The Thursday Colloquium THE ALGEBRA & GEOMETRY OF MODERN PHYSICS Academic year 2016/17 June 8th, 2017 Petr Vasko (WFUW) Correspondence between Chern-Simons theory and Wess-Zumino-Witten models Abstract: I will discuss various relations between CS and WZW theories at classical level. First, I show how the Wess-Zumino term arises as a gauge variation of Chern-Simons action, in case the three-manifold has a boundary. The main topic, however, will be to derive an equivalence between classical phase spaces of CS and WZW theories, following https://arxiv.org/abs/hep-th/0101170v1. The WZW model will be defined on a strip with "untwisted" boundary conditions while the corresponding CS theory lives on (disc x line) with two Wilson lines (in representations determined by boundary conditions at both ends of the strip) puncturing the disc. Brief comments about quantization might be mentioned if time allows. June 1st, 2017 Noémie Combe (Institut de Mathématiques de Marseille) A new cellulation for the n-th configuration space of R^2 and moduli space M(0,n) Abstract: The first point of this seminar is to introduce the configuration space and the moduli space M(0,n) for genus 0 curves with n distinct points. The second point will be to consider relations between the configuration space for R^2 to braid groups. We present as well the idea of a new cell decomposition for  M(0,n) which is reminiscent to Shabat-Voevodsky-Penner-Grothendieck cellulation for the Teichmuller space T_{g,n} for g>0. May 25th, 2017 Jacek Krajczok (WFUW) Wess-Zumino-Witten model, part II Abstract: During the talk I will introduce the Wess-Zumino-Witten model. I'll describe it's space of solutions and construct a pre-symplectic form on it. May 18th, 2017 Jacek Krajczok (WFUW) Wess-Zumino-Witten model Abstract: During the talk I will introduce the Wess-Zumino-Witten model. I'll describe it's space of solutions and construct a pre-symplectic form on it. April 27th, 2017 Piotr Surówka (Max Planck Institute for the Physics of Complex Systems) Symmetry, topology and hydrodynamics Abstract: I will show how hydrodynamics is modified if the underlying fluid constituents are massless Weyl fermions, which are anomalous at the quantum level. The non-dissipative nature of the modification allows one to construct a partition function which compactly describes the transport properties of the system.  I will explain how the anomalous properties can be understood in terms of heat kernels. April 20th, 2017 Helder Larraguivel (WFUW) Chern-Simons theory and Wilson lines, part V Abstract: Chern-Simons theory and Wilson lines, part V. April 6th, 2017 Helder Larraguivel (WFUW) Chern-Simons theory and Wilson lines, part IV Abstract: Chern-Simons theory and Wilson lines, part IV. March 30th, 2017 Helder Larraguivel (WFUW) Chern-Simons theory and Wilson lines, part III Abstract: Chern-Simons theory and Wilson lines, part III. March 16th, 2017 Jacek Krajczok (WFUW) Chern-Simons theory and Wilson lines, part II Abstract: Chern-Simons theory and Wilson lines, part II. March 9th, 2017 Hiroyuki Fuji (Kagawa University, Japan) Partial chord diagrams and matrix models Abstract: Concepts of fatgraphs and partial chord diagrams occur in many branches ofmathematics, including topology, geometry, and representation theory.During the last decade, some applications of these mathematical objects tothe research of the molecular biology have been reported. Among them, inparticular, a characterization by the genus in the fatgraph presentationof the RNA has been studied remarkably. In this talk, I will explain howthe concepts of fatgraphs and partial chord diagrams are applied to thestudy of the secondary structure of the RNA with.pseudoknots, andintroduce the matrix model that is invented by basic techniques of thequantum field theory. January 26th, 2017 Jacek Krajczok (WFUW) Chern-Simons theory and Wilson lines, part I Abstract: Chern-Simons theory and Wilson lines. December 1st, 2016 Helder Larraguivel (WFUW) Chern-Simons theory, III Abstract: Introduction to Chern-Simons theory. November 24th, 2016 Helder Larraguivel (WFUW) Chern-Simons theory, II Abstract: Introduction to Chern-Simons theory. November 17th, 2016 Helder Larraguivel (WFUW) Chern-Simons theory, I Abstract: Introduction to Chern-Simons theory. October 27th, 2016 Mariusz Tobolski (WFUW) Principal bundles - recap, II Abstract: Principal bundles. October 20th, 2016 Mariusz Tobolski (WFUW) Principal bundles - recap, I Abstract: Princiapl bundles. October 6th, 2016 Relaunch Relaunch Abstract: Organizational meeting. Academic year 2015/16 June 9th, 2016 Pawe³ Smoliñski (WFUW) Moment maps, IV Abstract: Having defined invariant vector fields and orbits of the