Topological Quantum Field Theory Seminar

Past sessions

Quantum differential equations, qKZ difference equations, and helices.

Quantum differential equations (qDEs) are a rich object attached to complex smooth projective varieties. They encode information on their enumerative geometry, topology and (conjecturally) on their algebraic geometry. In occasion of the 1998 ICM in Berlin, B.Dubrovin conjectured an intriguing connection between the enumerative geometry of a Fano manifold $X$ with algebro-geometric properties of exceptional collections in the derived category $D_b(X)$. Under the assumption of semisimplicity of the quantum cohomology of $X$, the conjecture prescribes an explicit form for local invariants of $QH^*(X)$, the so-called “monodromy data”, in terms of Gram matrices and characteristic classes of objects of exceptional collections. In this talk I will discuss an equivariant analog of these relations, focusing on the example of projective spaces. The study of the equivariant quantum differential equations for partial flag varieties has been initiated by V.Tarasov and A.Varchenko in 2017. They discovered the existence of a system of compatible qKZ difference equations, which have made the study of the quantum differential equations easier than in the non-equivariant case. I will establish relations between the monodromy data of the joint system of the equivariant qDE and qKZ equations for $\mathbb{P}^n$ and characteristic classes of objects of the derived category of T-equivariant coherent sheaves on $\mathbb{P}^n$.

Based on joint works with B.Dubrovin, D.Guzzetti and A.Varchenko.

Link invariants from finite crossed modules and a lifting of the Eisermann invariant

This talk is based on work with João Faria Martins (Univ. Leeds) [1] and several projects with students. I will describe the construction of an invariant of tangles and framed tangles which takes values in an arbitrary crossed module of finite groups. This involves the fundamental crossed module associated to a natural topological pair coming from a knot diagram, and a suitable class of morphisms from this fundamental crossed module to the chosen finite crossed module. Our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots [2-3], for which we also find a lifting. The Eisermann invariant detects information about a suitable choice of meridian and longitude in the knot complement boundary.

[1] João Faria Martins and Roger Picken: Link invariants from finite categorical groups, Homology, Homotopy and Applications, 17(2) (2015), 205–233; arXiv:1301.3803v2 [math.GT], arXiv:1612.03501v1 [math.GT]
[2] M. Eisermann: Knot colouring polynomials, Pacific J. Math. 231 (2007), no. 2, 305–336.
[3] M. Eisermann: Homological characterization of the unknot, J. Pure Appl. Algebra 177 (2003), no. 2, 131–157.

Roger-Picken-slides

An abstract theory of physical measurements

Since its early days, quantum mechanics has forced physicists to consider the interaction between quantum systems and classically described experimental devices — a fundamental tenet for Bohr was that the results of measurements need to be communicated using the language of classical physics.

Several decades of progress have led to improved understanding, but the tension between “quantum” and “classical” persists. Ultimately, how is classical information extracted from a measurement? Is classical information fundamental, as in Wheeler’s “it from bit”? In this talk, which is based on ongoing work [1], I approach the problem mathematically by considering spaces whose points are measurements, abstractly conceived in terms of the classical information they produce. Concretely, measurement spaces are stably Gelfand quantales [2] equipped with a compatible sober topology, but essentially their definition hinges on just two binary operations, called composition and disjunction, whose intuitive meanings are fairly clear. Despite their simplicity, these spaces have interesting mathematical properties. C*-algebras yield measurement spaces of “quantum type,” and Lie groupoids give us spaces of “classical type,” such as those which are associated with a specific experimental apparatus. The latter also yield a connection to Schwinger’s selective measurements, which have been recast in groupoid language by Ciaglia et al.

An interaction between the two types, providing a mathematical approach to Bohr’s quantum/classical split, can be described in terms of groupoid (or Fell bundle) C*-algebras as in [3]. I will illustrate the basic ideas with simple examples, such as spin measurements performed with a Stern–Gerlach apparatus.

References

1. P. Resende, An abstract theory of physical measurements (2021).
2. P. Resende, The many groupoids of a stably Gelfand quantale, J. Algebra 498 (2018), 197–210.
3. P. Resende, Quantales and Fell bundles, Adv. Math. 325 (2018), 312–374, MR3742593.

pedro-resende-slides

Second part of a double session, followed by a 20 minute discussion period.

