2019 Workshop on Deformation Theory and Homotopy Algebra

2019 Workshop on Deformation Theory and Homotopy Algebra, Southwest Jiaotong University-Emei Campus,May 6-11,2019.

On-Site Registration: 2:00pm-6:00pm, 5 May(Sunday), Hushan Hotel Openning Ceremony: 8:30am-9:00am, 6 May(Monday), No.5 conference room.

Title and Abstract

Chengming Bai (Chern institute of Mathematics)

Title: Deformations and their controlling cohomologies of $\mathcal O$-operators

Abstract: We establish a deformation theory of a kind of linear operators, namely, $\mathcal O$-operators in consistence with the general principles of deformation theories. On one hand, there is a suitable differential graded Lie algebra whose Maurer-Cartan elements characterize $\mathcal O$-operators and their deformations. On the other hand, there is an analogue of the Andr\'e-Quillen cohomology which controls the deformations of $\mathcal O$-operators. Infinitesimal deformations of $\mathcal O$-operators are studied and applications are given to deformations of skew-symmetric $r$-matrices for the classical Yang-Baxter equation. This is a joint work with Li Guo, Yunhe Sheng and Rong Tang.

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Kai Behrend(University of British Columbia )

Title：dg-manifolds form a category of fibrant objects

Abstract：This is a report on very recent work in progress (joint with H-Y Liao and P Xu) proving the theorem in the title. The purpose is to embed differentiable manifolds in a context where homotopy theory is possible: for example, the fibre products and intersections of differentiable manifolds, which do not exist in the category of differentiable manifolds, do exist as homotopy fibered products in the category of fibrant objects I will describe. Furthermore, deformation theory, by which I mean the homotopy theory of differential graded Lie algebras, is embedded in this context as well. The hope is that this will lead to a simpler context for derived differentiable topology than other more involved constructions. The proof is an application of the transfer theorem for curved L-infinity algebras.

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Yuri Berest (Cornell University)

Title: Derived algebraic groups

Abstract: Classically, an affine algebraic group (or affine group scheme) G over

a field k is defined by its functor of points, which is a representable group-valued functor on the category of commutative k-algebras. It is known that the natural (degreewise) extension of G to the category of simplicial commutative algebras has, unfortunately, poor homotopical properties: in particular, it is not a homotopy invariant functor. In the literature, going back to the classical work of R.Swan and F.Waldhausen, one can find different ways to remedy this problem by giving different homotopical approximations of G. In this talk, we address the question of how to construct and compare such approximations. This question is motivated by

the recent work of S. Galatius and A. Venkatesh on derived Galois deformation rings

and my joint work with A.C.Ramadoss and W.-k.Yeung on representation homology of simplicial groups and topological spaces.

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Francesco Bonechi (INFN)

Title: Poisson structures on differentiable stacks

Abstract: Quasi Poisson groupoids are bivector fields on Lie groupoids that are compatible with the groupoid multiplication and satisfy the Jacobi identity in a homotopical sense. We discuss their properties with respect to Morita equivalence. More generally, we introduce the graded 2-Lie algebra of polyvector fields and its Morita invariance. This is a joint work with N.Ciccoli, C.Laurent-Gengoux and P.Xu.

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Alberto Cattaneo(Universität Zürich)

Title: An Introduction to the BV-BFV Formalism

Abstract: The BV-BFV formalism unifies the BV formalism (which deals with the problem of fixing the gauge of field theories on closed manifolds) with the BFV formalism (which yields a cohomological resolution of the reduced phase space of a classical field theory). I will explain how this formalism arises and how it can be quantized.

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Zhuo Chen (Tsinghua University)

Title: SH Leibniz algebra from homotopic embedding tensors

Abstract: The embedding tensor appears in the gauging procedure of supergravity theories. Kotov and Strobl prove that there exists a correspondence between embedding tensors and Leibniz algebras, and show that the associated tensor hierarchy only depends on the corresponding Leibniz algebra. In this talk, we consider a homotopic version of embedding tensors in the context of dg geometry and derive SH(strongly homotopy) Leibniz algebras from homotopic embedding tensors. Several interesting examples will be discussed. This is a joint work with M.Xiang and T.Zhang.

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Ralph Kaufmann(Purdue University)

Title: Deformations, Hopf algebras and Feynman categories.

Abstract: There are several Hopf algebras arising from in physics, number theory and geometry whose existence can be traced back to simplicial structures, co-operads and Feynman categories. We review these constructions which yield the Hopf algebras governing multi-zeta values, chains on double loop spaces and renomalization. We then discuss deformations in this context. They first appear in a non-commutative setting. Here the Hopf algebras can be seen as a limit point of a deformation of bi--algebras in a formal parameter q. A second deformation appears, when considering more general multiplications in the bi--algebra. This leads to considerations of filtration and associated graded objects. The result here is that the under given conditions the general Hopf/bi--algebras are deformations of a quotient of

a free graded construction.