coadjoint action I will focus on reduction of the group manifold. If time permits I will present Marsden-Weinstein reduction with examples. June 2nd, 2016 Aleksander Strzelczyk (WFUW) Moment maps, III Abstract: Having defined the moment map we will proceed to establish a link between the action of Lie algebra and the moment map. This will lead us to the cocycle condition, supported by examples. May19h, 2016 Aleksander Strzelczyk (WFUW) Moment maps, II Abstract: Having defined the moment map we will proceed to establish a link between the action of Lie algebra and the moment map. This will lead us to the cocycle condition, supported by examples. May12h, 2016 Piotr Kucharski (WFUW) Moment maps, I Abstract: I will give an introductory talk setting the stage for the notion of the moment map. It will contain mainly definitions, examples and intuitions, which will hopefully serve for the following talks. Apr 28th, 2016 Jacek Krajczok (WFUW) Smooth action of a Lie group and the existence of a smooth manifold structure on the orbit space Abstract: During the lecture, we will be dealing with a smooth action of a Lie group on a smooth manifold. Our goal is a theorem which states that under additional assumptions on the action (it must be proper and free) there exists only one smooth manifold structure on the space of orbits such that the quotient map M -> M/G is a submersion. We will also prove some additional results concerning manifolds in general and those with a group action. Apr14th, 2016 Mariusz Tobolski (WFUW) Principal bundles and gauge theory, IV Abstract: We conclude our discussion of gauge theories in the language of principal bundles. The concept of gauge transformation will be introduced and some results highlighted in previous lectures proved. Apr 7th, 2016 Mariusz Tobolski (WFUW) Principal bundles and gauge theory, III Abstract: Physical gauge theories can be described in the language of principal bundles. Connections on a bundle and local gauge transformations are then of most importance. After finishing our discussion of local sections and transition mappings from the previous lecture, we will focus on understanding gauge transformations using connection forms (potentials). The case of Dirac monopole will be given as an example. Mar 31st, 2016 Mariusz Tobolski (WFUW) Principal bundles and gauge theory, II Abstract: Physical gauge theories can be described in the language of principal bundles. Connections on a bundle and local gauge transformations are then of most importance. After finishing our discussion of local sections and transition mappings from the previous lecture, we will focus on understanding gauge transformations using connection forms (potentials). The case of Dirac monopole will be given as an example. Mar 17th, 2016 Mariusz Tobolski (WFUW) Principal bundles and gauge theory, I Abstract: I will introduce the category of (smooth) principal bundles. The motivation behind this concept comes from both mathematics (classification of fibre bundles) and physics (gauge theory). The difference between local and global triviality of the principal bundle will be discussed. I will also talk about transition functions and the reconstruction theorem. Mar 10th, 2016 Tomasz Smo³ka (WFUW) Lie groups: from algebra to differential geometry, V Abstract: I will continue my presentation about properties of the Logarithmic Derivative. Next, I will also prove Automatic Smoothness Theorem for Lie Groups. Mar 3rd, 2016 Tomasz Smo³ka (WFUW) Lie groups: from algebra to differential geometry, IV Abstract: I will continue my presentation about properties of the Logarithmic Derivative. Next, I will also prove Automatic Smoothness Theorem for Lie Groups. Feb 25th, 2016 Tomasz Smo³ka (WFUW) Lie groups: from algebra to differential geometry, III Abstract: I will present basic properties of the Exponential Function. It is a natural generalization of the matrix exponential map, which is obtained for Lie group GL_{n}(R). I will also talk about the Logarithmic Derivative. Jan 28th, 2016 Jacek Krajczok (WFUW) Lie groups: from algebra to differential geometry, II Abstract: Lie groups: from algebra to differential geometry, II Jan 21th, 2016 Kazunobu Maruyoshi (Imperial College, London) Anomalies of class S_k theories via 6d Abstract: Class S_k theories are four-dimensional N=1 superconformal field theories (SCFTs) whose origin is supposed to be the C^2/Z_k orbifold of N=(2,0) theory in six dimensions by a Riemann surface compactification. We consider the 't Hooft anomalies, which are one of the most accessible physical quantities, of these theories. We show that the anomaly polynomial of the former is obtained from that of the latter by the integral over the