The Algebra & Geometry of Modern Physics
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A Khovanov bicomplex of the colored Jones polynomial
In this talk, first, we review the definition of Khovanov homology for uncolored Jones polynomials. Second, we recall what “colored” Jones polynomials are. Third, we specify a boundary map between complexes of Viro’s definition of the Khovanov homology producing a bicomplex whose graded Euler characteristic is a colored Jones polynomial.
Knot homology via string theory
In this talk, I would like to discuss the categorification of the quantum knot invariants via string dualities. By the recent developments, the categrification of the knot invariant becomes more tractable in the string duality. 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.
What the functor is a superfield?
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!
Problems in asymptotic analysis triggered by quantum integrable models
This talk aims at giving a brief outline of the various problems in asymptotic analysis that arise in the course of the study of certain correlation functions in so‐called quantum integrable systems. 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.
Some remarks on non-planar Feynman diagrams - Quasimodularity and physics of exotic smooth R4
Krzysztof Bielas: Some remarks on non-planar Feynman diagrams
Non-planar Feynman diagrams are interesting due to various reasons. The most recent is the new tools for calculating scattering amplitudes in N=4 SYM. 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.
Jerzy Król: Quasimodularity and physics of exotic smooth R4
In the 1980-ties mathematicians established the existence of two infinite families (uncountably many each) of distinct small and large exotic smooth structures on the topological trivial R^4. This phenomenon is possible only in dim. 4 - for other n we have precisely one smooth structure on every R^n. Since then there were rather substantial effort for deriving physical effects driven by these exotic R^4. Small exotic R^4 happened to be connected with the codimension-1 foliations of some 3-manifold and hence, via surgery along a link, of S^3. 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. In the talk I will discuss the quasimodularity issue of exotic small R^4 from the perspective of some susy YM theories on this exotic backgrounds: 1. N=2 SYM and also the effective low energy SW theory, and 2. N=4 topologically twisted (Witten-Vafa) SYM. One indeed finds for the correlation function on small exotic R^4 the expressions which are functions of Eisenstein E_2 series, and for the large case, the quasi-modular 3/2 forms (so called mock theta function of order 7, as proposed originally by Ramanujan). I will discuss possible physical meaning of the results.
Chern-Simons theory, quantum knot invariants, and volume conjectures (Part I)
I will give an introductory talk on relations between quantum knot invariants and Chern-Simons theory. The seminal paper by Witten showed that Chern-Simons theory provides a natural framework for the study of 3-manifolds and knot invariants. I will explain how the path integral formulations of invariants in Chern-Simons theory has led to rigorous formulations in mathematics. In addition, I will mention about the volume conjecture which relates quantum knot invariants to geometry of knot complements in S^3. The talk is supposed to be accessible to both mathematicians and physicists.
Quantum (field) theory as a functor, part III
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.) into the algebraic category of vector spaces (possibly with additional structure). 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. 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., and spanning a remarkable wealth of topics and ideas - from topological invariants generalising the Jones polynomial, all the way to the categorial quantisation programme for two-dimensional Conformal Field Theory and the state-sum models of Quantum Gravity. In these lectures, we present an elementary introduction to the axiomatics of TQFT (Part I), discuss its applications to the study of low-dimensional topology (Part II), and indicate its field-theoretic realisations and generalisations furnished by higher geometric structures that define lagrangean models of dynamics of topologically charged objects (Part III).
Quantum (field) theory as a functor, part II
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.) into the algebraic category of vector spaces (possibly with additional structure). 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. 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., and spanning a remarkable wealth of topics and ideas - from topological invariants generalising the Jones polynomial, all the way to the categorial quantisation programme for two-dimensional Conformal Field Theory and the state-sum models of Quantum Gravity. In these lectures, we present an elementary introduction to the axiomatics of TQFT (Part I), discuss its applications to the study of low-dimensional topology (Part II), and - time permitting (Part III?) - indicate its field-theoretic realisations and generalisations furnished by higher geometric structures that define lagrangean models of dynamics of topologically charged objects.
Quantum (field) theory as a functor, part I
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.) into the algebraic category of vector spaces (possibly with additional structure). 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. 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., and spanning a remarkable wealth of topics and ideas - from topological invariants generalising the Jones polynomial, all the way to the categorial quantisation programme for two-dimensional Conformal Field Theory and the state-sum models of Quantum Gravity. In these lectures, we present an elementary introduction to the axiomatics of TQFT (Part I), discuss its applications to the study of low-dimensional topology (Part II), and - time permitting (Part III?) - indicate its field-theoretic realisations and generalisations furnished by higher geometric structures that define lagrangean models of dynamics of topologically charged objects.
Categories & all that, part II...
While the path integral is rarely mathematically well-defined, it is usually assumed to have some useful properties like sewing laws (relating the integral over a domain which decomposes into two subdomains to path integrals over the subdomains). These were included by Atiyah into the definition of the topological quantum field theory as a functor on the category of cobordisms. This is one of the many ways the modern mathematical language of categories and functors becomes relevant to physicists. In this lecture we will continue the introduction to categories, moving into the direction of the ones used in topological quantum field theories. We will show how various notions from different areas of mathematics get unified in the categorial framework. We will discuss categories with multiplication and some geometric functors (like homology or K-theory) arising whenever global effects of the spacetime play a role.