Manual Smoothness theorem for differential BV algebras

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However in this talk we will discuss predictions M-theory makes about the non-perturbative behaviour of quantum field theory. In particular we will discuss examples where there is a hidden extra dimension that opens up at strong coupling. This generating function thus enjoys the topological recursion, and we demonstrate that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition.

Based on recent joint paper with J. Andersen, P.

Norbury, and R. Penner, arXiv: In the first 45 minutes, I will talk about the relationships between quantum physics and geometry mentioning my research in the colloquium-style. Using quantum knot invariants and Seiberg-Witten theory, I will explain how physics sheds new light on the concept of "quantization of geometry". The rest of time will be set up for answering questions and explanations of more details.

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The basic building block of the so-called spin-foam models, a covariant version of LQG, is the 4-simplex amplitude that gives rise to a discrete quantum geometry. I will derive this 4-simplex amplitude and analyse its semi-classical properties using spinor techniques. In order to do so we will have to introduce SU 2 -coherent states in the sense of Perelomov which are closely connected to the geometric interpretation of spinors.

Both are special cases of "TFTs with defects" which we shall study in some detail. In particular we will discuss a construction of new TFTs by covering worldsheets with appropriate networks of defect lines. If these defect lines encode the action of an orbifold group, then the new TFT precisely recovers the orbifold theory.

However, there are also other allowed defect networks, and such "non-group symmetries" lead to interesting relations between TFTs. Much of this is based on joint work with Ingo Runkel. To this end, representations of a Clifford algebra will be introduced, the Clifford group and its Pin and Spin subgroups will be defined and related to the orthogonal group of the quadratic space.

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Finally, the spin representation of the Clifford algebra will be constructed, and — time permitting — its distinguished place in the representation theory of the Clifford algebra will be indicated. An explicit construction of Cartan's geometric spinors associated with an isotropic subspace in a real pseudoeuclidean space will be detailed and related to Chevalley's algebraic spinors. The full classification of finite dimensional Clifford algebras over real and complex numbers will be introduced. Certain periodicities of finite dimensional Clifford algebras will play a key role in filling in the so-called Clifford chessboard and further in completing the classification theorem.

Notions such as the canonical element, the center, the antic enter and the canonical tensor product of Clifford algebras will be introduced and their properties will be studied, to an extent depending on the time available. Then I will assume finite-dimensionality of the underlying vector space in order to describe the linear vector space structure of Clifford Algebra and define its canonical element.

October 9th, Karol K. The analysis of the large-N asymptotic behaviour of these integrals is of interest to the description of the continuum limit of the integrable model. Such a complication in the analysis is due to the lack of dilation invariance of the exponential of the two-body interaction. In this talk, I shall discuss the main features of the method of asymptotic analysis which we have developed.

This is a joint work with G. Spinors are associated with subspaces in quadratic spaces and can be studied mathematically from representation theoretic or geometric perspective. The former one has to do with Clifford algebras and their representations, Spin groups, etc.

Spinors are also abundant in physics: they are used for the modelling of Fermi fields, the spin-statistics theorem is one of the pillars of quantum field theory, they are intimately related to supersymmetry. Apart from the analysis of spinors, in this semester we will also hear several talks by excellent guests on related or unrelated topics. In this seminar I will present both perspectives: vector bundles and gauge theory and show how they are related to each other.

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May 29th, Piotr M. We will review construction of instantons BPST solution, ADHM construction , properties of their moduli spaces, and discuss some of their applications. May 8th, Elizabeth Gasparim Unicamp, Campinas, Brasil The counting of instantons and BPS states Abstract: I will review classical results about existence of instantons on curved 4 manifolds, then describe their counting via Nekrasov partition function. Next I will explain a trick we made up make partition functions for singular varieties, which I will apply for both instant on and BPS state counting.

The concept of equivariant descent of a geometric object such as, e. We will recall classical constructions and some interesting examples. Feb 27th, Karol Palka IMPAN Quick introduction to fibre bundles, I Abstract: We will discuss basic notions and properties of fibre bundles and their maps, including principal and associated bundles.

By the recent developments, the categrification of the knot invariant becomes more tractable in the string duality.