Riemann surface, which gives a check of the statement above and interestingly leads to a proposal of the anomalies of strongly coupled N=1 SCFTs. This talk is based on the collaboration with I. Bah, A. Hanany, and Y. Tachikawa. Jan 14th, 2016 Jacek Krajczok (WFUW) Lie groups: from algebra to differential geometry, I Abstract: Lie groups: from algebra to differential geometry, I Jan 7th, 2016 Aleksander Strzelczyk (WFUW) Functorial quantisation - a case study I, cntd. 2 Abstract: Functorial quantisation - a case study I, cntd. 2 Dec 10th, 2015 Motohico Mulase (Univeristy of California, Davis) Opers and quantum curves through Gaiotto's conjecture Abstract: This talk aims at explaining our recent solution of Gaiotto's conjecture. The conjecture is about the precise construction of opers from a flat family of connections associated with the Hitchin component of the moduli space of Higgs bundles. It has been solved very recently in collaboration with Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo and Andrew Neitzke. This theorem leads us to developing a mathematical theory of quantum curves for Hitchin spectral curves. In this talk, the result of our joint work on the conjecture is outlined. Then we give a holomorphic formula for non-Abelian Hodge correspondence for the Hitchin components. Dec 3rd, 2015 Aleksander Strzelczyk (WFUW) Functorial quantisation - a case study I, cntd. Abstract: Functorial quantisation - a case study I, cntd. Nov 26th, 2015 Aleksander Strzelczyk (WFUW) Functorial quantisation - a case study I Abstract: Functorial quantisation - a case study I Nov 19th, 2015 Mariusz Tobolski (WFUW) The canonical description of a lagrangean field theory, VI Abstract: The canonical description of a lagrangean field theory, VI Nov 12th, 2015 Mariusz Tobolski (WFUW) The canonical description of a lagrangean field theory, V Abstract: The canonical description of a lagrangean field theory, V Nov 5th, 2015 Mariusz Tobolski (WFUW) The canonical description of a lagrangean field theory, IV Abstract: The canonical description of a lagrangean field theory, IV Oct 29th, 2015 Maciej Kolanowski (WFUW) The canonical description of a lagrangean field theory, III Abstract: The canonical description of a lagrangean field theory, III Oct 22nd, 2015 Maciej Kolanowski (WFUW) The canonical description of a lagrangean field theory, II Abstract: The canonical description of a lagrangean field theory, II Oct 15th, 2015 Mariusz Tobolski (WFUW) The canonical description of a lagrangean field theory, I Abstract: The canonical description of a lagrangean field theory, I Academic year 2014/15 May 21st, 2015 Piotr Su³kowski (WFUW) Boson-fermion correspondence Abstract: 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. May 7th, 2015 Pawe³ Nurowski (CFT) Spinors in action: a few examples, II Abstract: Spinors in action: a few examples April 23rd, 2015 Pawe³ Nurowski (CFT) Spinors in action: a few examples Abstract: Spinors in action: a few examples  April 16th, 2015 Piotr Kucharski (WFUW) Knots, BPS states, and algebraic curves Abstract: 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. April 9th, 2015 Andrzej Trautman (WFUW) Excerpts from my lectures on spinors given in Warsaw and in Trieste, II Abstract: 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. March 26th, 2015 Andrzej Trautman (WFUW) Excerpts from my lectures on spinors given in Warsaw and in Trieste, I Abstract: 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. March 19th, 2015 Henryk ¯o³±dek (MIMUW) The geometric Langlands program and electromagnetic duality, III Abstract: 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. March 12th, 2015 Neil Lambert (King's College London & WFUW) Konwersatorium L. Infelda: M-theory and hidden spacetime in quantum field theory Abstract: M-Theory arises as non-perturbative formulation of string theory and as such may be viewed as a complete quantum theory of particle physics, unified with gravity. 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. March 5th, 2015 Leonid Chekhov (Aarhus University) Gaussian means, discretizations of moduli spaces, and cohomological field theories Abstract: 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. February 26th, 2015 Henryk ¯o³±dek (MIMUW) The geometric Langlands program and electromagnetic duality, II Abstract: 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. January 22nd, 2015 Henryk ¯o³±dek (MIMUW) The geometric Langlands program and electromagnetic duality Abstract: 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. January 15th, 2015 Symposium: Faculty of Physics and Faculty of Mathematics, Informatics & Mechanics Welcome symposium at the Ochota campus Abstract: Welcome symposium