Plenary talks

Abstract: Chern-Simons theories with gauge supergroups appear naturally in string theory and they possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. In this talk we show the recent develop of the combinatorial quantisation of supergroup Chern- Simons theories for punctured Riemann surfaces of arbitrary genus. In our recent work we construct the algebra of observables, study their representations and applications to the construction of 3-manifold invariants via Heegaard splittings. In case of GL(1|1), which is worked out explicitly, we recover the invariants of lens spaces and more general Seifert fibered 3-manifolds found by Rozansky and Saleur using Dehn surgery.

Slides/Notes: to appear.

Abstract: This talk will discuss recent developments in the relation connecting topics in classical hierarchies of differential equations, combinatorics, and quantum spin chains and lattices.

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Abstract: I will outline a recent construction of a BFV algebra (of "brane currents"). This involves an exotic AKSZ construction, which is then "zero-locus" reduced. This BFV algebra explains the emergence of higher geometric (exceptional generalised geometric) structures on the worldvolume of any brane, similarly to how Hitchin-Gualtieri generalised geometry appears on the worldvolume of strings.

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Abstract: In this talk I will present the application of the AKSZ construction to the BFV description of the reduced phase space of the Palatini-Cartan theory obtaining a BV theory with a partial implementation of the torsion-free condition already on the space of fields. The AKSZ construction is a procedure to build a classical BV-BFV theory on cylindrical manifolds. We thus obtain a description of General Relativity in the Palatini-Cartan formalism, which encodes symmetries and which is compatible with the presence of a boundary. This is a joint work with A. S. Cattaneo and M. Schiavina.

Slides

Abstract: In this talk I will discuss the problem of quantization on manifolds with boundary, in the accessible context of scalar field theory. With classical constraints placed upon the boundary, this procedure is understood and yields a BD_0 quantization of the bulk observables, while quantizing the boundary in a compatible way requires the use of the BV-BFV formalism. I will present some ongoing work with Ödül Tetik in understanding this procedure at the level of factorization algebras and factorization homology.

Slides with notes

Abstract: Observables in generic CFTs carrying a large global charge Q can be studied systematically in an expansion in 1/Q. In this talk I will discuss the d=3 Wilson-Fisher O(2N) CFT in a large-charge sector, where also N is taken to be large. The large-charge expansion of critical exponents of this model turns out to be an asymptotic series with double factorial growth. Resurgent methods can be applied to obtain an unambiguous semi-classical reconstruction of this expansion, containing non-perturbative contributions and allowing extrapolation to small charge regimes. The precise non-perturbative structure of the expansion is captured by a worldline path integral on the sphere. In this geometrical picture, the large-charge result is reproduced via saddle-point expansion around (unstable) sphere geodesics, while Borel ambiguities are related to Morse indexes introduced by their negative modes deformations.

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Abstract: For some decades, deep connections have been forming among enumerative geometry, complex geometry, intersection theory and integrable systems. In 1990, Witten formulated his celebrated conjecture that predicts that the generating series of intersection numbers of psi-classes is a tau function of the KdV hierarchy, which was first proved by Kontsevich making use of a cell decomposition of a combinatorial model of the moduli space of curves by means of certain ribbon graphs which are Feynman graphs of a cubic hermitian matrix model with an external field. In 2007, Chekhov, Eynard and Orantin introduced a procedure that associates a family of differentials to a Riemann surface with some extra data, which we call spectral curve. This tool naturally fits in numerous algebro-geometric contexts, helping build relations among them. In the Witten--Kontsevich case, the Airy curve allows to build the connection with the 4 mentioned areas. After an introduction to topological recursion in general, the Witten--Kontsevich case in particular, I will introduce more general structures which help organising the intersection theory of the moduli space of curves: cohomological field theories. I will relate them to topological recursion in a large framework and I will describe a recent work with R. Belliard, S. Charbonnier and B. Eynard, in which we extend both this relation and the Witten--Kontsevich instance to intersection numbers with Witten’s r-spin class, allowing us to complete the connections to the 4 mentioned areas in the context of Witten’s generalised conjecture for $r \geq 2$.

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Abstract: The classical BV formalism is a homological construction of the reduced space of observables for a Lagrangian theory with infinitesimal symmetries. It amounts to taking a homotopy quotient of the space of solutions of the Euler-Lagrange equations by infinitesimal symmetries of the system. The goal of this talk will be to present a derived geometric construction of a space of BV complexes associated to a given Lagrangian. We will define this space of BV complexes to be the space of such quotient with some extra conditions involving the shifted symplectic structures. We will then explain to what extent the objects of that space resemble the usual construction of the classical BV complex.

Slides

Abstract: Generalized and Exceptional geometries are a natural framework for study of dynamics of extended objects such as strings and branes.

In this talk I will discuss how we can see these extended geometries using G-structures. I will furthermore introduce G-algebroids as an algebraic structure generalizing Courant algebroids omnipresent in Generalized Geometry. This class of algebroids is useful as they encode the gauge structure of supergravity and M theory in various instances.

This talk is based on joint work with M. Bugden, F. Valach, and D. Waldram.

Notes

Abstract: From the Cobordism hypotesis we know that providing a (fully extended) n-dimensional topological field theory is equivalent to finding fully n-dualizable objects in the target category. Hence, the problem of defining a TFT can be limited to examining what turns out to be a very algebraic condition in the chosen target category. One interesting target is the higher Morita category. I will present results regarding dualizability in these categories as well as work-in-progress concerning even higher dualizability.

