Gauge invariance: quantization and geometry

9 Sept 2024, 08:05 → 12 Sept 2024, 18:00 Europe/Zurich

Mons

A workshop in memory of Igor Batalin

 

The principle of gauge invariance and the quantum paradigm are two cornerstones of modern fundamental physics and geometry. The intricate relation of the two underlies various challenges in the physics of fundamental interactions and modern developments in mathematics ranging from quantum gravity and topological effects in quantum physics to homological algebra, supergeometry and derived geometry. The by now classical works of Igor Batalin with E. Fradkin and G. Vilkovisky targeted the very compatibility of gauge invariance and quantisation, resulting in a very general mathematical formalism which by now is considered as a far reaching generalisation of the usual Lagrangian/Hamiltonian framework at both the classical and the quantum level. The applications of the approach of Batalin, Fradkin and Vilkovisky expanded far beyond the original scope of gauge theory quantisation and the approach itself is more and more considered as a proper language to define quantum gauge field theories, to look for new theories of fundamental interactions and to study geometrical structures.

 

In addition to full-scale research talks the workshop will include poster presentations by young participants. The interested participants should submit their abstracts to the organizers by e-mail no later than August 15th.

---

Scientific committee:

• A. Barvinsky (Lebedev Physics Institute, Russia)
• A. Cattaneo (University of Zurich, Switzerland)
• M. Henneaux (Collège de France & International Solvay Institutes, Belgium)
• R. Kallosh (Stanford University)
• N. Reshetikhin (Tsinghua Yau Mathematical Sciences Center, China)
• A. Schwarz (University of California, Davis)
• S. Shatashvili (Trinity College Dublin, Ireland)

 

---

Organizers:

• G. Barnich (Université libre de Bruxelles, Belgium)
• K. Bering (Masaryk University, Brno, Czechia)
• N. Boulanger (Université de Mons, Belgium)
• M. Grigoriev (Université de Mons, Belgium)
• M. Markov (Université de Mons, Belgium)
• Z. Skvortsov (Université de Mons, Belgium)
  
  
  
  
  

Monday 9 September

09:50 → 10:00 Opening

10:00 → 11:00 Comments on selected aspects of the antifield formalism

Speaker: Marc Henneaux (International Solvay Institutes, Belgium)

Henneaux-BatalinMons202409.pdf

11:00 → 11:30 Coffee break

11:30 → 12:30 Fedosov product on differential forms and flat connections on differential graded manifolds

The Hochschild-Kostant-Rosenberg theorem relates negative cyclic homology of the functions on a differential graded manifold M to the de Rham complex of M. The natural A-infinity structure on negative cyclic homology corresponds to an A-infinity structure on the de Rham complex called the Fedosov product. (This product agrees with the usual one if either argument is closed, but it is not graded symmetric in general.) In this talk, we study the analogue of flat connections for this product: these are in bijection with flat connections for the usual product on the classical locus but appear to be different in general. The non-commutative analogue of the Gauss-Manin connection is our main example.

Speaker: Ezra Getzler (Northwestern University, USA)

12:30 → 13:45 Lunch

13:45 → 14:45 $L_\infty$ morphisms for higher BV operators

I will introduce classical and quantum "thick morphisms" of supermanifolds. These are differential-geometric constructions that provide L-infinity morphisms for homotopy analogs of Poisson brackets; in particular, for homotopy brackets generated by higher-order Batalin-Vilkovisky type operators. They generalize ordinary smooth maps of supermanifolds with pullbacks on functions that are non-linear maps.

Speaker: Ted Voronov (Manchester University, UK)

Barvinsky_talk.pdf

14:45 → 15:45 Generalized connections for Courant algebroid in sigma models and gravity

Speaker: Larisa Jonke (Rudjer Boskovic Institute, Zagreb)

15:45 → 16:15 Coffee break

16:15 → 17:15 Covariant phase space and $L_\infty$ algebras

Motivated by the problem of giving a Hamiltonian description of nonlocal field theories such as string field theory, we propose a formula for the phase space symplectic structure of a generic Lagrangian field theory expressed in BV form.

Speaker: Ted Erler (Institute of Physics of the Czech Academy of Sciences)

17:15 → 17:30 My interaction with Igor Batalin

Speaker: Klaus Bering (Masaryk University, Brno)

18:00 → 20:00 Reception

Tuesday 10 September

10:00 → 11:00 BV pushforward and applications

In the BV formalism the spaces of fields are presented as complexes whose cohomology returns the physical content. Different but equivalent complexes may be used, which turns out to be important conceptually and in practice. One useful operation is that of a partial integration (BV pushforward), which produces a chain map that, under some assumptions, is a quasiisomorphism. This has several applications: construction of observables (often as $L_\infty$ representations), renormalization à la Wilson, highly nontrivial equivalences of theories. I will discuss some examples.

