Schedule

Arrival: Sunday September 7 

Scientific Program: Monday September 8 - Friday September 12 (approximately noon)

Titles and Abstracts (in alphabetical order)

Francesco Bonechi

Title: Color-kinematics duality and double copy theories in BV, part 2

Abstract: In part 1 we will review the color-kinematics duality for gauge theories in scattering amplitudes and explain 

how it manifest itself for the case off-shell duality in BV formalism. In part 2 we discuss the relation to gravity via double copy constructions. In both parts we review some simple examples of these phenomena. 

Slides

Eugenia Boffo

Title: BCOV in the Large Hilbert Space 

Abstract: The B model of topological strings has a "second quantization", developed initially by Bershadsky-Ceccotti-Ooguri-Vafa. This is a field theory whose field equations describe deformations of the complex structure of a Calabi-Yau manifold. In the follow-up, several authors have refined the BCOV setup, shedding further light on it by means of the algebraic-geometric machinery proper to the BV framework.

However, BCOV is still quite peculiar: to mention some unusual aspects, the space of fields is the kernel of an operator inside a cochain complex and the BV structure is just Poisson, not symplectic. In this talk, based on arXiv:2506.02983 with Hulík and Sachs, we suggest another viewpoint on the subject, borrowing constructions from supergeometry, to provide a new perspective on BCOV's unique features. The corresponding sigma model is also suggested.

Alejandro Cabrera

Title: Overviewing instanton-like contributions in the Poisson Sigma Model

Abstract: In this talk, we shall provide an overview of the study of oscillatory observable computations in the PSM, its relation to quantization and the underlying (local) symplectic groupoid G. Starting with the source given by a disc with 3 insertions at the boundary, the main result is that the leading term in these computations is given by underlying smooth 2-simplices on G. We detail the proof of this result and discuss some of its consequences. Finally, we comment on pushing these computations further, their meaning in terms of Fourier Integral Operators and failure of associativity for "big" parameters (this last is based on joint work with R. Fernandes).

Slides

David Carchedi

Title: Derived Differential Geometry and BV

Abstract: We will explain a rigorous formulation of derived differential geometry and directly link it to dg-manifolds and BV. 

Miquel Cueca

Title: Q-manifolds over differentiable stacks

Abstract: Motivated by the perspective that differentiable stacks can be understood as the Morita class of a Lie groupoid, I will present a Morita-invariant framework for graded Lie groupoids equipped with a homological vector field. This approach not only unifies but also clarifies structures arising in geometry and mathematical physics, and I will illustrate it with natural examples linked to shifted Poisson and symplectic structures. The results I will discuss are part of a joint work with Daniel Alvarez.

Filippo Fila-Robattino

Title: BV/BFV description of N=1,D=4 SuGra

Abstract: The BV–BFV formalism provides a cohomological description of gauge theories on manifolds with boundary, pairing a bulk BV theory with compatible boundary BFV data. A persistent obstruction in gravity and supergravity is that the naive boundary presymplectic form has a singular kernel, so the theory is not directly BFV-extendable.

In this talk I will present the analysis for minimal N=1, D=4 supergravity in the first order formalism. I will begin by describing the boundary structure, and identifying the phase space as the zero locus of the constraints generating diffeomorphisms, local Lorentz transformations, and local supersymmetry. I will then explain the construction of a full off-shell BV action, which necessarily includes rank-2 antifield terms to achieve the classical master equation. Finally, I will briefly introduce the BV pushforward procedure, which allows to eliminate degeneracies to produce a reduced, BFV-extendable BV theory.

Ezra Getzler

Title: The Moyal bracket and the BV cohomology of the spinning particle

Abstract: 

The spinning particle is the one-dimensional reduction of the Neveu-Schwartz-Ramond superstring. It consists of a supersymmetric particle moving in a one-dimensional supergravity background, and its quantization is the Hilbert superspace of harmonic spinors. (These models are classified by N, the number of copies of fermionic fields. In this talk, N=1. The extension to N=2 is work in progress with Ivo.) It is actually an AKSZ model (so a generalization of one-dimensional Chern-Simons), and so associated to a differential graded symplectic supermanifold, by which we mean a pair (ω,Q), where ω is a(n exact) symplectic form and Q is an odd function of degree 1. The cohomology of the ring of functions of this supermanifold with differential the Poisson bracket  with Q determines the classical BV cohomology of the spinning particle, so is important for understanding perturbative BV quantization of this model. I calculated this cohomology in earlier work for N=1, and showed that it is somewhat bizarre, with two series of cohomology classes in arbitrary negative degrees, each a copy of the functions on the target manifold.

In the study of quantum BFV, we should instead consider the Moyal bracket on the target, and lift Q to an element Q satisfying [Q,Q]=0. The cohomology of the differential [Q,-] is the Moyal cohomology of the differential graded symplectic supermanifold. (This lift corresponds to the choice of a Spinc structure on the target manifold.) In this talk, I prove that the Moyal cohomology, unlike the Poisson cohomology, is well-behaved: in the spectral sequence from Poisson to Moyal cohomology, the extra cohomology classes of negative degree cancel each other pairwise at the E1 page.

