GAP XIV — Sheffield

Schedule

Note: On the afternoon of Friday the 12th, from 15:00 to 17:00, there will be two additional informal talks by Ted Voronov.

Downloads: Schedule and Abstracts.

Slides. Slides are available for a number of talks. Check for links below each abstract.

Mini-courses

Kirill Mackenzie (University of Sheffield):
Multiple Lie theory

Multiple Lie theory is the application of the methods of Lie group and Lie algebra theory to double, triple and n-fold structures arising in differential geometry.

Lie groups owe their power and importance to two principal properties: they can be linearized by differentiation to Lie algebras, which admit algebraic classifications, and they embody the most immediately recognizable and important notion of symmetry.

Lie groupoids extend this paradigm to the symmetry properties of bundle structures. The elements of Lie groupoids are abstract `elements of length' and the elements of Lie algebroids can be regarded as abstract tangent vectors — they thus extend the basic tools of standard differential geometry, and many of the basic processes of standard differential geometry are encompassed by the Lie theory of Lie groupoids and Lie algebroids.

Double Lie groupoids capture the corresponding idea of an abstract `element of area'. Here `double' is used in the categorical sense, which goes back to Ehresmann: a double groupoid is a groupoid object in the category of groupoids. The construction of the double Lie algebroid of a double Lie groupoid S requires two differentiation processes but is otherwise straightforward: if S consists of quadruples of points from a manifold M then the double Lie algebroid A2S is the iterated tangent bundle T2M and if S is a symplectic double groupoid then A2S is the cotangent double of the associated Lie bialgebroid.

The abstract concept of double Lie algebroid is nontrivial and developed from the duality theory of double vector bundles. Double Lie algebroids may be regarded as abstract forms of the double (= iterated) tangent bundle, but also generalize the classical Drinfeld double of a Lie bialgebra; Drinfeld's use of the word `double' is a priori unrelated to Ehresmann's.

Double Lie structures arise naturally in Poisson geometry. The cotangent manifold of a Poisson Lie group possesses a Lie groupoid and a Lie algebroid structure and these are compatible in the categorical sense. This is an example of an LA--groupoid; these stand midway between double Lie groupoids and double Lie algebroids: differentiating the cotangent LA--groupoid of a Poisson Lie group gives the classical Drinfeld double of the Lie bialgebra.

The lectures will give an overview of double Lie groupoids and double Lie algebroids, and their use in Poisson geometry. If time permits I will say something about multiple duality for n-fold vector bundles, n⩾3. Some familiarity with the basic theory of Lie groupoids, Lie algebroids and Poisson geometry will be assumed.

References:

KM, General Theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, no. 213, Cambridge University Press, 2005, xxxiv + 501 pages.

KM, "Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids". Journal für die Reine und Angewandte Mathematik. (Crelle's Journal), 658, (2011), 193–245.

Th. Th. Voronov, "Q-Manifolds and Mackenzie Theory". Comm. Math. Phys. 315, (2012), 279–310.

A. Gracia-Saz, M. Jotz Lean, KM and R. Mehta, "Double Lie algebroids and representations up to homotopy." arXiv:1409.1502

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Jonathan Pridham (Edinburgh):
Derived algebraic geometry with a view to quantisation

I will present a relatively concrete formulation for derived algebraic geometry, closely related to other contexts such as higher Lie groupoids. I will then explain how to formulate shifted symplectic and Poisson structures in this setting, and say something about quantisation.

[PDF 1] [PDF 2] [PDF 3]

Maxim Zabzine (Uppsala):
Equivariant localization in infinite dimensional setting

I will give the pedagogical introduction to the use of the Atyiah-Bott formula in infinite dimension setting, especially in the context of supersymmetric gauge theories. I will start by reviewing the finite dimensional Atyiah-Bott formula and I will outline its proof in the context of supergeometry. As the main infinite dimensional example I will present the calculation of the partition function for Chern-Simons theory. If time allows I will briefly review the latest advances in the field of supersymmetric localization.

Plenary talks

Iakovos Androulidakis (National and Kapodistrian University of Athens, Greece):
Singular foliations and the Baum-Connes conjecture

Among other things, the Baum-Connes conjecture is a means for the calculation of the K-theory of the C*-algebra associated with a geometric situation: a group, a groupoid, a regular foliation. Such a calculation is important because (in principle) it provides information about the representation theory of the situation involved.

On the other hand, in earlier work with G. Skandalis, we associated a C*-algebra to any singular foliation. In this lecture we will discuss the calculation of the K-theory group for this C*-algebra in a very large class of singular foliations, and give explicit calculations in specific examples. Based on these calculations, we will discuss the formulation of the Baum-Connes conjecture for this class of singular foliations.

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Dorothea Bahns (Mathematical Institute, Georg August University Göttingen):
The Sine Gordon Model in Perturbative Algebraic Quantum Field Theory

We study the Sine Gordon model (with hyperbolic signature) in the framework of perturbative algebraic quantum field theory, i.e. in a purely algebraic setting, without using Hilbert or Krein spaces. This is joint work with Kasia Rejzner (York).