Statistical Interpretation in the Schwinger’s picture of Quantum Mechanics

In this talk I will illustrate some ideas about the statistical interpretation in the Schwinger’s picture of Quantum Mechanics. After a brief introduction on the postulates assumed in this framework, I will recall the basic ingredients of Connes’ non commutative integration theory. This language allows me to define, on one hand quantum measures on the groupoid associated with the quantum systems, and on the other weights on the corresponding groupoid von-Neumann algebra. In particular, quantum measures are a generalization of measures on sigma-algebras which is suited for the description of interference phenomena. Then, the final part of the talk will be devoted to the statistical interpretation associated with both situations.

fabio-di-cosmo-slides

First part of a double session, followed by a 20 minute break for coffee and discussion, before the second speaker, Pedro Resende.

Skein Lasagna modules of 2-handlebodies

Morrison, Walker and Wedrich recently defined a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. We will discuss recent joint work with Ciprian Manolescu on computing the "skein lasagna module," a basic part of MWW's invariant, for a certain class of 4-manifolds.

Ikshu-Neitalath-slides

Bulk-boundary correspondences with factorization algebras

Factorization algebras provide a flexible language for describing the observables of a perturbative QFT, as shown in joint work with Kevin Costello. Those constructions extend to a manifold with boundary for a special class of theories. I will discuss work with Eugene Rabinovich and Brian Williams that includes, as an example, a perturbative version of the correspondence between chiral ${\rm U}(1)$ currents on a Riemann surface and abelian Chern-Simons theory on a bulk 3-manifold, but also includes a systematic higher dimensional version for higher abelian CS theory on an oriented smooth manifold of dimension $4n+3$ with boundary a complex manifold of complex dimension $2n+1$. Given time, I will discuss how this framework leads to a concrete construction of the center of higher enveloping algebras of Lie algebras, in work with Greg Ginot and Brian Williams.

Owen Gwilliam notes

Note the unusual time.

Introduction to the Hecke category and the diagonalization of the full twist

The group algebra of the symmetric group has a large commutative subalgebra generated by Young-Jucys-Murphy elements, which acts diagonalizably on any irreducible representation. The goal of this talk is to give an accessible introduction to the categorification of this story. The main players are: Soergel bimodules, which categorify the Hecke algebra of the symmetric group; Rouquier complexes, which categorify the braid group where Young-Jucys-Murphy elements live; and the Elias-Hogancamp theory of categorical diagonalization, which allows one to construct projections to "eigencategories."

Ben-Elias-slides

Semisimple topological field theories in even dimensions

A major open problem in quantum topology is the construction of an oriented 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. More generally, how much manifold topology can a TQFT see?

In this talk, I will answer this question for semisimple field theories in even dimensions — I will sketch a proof that such field theories can at most see the stable diffeomorphism type of a manifold and conversely, that if two sufficiently finite manifolds are not stably diffeomorphic then they can be distinguished by semisimple field theories. In this context, 'semisimplicity' is a certain algebraic condition applying to all currently known examples of vector-space-valued TQFTs, including 'unitary field theories’, and 'once-extended field theories' which assign algebras or linear categories to codimension 2 manifolds. I will discuss implications in dimension 4, such as the fact that oriented semisimple field theories cannot see smooth structure, while unoriented ones can.

Throughout, I will use the Crane-Yetter field theory associated to a ribbon fusion category as a guiding example.

This is based on arXiv:2001.02288 and joint work in progress with Chris Schommer-Pries.