Maxim Grigoriev 

Title: Gauge PDEs on manifolds with (asymptotic) boundaries 

Abstract: 

Gauge PDEs are BV-extensions of usual PDEs defined intrinsically and can also be considered as a generalisation of non-Lagrangian AKSZ sigma-models.  Gauge PDEs behave well when restricting to submanifolds/boundaries, giving a well-defined gauge PDE therein which describes all the data induced on the boundary by the bulk gauge theory. I plan to give a short overview of the gauge PDE formalism including the so-called weak generalisation and presymplectic structures. In the holographic context we are typically dealing with asymptotic boundaries as e.g. in asymptotically AdS (flat) spacetimes, in which case the naive pullback to the boundary doesn't work. It turns out that there is a natural reformulation of general relativity adapted to the conformal compactification in such a way that the boundary gauge PDE is well-defined. The same applies to generic gauge fields coupled to GR. In this approach one can obtain a gauge theory describing asymptotic  boundary dynamics of gauge theories coupled to gravity. In the case of asymptotically AdS gravity this gives a gauge-theoretical version of the celebrated Fefferman-Graham construction and its extension to general gauge theories. The formalism is especially useful in studying asymptotic symmetries and their associated charges. For instance, the celebrated Bondi-Metzner-Sachs symmetries of the asymptotically-flat GR can be systematically derived in this approach.

Fabian Hahner

Title: From superspace to twisted supergravity

Abstract: I will present a geometric perspective on superspace geometry that proves fruitful for studying twisted supergravity. For eleven-dimensional supergravity this technique allows for a uniform description of the theory in terms of homotopy Poisson-Chern-Simons theory on appropriate superspaces. If time permits, I will further discuss some relations to conformal supergravity and the structure of higher symmetries in twisted supergravity. 

Slides

Roberta Iseppi

Title: The BV Construction in NCG: Towards the Infinite-Dimensional Case

Abstract: The Batalin-Vilkovisky (BV) formalism plays a fundamental role in the quan-

tization of gauge theories, yet its rigorous formulation within the framework of

noncommutative geometry (NCG) remains an open challenge. After having es-

tablished the construction for finite spectral triples, the next step is to extend it

to the infinite-dimensional setting. In this talk, I will present recent progress in

this direction, based on joint work with T. Krajewski and C. Perez-Sanchez. We

focus on the BV construction for Chern-Simons theory, formulated in terms of

cyclic cocycles on infinite-dimensional noncommutative *-algebras, and discuss

key insights and obstacles encountered along the way. 

Slides

Shuhan Jiang

Title: Jet models of symplectic dg manifolds.

Abstract: In this talk, we introduce the jet space of a Poisson dg manifold M of degree k and prove that its structure sheaf yields a semi-free resolution of that of M as P_{k+1}-algebras. We then show that a suitable choice of formal exponential map produces a curved cyclic L_\infty[1] algebra of degree k modeling M. This leads to a natural framework for globalizing AKSZ theories over arbitrary moduli spaces of classical solutions. This talk is based on joint work in progress with Alberto S. Cattaneo.

Slides

Julian Kupka

Title: Generalised Geometry and Higher Fermion Terms in N=1, D=10 Supergravity 

Abstract: In this talk, we lay the foundation for the BV formalism of N=1, D=10 supergravity. We show that generalised geometry provides a natural and elegant formulation of such supergravities coupled to Yang-Mills multiplets that extends to higher-fermion terms in both the action and supersymmetry transformations. We introduce the language of Courant algebroids and explain how they encapsulate ten-dimensional supergravities in a geometrical framework, without the need for superspace or any supercovariant constructions.

Based on joint work with Fridrich Valach and Charles Strickland-Constable. 

Julian

Du Pei

Title: Gauge theory and skein modules

Abstract: In this talk, we will study 3-manifold skein modules by embedding them into the space of BPS states of 4d N=4 gauge theories. We give an explicit algorithm to determine the dimensions as well as the list of generators for general gauge group when the three-manifold has reduced holonomy. We test this result by relating it to the geometry of the Hitchin moduli space. As a by-product, we explain why the dimension of skein modules doesn't have a TQFT-like behavior. 

Katarina Rejzner

Title: Perturbative algebraic quantum field theory with boundary

Abstract: In this talk, I will begin by outlining the formalism of perturbative algebraic quantum field theory (pAQFT)—a mathematically rigorous approach to constructing interacting quantum field theories on Lorentzian manifolds. Building on this foundation, I will explain how the framework can be extended to contexts where a boundary is present, including treatment of asymptotic ‘boundaries at infinity’. This generalization involves a subtle interplay between the Batalin–Vilkovisky (BV) formalism—a homological algebra tool used to handle gauge symmetries—and techniques from functional analysis, which are necessary to manage the infinite-dimensional topology of the configuration spaces involved. Ultimately, this talk will illustrate how combining these mathematical frameworks allows us to handle boundary effects within pAQFT, thereby broadening the reach of perturbative algebraic methods in mathematical QFT. This talk is based on a variety of joint projects with Michele Schiavina, Chris Fewster, Daan Janssen and Eli Hawkins.

