Titles and abstracts for the two parallel seminars (starting March 18)

11 março 2024, 09:54 • Gonçalo Oliveira

Title: The BV quantization of gauge theories
Abstract: The quantization of Yang–Mills theories by 't Hooft and Veltman is one of the crowning achievements of 20th century physics, one with deep real world import, since it made the Standard Model possible. In this learning seminar for mathematicians, our goal is to understand the construction of quantum Yang–Mills theory as a BV theory, and to sketch a proof of Costello's theorem:
Theorem: For any semisimple Lie group G as a gauge group, Yang–Mills theory on R^4 admits an essentially unique BV quantization.

To this end, we will discuss the following topics:

• Classical BV theory: the derived critical locus, -1-shifted symplectic structures, the classical master equation.
• Quantum BV theory in finite dimensions: the quantum master equation.
  
• Analytical challenges in infinite-dimensions. Effective field theories and homotopical RG flow.
  
• The definition of a BV quantization. Obstructions to quantization.
  
• Feynman diagrams.
  
• The simplicial set of quantizations.
• The cohomological form of the obstruction to quantizing Yang–Mills theory.
• The vanishing of these cohomology groups. The contractibility of the simplicial set of quantizations.
  

• Elliott, Yoo, Williams - Asymptotic freedom in the BV formalism.
• Costello - Renormalization and effective field theory.
• Costello and Gwilliam - Factorization algebras in quantum field theory (2 volumes)
• Mnev - Quantum field theory: Batalin–Vilkovisky formalism and its applications.