The Geometry of Forms on Supermanifolds

In this talk, I will discuss the main features of the geometry of forms on supermanifolds. In particular, I will introduce differential and integral forms by means of a double complex construction, that allows one to easily compute their cohomology. Further, if time permits, I will discuss some peculiar aspects of the geometry of forms on complex or algebraic supermanifolds and the degeneration of the related Hodge-to-de Rham spectral sequence.

On the Fukaya-Morse A-infinity category

An interesting class of supermanifolds is the dg ("differential graded") manifolds. In the talk I will explain how a natural notion of pushforward of dg manifolds helps to understand an enumerative problem in Morse theory. I will explain the construction of the Fukaya-Morse category of a Riemannian manifold X -- an A-infinity category where the higher composition maps are given in terms of numbers of embedded trees in X, with edges following the gradient trajectories of certain Morse functions. I will give simple examples and explain different approaches to understanding the structure and proving the quadratic relations on the structure maps -- (1) via homotopy transfer, (2) effective field theory ("second quantization") approach, (3) topological quantum mechanics ("first quantization") approach. The talk is based on a joint work with O. Chekeres, A. Losev and D. Youmans, arXiv:2112.12756.

Super Stable Maps of Genus Zero: Moduli spaces and Normal Bundles

Super J-holomorphic curves are maps from a super Riemann surface to an almost Kähler manifold satisfying a Cauchy-Riemann equation. Super stable maps are supergeometric generalizations of stable maps appearing naturally in the compactification of the moduli space of super J-holomorphic curves. In this talk I will explain how the moduli spaces of super stable maps of genus zero extend the moduli spaces of non-super stable maps and explain properties of the normal bundles of the inclusion.

Closed Supersymmetric String Field Theories: Existence of Vertices (online)

String field theory is a first-principle formulation of string theory based on a spacetime action. One of the main ingredients in the construction of a string field theory is its vertices. In this talk, we start off with a brief review of string field theory and the crucial rule played by string vertices. We then discuss the question of the existence of these vertices in the context of closed bosonic-string field theory, and provide a geometric proof of its existence, based on the work of Kevin Costello. We then turn to the question of string vertices in the context of supersymmetric string theories and explain our geometric proof of the existence of their vertices. We end the presentation with a list of interesting questions for further exploration. This talk is based on arXiv:1911.04343.

Mysterious Triality

Mysterious duality was discovered by Iqbal, Neitzke, and Vafa in 2001 as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series E_k. It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics which gives rise to the same E_k symmetry pattern. I will present a sequence of topological spaces, starting with the four-sphere S^4, and then forming its iterated cyclic loop spaces L_c^k S^4, within which we will see the E_k symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its E_k symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space L_c^k S^4 is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of (11-k)-dimensional supergravity as the defining equations of the Sullivan minimal model of L_c^k S^4. This gives an explicit duality between rational homotopy theory and physics. Thereby, Iqbal, Neitzke, and Vafa's mysterious duality between algebraic geometry and physics is extended to a triality involving algebraic topology, with the duality between topology and physics made explicit, i.e., demystified. The mystery is now transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. This is a report on the works, arXiv:2111.14810 [hep-th] and arXiv:2212.13968 [hep-th], with Hisham Sati.

State sum Homotopy QFTs

Homotopy quantum field theories (HQFTs) generalize topological quantum field theories (TQFTs). The idea is to use TQFTs' techniques to study principal fiber bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space X. In particular, such an HQFT associates a scalar invariant under homotopies to each map from a closed manifold to X. It is well-known that groups are algebraic models for 1-types. Generalizing groups, crossed modules model 2-types. In this talk, I will introduce the notion of a crossed module graded fusion category which generalizes that of a group graded fusion category. Then, using such categories, I will construct a 3-dimensional state-sum HQFT with a 2-type target. This is joint work with Kursat Sozer.

On E7 exceptional geometry

Various aspects of string theory (especially those pertaining to its massless sector) can be conveniently described using the so-called generalised geometry. Embedding the setup into the framework of Courant algebroids, one obtains in addition a useful handle for the study of dualities. The corresponding story in the case of (reductions of) M-theory, called exceptional generalised geometry due to the presence of exceptional U-duality groups, is much less developed. Only recently it was understood what is the right generalisation of Courant algebroids in this context. However, due to issues related to the dual graviton, this understanding works only for groups with rank<7. I will describe what seems to be the correct type of structure for the case n=7, and how it relates to the Poisson-Lie U-duality and consistent truncations of M-theory.

A once-extended TQFT categorifying Quinn's finite total homotopy TQFT

Quinn's Finite Total Homotopy TQFT is a TQFT (topological quantum field theory) defined for any dimension, $n$, of space, and that depends on the choice of a homotopy finite space, $B$. For instance, $B$ can be the classifying space of a finite group or of a finite 2-group. I will report on recent joint work with Tim Porter on once-extended versions of Quinn's Finite total homotopy TQFT, taking values in the (symmetric monoidal) bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, thus in particular it yields (0,1,2)-, (1,2,3)- and (2,3,4)-extended TQFTs, any time we are given a homotopy finite space $B$. I will show how to compute these once-extended TQFTs for the case when $B$ is the classifying space of a finite strict omega-groupoid, represented by a crossed complex. I will present several open problems. The talk will be accessible to non-specialists. References: Faria Martins J, Porter T : A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids. arXiv:2301.02491 [math.CT].

