THE ALGEBRA & GEOMETRY OF MODERN PHYSICS
Organisers
Karol Palka ( Instytut Matematyczny PAN , e‐mail: k.palka at impan.pl )
Piotr Sułkowski ( Instytut Fizyki Teoretycznej, WFUW , e‐mail: psulkows at fuw.edu.pl )
Rafał R. Suszek ( Katedra Metod Matematycznych Fizyki, WFUW , e‐mail: suszek at fuw.edu.pl )
Abstract - 2018/2019 - Introduction to supergeometry
Contents:
1. Linear superalgebra (supervector spaces, superalgebras, Lie superalgebras and their modules)
2. Manifolds in the language of sheaves and functors (a brief intro to category theory and sheaves, locally ringed spaces, the Yonneda Lemma and the functor of points)
3. Elementary supergeometry (superspaces, supermanifolds, superpoints in a supermanifold, the local description of supermanifold morphisms, the tangent sheaf and its dual, the odd tangent bundle)
4. Lie supergroups (group objects in sMan, the tangent Lie superalgebra of a Lie supergroup, super Harish-Chandra pairs, Lie-supergroup actions and homogeneous spaces, the action of |R^{0|1} on a supermanifold)
5. Mathematical modelling of classical fermions.
The basic reference and metronome in our mini-course are Fioresi's "Supergeometry Lectures" (the Zuerich version), available online under the link: https://www.dm.unibo.it/~fioresi/. The points that are missing on Fioresi's list can be found in alternative sources:
- the tangent sheaf and its dual and the odd tangent bundle - in Dumtirescu's "Superconnections and parallel transport" and in Hohnhold et al.'s "Differential forms and 0-dimensional supersymmetric field theories";
- the action of |R^{0|1} on a supermanifold - in Hohnhold et al.'s "Differential forms and 0-dimensional supersymmetric field theories";
- Mathematical modelling of classical fermions - in Lecture 1 of Freed's "Five Lectures on Supersymmetry".
All, or at least a good proportion of the above can also be found in Varadarajan's "Supersymmetry for Mathematicians: An Introduction" (Chapters 3,4 and 7).
Abstract - 2017/2018 - Supersymmetry and index theorems - physical and mathematical perspectives
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
[1] M. Nakahara,
Geometry, Topology and Physics,
Institute of Physics Publishing, 2003.
[2] L. A. Takhtajan,
Quantum Mechanics for Mathematicians,
American Mathematical Society, 2008.
[3] O. Alvarez,
Lectures on Quantum Mechanics and the Index Theorem
[4] J. M. Rabin
Introduction to Quantum Field Theory for Mathematicians
[5] L. Alvarez-Gaume
Topics in String Theory and Quantum Gravity
Les Houches Summer School on Gravitation and Quantization (1992)
arXiv: hep-th/9212006
[6] K. Hori et al. (Eds.),
Mirror symmetry,
Clay Mathematics Monographs, 2003.
Abstract - 2015/2016/2017
The programme of this year’s edition of our colloquium — newly turned a student proseminar on mathematical physics
and to be organised under the auspices of the Department of
Mathematical Methods in Physics (Katedra Metod Matematycznych Fizyki)
from now onwards — is dedicated to a systematic study and elucidation
of the geometric and functorial quantisation of a three-dimensional
topological lagrangean gauge field theory with defects, known as the
Chern—Simons theory with Wilson lines. This quantisation scheme,
originating in the seminal works of Kostant, Segal, Witten, Atiyah and
other distinguished members of the mathematical-physical community in
the second half of the twentieth century, can be understood as an
explicit (geometro-cohomological) implementation of Segal’s programme
of categorial quantisation and as such offers unique and deep
conceptual insights into the very idea of quantisation and quantum from
the comfortable vantage point of a topological field theory, with its
finite number of non-dynamical degrees of freedom ensuring
controllability, and safely rid of the purely technical complications
in which more phenomenologically relevant models abound and sometimes
drown, and which render their formal treatment involved and hence
oftentimes less tractable and transparent. Thus, our proseminar may
well be regarded as an appetiser to be served prior to any standard
lecture course in quantum theory.
