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The Algebra & Geometry of Modern Physics

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room 2.23, Pasteura 5 at 16:15

Antonia Zipfel (IFT WFUW)

Spinors and Quantum Geometry

Loop Quantum Gravity (LQG) aims at a manifestly background independent quantization of General Relativity. 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.

room 2.23, Pasteura 5 at 16:30

Nils Carqueville (Vienna University)

Two-dimensional topological field theories with defects, and their symmetries

In this talk we first recall the functorial definition of closed and open/closed TFTs, as well as their equivalent algebraic descriptions (in terms of Frobenius algebras and Calabi-Yau categories). 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.

room 2.23, Pasteura 5 at 16:15

Rafał R. Suszek (KMMF WFUW)

Introduction to Clifford algebras V: Enter the Spinor (II)

The aim of the talk is to formulate – after Chevalley – an algebraic concept of a spinor associated with a given quadratic space and its Clifford algebra. 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. 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.

room 2.23, Pasteura 5 at 16:15

Rafał R. Suszek (KMMF WFUW)

Introduction to Clifford algebras V : Enter the spinor

The aim of the talk is to formulate an algebraic concept of a spinor associated with a given quadratic space and its Clifford algebra. 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. 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.

room 2.23, Pasteura 5 at 16:15

Mariusz Tobolski (WFUW)

Introduction to Clifford algebras IV

This talk will complete our algebraic introduction to Clifford algebras. 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.

room 2.23, Pasteura 5 at 16:15

Paweł Ciosmak (MIMUW)

Introduction to Clifford algebras III

The third talk on Clifford algebras will be devoted to the finite dimensional case. 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.

room 2.23, Pasteura 5 at 16:15

Piotr Suwara (MIMUW)

Introduction to Clifford Algebras II

I will introduce common tools for studying the Clifford Algebras, namely Z2-gradation, direct decomposition and involution morphism. 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.

room 2.23, Pasteura 5 at 16:15

Magdalena Zielenkiewicz (MIMUW)

Introduction to Clifford algebras

I will give an introductory lecture on Clifford algebras, including: the definitions, the proof of the existence and the uniqueness of Clifford albegras over a given vector space with an inner product, and some of the basic properties.

room 2.23, Pasteura 5 at 16:15

Karol K. Kozłowski (CNRS, IMB, UB Dijon)

Large-N asymptotic expansion of multiple integrals related to the quantum separation of variables method

The scalar products and certain correlation functions of models solvableby the quantum separation of variables can be expressed in terms of$N$-fold multiple integrals which can be thought of as the partitionfunction of a one dimensionalgas of particles trapped in an externalpotential $V$ and interacting through repulsive two-body interactions ofthe type $\ln \big[ \sinh[\pi \omega_1(\lambda-\mu)]\cdot \sinh[\pi \omega_2(\lambda-\mu)] \big]$. The analysis of the large-$N$ asymptotic behaviour ofthese integrals is of interest to the description of the continuum limitof the integrable model. Although such partition functions present certainstructural resemblances with those arising in the context of the so-called$\beta$-ensembles, their large-$N$ asymptotic analysis demands theintroduction of several new ingredients. Such a complication in theanalysis is due to the lack of dilation invariance of the exponential ofthe two-body interaction. In this talk, I shall discuss the main featuresof the method of asymptotic analysis which we have developed. The methodutilises large-deviation techniques on the one hand and theRiemann--Hilbert problem approach to truncated Wiener-Hopfsingular-integral equations on the other hand. This is a joint work withG. Borot (Max-Planck Institut, Bonn, Germany) and A. Guionnet (MIT,Boston, USA).

room 2.23, Pasteura 5 at 16:15

Karol Palka, Piotr Sułkowski, Rafał R. Suszek

Organizational meeting

The main theme we are going to focus on in this semester are spinors and their appearance and role in mathematics and physics. 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.; the latter approach focuses on the behaviour of spinors under the action of Lie groups. 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.

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