Proseminar "The Algebra & Geometry of Modern Physics"
|Kod Erasmus / ISCED:||
|Nazwa przedmiotu:||Proseminar "The Algebra & Geometry of Modern Physics"|
Fizyka i astronomia; seminaria (Lista S)
Fizyka, II stopień; przedmioty do wyboru
Fizyka; przedmioty prowadzone w języku angielskim
Przedmioty do wyboru dla doktorantów;
Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka
|Punkty ECTS i inne:||
(tylko po angielsku)
Academic year 2017/2018 – Supersymmetry and index theorems
Index theorems are one of the most important achievements of contemporary mathematics. They relate analytical and topological properties of manifolds, and can be regarded as sophisticated generalizations of the Gauss-Bonnet integral formula for the Euler characteristic of a two-dimensional surface. It turns that both the Gauss-Bonnet integral formula, as well as index theorems, can be interpreted and proven from the perspective of modern physics. Such interpretations and proofs are based on intricate properties of quantum mechanics and supersymmetry. This has been an important discovery of the last decades of the 20th century, which inspired new generations of theoretical physicists and mathematicians, and once again showed unity of physics and mathematics.
The aim of our colloquium in this academic year is to understand index theorems and their physical interpretation. In the first part of our adventure we will introduce and summarize relevant properties of quantum mechanics, path integrals, and supersymmetry. In the second part we will discuss mathematical meaning of index theorems. In the final set of our colloquia we will reveal how to prove and interpret index theorems from physics perspective. Our colloquium should be understandable to everyone familiar with basics of quantum mechanics; we do not assume prior knowledge of supersymmetry or index theorems.
Literature for this year
 M. Nakahara,
Geometry, Topology and Physics,
Institute of Physics Publishing, 2003.
 L. A. Takhtajan,
Quantum Mechanics for Mathematicians,
American Mathematical Society, 2008.
 O. Alvarez,
Lectures on Quantum Mechanics and the Index Theorem
 J. M. Rabin
Introduction to Quantum Field Theory for Mathematicians
 L. Alvarez-Gaume
Topics in String Theory and Quantum Gravity
Les Houches Summer School on Gravitation and Quantization (1992)
 K. Hori et al. (Eds.),
Clay Mathematics Monographs, 2003.
General introduction – long term perspective
The last three decades have witnessed an unprecedented development of algebraic and geometric methods in quantum field theory and string theory. These methods provide a natural framework for a rigorous formulation of the classical model with a topological sector or an extended symmetry, broaden our perspective on its quantisation, and lead to novel insights into the microscale structure of the emergent spacetime and the nature of physical fields thereon. Furthermore, they prove extremely effective in obtaining a combinatorial and algebraic description of the topology of (low‐dimensional) spaces and in establishing categorial correspondences between non‐homeomorphic spaces, the latter phenomenon being exemplified by the mirror symmetry of Calabi–Yau manifolds. In our colloquium, we intend to give proper credit to the said development by discussing a variety of topics from the forefront of mathematical physics inspired by string theory and some exactly solvable quantum field theories. More specifically, and as we shall motivate below, our discourse is envisaged to revolve around the topics from the following dynamical) list:
Rudimentary physical background and its mathematical underpinning
An exceptionally rich pool of ideas and an excellent testing ground for the mathematical methods that we want to explore is offered by two‐dimensional conformal field theory and topological field theories related to it. The physical models of interest, including the two‐dimensional non‐linear sigma model of the critical string, topological string theory, and the three‐dimensional Chern–Simons topological gauge field theory, describe extended objects carrying a topological charge to which a (higher) gauge field couples, classified by an appropriate variant of cohomology (sheaf, equivariant, relative etc.). A consistent formulation of their dynamics calls for the introduction of higher categorial structures that furnish a geometric realisation of the relevant cohomology classes, e.g., n‐gerbes and their morphisms. These yield a neat classification of (phases of) consistent such models and capture a great deal of structural information on morphisms between them, represented by groupoidal and more general categorial constructs, and on their consistent gauging. Quite remarkably, and importantly, they also canonically induce – through Gawędzki's cohomological transgression – a geometric quantisation of the models that naturally incorporates interaction processes represented by suitable cobordisms between Cauchy hypersurfaces. Thus, transgression constitutes an explicit implementation of Segal's idea of categorial quantisation that proposes to define a quantum field theory as a functor from the geometric category of decorated) cobordisms to the algebraic category of topological vector spaces. When phrased in the language of the quantum group‐theoretic modular tensor categories naturally associated with the Chern–Simons theory and related models, the abstract idea led Witten, Turaev, Reshetikhin, Viro et al. to the elucidation and a far‐reaching extension of Jones' construction of topological (knot) invariants for three‐dimensional manifolds. These results, in conjunction with the beautiful and intrinsically holographic relation between the Chern–Simons theory and a specific two‐dimensional conformal field theory (the Wess–Zumino–Witten sigma model), are at the very core of what structurally seems to be the most complex and complete realisation of Segal's idea so far, to wit, the categorial quantisation programme for two‐dimensional (rational) conformal field theory, due to Felder, Fröhlich, Fuchs, Schweigert, Runkel et al.