David-Reutter-slides

A groupoid-based perspective on quantum mechanics

In this talk, I will expound a point of view on the theoretical investigation of the foundations and mathematical formalism of quantum mechanics which is based on Schwinger’s “Symbolism of atomic measurement” [8] on the physical side, and on the notion of groupoid on the mathematical side. I will start by reviewing the “development” of quantum mechanics and its formalism starting from Schrödinger’s wave mechanics, passing through the Hilbert space quantum mechanics, and arriving at the $C^∗$-algebraic formulation of quantum mechanics in order to give an intuitive idea of what is the “place” of the groupoid-based approach to quantum theories presented here. Then, after (what I hope will be) a highly digestible introduction to the notion of groupoid, I will review two historic experimental instances in which the shadow of the structure of groupoid may be glimpsed, namely, the Ritz-Rydberg combination principle, and the Stern-Gerlach experiment. The last part of the talk will be devoted to building a bridge between the groupoid-based approach to quantum mechanics and the more familiar $C^∗$-algebraic one by analysing how to obtain a (possibly) non-commutative algebra out of a given groupoid. Two relevant examples will be discussed, and some comment on future directions (e.g., the composition of systems) will close the talk. The material presented is part of an ongoing project developed together with Dr. F. Di Cosmo, Prof. A. Ibort, and Prof. G. Marmo. In particular, the discrete-countable theory has already appeared in [1, 2, 3, 4, 5, 6, 7].

References

[1] F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo. Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics. Entropy, 11(22):1292 – 18, 2020.

[2] F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics. International Journal of Geometric Methods in Modern Physics, 17(04):2050054 (14), 2020.

[3] F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics IV: Composition and independence. International Journal of Geometric Methods in Modern Physics, 17(04):2050058 (34), 2020.

[4] F. M. Ciaglia, A. Ibort, and G. Marmo. A gentle introduction to Schwinger’s formulation of quantum mechanics: the groupoid picture. Modern Physics Letters A, 33(20):1850122–8, 2018.

[5] F. M. Ciaglia, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics I: Groupoids. International Journal of Geometric Methods in Modern Physics, 16(08):1950119 (31), 2019.

[6] F. M. Ciaglia, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics II: Algebras and Observables. International Journal of Geometric Methods in Modern Physics, 16(09):1950136 (32), 2019.

[7] F. M. Ciaglia, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics III: The Statistical Interpretation. International Journal of Geometric Methods in Modern Physics, 16(11):1950165 (37), 2019.

[8] J. Schwinger. Quantum Mechanics, Symbolism of Atomic Measurements. Springer-Verlag, Berlin, 2001.

Florio-Ciaglia-annotated-slides
Florio-Ciaglia-slides

Universal Symmetries of Gerbes and Smooth Higher Group Extensions

Gerbes are geometric objects describing the third integer cohomology group of a manifold and the B-field in string theory; they can essentially be understood as bundles of categories whose fibre is equivalent to the category of vector spaces. Starting from a hands-on example, I will explain gerbes and their categorical features. The main topic of this talk will then be the study of symmetries of gerbes in a universal manner. We will see that these symmetries are completely encoded in an extension of smooth 2-groups. In the last part, I will survey how this construction can be used to provide a new smooth model for the string group, via a theory of group extensions in $\infty$-topoi.

Severin-Bunk-slides

Manifolds with odd Euler characteristic and higher orientability

Orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: $k$-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is $2^{k+1}m$ for some integer $m$. Here we call a manifold $k$-orientable if the $i^{\rm{th}}$ Stiefel-Whitney class vanishes for all 0 < $i$ < $2^k$. This theorem is strict for $k=0,1,2,3$, but whether there exist 4-orientable manifolds with an odd Euler characteristic is a new open question. Such manifolds would have dimensions that are a multiple of 32. I discuss manifolds of dimension high powers of 2 and present the results of calculations on the cohomology of the second Rosenfeld plane, a special 64-dimensional manifold with odd Euler characteristic.

Renee-Hoekzema-slides

Multiple zeta values in deformation quantization

In 1997, Kontsevich gave a universal solution to the deformation quantization problem in mathematical physics: starting from any Poisson manifold (the classical phase space), it produces a noncommutative algebra of quantum observables by deforming the ordinary multiplication of functions. His formula is a Feynman expansion whose Feynman integrals give periods of the moduli space of marked holomorphic disks. I will describe joint work with Peter Banks and Erik Panzer, in which we prove that Kontsevich's integrals evaluate to integer-linear combinations of multiple zeta values, building on Francis Brown's theory of polylogarithms on the moduli space of genus zero curves.

Pym slides.pdf

Categorification of Verma Modules in low-dimensional topology

In this talk I will review the program of categorification of Verma modules and explain their applications to low-dimensional topology, namely to the construction of Khovanov invariants for links in the solid torus via a categorification of the blob algebra.

