Emergent Geometries from Strings and Quantum Fields

I will review some ideas from BV formalism: the equivariant extension of BV and BV push forward map. As an example, I will discuss 4D Yang-Mills theory and its different extensions. The talk is based on the joint work with F. Bonechi and A. Cattaneo.

The mathematical field of optimal transport interprets curvature in terms of the evolution of the entropy of probability distribution of particles. We will see that this is particularly natural for gravity compactifications: the warping function is unified with the internal Ricci tensor. Singularities of positive-tension objects are also naturally included in the formalism. As a concrete consequence, I will present several bounds on the masses of KK spin-two fields, with applications to the problems of scale separation and gravity localization in string theory.

I propose a conceptual frame to interpret the prolate differential operator, which appears in Communication Theory, as an entropy operator; indeed I write its expectation values as a sum of terms, each subject to an entropy reading by an embedding suggested by Quantum Field Theory. This adds meaning to the classical work by Slepian et al. on the problem of simultaneously concentrating a function and its Fourier transform, in particular to the "lucky accident" that the truncated Fourier transform commutes with the prolate operator. The key is the notion of entropy of a vector of a complex Hilbert space with respect to a real linear subspace, recently introduced by the author by means of the Tomita-Takesaki modular theory of von Neumann algebras. We consider a generalization of the prolate operator to the higher dimensional case and show that it admits a natural extension commuting with the truncated Fourier transform; this partly generalizes the one-dimensional result by Connes to the effect that there exists a natural selfadjoint extension to the full line commuting with the truncated Fourier transform.

The stringor bundle has been anticipated by pioneering work of Stephan Stolz and Peter Teichner, as a string-theoretic analog of the spinor bundle. In this talk, I will explain a neat construction of the stringor bundle as an associated 2-vector bundle. The main ingeredients are a new model for the string 2-group and a representation of that 2-group on a von Neumann algebra.

In the Batalin–Vilkovisky formalism, gauge conditions are expressed as Lagrangian submanifolds in the space of fields and antifields. We discuss a way of patching together gauge conditions over different parts of the space of fields, and apply this method to extend the light-cone gauge for the superparticle to a conic neighbourhood of the forward light-cone in momentum space. This is joint work with Sean Pohorence (doi.org/10.1090/pspum/103.2)

This is joint work with Rosa Maria Miró Roig, arXiv preprint arXiv:2306.05338. Let X be a projective complex K3 surface; Mukai proved that the algebraic space parametrising simple sheaves of fixed rank and Chern classes on X has a natural holomorphic simplectic structure. In a previous joint work, we showed that generalised syzygy bundles provide a natural way to relate moduli spaces of simple bundles with different ranks and Chern classes over an arbitrary smooth projective variety of dimension at least 2.. In this paper, we show that the same construction maps Lagrangian subspaces (which we assume to be smooth and locally closed) to other Lagrangian subspaces. This leads to a recursive explicit construction, starting for instance by taking any ample line bundle generated by its global sections. We also give another a different method to produce simple bundles from simple bundles by using extensions instead of syzygies, and show that this also maps Lagrangians to Lagrangians.

We consider noncommutative principal bundles which are equivariant for a triangular Hopf algebra and analyze an associated (noncommutative) gauge groupoid as well as an Atiyah sequence of braided infinite dimensional Lie algebras which are related to gauge transformations acting on connections. From this sequence we derive a Chern-Weil homomorphism and braided Chern-Simons terms. We present explicit examples over noncommutative spheres.

In this talk, I will outline an answer to this question for a broad class of TQFTs fulfilling a certain representation theoretic property. I will discuss various implications, such as the fact that all oriented semisimple 4d TQFTs cannot see smooth structure, while there are unoriented ones which can. I will explain how these results suggest to think of TQFTs as appropriately "dual" to manifolds, and how they lead to classification schemes for certain classes of TQFTs which are "dual" to surgery theoretic classifications of manifolds. If time permits, I will explain such a classification of linear once-extended 4-dimensional TQFT in terms of certain group theoretical data and bordism invariants, and comment on higher-dimensional variants. The first part of this talk is based on arXiv:2001.02288 and arXiv:2206.10031, the latter joint work with Christopher Schommer-Pries. Some of the results in the second part of the talk are ongoing joint work with Christopher Schommer-Pries and Noah Snyder, and with Theo Johnson-Freyd.

