Juvitop Seminar
Fall 2021
In Fall 2021, Juvitop was about Factorization Algebras and the Quantum Noether Theorem.-
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Factorization Algebras, Symmetries, and Quantization
Araminta Amabel
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Appendix A.3 of Costello-Gwilliam Volume 1
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Appendix A.1 of Costello-Gwilliam Volume 2
Natalie Stewart
References:
In this talk, I'll give an overview of (ordinary) Lie algebras, the universal enveloping algebra of a Lie algebra, the Chevalley-Eilenberg (co)algebras of a Lie algebra, and Lie algebra (co)homology. We will also discuss dg Lie algebras, L_∞-algebras, and the Chevalley-Eilenberg (co)algebras of an L_∞-algebra.
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Chapter 3.1 of Costello-Gwilliam Volume 2
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Lurie's DAG X Derived Algebraic Geometry X: Formal Moduli Problems
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Max Planck Institute Learning seminar on deformation theory
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Chapter 4 of Keller's Chern-Simons Theory and Equivariant Factorization Algebras
Ishan Levy
References:
I will discuss the notion of a formal moduli problem, and state the result of Lurie-Pridham. Then, I will explain how one can think of the functor Bg assigning a lie algebra to a formal moduli problem, and give an example.
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Wyatt Reeves
I plan to give a brief overview of Lagrangian and Hamiltonian mechanics, classical Noether's theorem, gauge theories, and quantum mechanics.
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Chapter 4 of Costello-Gwilliam Volume 2
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Master Class The BV formalism in classical and quantum field theories
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Twisted QFT Lecture 1 Second Half
Natalia Pacheco-Tallaj
References:
In this talk, I will introduce the notion of a derived critical locus of the action functional S(phi) of a field theory. We will first compute the derived critical locus in the toy case of a finite dimensional space of fields using only homological algebra, and then use this as a guide to construct a model of the derived critical locus that works for infinite dimensional spaces of fields: an elliptic formal moduli problem (local L_∞ algebra) that encodes the solutions to the equations of motion near a given base solution phi_0. This will allow us to make sense of Costello and Gwilliam's definition of a classical field theory.
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Cameron Krulewski
In this talk we’ll look at some examples of field theories, including the example of the free scalar, which we’ve seen a couple times before, as well as an interacting theory and a few examples of gauge theories. Our goal will be to get an idea of how one can do perturbation theory in Costello and Gwilliam’s framework and to consider how some of these examples fit into a general construction called a cotangent theory.
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Chapter 4 of Ginot's notes
Zihong Chen
References:
I will introduce the notion of factorization algebra formulated using Weiss topology. We'll see some short examples of how to construct factorization algebras such as the Sym construction and the factorizing envelope, as well as some explicit examples on R and S^1.
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Dylan Wilson
Let's do some examples. Associative algebras, E_d-algebras, enveloping algebras, algebras + "Hamiltonian", and vertex algebras if there's time.
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Jae Hee Lee
I will start by giving a lightning review of the physical concept of quantizing a Lagrangian field theory via path integrals. Then I will describe the corresponding algebraic problem of (deformation) quantizing the algebra of classical observables. We will focus on the free field theory example, and how the factorizing envelope construction for the Heisenberg algebra gives a solution in this case.
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Leon Liu
This week we revisit Noether's theorem in classical field theory and interpret it under Costello-Gwilliam's factorization algebra formalism. After reviewing how Noether's theorem works in the Hamiltonian formalism of classical mechanics, we define symmetries of a classical field theory in the factorization algebra setting and state and possibly sketch the corresponding Noether's theorem.
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- Costello-Gwilliam Volume 2, Ch. 13.2-13.3.
Araminta Amabel
References:
Last week we learned how to talk about a Lie algebra acting on a classical field theory. This week, I'll discuss how such an action interacts with the quantization process. We'll see obstructions to quantizing in a way compatible with the action. This will come up again in the quantum version of Noether's theorem next week.
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Sanath Devalapurkar
I will explain the tools (BD-algebras, Rees constructions viewed as quantization) needed to state the quantum Noether theorem, and then (obviously) state the result. I'll end by relating this to the classical Noether theorem, and give an interpretation via Noether currents.
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NO TALK - Thanksgiving
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Sanath Devalapurkar
In this talk, I will give two examples of the Noether theorems. The first will be on classical mechanics, which in fancy terms can be understood as sigma-models of maps from R to the cotangent space of V for V = vector space. We will see that the classical and quantum moment maps give the desired Noether currents. The next example will be the "beta-gamma system", which describes holomorphic maps Sigma -> T*(W) from a Riemann surface to a complex vector space W. Here, we will try to understand the Noether currents corresponding to the Kac-Moody and the Virasoro symmetries of the system.
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This seminar was organized by Araminta Amabel and Nat Pacheco-Tallaj.