Title and abstracts

GEOQUANT 2017

Mario Garcia Fernandez (ICMAT)

Title: Ricci flow, Killing spinors, and T-duality in generalized geometry

Abstract: In this talk we will overview recent work in arXiv:1611.08926, where we introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Strominger system. We then consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrate that solutions of the equations are exchanged under this symmetry. As applications, we give a mathematical explanation of the "dilaton shift" and prove that the Strominger system is preserved by heterotic T-duality, as defined by Baraglia and Hekmati.

Brian Hall (University of Notre Dame, USA) 
Title: Old and new unitarity results in "quantization commutes with reduction"

Abstract: A famous result of Guillemin and Sternberg says that in the context of geometric quantization of compact Kahler manifolds, there is a natural one-to-one and onto map between the “first quantize and then reduce” space and the “first reduce and then quantize” space. Guillemin and Sternberg, however, do not address the important question of the unitarity of this map. In earlier work with Will Kirwin, I showed that the Guillemin-Sternberg map is not unitary in general, indeed, not even asymptotically unitary as Planck’s constant tends to zero. On the other hand, Kirwin and I showed that if one includes half-forms in the quantization process, one still has a bijective Guillemin-Sternberg-type map, and this map is asymptotically unitary for small values of Planck’s constant. There remains a question of whether there are interesting cases in which the half-form corrected Guillemin-Sternberg map is actually exactly unitary. I will review my results with Kirwin and discuss new results with Benjamin Lewis giving examples where exact unitarity is achieved.

Rinat Mavlyavievich Kashaev (University of Geneva, Switzerland)

Title: Teichmüller TQFT: old and new formulations

Abstract: Based on the combinatorics of shaped ordered Delta triangulations, Teichmüller TQFT suggests mathematically precise de finitions and calculation recipes for quantum Chern-Simons theory with non-compact gauge groups PSL(2;R) and PSL(2;C). There exists two different formulations of the theory which we call old and new, and the relation between them is a topic of ongoing research. In this talk I will explain the equivalence of the two formulations in the case of integer homology spheres. The talk is based on joint works with Jørgen Ellegaard Andersen.  

Ryoichi Kobayashi (Nagoya Universit, Japan)
Title: A Quantization of Osserman's theory of algebraic minimal surfaces

Abstract: I propose a quantization scheme of Osserman's theory on algebraic minimal surfaces. Starting with an observation that the basic quantity in Osserman's theory is interpreted as a sort of partition function in which the limit h to 0 comes fi rst before summation, I will answer the question what happens if we perform summation first and then take the limit h to 0. We arrive at a geometry completely different from Osserman's. 

Leonid Polterovich 
Title: Quantum speed limit vs. classical displacement energy

Abstract: We discuss a link (in the context of the Berezin-Toeplitz quantization) between displacement energy, a fundamental notion of symplectic topology, and the quantum speed limit, a universal constraint on quantum-mechanical processes. Joint work with Laurent Charles.

Yuuji Tanaka (Nagoya University, Japan) 

Title: Vafa-Witten invariants for projective surfaces

Abstract: I will speak about work with Richard Thomas on Vafa-Witten invariants for projective surfaces such as motivation; de nitions; calculations in examples; and their matches with Vafa-Witten's original conjectures raised more than 20 years ago.

Theodore Voronov (University of Manchester, UK)
Title: Microformal geometry, Poisson manifolds and homotopy algebra

Abstract: Given two homotopy Poisson manifolds, it is necessary for some problems to construct L- in finity morphisms between the algebras of functions. Recall that L-infi nity morphisms are described geometrically as generally non-linear maps between Q-manifolds. Therefore, the problem, in particular, is to have some general construction giving non-linear mappings between algebras of functions viewed as in finite-dimensional supermanifolds. In the talk, I will describe the solution of this problem, which is as follows. There is a natural generalization of smooth maps of (super)manifolds, which are not maps in the ordinary sense but rather certain type canonical relations between the cotangent bundles. I call them "microformal morphisms" or "thick morphisms". They induce pullbacks of smooth functions, which are, in general, non-linear mappings. (More precisely, formal nonlinear differential operators.) Such nonlinear pullbacks have interesting properties, including the fact that their derivatives are algebra homomorphisms. In homotopy Poisson setting, they yield desired L-infi nity morphisms. Another application is a construction of adjoints for nonlinear operators on vector bundles. This can be applied to L-infi nity algebroids. Thick morphisms form a formal category which is a formal neigh- borhood of the ordinary category of supermanifolds. (To be more precise, there are two parallel constructions, for even and odd functions or "bosonic" and "fermionic" fields.) Finally, there is a quantum version of all that, where "quantum thick morphisms" are represented by oscillatory integral operators.