at the Ochota campus January 8th, 2015 Satoshi Nawata (IFT WFUW) Quantum Physics and Geometry Abstract: 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. December 11th, 2014 Antonia Zipfel (IFT WFUW) Spinors and Quantum Geometry Abstract: 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. December 4th, 2014 Nils Carqueville (Vienna University) Two-dimensional topological field theories with defects, and their symmetries Abstract: 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. November 27th, 2014 Rafa³ R. Suszek (KMMF WFUW) Introduction to Clifford algebras V: Enter the spinor (II) Abstract: 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. November 20th, 2014 Rafa³ R. Suszek (KMMF WFUW) Introduction to Clifford algebras V: Enter the spinor Abstract: 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. November 6th, 2014 Mariusz Tobolski (WFUW) Introduction to Clifford algebras IV Abstract: 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. October 30th, 2014 Pawe³ Ciosmak (MIMUW) Introduction to Clifford algebras III Abstract: 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. October 23rd, 2014 Piotr Suwara (MIMUW) Introduction to Clifford algebras II Abstract: 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. October 16th, 2014 Magdalena Zielenkiewicz (MIMUW) Introduction to Clifford algebras Abstract:  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. October 9th, 2014 Karol K. Koz³owski (CNRS, IMB, UB Dijon) Large-N asymptotic expansion of multiple integrals related to the quantum separation of variables method Abstract: The scalar products and certain correlation functions of models solvable by the quantum separation of variables can be expressed in terms of N-fold multiple integrals which can be thought of as the partition function of a one dimensionalgas of particles trapped in an external potential V and interacting through repulsive two-body interactions of the type ln{sinh[πω1(λ-μ)]·sinh[πω2(λ-μ)]}. 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. Although such partition functions present certain structural resemblances with those arising in the context of the so-called β-ensembles, their large-N asymptotic analysis demands the introduction of several new ingredients. 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. The method utilises large-deviation techniques  on the one hand and the Riemann–Hilbert problem approach to truncated Wiener–Hopf singular-integral equations on the other hand. This is a joint work with G. Borot (Max-Planck Institut, Bonn, Germany) and A. Guionnet (MIT, Boston, USA). October 2nd, 2014 Karol Palka, (IMPAN), Piotr Su³kowski (IFT WFUW), Rafa³ R. Suszek (KMMF WFUW) Organizational meeting Abstract:  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. Academic year 2013/14 June 5th, 2014 Piotr Kucharski (IFT WFUW) Bundles and Aharonov-Bohm effect Abstract: The Aharonov-Bohm effect provides a very nice example of a phenomenon that can be seen either from the mathematical and physical point of view. In this seminar I will present both perspectives: vector bundles and gauge theory and show how they are related to each other. May 29th, 2014 Piotr M. Hajac (IMPAN) The joy of principal bundles Abstract: The joy of principal bundles. May 15th, 2014 Piotr Su³kowski (IFT WFUW) Instantons revisited Abstract: This lecture is an introduction to physics and mathematics of instantons. We will review construction of instantons (BPST solution, ADHM construction), properties of their moduli spaces, and discuss some of their applications. May 8th, 2014 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. Apr 10th, 2014 Rafa³ R. Suszek (KMMF WFUW) Principal bundles, equivariant descent & the gauge principle, II Abstract: In a series of lectures, the applicability of the structure of a principal bundle with connection in the context of ﬁeld theory will be demonstrated. First, the deﬁnition of a principal bundle with a structural groupoid and a compatible connection will be presented, with view to establishing a natural description of ﬁeld theories with some of their global symmetries rendered local (or "gauged"). The concept of equivariant descent of a geometric object (such as, e.g., a tensor, a bundle or a higher structure) will be introduced and illustrated through the example of an equivariant bundle over a base acted upon by a group. Finally, a general gauge principle will be formulated for ﬁeld theories deﬁned in terms of lagrangean densities, and subsequently concretised in the physically interesting setting of the dynamics of topologically charged objects in external ﬁelds. This will, in particular, lead us to an algebroidal and cohomological classiﬁcation