The Atiyah class of a dg manifold $(\mathcal{M},Q)$ is the obstruction to the existence of an affine connection on the graded manifold $\mathcal{M}$ that is compatible with the homological vector field $Q$. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory. Using Kontsevich's famous formality theorem, Liao, Xu and I established a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold $(\mathcal{M},Q)$, there exists an $L_\infty$ quasi-isomorphism of dglas from an appropriate space of polyvector fields $\mathcal{T}_{\oplus,\operatorname{poly}}^{\bullet}(\mathcal{M})$ endowed with the Schouten bracket $[-,-]$ and the differential $[Q,-]$ to an appropriate space of polydifferential operators $\mathcal{D}_{\oplus,\operatorname{poly}}^{\bullet}(\mathcal{M})$ endowed with the Gerstenhaber bracket $\llbracket -,- \rrbracket$ and the differential $\llbracket m+Q,- \rrbracket$, whose first Taylor coefficient (1) is equal to the composition of the action of the square root of the Todd class of the dg manifold $(\mathcal{M},Q)$ on $\mathcal{T}_{\oplus,\operatorname{poly}}^{\bullet}(\mathcal{M})$ with the Hochschild--Kostant--Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we proved the Kontsevich--Shoikhet conjecture: a Kontsevich--Duflo type theorem holds for all finite-dimensional smooth dg manifolds. This last result shows that, when understood in the unifying framework of dg manifolds, the classical Duflo theorem of Lie theory and the Kontsevich--Duflo theorem for complex manifolds are really just one and the same phenomenon.

A Hermitian metric on a complex manifold is called balanced if its fundamental form is co-closed. In the talk I will review some general results about balanced metrics and the Hull-Strominger system. In particular, I will show that the Fu-Yau solution to this system on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds.

Recent works on Jackiw-Teitelboim gravity (e.g. Saad-Shenker-Stanford) suggest that moduli spaces of hyperbolic $0$-metrics on surfaces with boundary carry canonical symplectic structures. Here $0$-metrics are metrics which have a singularity $y^-2$ at the boundary, where $y$ is a boundary defining function. An example of such a metric is the standard hyperbolic metric on the Poincaré half-plane.

Sen's action for a p-form gauge field with self-dual field strength coupled to a spacetime metric involves an explicit Minkowski metric and the presence of this raises questions as to whether the action is coordinate independent and whether it can be used on a general spacetime manifold. A natural generalisation of Sen's action is presented in which the Minkowski metric is replaced by a second metric on spacetime. The theory is covariant and can be formulated on any spacetime. The theory describes a physical sector, consisting of the chiral p-form gauge field coupled to the dynamical metric g, plus an auxiliary sector consisting of a second chiral p-form and the second metric. The resulting theory is covariant and can be formulated on any spacetime. A spacetime with two metrics has some interesting geometry and some of this is explored here and used in the construction of the interactions. The action has two diffeomorphism-like symmetries, one acting only on the physical sector and one acting only on the auxiliary sector, with the spacetime diffeomorphism symmetry arising as the diagonal subgroup.

I will present an algorithm to construct mirror and more general SL(2,Z) duals of 3d N=4 quiver theories and of their 4d uplifts. The algorithm uses a set of basic duality moves and the properties of the duality-walls providing a generalisation of the Kapustin-Strassler local dualisation to the non-abelian case. All the basic duality moves can be derived by iterative applications of Seiberg-like dualities, hence our algorithm implies that mirror and SL(2,Z) dualities can be derived assuming only Seiberg duality. I will also discuss the case of bad theories, where the dualisation algorithm allows us to extract non-trivial information on the quantum moduli space.

