[Paper Review] Phase Space and Quantization of 2D BF Theory Coupled to 1D Quantum Mechanics

Seyed Faroogh Moosavian, Yehao Zhou|arXiv (Cornell University)|Nov 12, 2021

Black Holes and Theoretical Physics4 citations

TL;DR

This paper studies the phase space and quantization of 2D BF theory with GL_N gauge group coupled to 1D quantum mechanics with GL_K global symmetry, constructing the classical and quantum algebras of gauge-invariant observables. It identifies the large-N limit of the phase space as a normal affine variety and shows that the quantized algebra matches the Coulomb branch of a 3D N=4 quiver gauge theory, with deformation quantization realizing the Yangian structure via a coproduct, providing a finite-N refinement of twisted holography.

ABSTRACT

We study ring of functions on the (classical and quantized) phase space of 2-dimensional BF theory with the gauge group $\mathrm{GL}_N$ coupled to a 1-dimensional quantum mechanics with global symmetry $\mathrm{GL}_K$. These functions are gauge-invariant local observables of the coupled system. We first construct the classical phase space of this system and describe its ring of functions and their large-$N$ limit. We next compute the Hilbert series of these algebras for finite-$N$ and also in the large-$N$ limit. We then study the quantization of this phase space and the deformation quantization of its ring of functions, elaborate its relation to the Yangian, and construct its coproduct. Finally, we identify these quantized algebras with the quantized Coulomb branch algebras of certain 3-dimensional $\mathcal{N}=4$ quiver gauge theories.

Motivation & Objective

• To characterize the classical phase space of 2D BF theory coupled to 1D quantum mechanics with GL_K global symmetry, identifying its algebra of gauge-invariant functions.
• To compute the Hilbert series of the classical and quantum algebras for finite N and in the large-N limit.
• To study the deformation quantization of the phase space and its relation to the Yangian algebra, including the construction of a coproduct.
• To identify the quantized algebra with the Coulomb-branch algebra of a 3D N=4 quiver gauge theory.
• To provide a finite-N formulation of twisted holography by realizing the large-N Yangian as a quotient of the finite-N algebra by an ideal.

Proposed method

• Construct the classical phase space M(N,K) as a normal affine variety parameterized by (B, ψ, ψ̄), with dimension 2NK.
• Identify the ring of functions C[M(N,K)] generated by Tr(B^n) and ψ̄B^nψ, showing that M(N,1) ≅ A^{2N} and using morphisms η_ab to embed M(N,K) into a product of K^2 copies of M(N,1).
• Define a Poisson structure on M(N,K) with canonical brackets {ψ_ia, ψ̄_bj} = δ_ab δ_ij and {B_mn, B_pq} = δ_pn B_mq - δ_mq B_pn.
• Perform deformation quantization to obtain the quantum algebra C_ħ[M(N,K)], showing it carries a coproduct and realizes the Yangian of gl_K.
• Use geometric representation theory to identify the quantized algebra with the Coulomb branch of a 3D N=4 quiver gauge theory via localization on the affine Grassmannian.
• Apply localization techniques on P^{n-1} to compute K-theoretic invariants and relate the convolution action of line bundles to the Jing operator S^q_m, establishing a geometric realization of Macdonald polynomials.

Experimental results

Research questions

• RQ1 How is the classical phase space of 2D BF theory coupled to 1D quantum mechanics structured as an algebraic variety for finite N?
• RQ2 What is the Hilbert series of the algebra of gauge-invariant observables in the finite-N and large-N limits?
• RQ3 How does the deformation quantization of the phase space realize the Yangian algebra of gl_K, and what is the role of the coproduct?
• RQ4 How is the quantized algebra related to the Coulomb branch of a 3D N=4 quiver gauge theory?
• RQ5 Can the large-N Yangian limit be understood as a quotient of the finite-N algebra, enabling a finite-N formulation of twisted holography?

Key findings

• The classical phase space M(N,K) is a normal affine variety of dimension 2NK, with M(N,1) ≅ A^{2N} and M(N,K) embedded into a product of K^2 copies of M(N,1) via morphisms η_ab.
• The ring of functions on M(N,K) is generated by Tr(B^n) and ψ̄B^nψ, with Tr(B^n) dual to gravitons and det(B^n) to giant gravitons in the holographic bulk.
• The Poisson structure on M(N,K) is defined by canonical brackets among ψ, ψ̄, and B, and the deformation quantization leads to a quantum algebra with a coproduct isomorphic to the Yangian of gl_K.
• The quantized algebra C_ħ[M(N,K)] is identified with the Coulomb-branch algebra of a 3D N=4 quiver gauge theory via geometric Satake and localization on the affine Grassmannian.
• The convolution action of line bundles on the affine Grassmannian realizes the Jing operator S^q_m, and the character of the structure sheaf on the orbit closure Gr^{Nω_1} computes the Macdonald polynomial H_{(k^N)}(x;q).
• The Hilbert series of the finite-N algebra and its large-N limit are computed, showing convergence to the Yangian in the large-N limit.