of obstructions to the gauging, aka gauge anomalies. Apr 3rd, 2014 Rafa³ R. Suszek (KMMF WFUW) Principal bundles, equivariant descent & the gauge principle, I Abstract: In a series of lectures, the applicability of the structure of a principal bundle with connection in the context of ﬁeld theory will be demonstrated. First, the deﬁnition of a principal bundle with a structural groupoid and a compatible connection will be presented, with view to establishing a natural description of ﬁeld theories with some of their global symmetries rendered local (or "gauged"). The concept of equivariant descent of a geometric object (such as, e.g., a tensor, a bundle or a higher structure) will be introduced and illustrated through the example of an equivariant bundle over a base acted upon by a group. Finally, a general gauge principle will be formulated for ﬁeld theories deﬁned in terms of lagrangean densities, and subsequently concretised in the physically interesting setting of the dynamics of topologically charged objects in external ﬁelds. This will, in particular, lead us to an algebroidal and cohomological classiﬁcation of obstructions to the gauging, aka gauge anomalies. Mar 20th, 2014 Pawe³ Urbañski (KMMF WFUW) Connections in the sense of Ehresmann, II Abstract: Powi±zanie w sensie Ehresmanna na wi±zce w³óknistej jest dystrybucj± (horyzontaln±) dope³niaj±c± dystrybucjê wektorów pionowych. Przedyskutujê ró¿ne sposoby reprezentacji tej dystrybucji oraz podstawowych konstrukcji z ni± zwi±zanych (podniesienie horyzontalne, pochodna kowariantna, krzywizna). Nastêpnie omówiê koneksje zgodne z zastan± struktur± rozw³óknienia (wi±zki wektorowej, wi±zki wektorowej z metryk±, wi±zki stycznej i kostycznej, wi±zki g³ównej). Mar 13th, 2014 Pawe³ Urbañski (KMMF WFUW) Connections in the sense of Ehresmann Abstract: Powi±zanie w sensie Ehresmanna na wi±zce w³óknistej jest dystrybucj± (horyzontaln±) dope³niaj±c± dystrybucjê wektorów pionowych. Przedyskutujê ró¿ne sposoby reprezentacji tej dystrybucji oraz podstawowych konstrukcji z ni± zwi±zanych (podniesienie horyzontalne, pochodna kowariantna, krzywizna). Nastêpnie omówiê koneksje zgodne z zastan± struktur± rozw³óknienia (wi±zki wektorowej, wi±zki wektorowej z metryk±, wi±zki stycznej i kostycznej, wi±zki g³ównej). Mar 6th, 2014 Karol Palka (IMPAN) Quick introduction to fibre bundles, II Abstract: We will discuss basic notions and properties of fibre bundles and their maps, including principal and associated bundles. We will recall classical constructions and some interesting examples. Feb 27th, 2014 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. We will recall classical constructions and some interesting examples. Jan 23rd, 2014 Noboru Ito (Waseda Institute of Advanced Study & University of Bonn) A Khovanov bicomplex of the colored Jones polynomial Abstract: 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. Jan 16th, 2014 Hiroyuki Fuji (Tsinghua University & MPI Bonn) Knot homology via string theory Abstract: 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. Jan 9th, 2014 Andrew Bruce (IMPAN) What the functor is a superfield? 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, 2013 Karol K. Koz³owski (CNRS, IMB, UB Dijon) Problems in asymptotic analysis triggered by quantum integrable models Abstract: 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. Dec 12th, 2013 Krzysztof Bielas (Uniwersytet ¦l±ski, Katowice) Some remarks on non-planar Feynman diagrams Abstract: 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. Dec 12th, 2013 Jerzy Król (Uniwersytet ¦l±ski, Katowice) Quasimodularity and physics of exotic smooth R4 Abstract: 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. Dec 5th and 6th, 2013 Satoshi Nawata (NIKHEF, Amsterdam) Chern–Simons theory, quantum knot invariants, and volume conjectures (Parts I and II) (lecture notes) Abstract: 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. Nov 7th, 14th and 28th, 2013 Rafa³ R. 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.) 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). Oct 24th, 2013 Motohico Mulase (University of California, Davis) Quantization of the spectral curves for Hitchin fibrations via Eynard–Orantin theory Abstract: Quantum curves are a magical object. It is conjectured that they capture the information of quantum topological invariants in an effective and compact way. The relation between quantum curves and the Eynard–Orantin theory was first exemplified in an influential paper of Gukov and Su³kowski. 