity with cosmological constant and their homogeneous deformations," Classical and Quantum Gravity 30 (2013) no. 12, 125014, arXiv:1211.2581 [gr-qc]. [8] E. Hackett-Jones and G. Moutsopoulos, \Quantum mechanics of the doubled torus," JHEP 0610 (2006) 062, arXiv:hep-th/0605114 [hep-th]. [9] A. Chatzistavrakidis, F. F. Gautason, G. Moutsopoulos, and M. Zagermann, “Effective actions of non-geometric fivebranes," Phys.Rev. D89 (2014) 066004, arXiv:1309.2653 [hep-th]. [10] Y. Mitsuka and G. Moutsopoulos, \No more CKY two-forms in the NHEK," Class.Quant.Grav. 29 (2012) 045004, arXiv:1110.3872 [gr-qc]. [11] G. Moutsopoulos and P. Ritter, \An Exact Conformal Symmetry Ansatz on Kaluza-Klein Reduced TMG," Gen.Rel.Grav. 43 (2011) 3047{3063, arXiv:1103.0152 [hep-th]. [12] F. Jugeau, G. Moutsopoulos, and P. Ritter, \From accelerating and Poincare coordinates to black holes in spacelike warped AdS3, and back," Class.Quant.Grav. 28 (2011) 035001, arXiv:1007.1961 [hep-th]. Mathematics Department, South Campus, Bogazici University, Istanbul, Turkey.
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