# Recent seminars

## 25/06/2021, Friday, 17:00–18:00 Europe/Lisbon — Online

Fabian Haiden, Mathematical Institute, University of Oxford
Categorical Kähler Geometry

This is a report on joint work in progress with L. Katzarkov, M. Kontsevich, and P. Pandit. The Homological Mirror Symmetry conjecture is stated as an equivalence of triangulated categories, one coming from algebraic geometry and the other from symplectic topology. An enhancement of the conjecture also identifies stability conditions (in the sense of Bridgeland) on these categories. We adopt the point of view that triangulated (DG/A-infinity) categories are non-commutative spaces of an algebraic nature. A stability condition can then be thought of as the analog of a Kähler class or polarization. Many, often still conjectural, constructions of stability conditions hint at a richer structure which we think of as analogous to a Kähler metric. For instance, a type of Donaldson and Uhlenbeck-Yau theorem is expected to hold. I will discuss these examples and common features among them, leading to a tentative definition.

### Video

Haiden slides.pdf

## 18/06/2021, Friday, 17:00–18:00 Europe/Lisbon — Online

Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
Higher S-matrices

Each fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter.

### Video

Theo-Johnson-Freyd-slides

## 04/06/2021, Friday, 17:00–18:00 Europe/Lisbon — Online

Ingmar Saberi, Ludwig Maximilian University, Munich
Twists of supergravity theories via algebraic geometry

Twists of supersymmetric field theories have been the source of an enormous amount of new mathematics, including (just for example) Seiberg-Witten theory and mirror symmetry. It is reasonable to expect that twists of supergravity theories will exhibit even richer structure, but they remain comparatively unexplored, largely due to their intricacy. For example, a holomorphically twisted version of the AdS/CFT correspondence was proposed by Kevin Costello and Si Li, motivated by constructions in topological string theory and the work of Bershadsky-Cecotti-Ooguri-Vafa; Costello and Li conjectured a connection between their version of BCOV theory and the type IIB supergravity theory, but did not verify this connection directly. I will show that the pure spinor superfield technique, which has been known for some time in the physics literature, can be used to elegantly and economically construct supersymmetric theories, as well as to swiftly compute their twists. In each case, the resulting structures are governed by the classical algebraic geometry of certain affine varieties. If time permits, I'll discuss the examples of type IIB supergravity and eleven-dimensional supergravity in some detail.

### Video

Ingmar-Lisbon-notes.pdf

## 28/05/2021, Friday, 17:00–18:00 Europe/Lisbon — Online

Dmitry Melnikov, International Institute of Physics, Natal
Entanglement and complexity from TQFT

In the 1990s Aravind proposed that topological links can be used to classify different patterns of quantum entanglement. One way this connection can be investigated is through an appropriate quantum mechanical definition of knots. I will start from the category theory definition of a TQFT and derive a relation between measures of quantum entanglement and topological invariants of links. We will see how patterns of quantum entanglement emerge in the TQFT picture. Meanwhile, complexity is a complementary measure of quantum correlations. In the TQFT case, it can also be related to topological invariants. I will discuss a few definitions of complexity for several families of knots and links.

### Video

dmitry-melnikov-slides.pdf

## 21/05/2021, Friday, 17:00–18:00 Europe/Lisbon — Online

Bruce Bartlett, Stellenbosch University
Asymptotics of the classical and quantum $6j$ symbols

The classical (resp. quantum) $6j$ symbols are real numbers which encode the associator information for the tensor category of representations of $SU(2)$ (resp. the quantum group of $SU(2)$ at level $k$). They form the building blocks for the Turaev-Viro $3$-dimensional TQFT. I will review the intriguing asymptotic formula for these symbols in terms of the geometry of a Euclidean tetrahedron (in the classical case) or a spherical tetrahedron (in the quantum case), due to Ponzano-Regge and Taylor-Woodward respectively. There is a wonderful integral formula for the square of the classical $6j$ symbols as a group integral over $SU(2)$, and I will report on investigations into a similar conjectural integral formula for the quantum $6j$ symbols. In the course of these investigations, we observed and proved a certain reciprocity formula for the Wigner derivative for spherical tetrahedra. Joint with Hosana Ranaivomanana.