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The Algebra & Geometry of Modern Physics

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room 2.23, Pasteura 5 at 16:15

Piotr Sułkowski (WFUW)

Boson-fermion correspondence

In two dimensions there is a remarkable correspondence between certain quantum bosonic and fermionic theories. Originally it has been found in physics as an equivalence of a theory of a (massive) Thirring model (i.e. an interacting Dirac fermion) and the so-called bosonic sine-Gordon model, and it provides a prototype example of dualities in quantum field theories and string theory. Mathematically this correspondence gives rise to an isomorphism between certain vector spaces (i.e. bosonic and ferminic Fock spaces), which respectively form representations of infinite-dimensional Heisenberg and Clifford algebras; also beautiful and surprising links with representation theory, soliton equations, integrable hierarchies, etc. arise from this correspondence. In this talk this correspondence and some of its consequences will be reviewed.

room 2.23, Pasteura 5 at 16:15

Paweł Nurowski (CFT)

Spinors in action: a few (more) examples, II

Spinors in action: a few examples, II

room 2.23, Pasteura 5 at 16:15

Paweł Nurowski (CFT)

Spinors in action: a few examples

room 2.23, Pasteura 5 at 16:30

Piotr Kucharski (WFUW)

Knots, BPS states, and algebraic curves

Starting from a wide and pedagogical introduction about the intersection of knot theory and strings, I will present a new, missing link in the triangle made of knots, BPS states and algebraic curves.

room 2.23, Pasteura 5 at 16:15

Andrzej Trautman (WFUW)

Excerpts from my lectures on spinors given in Warsaw and in Trieste, II

A little of history; complements of algebra; definition of Spin groups: algebra vs topology; two periodicities of properties of Clifford algebras; complex conjugation and antiparticles; the Chevalley theorem and the Brauer-Wall group: spinorial clock; Radon-Hurwitz numbers and vector fields on spheres; spinors on manifolds: two approaches; (S)pin structures; tensor bundles are natural; spinor bundles are not; triviality of spinor bundles of spheres; covariant differentiation of spinor fields; Dirac operator: Schroedinger's formula, spectrum on spheres.

room 2.23, Pasteura 5 at 16:15

Andrzej Trautman (WFUW)

Excerpts from my lectures on spinors given in Warsaw and in Trieste, I

room 2.23, Pasteura 5 at 16:15

Leonid Chekhov (Aarhus University)

Gaussian means, discretizations of moduli spaces, and cohomological field theories

We begin with combinatorial construction establishing the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model (or, in other words, chord diagrams) and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces of Riemann surfaces with holes given by (quasi)polynomials $N_{g,s}(P_1,\dots,P_s)$ where $(P_1,\dots,P_s)\in\mathbb R_+^s$ are (fixed) perimeters of holes. 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. Using the topological recursion, we find explicit recurrent relations for general $s$-loop means in all genera proving simultaneously that the corresponding simple linear combinations of ancestor invariants are nonnegative integers. Based on recent joint paper with J.E.Andersen, P.Norbury, and R.C.Penner, arXiv: 1501.05867.

room 2.23, Pasteura 5 at 16:15

Henryk Żołądek (MIMUW)

The geometric Langlands program and electromagnetic duality, II

The Langlands program predicts a correspondence between finite dimensional representations of Galois groups of extensions of number fields and automorphic representations of groups over fields of adeles. The geometric analogy of this program constructs so-called Hecke eigensheaves associated with flat bundles over Riemann surfaces. Kapustin and Witten have described the Geometric Langlands program by compactifying on a Riemann surface certain version of supersymmetric Yang-Mills theore. The supersymmetries of their model involve spinors from a space of some representation of a Clifford algebra associated with a twisted embedding of the groups Spin(4) into Spin(4)xSpin(6). The analogues of flat bundles are played by so-called electric zerobranes and the analogues of the Hecke operators are so-called 't Hooft operators acting on magnetic branes. In the first talk I will present mathematical introduction to the geometric Langlands program.

room 2.23, Pasteura 5 at 16:15

Henryk Żołądek (MIMUW)

The geometric Langlands program and electromagnetic duality

room 2.23, Pasteura 5 at 16:15

Satoshi Nawata (IFT, WFUW)

Quantum Physics and Geometry

This talk is a demo talk for my faculty interview. 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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