What makes it a sight more than just that is the exceptional breadth and depth of the physical as well as mathematical context in which the Chern—Simons model figures prominently and naturally: from higher geometry and gauge theory in the sense of Baez, required by the model’s very definition in the classical régime and determining its quantisation through the so-called cohomological transgression, via the first-order formalism in classical lagrangean field theory in the sense of Tulczyjew, Gawędzki, Kijowski and Szczyrba, the theory of the moment map and the Marsden—Weinstein reduction, all the way to an equivariant Kähler quantisation of the classical model with a hands-on application of (well-defined) path-integral techniques; from advanced group and quantum-group (representation) theory and higher-category and higher-cohomology theory, organising the model’s field and defect content and dictating, among other things, its generalised orbifolding, via the symplectic theory of moduli spaces of flat principal bundles over graph-decorated Riemann surfaces, all the way to topological invariants generalising the celebrated HOMFLY-PT polynomial, and — last but not least, most astonishingly and inspiringly — two-dimensional conformal field theory with a chiral-current extension of its gauge-symmetry (Virasoro) algebra. Needless to say, a context of such richness and gravity exceeds by far the scope of any one-year student proseminar, and yet we are confident and determined that a substantial proportion of the topics enumerated above, alongside some of their modern frontline applications, can and will be discussed in a rigorous and meaningful manner in the course of our one-year programme, with view to giving the interested student a good understanding of the subject matter and an acquaintance with the formal tools involved requisite for further advancement of the study thus initiated and — hopefully, and most importantly — an original application thereof in her or his future research.
It is our expectation and aim that the colloquium have a successful continuation, in its current character of a student proseminar, in the years to come, invariably serving as a natural platform of study and discussion of the many and diverse aspects of the vast realm of mathematical physics sensu largo.
What makes it a sight more than just that is the exceptional breadth and depth of the physical as well as mathematical context in which the Chern—Simons model figures prominently and naturally: from higher geometry and gauge theory in the sense of Baez, required by the model’s very definition in the classical régime and determining its quantisation through the so-called cohomological transgression, via the first-order formalism in classical lagrangean field theory in the sense of Tulczyjew, Gawędzki, Kijowski and Szczyrba, the theory of the moment map and the Marsden—Weinstein reduction, all the way to an equivariant Kähler quantisation of the classical model with a hands-on application of (well-defined) path-integral techniques; from advanced group and quantum-group (representation) theory and higher-category and higher-cohomology theory, organising the model’s field and defect content and dictating, among other things, its generalised orbifolding, via the symplectic theory of moduli spaces of flat principal bundles over graph-decorated Riemann surfaces, all the way to topological invariants generalising the celebrated HOMFLY-PT polynomial, and — last but not least, most astonishingly and inspiringly — two-dimensional conformal field theory with a chiral-current extension of its gauge-symmetry (Virasoro) algebra. Needless to say, a context of such richness and gravity exceeds by far the scope of any one-year student proseminar, and yet we are confident and determined that a substantial proportion of the topics enumerated above, alongside some of their modern frontline applications, can and will be discussed in a rigorous and meaningful manner in the course of our one-year programme, with view to giving the interested student a good understanding of the subject matter and an acquaintance with the formal tools involved requisite for further advancement of the study thus initiated and — hopefully, and most importantly — an original application thereof in her or his future research.
It is our expectation and aim that the colloquium have a successful continuation, in its current character of a student proseminar, in the years to come, invariably serving as a natural platform of study and discussion of the many and diverse aspects of the vast realm of mathematical physics sensu largo.
Abstract - 2014/2015
The main theme we are going to focus on in this academic year 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.
Abstract - general
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:
- Bundles, sheaves, (co)homology theories, category theory and their rôle in physics
- Quantum field theory as a functor – from topological to conformal field theory, and beyond
- Knot theory and quantum fields
- Invariants of complex and algebraic varieties motivated by quantum field theory and string theory (Gromov–Witten and Gopakumar–Vafa invariants, Donaldson–Thomas invariants, etc.)
- Higher categories in physical models – geometric realisations, categorified symmetries, geometric quantisation through transgression, and categorial holography
- T‐duality and mirror symmetry – topological string theory and the Fourier–Mukai transform
- Random‐matrix models and topological recursion
- The spectral non‐commutative geometry of the two‐dimensional superconformal field theory
- The geometric Langlands programme and quantum field theory
- Vertex Operator Algebras – from lattice CFTs to the Moonshine, and beyond
- The geometry and physics of generalised Ricci flows
- Integrability in classical and quantum field theory
- Exact results in (extended) SUSY theories, BPS counting, string theory and M‐theory realisations
- Quantum field theories with defects
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.
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.
Rules
The standard and preferred form of presentation at the Colloquium is a blackboard talk. A single session of the Colloquium is split into two parts, each of an approximate duration of 45 minutes. The two parts are separated by a coffee break of ca. 15 minutes. The last quarter of an hour of the session is typically dedicated to discussion.
The speaker is requested to kindly provide the organisers with a copy (preferrably electronic) of his or her notes for the talk or, alternatively, to authorise (upon correction, whenever deemed necessary) those written up by a member of the audience. In either case, the notes will be subsequently placed in the Colloquium's open-access database linked to the Notes section.
Participants of the Colloquium are invited to ask pertinent questions and engage in brief exchanges of views during the talk. Questions and remarks whose nature requires extensive elaboration or a major departure from the main topic of the talk will be relegated to the discussion time. There will be, circumstances permitting, an additional opportunity to discuss with invited speakers during an informal dinner with the speaker.
Last updated on June 4th, 2019