The material presented spreads along several collaborations with Abel Lacabanne, and Grégoire Naisse.

pedro-vaz-slides.pdf

Klein TQFT and real Gromov-Witten invariants

In this talk I will explain how the Real Gromov-Witten theory of local 3-folds with base a Real curve gives rise to an extension of a 2d Klein TQFT. The latter theory is furthermore semisimple which allows for complete computation from the knowledge of a few basic elements which can be computed explicitly. As a consequence of the explicit expressions we find in the Calabi-Yau case, we obtain the expected Gopukumar-Vafa formula and relation to SO/Sp Chern-Simons theory.

Penka-Georgieva-slides

Cyclotomic expansions of the $gl_N$ knot invariants

Newton’s interpolation is a method to reconstruct a function from its values at different points. In the talk I will explain how one can use this method to construct an explicit basis for the center of quantum $gl_N$ and to show that the universal $gl_N$ knot invariant expands in this basis. This will lead us to an explicit construction of the so-called unified invariants for integral homology 3-spheres, that dominate all Witten-Reshetikhin-Turaev invariants. This is a joint work with Eugene Gorsky, that generalizes famous results of Habiro for $sl_2$.

Anna-Beliakova-slides.pdf

Two dimensional topological field theories and partial fractions

This talk is based on joint work with M. Khovanov and Y. Kononov. By evaluating a topological field theory in dimension $2$ on surfaces of genus $0,1,2$, etc., we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.

victor-ostrik-slides.pdf

Projecto FCT UIDB/04459/2020.

$3d$ A and B models and link homology

I will discuss some current work (with Garner, Hilburn, Oblomkov, and Rozansky) on new and old constructions of HOMFLY-PT link homology in physics and mathematics, and new connections among them. In particular, we relate the classic proposal of Gukov-Schwarz-Vafa, involving M-theory on a resolved conifold, to constructions in $3d$ TQFT's. In the talk, I will focus mainly on the $3d$ part of the story. I'll review general aspects of $3d$ TQFT's, in particular the "$3d$ A and B models" that play a role here, and how link homology appears in them.

Tudor-Dimofte-slides.pdf

Projecto FCT UIDB/04459/2020.

Ellipsoidal billiards, extremal polynomials, and partitions

A comprehensive study of periodic trajectories of the billiards within ellipsoids in the $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of $d$ intervals on the real line. Classification of periodic trajectories is based on a new combinatorial object: billiard partitions.

The case study of trajectories of small periods $T$, $d \leq T \leq 2d$, is given. In particular, it is proven that all $d$-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates $d + 1$-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for $d = 3$.

The talk is based on the following papers:

• V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications Mathematical Physics, 2019, Vol. 372, p. 183-211.
• G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.

Dragovic-slides.pdf

The double-centralizer theorem in 2-representation theory and its applications

Finitary birepresentation theory of finitary bicategories is a categorical analog of finite-dimensional representation theory of finite-dimensional algebras. The role of the simples is played by the so-called simple transitive birepresentations and the classification of the latter, for any given finitary bicategory, is a fundamental problem in finitary birepresentation theory (the classification problem).

After briefly reviewing the basics of birepresentation theory, I will explain an analog of the double centralizer theorem for finitary bicategories, which was inspired by Etingof and Ostrik's double centralizer theorem for tensor categories. As an application, I will show how it can be used to (almost completely) solve the classification problem for Soergel bimodules in any finite Coxeter type.

marco-mackaay-slides.pdf

Projecto FCT UIDB/04459/2020.

Crossed modules, homotopy 2-types, knotted surfaces and welded knots

I will review the construction of invariants of knots, loop braids and knotted surfaces derived from finite crossed modules. I will also show a method to calculate the algebraic homotopy 2-type of the complement of a knotted surface $\Sigma$ embedded in the 4-sphere from a movie presentation of $\Sigma$. This will entail a categorified form of the Wirtinger relations for a knot group. Along the way I will also show applications to welded knots in terms of a biquandle related to the homotopy 2-type of the complement of the tube of a welded knots.

The last stages of this talk are part of the framework of the Leverhulme Trust research project grant: RPG-2018-029: Emergent Physics From Lattice Models of Higher Gauge Theory.

João Faria Martins slides

Projecto FCT UIDB/04459/2020.

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