Slides/Notes

Abstract: I review recent progress made in proving the AdS/CFT correspondence in three dimensions, utilising a powerful new worldsheet description of the so-called "tensionless" limit of string theory on AdS_3.

Slides

Abstract: Topological quantum field theories, as defined by Atiyah, are symmetric monoidal functors from a bordism category to vector spaces. The bordism category used by Atiyah is the homotopy category of a higher category of bordisms with diffeomorphisms and their isotopies as higher morphisms. Functors from the two dimensional higher bordism category to an appropriated 2-category of linear categories are one axiomatisation of modular functors appearing in conformal field theory. My talk will be concerned with the structure present on the category that a modular functor assigns to the circle. More precisely, we will show that it admits a balanced braided Grothendieck-Verdier structure, a generalisation of the concept of a ribbon category, introduced by Boyarchenko and Drinfeld. This turns out to be a consequence of a classification of cyclic algebras over the framed little disk operad in terms of balanced braided Grothendieck-Verdier categories. The talk is based on joint work with Lukas Woike.

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Abstract: We briefly review the formulation of Yang-Mills respectively General Relativity about Minkowski spacetime in terms of a differential graded Lie algebra. The associated L-infinity minimal model brackets, given by a sum of cubic trees, coincide with the tree scattering amplitudes. Geometrically, the amplitudes are sections of a sheaf on a variety of complex momenta, with residues that factor into lower-point amplitudes. In our setting, this factorization follows from an application of the homological perturbation lemma. Using Hartogs extension for complete intersections, the factorization yields a simple recursive characterization of the amplitudes, independent of their original definition in terms of cubic trees. It is similar to BCFW recursion but does not invoke BCFW's trick of shifting momenta. Arxiv 1812.06454.

Slides

Abstract: We recall the close relation of (twisted) modular operads to algebras considered in string theory (quantum homotopy algebras). We show how the representation of Feynman transform of the modular operad is naturally characterized by a particular solution of the quantum master equation on the Batalin-Vilkovisky algebra of functions. In the case of quantum L-infinity algebra, it is known that this space of functions is equipped with a product. However, in the general case, modular operads don't provide any structure to induce such a product. We introduce a new additional structure on the modular operads, the connected sum. The graded commutative product on the space of functions together with BV bracket and Laplacian leads to Beilinson-Drinfeld algebra, "cousin" of the Batalin-Vilkovisky algebra. This talk is based on joint work with M. Doubek, B. Jurco, and J. Pulmann.

Slides

Abstract: A chiral boundary condition for Chern-Simons theory requires the connection one-form to be of degree (1, 0). We find a propagator compatible with this boundary condition and its generalization, given by a generalized metric on the Lie algebra. As an application, we get a 1-loop renormalization group flow of this boundary condition for Chern-Simons, and more generally for the Courant sigma-model, rediscovering the generalized Ricci tensor. This is a report on a joint work with Pavol Ševera and Donald R. Youmans, arXiv:2009.00509.

Notes

Abstract: A factorization algebra is a cosheaf-like object meant to model the structure encoded in the observables of a field theory. Indeed, Costello and Gwilliam have constructed, for every perturbative quantum Batalin-Vilkovisky theory on a manifold M, a factorization algebra of observables on M. In this talk, we discuss extensions of the results of Costello and Gwilliam to bulk-boundary systems on manifolds with boundary. We will focus in particular on two theories on the half-line: topological mechanics and one-dimensional BF theory, each endowed with a natural boundary condition. If time permits, we will discuss work-in-progress on the analogous results for BF theory in spacetime dimension greater than 1.

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Abstract: Verma modules define conformal blocks in 2-dimensional CFT, and are related to the quantisation of moduli spaces of logarithmic connections on the sphere. In this talk we will review this story, and introduce generalised modules to define irregular versions of conformal blocks: these are related to the quantisation of moduli spaces of irregular meromorphic connections.

Notes

Abstract: I will give an overview of some recent developments concerning special types of integrable deformations of superstrings that go under the name of eta (or Yang-Baxter) deformations. I will explore the fate of Weyl invariance under these deformations and analyse the worldsheet scattering theory of the deformed models.

Slides

Abstract: Mathematically, one may conceptualize field theory as the study of geometric structures pertinent to quantization on moduli spaces of solutions of certain PDE's on manifolds. In general, this is quite difficult, not least because these moduli spaces tend to be highly singular. One approach is the BV formalism, which does not aim to describe the entire moduli space, only the formal neighbourhood of a given solution using the tools of derived deformation theory and shifted symplectic geometry. In geometric analysis and symplectic topology on the other hand, the moduli spaces are accessed globally, but at the cost of having to achieve smoothness (transversality) by perturbing the PDE, destroying much of the relevant structure. This talk will propagate a rather novel kind of mathematical objects called derived C^{oo}-stacks -analogues of the objects studied in derived algebraic geometry- that fully capture the global geometry of moduli space of elliptic PDE's, including derived symplectic structures, orientations and virtual fundamental classes.

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Abstract: The perturbative Chern-Simons partition function is an invariant of framed 3-manifolds that appears in various guises in the literature, and depends on the choice of a reference flat connection. Most constructions require certain restrictions either on the 3-manifold or the reference flat connection. In the Batalin-Vilkovisky formalism such assumptions are not necessary. I will explain the construction of this invariant and comment on some ongoing work concerning the dependence on the reference flat connection. In particular, I will show that the BV cohomology class of the partition function descends to the moduli space of flat connections.

Slides