Equivariant cohomology and equivariant integrals have numerous uses in mathematics and physics. We will review applications in the context of Sasakian geometry, where it played an important role in the formulation of volume extremization for Sasaki-Einstein manifolds, and then we will discuss some new applications motivated by holography. We will argue that this is the appropriate mathematical framework to study universal properties of supersymmetric solutions of supergravity. Time permitting, we will also discuss equivariant cohomology in the context of supersymmetric field theories.

As a toy model for a background independent formulation of string theory we consider the BRST quantisation of the relativistic spinning particle. I will describe how this can be viewed as instance of a more general framework where a BV field theory in space-time and indeed the geometry itself emerges from (almost) purely algebraic data.

The study of topological defects in quantum field theory has seen a wealth of activity recently leading to many interesting insights, for example the explicit realisation of non-invertible topological defects in higher dimensional QFTs via the gauging of higher form symmetries. In this talk, I would like to focus on two-dimensional examples, where such defects and their properties have been investigated for some time already. I would like to show some of the structural insights into 2d CFT one can gain from topological defects and from their gauging. In this way, the well-understood two-dimensional case might serves as a source of ideas and as a test-case for higher dimensional constructions.

In this talk, we plan to give a summary on how the geometry of the so-called symplectic groupoid G integrating a Poisson manifold M can be seen to be "emergent" from certain quantum field theory computations. More specifically, we consider the Poisson Sigma model (PSM) associated to M on a 2-disc and certain expectation value E, analogous to a well known Cattaneo-Felder one, now associated with a pair of high-frequency (as $h \to 0$) oscillatory functions on M. We then explain how, in the semiclassical limit $h \to 0$, the fields giving the leading contribution to E must satisfy a non-linear PDE which generalizes the Maurer-Cartan equation for Lie algebras and groups. Finally, we prove an existence and classification theorem for this PDE in which the geometry of G "emerges".

In recent years, increasing attention has been drawn to the behaviour of field theory with local (gauge) symmetries on manifolds with boundary. On the physics side, this has led to the discovery of a relation between perturbative scattering results (e.g. Weinberg's "soft photon" theorems) and certain "soft" gauge symmetries, with their associated conserved charges. However, on the mathematics side we have been lacking an exhaustive explanation of these phenomena.

We describe how the BV-AKSZ construction can be extended to generic local gauge field theories including non-topological and non-diffeomorphism-invariant ones. The minimal formulation of this sort has a finite-dimensional target space which is a pre Q-manifold equipped with a compatible presymplectic structure. The nilpotency condition for the homological vector field is replaced with a presymplectic version of the classical BV master equation. Such a formulation can be constructed, at least locally, starting from the usual jet-bundle BV description. Other way around, given such a presymplectic BV-AKSZ formulation, it defines a standard jet-bundle BV formulation by taking a symplectic quotient of the space of AKSZ supermaps. The construction is illustrated by a number of examples including (conformal) gravity and YM theory.

In this talk, we will give an overview of (-1)-shifted derived Poisson manifolds in the C^\infty-context, and discuss the quantization problem. We describe the obstruction theory and prove that the linear (-1)-shifted derived Poisson manifold associated to any L_\infty-algebroid admits a canonical BV_\infty quantization. This is a joint work with Kai Behrend and Matt Peddie.

In the case of singular coisotropic reduction the appropriate framework is the cohomological one of Batalin, Fradkin and Vilkovisky (BFV). In contrast to textbook knowledge about regular reduction, this generically leads to an infinite tower of ghosts. We show that this tower can be given the structure of decorated trees and that, in particular, this leads to recursive formulas for the cohomological BFV charge and the BFV Hamiltonian. The method can be also applied to the BV setting of Kazhdan and Felder and we show that it leads to a highly simplified way of obtaining BV extensions. This is joint work with A. Hancharuk and C. Laurent-Gengoux.