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In particular from the framework of the type IIB superstring theory, the knot homologies can be interpreted manifestly using the Landau-Ginzburg model and the matrix factorization. In this talk, some kinds of the categorifications of quantum knot invariants will be discussed using the above framework. Abstract: Physicists are usually quite happy to formally manipulate the mathematical objects that they encounter without really understanding the structures they are dealing with.

It is then the job of the mathematician to try to make sense of the physicists manipulations and give proper meaning to the structures. Confusing physicists is not the job of mathematicians, however mathematicians are good at it! In this talk we will uncover the structure of Grassmann odd fields as used in physics. For example such fields appear in quasi-classical theories of fermions and in the BV—BRST quantisation of gauge theories. To understand the structures here we need to jump into the theory of supermanifolds.

However we find that supermanifolds are not quite enough! We need to deploy some tools from category theory and end up thinking in terms of functors! Dec 19th, Karol K. I shall mostly organise the lecture around the example of the two-dimensional Ising model. There, the correlation functions are expressed in terms of Toeplitz or Fredholm determinants. These explicit examples will allow me, first, to discuss the principal difficulties in the analysis of correlation functions and, second, to present the tools that have been developed for tackling these problems.

I shall conclude by discussing a few examples of representations for the correlation functions in more complex quantum integrable models. This will allow me to provide a succinct introduction to the problems that are currently being investigated. Since these methods are applied in general only to planar sector of the theory, the question arises whether it is possible to extend these tools to non-planar diagrams and what the obstacles are.

In particular, it is interesting whether categorical structures could allow to give more insight into interplay between planar and non-planar diagrams.

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  6. This phenomenon is possible only in dim. Connes-Moscovici approach gives the interpretation for the universal Godbillon-Vey class of the foliation as the Eisenstein second series. This is quasimodular object rather than modular one. I will discuss possible physical meaning of the results. I will explain how the path integral formulations of invariants in Chern—Simons theory has led to rigorous formulations in mathematics. The talk is supposed to be accessible to both mathematicians and physicists. Suszek KMMF WFUW Quantum field theory as a functor Parts I, II and III lecture notes Abstract: An abstraction of the basic structural pattern underlying any attempt at quantising a physical model yields a functor from a geometric category modelling the spacetime propagation and interactions of the physical entities particles, strings etc.

    This very general observation may — under favourable circumstances — lead to highly nontrivial insights and concrete computational results concerning the physical theory and the ambient geometry itself.

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    Emblematic of this line of thought is the development of the Topological Quantum Field Theory, having its origin in the pioneering works of Segal, Witten, Atiyah, Turaev et al. It is conjectured that they capture the information of quantum topological invariants in an effective and compact way.

    In the first part of this talk, for introduction, I will present the simplest and mathematically elegant example of quantum curves and the Eynard-Orantin formalism based on the Catalan numbers. This part is based on my joint paper with P. In the second part I will explain the construction of quantization of the spectral curves appearing in the theory of Hitchin fibrations. The moduli space of generalized deformations of a Calabi-Yau hypersurface is computed in terms of the Jacobian ring of the defining polynomial.

    The fibers of the tangent bundle to this moduli space carry algebra structures, which are identified using subalgebras of a deformed Jacobian ring. Associated to a differential BV algebra are two differential graded Lie algebras: we call one classical and the other, which contains a formal h-bar parameter, quantum. The classical dgLa is always smooth formal. In this paper, we give necessary and sufficient conditions for the quantum dgLa to be smooth formal.

    These conditions are equivalent to the degeneration of a version of the noncommutative Hodge to de Rham spectral sequence. References added. Thermodynamic interpretation of quantum error correcting criterion. Shanon's fundamental coding theorems relate classical information theory to thermodynamics. More recent theoretical work has been successful in relating quantum information theory to thermodynamics.

    For example, Schumacher proved a quantum version of Shannon's classical noiseless coding theorem. In this note, we extend the connection between quantum information theory and thermodynamics to include quantum error correction. There is a standard mechanism for describing errors that may occur There are well known necessary and sufficient conditions for a quantum code to correct a set of errors.

    We study weaker conditions under which a quantum code may correct errors with probabilities that may be less than one. We work with stabilizer codes and as an application study how the nine qubit code, the seven qubit code, and the five qubit code perform when there are errors on more than one qubit. As a second application, we discuss the concept of syndrome quality and use it to suggest a Let TA denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then TA is a differential Batalin-Vilkovisky algebra.

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