Katarzyna Grabowska (University of Warsaw):
Mechanics on graded bundles

Lagrangian description of analytical mechanics in its very classical version is based on the geometry of the tangent bundle. The tangent bundle has a very rich structure. It is of course a vector bundle equipped with a Lie bracket of sections, which makes it a canonical example of a Lie algebroid. This structure alone can be described in many ways: as a bracket of sections, as a linear Poisson (symplectic in this case) structure on the dual or as a certain double vector bundle morphism. Using this broader perspective of algebroids one can reduce Lagrangian mechanics with respect to symmetries even if the result of the reduction is not on a tangent bundle any more. All of this is not directly applicable to the higher order mechanics, i.e. for the case when Lagrangians depend on jets of curves of order higher than one. In this case the broader perspective is provided by the theory of graded bundles, i.e. manifolds equipped with an action of multiplicative monoid of real numbers. In my talk I shall present the basics of the theory of graded bundles and its application to higher order mechanics. For this I shall need a concept of a weighted algebroid which is an aglebroid compatible with the structure of graded bundle. Using graded bundles and a bit of affine geometry one can provide Lagrangian as well as Hamiltonian versions of mechanics of higher order systems, including reduction with respect to symmetries.

The talk will be based on series of papers by Andrew J. Bruce, J. Grabowski, K. Grabowska and M. Rotkiewicz on the geometry of graded bundles, in particular on `Linear duals of graded bundles and higher analogues of (Lie) algebroids', J. Geom. Phys. 101, (2016), 71-99 and `Higher order mechanics on graded bundles', J. Phys. A: Math. Theor. 48 (2015).

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Hovhannes Khudaverdian (The University of Manchester):
Higher order Koszul brackets

For an arbitrary manifold M, we consider supermanifolds ΠTM and ΠT^∗M, where Π is the parity reversion functor. The space ΠT^∗M possesses canonical odd Schouten bracket and space ΠTM posseses canonical de Rham differential d. An arbitrary even function P on ΠT^∗M such that [P,P]=0 induces a homotopy Poisson bracket on M, a differential, dP on ΠT^∗M, and higher Koszul brackets on ΠTM. (If P is fiberwise quadratic, then we arrive at standard structures of Poisson geometry.) Using the language of Q-manifolds and in particular of Lie algebroids, we study the interplay between canonical structures and structures depending on P. Then using just recently invented theory of thick morphisms we construct a non-linear map between the L∞ algebra of functions on ΠTM with higher Koszul brackets and the Lie algebra of functions on ΠT^∗M with the canonical odd Schouten bracket. The talk is based on the work with Th.Voronov

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Yvette Kosmann-Schwarbach (École Polytechnique):
Topics in quasi-Poisson geometry

This survey of quasi-Poisson geometry will include the g-quasi-Poisson Lie groups recently defined by Ševera and Valach (arXiv:1604.07164). An unpublished result of Valach shows how to determine the corresponding infinitesimal structure in terms of a suitably defined big bracket.

Camille Laurent-Gengoux (Université de Lorraine):
Global action-angle variables in Poisson geometry

Action-angle theorems do make sense in Poisson geometry, especially when one considers non-commutative integrable systems with compact fibers. We give examples, obstructions, and explain the geometric meaning of those. Joint work with Rui Fernandes and Pol Vanhaecke.

Jae-Suk Park (IBS-CGP, POSTECH):
Homotopy probability theory of A∞-algebra

The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. Conversely, a homotopy algebra gives rises to an appropriate type of homotopy probability spaces such that the laws of random variables may also serve to provide invariants of the corresponding algebraic homotopy types.

Homotopy probability theory of A∞ algebra leads to a homotopy functor from the category of A∞-algebras to the category of pro-unipotent affine monoid schemes over a field such that the laws of random variables of the probability theory of given A∞ algebra corresponds to the algebra of functions on the attached monoid scheme. This also generalise Chen's theory of iterated integrals of CDGA of differential forms to arbitrary A∞-algebra as correlation functions of non-commutative homotopy probability space.

Fani Petalidou (Aristote University of Thessaloniki):
Poisson brackets with prescribed familly of functions in involution

It is well known that functions in involution with respect to Poisson brackets have a privileged role in the theory of completely integrable systems and the fundamental problem of this theory is the construction of functions in involution. In this talk, we will present some of our results on the study of the inverse, so to speak, problem. By developing a technique analogous of that presented in [1] we construct Poisson brackets which have a given family of functions in involution. Our approach allows us to deal with bi-hamiltonian structures constructively and therefore allows us to also deal with the complete integrable systems that arise in such a framework.

[1] P. A. Damianou and F. Petalidou, Poisson brackets with prescribed Casimirs, Canad. J. Math. 64 (2012), 991-1018.