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. Su³kowski and an earlier paper with Dumitrescu et al.) In the second part I will explain the construction of quantization of the spectral curves appearing in the theory of Hitchin fibrations. The main theorem is that the Eynard–Orantin theory indeed provides a mechanism of constructing the canonical generator of a D‐module on an arbitrary compact Riemann surface. The semi‐classical approximation of this D‐module coincides with the Hitchin spectral curve. (This part is based on my joint work with O. Dumitrescu of Hannover.) The talk will be given in an elementary and pedagogical language. Oct 17th, 2013 Micha³ Heller (University of Amsterdam & NCBJ) A brief introduction to the AdS/CFT correspondence and its applications Abstract: One of the most profound developments in the contemporary high energy physics is the discovery of the AdS/CFT correspondence (or more generally the gauge‐gravity duality), which states exact equivalence between a class of string theory vacua and certain quantum field theories. I will present the original Maldacena's argument for the existence of the AdS/CFT correspondence, as well as various qualitative and quantitative indications why this conjecture seems to be correct. If time permits, I will discuss recent research trends in this field. Oct 10th and 31st, 2013 Karol Palka (IMPAN) Categories and all that... (Parts I and II) Abstract: 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. The lecture will be an introduction to categories and functors. 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. Oct 3rd, 2013 Piotr Su³kowski (IFT WFUW) & Rafa³ R. Suszek (KMMF WFUW) The algebra and geometry of modern physics - an introduction Abstract: In this introductory lecture, we shall give a selective overview of recent developments of algebraic and geometric methods of mathematical physics inspired by (topological) quantum field theory and string theory, and delineate the main scientific objectives of the Colloquium. Academic year 2012/13 Mar 21st and 22nd, and Apr 4th and 5th, 2013 Karol K. Koz³owski (CNRS, IMB, UB Dijon) Asymptotic behaviour of β‐ensembles (lecture notes) Abstract: Random matrices first arose in the works of Wishart in the late 1920s. Later, in the mid‐1950s, Wigner introduced them as a tool for modelling spectra of heavy nuclei. Since then, these objects have appeared in many other branches of mathematics and physics and have been observed to have many more applications, such as to combinatorial problems of counting various types of graphs. One may consider various types of random matrices, the simplest model having a single random matrix satisfying certain symmetry conditions such as being real, symmetric or hermitian. The study of statistical properties of such random matrix ensembles boils down to the analysis of certain N‐fold integrals, N being the size of the square matrix. In fact, these integrals belong to the class of so‐called β‐ensemble integrals, which can be thought of as representing the partition function of a one‐dimensional Coulomb gas subject to an external potential V. Indeed, at specific values of the parameter β, to wit, β=1,2,4, one recovers the non‐trivial part of partition functions associated with the so‐called matrix model subordinate to the symmetric (the orthogonal ensemble), hermitian (the unitary ensemble) and hermitian self‐dual (the symplectic ensemble) random matrices. Most of the information of interest to matrix models (and β‐ensembles) is related to the large‐N behaviour. Therefore, its extraction demands analysing the N→∞ behaviour of N‐fold integrals. Introducing modern methods allowing one to carry out this sort of asymptotic analysis is the very purpose of these lectures.I shall commence by introducing the β‐ensembles and discussing their relation, for β=1,2,4, to the aforementioned matrix ensembles. I shall then make a few heuristic connections between partition functions of matrix models and enumeration problems of graphs drawn on surfaces with prescribed genera . Next, I shall proceed by entering deeper into the subject; namely, by introducing the various concepts and tools allowing one to set a convenient framework for carrying out the large‐N asymptotic analysis. I shall begin by discussing certain general topological properties of spaces of probability measures over Polish spaces. This will enable me to introduce the concept of large deviations for sequences of probability measures as well as to prove various crucial theorems relevant to these constructions. At that very point, I shall be able to explain how the framework can be utilised as a powerful tool for computing the leading large‐N asymptotic behaviour of the β‐ensemble integrals. Time permitting, I shall briefly expand upon techniques allowing one to compute subdominant large‐N asymptotics. Last updated November 4th, 2015