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Andrey Lazarev (Lancaster University )

Title: Homotopy theory of monoids

Abstract: I will explain how the category of discrete monoids models the homotopy category of connected spaces. This correspondence is based on derived localization of associative algebras and could be viewed as an algebraization result, somewhat similar to rational homotopy theory (although not as structured). Time permitting, I will describe one application of this circle of ideas, namely a simple proof of a generalization of Adams’s cobar construction to general nonsimply connected spaces obtained recently by Hess-Tonks and Rivera-Zeinalian using different methods. This is a joint with J. Chuang and J. Holstein.

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Si Li (Tsinghua University )

Title: Loday-Quillen-Tsygan Theorem and Large N gauge theory

Abstract: We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira- Spencer gravity. The 1st order deformation is realized by the Loday-Quillen-Tsygan Theorem on the Lie algebra cohomology of large N matrices. We show that the dynamics of Kodaira-Spencer gravity is fully recovered from this large N holomorphic Chern-Simons theory. This gives a construction of quantum open-closed B-model. We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological B-model. At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type I version of Kodaira-Spencer theory in complex dimension 5 and show that it can only be coupled to holomorphic Chern-Simons theory with gauge group SO(32) at quantum level. This coupled system is conjectured to be a supersymmetric localization of type I string theory.

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Mathieu Stienon(Pennsylvania State University)

Title: Atiyah class and Todd class of dg manifolds

Abstract: Exponential maps arise naturally in the contexts of Lie theory and of smooth manifolds. The infinite jets of these exponential maps are related to the Poincaré--Birkhoff--Witt isomorphism and the complete symbols of differential operators. We will discuss how these exponential maps can be extend to the context of dg manifolds. As an application, we will describe a natural L-infinity structure associated with the Atiyah class of a dg manifold.

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Raphael Ponge (Sichuan University)

Title: Cyclic homology of crossed-product algebras

Abstract: There is a great amount of work on the cyclic homology of crossed-product algebras, by Baum-Connes, Brylinski-Nistor, Connes, Crainic, Elliott-Natsume-Nest, Feigin-Tsygan, Getzler-Jones, Nest, Nistor, among others. However, at the exception of the characteristic map of Connes from the early 80s, we don’t have explicit chain maps that produce isomorphisms at the level of homology and provide us with geometric constructions of cyclic cycles in the case of group actions on manifolds or varieties.

The aim of this talk is to present the construction of explicit quasi-isomorphisms for crossed products associated with actions of discrete groups. Along the way we recover and clarify various earlier results (in the sense that we obtain explicit chain maps that yield quasi-isomorphisms). In particular, we recover the spectral sequences of Feigin-Tsygan and Getzler-Jones, and derive an additional spectral sequence.

In the case of group actions on manifolds we have an explicit description of cyclic homology and periodic cyclic homology. In the finite order case, the results are expressed in terms of what we call ``mixed equivariant homology”, which interpolates group homology and de Rham cohomology. This is actually the natural receptacle for a cap product of group homology with equivariant cohomology. As a result taking cap products of group cycles with equivariant characteristic classes naturally gives to a geometric construction of cyclic cycles. For the periodic cyclic homology we recover earlier results of Connes and Brylinski-Nistor via a Poincar\'e duality argument. For the non-periodic cyclic homology the results seem to be new. In the infinite order case, we fix and simplify the misidentification of cyclic homology by Crainic.

In the case of group actions on smooth varieties we obtain the exact analogues of the results for group actions on manifolds. In particular, in the special case of finite group actions on smooth varieties we recover recent results of Brodzki-Dave-Nistor via the construction of an explicit quasi-isomorphism.

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Junwu Tu(ShanghaiTech University)

Title: The Bogomolov-Tian-Todorov Theorem of cyclic A-infinity algebras

Abstract: Let A be a finite-dimensional smooth unital cyclic A-infinity algebra. Assume furthermore that A satisfies the Hodge-to-de-Rham degeneration property. We prove the non-commutative analogue of the Bogomolov-Tian-Todorov theorem: the deformation functor associated with the differential graded Lie algebra of Hochschild cochains of A is smooth. Furthermore, the deformation functor associated with the DGLA of cyclic Hochschild cochains of A is also smooth.

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Yu Ye(University of Science and Technology of China )

Title: Clifford deformation and noncommutative quadrics

Abstract: In this talk, we will introduce Clifford deformations for Koszul Frobenius algebras, which correspond to quadric hypersurface algebras via Koszul duality. It turns out that a quadric ypersurface is a noncommutative isolated singularity if and only if the corresponding Clifford deformation algebra is semi-simple as a $Z_2$-graded algebra. We also recover Knorrer's Periodicity theorem for quadric hypersurfaces without using matrix factorizations. This is a joint work with Jiwei He.

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Guodong Zhou(East China Normal University )

Title: Cohomology of lower dimensional Poisson algebras with applications to Happel's question

Abstract: We will talk about two results about computation of Poisson cohomology. In the first part, following P. Monnier, we compute the Gerstenhaber algebra structure over the Poisson cohomology of certain isolated singularities in dimension two. In the second part, we show how to use the interplay between Poisson cohomology and Hochschild cohomology to construct counter-examples to Happel's question. This talk is based on ongoing joint work with Zi-Hao Qi.