The simple geometric idea underlying the physical models of interest, which is that of a functorial mapping of cobordisms (or `world‐volumes') between spatially extended boundary cycles, decorated by cells of a geometrically realised (higher) category, into an algebraic category of Hilbert spaces has far‐reaching consequences for the structure of the emergent geometry of the covariant configuration bundle (or `target space') that supports the said realisation of the category. This is particularly well‐understood in the setting of string theory in which we find two‐dimensional cobordisms (or `world‐sheets') between boundary loops, decorated by cells of the bicategory of abelian bundle gerbes with connection over the target space of the two‐dimensional field theory. Owing to the geometric nature of the physical model, an effective description of the (dynamical) geometry of the target space may be extracted from the spectrum of string excitations. The description is in terms of an algebra of functions on the target space, in the spirit of the Gelfand–Naimark theorem. Here, an in‐depth study of the quantised theory opens avenues to significant conceptual departures from the riemannian paradigm for the geometry of the target space, as `probed' by the loops, towards essentially stringy geometries. Among the latter, we find – on the one hand – generalised orbispaces of the loop‐mechanical duality groups (e.g., the so‐called T‐folds discovered by Hull), modeled on riemannian geometry only locally, and – on the other hand – incarnations of Connes' spectral non‐commutative geometry encoded – à la Fröhlich–Gawędzki – by the operator‐algebraic content of the superconformal field theory of the superstring.
(tylko po angielsku)
The literature for current topics will be suggested by the lecturer. Independently of such suggestions, participants of the colloquium are urged to consult any of the following references:
● B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, University Lecture Series vol. 21, American Mathematical Society, 2001.
● J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics vol. 107, Birkhäuser, 1993.
● A. Connes, Noncommutative Geometry, Academic Press, 1994.
● P. Deligne, et al. (Eds.), Quantum Fields and Strings: A Course For Mathematicians, vols. I,II, American Mathematical Society, 1999.
● P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer, 1997.
● G. Folland, Quantum Field Theory, American Mathematical Society, 2008.
● T. Frankel, The Geometry of Physics, Cambridge University Press, 1997.
● E. Frenkel, ``Lectures on the Langlands Program and Conformal Field Theory'', In: Frontiers in Number Theory, Physics, and Geometry II, Springer, 2007, pp. 387–533 [arXiv:hep-th/0512172].
● J. Fröhlich and K. Gawędzki, ``Conformal Field Theory and Geometry of Strings'', In: Vancouver 1993, Proceedings, Mathematical Quantum Theory I: Field Theory and Many-Body Theory, J. Feldman, R. Froese and L.M. Rosen (Eds.), CRM Proceedings & Lecture Notes vol. 7, American Mathematical Society, 1994, pp. 57–97 [arXiv:hep-th/9310187].
● J.A. Fuchs, Aﬃne Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory, Cambridge University Press, 1992.
● K. Gawędzki, ``Conformal ﬁeld theory: A case study'', 1999 [hep-th/9904145].
● K. Hori et al., Mirror symmetry, Clay Mathematics Monographs, 2003.
● S. Hu, Lecture Notes on Chern–Simons–Witten Theory, World Scientiﬁc, 2001.
● J. Lurie, ``On the Classiﬁcation of Topological Field Theories'', In: Current Developments in Mathematics Volume 2008, 2009, pp. 129–280 [arXiv:0905.0465].
● S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics vol. 5, Springer, 1971.
● M. Mariño, Chern-Simons Theory, Matrix Models, and Topological Strings, Oxford University Press, 2005.
● M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, 2003.
● H. Sati and U. Schreiber (Eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics vol. 83, American Mathematical Society, 2011.
● G.B. Segal, ``The Deﬁnition of Conformal Field Theory'', In: Symposium On Topology, Geometry And Quantum Field Theory (Segalfest), G.B. Segal and U. Tillmann (Eds.), London Mathematical Society Lecture Note Series vol. 308, Cambridge University Press, 2004, pp. 421–575.
● L. Takhtajan, Quantum Mechanics for Mathematicians, American Mathematical Society, 2008.
● R. Ticciati, Quantum Field Theory for Mathematicians, Cambridge University Press, 1999.
● V.G. Turaev, Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics vol. 18, Walter de Gruyter, 1994.
● N.M.J. Woodhouse, Geometric Quantization, Oxford University Press, 1992.
Zajęcia w cyklu "Semestr letni 2022/23" (w trakcie)
|Okres:||2023-02-20 - 2023-06-18||
zobacz plan zajęć
Seminarium, 30 godzin, 30 miejsc
|Koordynatorzy:||Karol Palka, Piotr Sułkowski, Rafał Suszek|
|Prowadzący grup:||(brak danych)|
|Lista studentów:||(nie masz dostępu)|
Zaliczenie na ocenę
Seminarium - Zaliczenie na ocenę
Właścicielem praw autorskich jest Uniwersytet Warszawski.