Ivo Sachs

Title:  Integrating In

Abstract: I will formulate the embedding of the state space of Yang-Mills theory into a sub-module of vertex operator algebra of the spinning world line as a quasi isomorphism. In the process we encounter  a number auxiliary fields, which are "integrated in" to enforce the quasi isomorphism. On the resulting submodule we then compute the pull-back of the world line correlations functions to super moduli space which gives a generalization of Yang-Mills theory.

Ingmar Saberi

Title: TBA

Abstract: TBA

Michele Schiavina

Title: Local BV-BFV formalism, revisited 

Abstract: The classical BV formalism is encoded by a -1 symplectic Hamiltonian dg manifold $(X,\omega, S,Q)$ where the symplectic form induces a 1-Poisson algebra, and the solution of the classical master equation S is the hamiltonian function of the cohomological vector field Q.

The classical BFV formalism is encoded by a 0 symplectic Hamiltonian dg manifold $(X_\partial,\omega_\partial, S_\partial,Q_\partial)$ where the symplectic form induces a 0-Poisson algebra, and the solution of the classical master equation S_\partial is the hamiltonian function of the cohomological vector field Q_\partial.

When thinking about Lagrangian field theory it is often convenient to formulate the above data in terms of local forms, borrowing language and constructions from the variational bicomplex. When doing so, one discovers that neither the Hamiltonian condition between $S$ and $Q$ nor the classical master equation hold exactly anymore. This fact can be used effectively to link different-codimension field-theoretic data, an effort that goes under the name of BV-BFV formalism.

In this talk, I wish to use the variational complex to its full extent to obtain a local form version of the BV-BFV formalism. 

We find that one is then able to naturally build $L_\infty$ algebras to control various choices involved in the definition of higher codimension BV data, and reformulate the whole formalism in terms of 1) Hamiltonian triples and 2) multisymplectic geometry.

This is a joint work with Jonas Schnitzer. If time permits I will comment on how one can embed this construction within the perturbative algebraic field theory approach to quantisation.

Jonas Schnitzer

Title: Quantization of the Moment Map 

Abstract: If a Lie group acts on a Poisson manifold by Hamiltonian symmetries there is a well-understood way to get rid of unnecessary degrees of freedom and pass to a Poisson manifold of a lower dimension. This procedure is known as Poisson-Hamiltonian reduction. There is a similar construction for invariant star products admitting a quantum momentum map, which leads to a deformation quantization of the Poisson-Hamiltonian reduction of the classical limit.

The existence of quantum momentum maps is only known in very few cases, like linear Poisson structures and symplectic manifolds. The aim of this talk is to fill this gap and show that there is a universal way to find quantized momentum maps using so-called adapted formality morphisms which exist, if one considers nice enough Lie group actions. This talk is based on \url{https://arxiv.org/abs/2502.18295} with Chiara Esposito, Ryszard Nest and Boris Tsygan.

Vivek Shende

Title: Holomorphic curves and Chern-Simons: towards a pertubative approach 

Abstract:

 I will first review the skein-valued curve counting theory due to Ekholm and myself, which is a rigorous mathematical account of how to define invariant counts of holomorphic curves with Lagrangian boundary conditions in Calabi-Yau 3-folds.  The basic point is that the obstructions to invariance match the skein relations for the fundamental Wilson line in su(n) Chern-Simons theory, so we may count the holomorphic curves “as if” their boundary introduced such a Wilson line into a Chern-Simons theory on the Lagrangian brane.  (This is a rigorous mathematical shadow of Witten’s belief that the corresponding strings do introduce such lines.)

I will then sketch a new approach to the same problem, intended to make more direct contact with the Chern-Simons theory (rather than just its defects).  Some familiar characters will appear: the propagator, Jacobi diagrams, a formula closely resembling the Kontsevich integral…

The speaker’s motivation for giving the talk is the hope that the audience will recognize and explain certain ad-hoc constructions met along the way.

Jakob Ulmer

Title: Open-Closed String Field Theory from Calabi-Yau Categories

Abstract: 

I recollect how to build the (large N) open-closed SFT from a Calabi-Yau category and an object, in the language of BV algebras. I emphasise the proposed role of SFT in enumerative geometry and mention some contributions of the speaker. Time permitting I comment on the relation to Costello-Li’s approach to BCOV theory and comment on ideas’s for a categorical version of Costello-Gaiotto’s "Twisted Holography“ program.

Fridrich Valach

Title: BV formulation of 10D N=1 supergravity

Abstract:

We present a full, background independent Batalin-Vilkovisky description of 10D N=1 supergravity coupled to Yang-Mills multiplets.

Slide

Maxim Zabzine

Title: Color-kinematics duality and double copy theories in BV, part 1

Abstract: 

In part 1 we will review the color-kinematics duality for gauge theories in scattering amplitudes and explain 

how it manifest itself for the case off-shell duality in BV formalism. In part 2 we discuss the relation to gravity via double copy constructions. In both parts we review some simple examples of these phenomena.