Modular Functors and Factorization Homology

A modular functor is a system of mapping class group representations on vector spaces. These vector spaces are called conformal blocks and required to be compatible with the gluing of surfaces. Modular functors are closely related to three-dimensional topological field theory and play an important role in the representation theory of quantum groups and conformal field theory. In my talk, I will recall some classical constructions of modular functors and explain the approach to modular functors via cyclic and modular operads. More precisely, I will characterize modular functors as cyclic algebras over the framed little disks operad (up to coherent homotopy) subject to a condition that can be expressed in terms of factorization homology. (The talk is based on different joint works with Adrien Brochier and Lukas Müller.)

Quantum Principal Bundles over Projective Bases

In non commutative geometry, a quantum principal bundle over an affine base is recovered through a deformation of the algebra of its global sections: the property of being a principal bundle is encoded by the notion of Hopf Galois extension, while the local triviality is expressed by the cleft property. The quantum algebra of the base space is realized as suitable coinvariants inside the global sections of the quantum principal bundle. We want to examine the case of a projective base X in the special case X=G/P, where G is a complex semisimple group and P a parabolic subgroup. We will substitute the coordinate ring of X with the homogeneous coordinate ring of X with respect to a projective embedding, corresponding to a line bundle L obtained via a character of P. The quantization of the line bundle will come through the notion of quantum section and the quantizations of the the base (a quantum flag) will be obtained as semi-coinvariants. The quantization of G will then be interpreted as the quantum principal bundle on the quantum base space X, obtained via a quantum section.

Noncommutative Geometry of Quantum Flag Manifolds

We begin by briefly recalling the simple Lie algebra sln, its universal enveloping algebra U(sln), and the classification of its finite-dimensional simple modules. We then progress to the work of Drinfeld, Jimbo, and the Leningrad school, who, motivated by the theory of integral systems, introduced a seminal q-deformation of U(sln) in the 1980s. Dually, we have the q-deformed coordinate algebra Oq(SUn) and its distinguished sub-algebras known as quantum flag manifolds Oq(G/L). We show that Lusztig's celebrated quantum root vectors give a noncommutative tangent space for each Oq(G/L), and that from this one can construct a direct q-deformation of the anti-holomorphic Dolbeault complex ofG/L. Finally, we show that our work recovers the celebrated Heckenberger–Kolb calculus for the special case of the quantum Grassmannians.

Bimodule connections for line bundles over irreducible quantum flag manifolds.

Connections on vector bundles are a fundamental tool in classical differential geometry. When we generalize their definition to noncommutative differential geometry, we are led to consider bimodule connections. In this talk I will present a construction of bimodule connections for line bundles over a large class of quantum homogeneous spaces, namely the irreducible quantum flag manifolds. Generalizing the work of Beggs and Majid for the Podles sphere, we realize bimodule connections as associated to a principal connection for the Heckenberger-Kolb calculus . Time allowing, I will review explicit presentations of the associated bimodule maps first in terms of generalised quantum determinants and then in terms of Takeuchi' s categorical equivalence for relative Hopf modules.

Jet functors and differential operators in noncommutative geometry

We construct an N-indexed family of endofunctors on the category of left modules over a unital associative algebra equipped with a differential calculus. These functors generalize the jet functors on vector bundles from classical differential geometry. In particular, our construction coincides with the classical jet functor on vector bundles when the algebra is the smooth functions on a manifold and the calculus is generated by the classical exterior derivative. We show that our jet functors gives rise to a category of linear differential operators between left modules, that these satisfy many properties one might expect, and that most maps which are expected to be differential operators (connections, differentials, partial derivatives), indeed are. We also discuss representatibility, symbols, and define a notion of vector field and Lie bracket in this setting. Joint work with M. Mantegazza and H. Winther.

The Euclidean contour rotation in quantum gravity

The talk will discuss the rotation of the contour of functional integration in quantum gravity from Lorentzian geometries to Euclidean geometries. It uses the usual framework of an action for fields on a manifold but under the assumption that this is a low-energy approximation only. There is an explanation of the relation between the spectral triple formulation of gravity and the standard model in the Lorentzian framework and the Euclidean framework, explaining some features of the Euclidean framework. It is hoped that these formulas will provide exact mathematical results when applied to theories that are fully non-commutative, though this is beyond the scope of this talk.

Bootstrapping Dirac ensembles

In this talk, I shall explain my recent work unraveling connections between NCG and random matrix theory by highlighting techniques we have employed so far. In some cases, one can apply the Coulomb gas method to find the empirical spectral distribution and rigorously prove existence of phase transitions. More recently, we applied the newly developed bootstrap technique to find the moments of certain multi-trace and multi-matrix random matrix models suggested by noncommutative geometry. Using bootstrapping, we are able to find the relationships between the coupling constants of these models and their second moments. Using the Schwinger-Dyson equations, all other moments can be expressed in terms of the coupling constant and the second moment. Explicit relations for higher moments are obtained. Finally I shall explain the double scaling limit of Dirac ensembles and its relation to Liouville quantum gravity. The talk will be a general overview of techniques we have used so far and should be accessible to a larger audience.

Quantum all the way: form spacetime to symmetries to observers.

Quantum gravity requires a quantum spacetime, which in turn requires quantum symmetries. This was the parallel development of noncommutative geometry and Hopf algebras/quantum groups. I will describe this with the examples of some Lie-algebra type noncommutative geometry, and add a further element: the need to have quantum observers in the theory.