N. Poncin, J. Grabowski, S. Kwok, V. Salnikov, K. Grabowska.

Minicourse - On the physical background of BRST/BV

Minicourse - On the physical background of BRST/BV During the first three talks I shall mainly focus on the definition of path integral quantization in mechanics and field theory.
Talk 1. Classical mechanics: Lagrangian and Hamiltonian formalism. Basics of quantum mechanics.
Talk 2. Examples of the free particle and of the harmonic oscillator. Idea of Feynman integral.
Talk 3. Feynman Green's function, quantization.
Talk 4.
Talk 5. Only Alessandro knows where this goes
Talk 6. Maybe... (It was about quantizing Maxwell)
Talk 7. Actually he does and he does not - that's all quantum now. (And this one was about principal bundles)
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Minicourse - Microformal geometry

Minicourse - Microformal geometry In search of constructions that may give L-infinity morphisms for algebras of functions on homotopy Poisson (or odd Poisson) manifolds, I discovered a generalization of familiar pullbacks with respect to smooth maps. Surprisingly, these new pullbacks are *nonlinear*, actually formal, mappings of the vector spaces of functions given by formal nonlinear differential operators. As nonlinear, they cannot be algebra homomorphisms in contrast with the ordinary pullbacks; however, their derivatives at each point are algebra homomorphisms.
Underlying these "nonlinear pullbacks", there is a formal category (actually, there are two parallel versions of such a category giving pullbacks of even and odd functions), which is a formal neighborhood of the semi-direct product of the usual category of (super)manifolds with algebras of smooth functions. Morphisms in this formal category --- "microformal" or "thick" morphisms of (super)manifolds, as I call them, --- are formal canonical correspondences between the cotangent bundles and are described by formal generating functions. (A close non-formal category based on germs of symplectic manifolds was introduced for different purposes by Cattaneo--Dherin--Weinstein, but their theory does not have pullbacks.) For homotopy Poisson supermanifolds, nonlinear pullbacks of functions induced by Poisson thick morphisms (which are easy to define) are indeed L-infinity morphisms for the corresponding algebra structure. Another application is related with vector bundles. By generalizing ordinary maps to thick morphisms, it is possible to have *adjoints for nonlinear operators* on vector bundles. This has applications to L-infinity (bi)algebroids. The most recent development is related with a *quantum version* of this theory. Namely, it is possible to define "quantum thick morphisms", which are particular oscillatory integral operators, so that "classical" thick morphisms arise as their limits as Planck's h goes to zero. This is work in progress.
Plan of lectures (tentative):
1. Thick morphisms of (super)manifolds and nonlinear pullbacks of functions;
2. Applications to homotopy Poisson structures, vector bundles and L-infinity (bi)algebroids;
3. Quantum thick morphisms.
References:
arXiv:1409.6475, arXiv:1411.6720, arXiv:1506.02417, arXiv:1512.04163.
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Kirillov and Jacobi structures up to homotopy.

Kirillov and Jacobi structures up to homotopy. In this talk I will present the notion of a homotopy Kirillov structure on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we have a homotopy Jacobi structure. These structures are understood furnishing the module of sections with an L_\infty-algebra; which is a 'higher' or 'homotopy' version of a Lie algebra. The listener is only expected to have some rudimentary familiarity with Poisson and hopefully Jacobi manifolds. No prior knowledge of L_\infty-algebras will be assumed and our treatment of supermanifolds will be elementary. The talk will be based on the preprint 'Jacobi structures up to homotopy' (arXiv:1507.00454 [math.DG]) which is joint work with Alfonso Tortorella, Universita degli Studi di Firenze, Italy.
Slides
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Graded geometry in gauge theories and beyond (part 1)

Graded geometry in gauge theories and beyond (part 1) We study graded geometric constructions appearing naturally in the context of gauge theories. We introduce the language of Q-bundles convenient for description of symmetries of sigma models. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary Q-manifolds introducing thus the concept of equivariant Q-cohomology.
This notion turns out to be useful for analysis of such theories as the (twisted) Poisson sigma model and the Dirac sigma model. We obtain these models by a gauging-type procedure of the action of a group related to Lie algebroids and n-plectic manifolds. We also show that the Dirac sigma model is universal in space-time dimension 2. On top of applications to gauge theories (time permitting) I will comment on a possible definition of equivariant cohomology for Courant algebroids.
This is a joint work with Thomas Strobl, and in part with Alexei Kotov.
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Graded geometry in gauge theories and beyond
(part 2: multiple gradings and SUSY)

Graded geometry in gauge theories and beyond
(part 2: multiple gradings and SUSY) In this second part of the talk I am going to address the question of generalizing the results of the first part to "supersymmetric" theories. This is mostly work in progress related to introduction of multiple gradings in the context of the Poisson sigma model and the Chern-Simons theory. The global goal is to approach a clean construction of an appropriate multigraded analog of the AKSZ procedure. I will also provide some (rather toy model) examples.
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Variations on the BRST construction

Variations on the BRST construction The classical BRST construction provides a homological framework for symplectic reduction. After recalling how this works in the simplest setting, I will outline a generalization - known as the BFV construction - to arbitrary coisotropic submanifolds and its use in deformation theory. I will finish with some thoughts on Poisson submanifolds. abstract
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Representations of compact supergroups and SUSY structures

Representations of compact supergroups and SUSY structures In this talk we present some results on the Peter-Weyl theorem for the special examples of $S^{1|1}$ and $S^{1|2}$ and the connection with SUSY-structures.
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Noncommutative analogues of moduli spaces via brick manifolds

Noncommutative analogues of moduli spaces via brick manifolds I shall talk about a remarkable series of algebraic varieties that resemble the Deligne-Mumford compactifications of moduli spaces of curves of genus zero with marked points. They admit three equivalent descriptions: as "brick manifolds" recently defined by Escobar, as toric varieties of Loday's realisations of associahedra, and as De Concini-Procesi wonderful models of certain subspace arrangements. The talk will introduce, on a level accessible to a general mathematics audience, these three set-ups from scratch, and outline remarkable properties behind those spaces. It is based on a joint work with Sergey Shadrin and Bruno Vallette
http://arxiv.org/abs/1510.03261.
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Central extensions of mapping class groups from characteristic classes

Central extensions of mapping class groups from characteristic classes I will discuss a functorial construction of extensions of mapping class groups of smooth manifolds which are induced by extensions of (higher) diffeomorphism groups via the group stack of automorphisms of manifolds equipped with higher degree topological structures. The problem of constructing such extensions arises naturally in the study of topological quantum field theories, in particular in 3d Chern-Simons theory. Based on joint work with Domenico Fiorenza and Urs Schreiber.
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