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Arkady Vaintrob (Oregon):
Deformations and normal forms of Q-manifolds and related structures

A Q-manifold is a supermanifold with a homological vector field. Numerous geometric and algebraic structures can be described and studied in terms of homological vectors fields. We will discuss general results about deformations and normal forms of Q-manifolds and will show how they can be used in the study of Lie algebroids, Poisson manifolds, bialgebroids, Dirac manifolds, actions up to homotopy, and related structures.

Luca Vitagliano (University of Salerno):
L∞-algebroids and BV∞-algebras

We discuss left and right representations of L∞-algebroids and the associated Cartan and Schouten calculi. We show that a right action of an L∞-algebroid on the trivial line bundle canonically determines a BV∞-algebra. In particular, there is a BV∞-algebra attached to every Poisson∞-manifold. This generalizes to the homotopy setting a result of Xu (P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999) 545–560).

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Theodore Voronov (University of Manchester):
Microformal geometry and homotopy algebras

I will speak about a natural generalization of smooth maps that makes possible to have pullbacks of functions that are, in general, non-linear mappings. (More precisely, formal nonlinear differential operators.) I came to it motivated by a concrete problem where I needed to construct an L-infinity morphism between particular homotopy Poisson brackets. As it has turned out, these new "nonlinear pullbacks" have applications ranging from homotopy Poisson (or homotopy Schouten) structures to vector bundles and algebroids. The "quantum version" of theory is related with oscillatory integral operators.

I plan to give three lectures, from which the first I will try to make self-contained and giving a "big picture" of the theory. The following two will contain more details and elaborate particular topics. The general plan is as follows:

1. "Thick morphisms" of (super)manifolds: construction and main properties;

2. Application to homotopy Poisson structures;

3. Further applications (to vector bundles and L-infinity (bi)algebroids). Quantum version.

References:

The "nonlinear pullback" of functions and a formal category extending the category of supermanifolds. arXiv:1409.6475

Microformal geometry. arXiv:1411.6720

Thick morphisms of supermanifolds and oscillatory integral operators. arXiv:1503.06542.

Quantum microformal morphisms of supermanifolds: an explicit formula and further properties. arXiv:1512.04163

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Contributed talks

David Carchedi (George Mason University):
Dg-manifolds as derived manifolds

Given two smooth maps of manifolds f:M→L and g:N→L, if they are not transverse, the fibered product M×_L N may not exist, or may not have the expected dimension. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In fact, every (quasi-smooth) derived manifold is locally of this form. In this talk, we briefly explain what derived manifolds ought to be, why one should care about them, and finally discuss some details of an on going project of ours with Dmitry Roytenberg to use differential graded manifolds to model them.

Josua Groeger (University of Cologne, Germany):
Holonomy in Supergeometry: Theory and Applications

Given a connection on a vector bundle, the set of parallel transport operators along loops at some fixed point is known as the holonomy group. The problem of finding a reasonable generalisation to supergeometry is nontrivial. I will first give a review on the two existing solutions to that problem, with an emphasis on the intricate way of how both approaches are equivalent. The resulting unified theory of super holonomy is very powerful. In the last part of my talk, I will focus on applications.

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Charlotte Kirchhoff-Lukat (University of Cambridge):
Natural lifts of Dorfman brackets in the context of double vector bundles

Dorfman brackets on general vector bundles constitute a natural non-antisymmetric generalisation of Lie algebroids which still satisfy a type of Jacobi identity. Both the 'standard' example, the Courant-Dorfman bracket on TM⊕T∗M, as well as the brackets from so-called 'exceptional generalised geometry', are used in string theory to encode infinitesimal gauge transformations (also called generalised diffeomorphisms).

General Dorfman brackets have not yet been extensively studied in the mathematical literature. We (CKL in collaboration with M. Jotz-Lean) aim to understand the construction principles and symmetries of Dorfman brackets on vector bundles of the form TM⊕E^∗ (E→M a general smooth vector bundle over a smooth manifold). In the context of double vector bundles, such brackets can be viewed as restrictions of the Courant-Dorfman bracket on the standard VB-Courant algebroid TE⊕T^∗E.

We show that each Dorfman bracket is equivalent to a lift of sections of TM⊕E^∗ to the linear sections of TE⊕T^∗E, which lifts it to the Courant-Dorfman bracket. We further demonstrate that symmetries and twistings of the Dorfman bracket on TM⊕E^∗ interact naturally with this lift.

This approach provides a new view on Dorfman brackets in terms of higher vector bundles and demonstrates a certain universality of the Courant-Dorfman bracket.

Dominik Ostermayr (University of Cologne):
Harmonic maps from super Riemann surfaces into complex projective superspaces: Twistor lifts and integrable systems

In their seminal paper, Eells and Wood classified isotropic harmonic maps from a Riemann surface into complex projective spaces via twistor lifts. This accounts for all harmonic maps if the source is a sphere. Moreover, Burstall showed that all non-isotropic harmonic 2-tori can be constructed by integrating commuting flows. I shall discuss the analogues of these two approaches for harmonic maps from a super Riemann surface into complex projective superspaces.

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