Speaker: Alberto Cattaneo (University of Zurich, Switzerland)

Mons. BV pushforward and applications.pdf

11:00 → 11:30 Coffee break

11:30 → 12:30 BV quantization in topological/holomorphic QFT

We discuss BV quantization that arises from topologically/holomorphically twisted quantum field theory. We illustrate some applications in topological/chiral algebraic index, topological B-model and mirror symmetry.

Speaker: Si Li (Tsinghua University, China)

Si-Li-Quantization.pdf

12:30 → 13:45 Lunch

13:45 → 14:45 Complete sets of invariant fields in gauge theories and gravity from purely virtual particles

Speaker: Damiano Anselmi (University of Pisa, Italy)

14:45 → 15:45 Correlation functions and the homological perturbation lemma

Speaker: Richard Szabo (Heriot-Watt University, Edinburgh, UK)

Mons24.pdf

15:45 → 16:15 Coffee break

16:15 → 17:15 Homotopy Algebra Perspective on Quantum Field Theory

I will review recent progress on understanding perturbative quantum field theory within the realm of homotopy algebras.

Speaker: Martin Wolf (University of Surrey, UK)

Wolf-Mons_Aug2024.pdf

17:30 → 19:00 Poster session

Wednesday 11 September

10:00 → 11:00 Non-topological boundary conditions of perturbative Chern-Simons and AKSZ models

In Witten's CS/WZW correspondence the chiral WZW model appears from the chiral boundary condition imposed on the CS theory.
I will describe how similar boundary conditions (mixed chiral-antichiral) of the CS theory and of its AKSZ analogs produce many other interesting examples and how they explain, in particular, Poisson-Lie T-duality. Unlike the CS/WZW correspondence, this generalization is at the moment only perturbative; on the other hand, the perturbative renormalization leads quickly to the (generalized) Ricci tensor and hopefully still hides many interesting things. The calculation is done using the Batalin-Vilkovisky method and the main tool is a CS propagator compatible with the chiral boundary conditions. Based on joint works with J. Pulmann, F. Valach, and D. Youmans.

Speaker: Pavol Severa (University of Geneva, Switzerland)

severa_mons-1.pdf

11:00 → 11:30 Coffee break

11:30 → 12:30 Gauge fixing in coset spaces and supergravities

Speaker: Antoine Van Proeyen (KU Leuven, Belgium)

Van-Proeyen-GaugeFixingCosets3.pdf

12:30 → 13:45 Lunch

13:45 → 14:45 BV actions for extended geometry

Speaker: Martin Cederwall (Chalmers University of Technology, Gothenburg, Sweden)

14:45 → 15:45 TBA

TBA

Speaker: Alexey Kotov (Hradec Králové University, Czech Republic)

15:45 → 16:15 Coffee break

16:15 → 17:15 Geometry of Dirac-BFV quantization and quantum cosmology

Speaker: Andrei Barvinsky (Lebedev Physics Institute, Moscow, Russia)

18:30 → 21:30 Social dinner

Thursday 12 September

10:00 → 11:00 A BV field theory from super moduli space

The BRST quantisation of the relativistic spinning particle pulls the path integral back to ist (super) moduli space and thereby interpolates between the space-time BV action and the BV-equation on super moduli space. This is perhaps the simplest toy model of super string field theory.

Speaker: Ivo Sachs (Ludwig Maximilian University of Munich, Germany)

11:00 → 11:30 Coffee break

11:30 → 12:30 From Gauge Theories to the octonionic Lie groupoid.

We first review how one can turn the standard sigma model in
to a gauge theory and under what conditions. This leads us to the notion
of singular Riemannian foliations (SRFs) and equivalence of gauge
theories leads to Morita equivalence of SRFs. This leads one looking at
the octonionic singular foliation on R^16. In this talk we provide a Lie
groupoid generating this foliation and show that it has the minimal
possible dimension.

Speaker: Thomas Strobl (Claude Bernard University Lyon, France)

12:30 → 13:45 Lunch

13:45 → 14:45 Monopoles, Dirac Strings and Generalised Symmetries

Speaker: Chris Hull (Imperial College London, UK)

BatalinTalk.pdf

14:45 → 15:45 Color-kinematics duality in BV formalism

Speaker: Maxim Zabzine (Uppsala University, Sweden)

15:45 → 16:15 Coffee break

16:15 → 17:15 On some applications of Igor’s work in String Theory

Speaker: Samson Shatashvili (Trinity College Dublin, Ireland)