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Notes on the Quantization of Tropological Yang Mills Theory

Emil Albrychiewicz, Andrés Franco Valiente, Viola Zixin Zhao

Abstract

We continue our investigation into anisotropic topological field theories which arise from a tropical limit of conventional isotropic topological field theories. We analyze both the TBF theory and the tropical analogue of 2D topological Yang-Mills theory (TrYM) through a direct path integral calculation which probes a deformed analytic torsion and also through canonical quantization. The explicit construction of the Hilbert space of TrYM theory demonstrates that the TrYM theory provides an example of a solvable field theory where anisotropy properties and topological invariance can simultaneously hold. We show that the partition function has an asymptotic limit, which verifies that the dimension of the moduli space of tropicalized flat connection on a Riemann surface of genus $g>1$ is precisely given by $(g-1) \operatorname{rank}(\mathfrak{g})$. We show that the interpretation of this result is that the random matrix model associated to the U$(N)$ TrYM is in fact a novel random matrix theory whose integration space is still the same space of hermitian matrices (similar to a GUE) however the Dyson index matches that of a GOE consistent with the usual intuition that tropicalization reduces down complex objects to their real counterparts.

Notes on the Quantization of Tropological Yang Mills Theory

Abstract

We continue our investigation into anisotropic topological field theories which arise from a tropical limit of conventional isotropic topological field theories. We analyze both the TBF theory and the tropical analogue of 2D topological Yang-Mills theory (TrYM) through a direct path integral calculation which probes a deformed analytic torsion and also through canonical quantization. The explicit construction of the Hilbert space of TrYM theory demonstrates that the TrYM theory provides an example of a solvable field theory where anisotropy properties and topological invariance can simultaneously hold. We show that the partition function has an asymptotic limit, which verifies that the dimension of the moduli space of tropicalized flat connection on a Riemann surface of genus is precisely given by . We show that the interpretation of this result is that the random matrix model associated to the U TrYM is in fact a novel random matrix theory whose integration space is still the same space of hermitian matrices (similar to a GUE) however the Dyson index matches that of a GOE consistent with the usual intuition that tropicalization reduces down complex objects to their real counterparts.
Paper Structure (10 sections, 109 equations, 4 figures)

This paper contains 10 sections, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Classical Geometry of the TBF and Anisotropic Yang-Mills Theory
  3. Essentials of Tropical Geometry and 2D TBF and 2D TrYM Theory
  4. The Jordan Star and Gauge Fixing Conditions
  5. Quantization of the TBF and Anisotropic Yang-Mills Theory
  6. The Partition Function and Deformed Analytic Torsion
  7. The TrYM Hilbert Space via Canonical Quantization
  8. 2D TrYM Partition Function and Holonomies on $\Sigma_{g,b}$
  9. Explicit Random Matrix Model for 2D TrYM: The $\beta=1$ GUE.
  10. Conclusions: Inching Towards Nonequilibrium String Theory

Figures (4)

  • Figure 1: The three perspectives of the underlying tropical geometry. The leftmost picture shows tropical projective space as a foliated $\mathbb{CP}^1$, the middle picture shows its equivalency to an infinite foliated cylinder known as the sleeve and the rightmost picture would be the quotient space that on would obtain from collapsing the space of leaves into the underlying tropical graph. All three perspectives are equivalent for the purposes of quantum field theory, we call the sleeve the covering space perspective and the one-dimensional tropical graph, the quotient space perspective.
  • Figure 2: The pair of pants surface. The leaves of the foliation are denoted as teal circles and the singular leaf appears in red. The boundaries $C_i$ are denoted via black circles.
  • Figure 3: A foliated Riemann surface of genus $g=2$ with $b=3$ boundary components.
  • Figure 4: The observables $\mathcal{O}$ would depend on the radial direction transverse to the leaves of the foliation but from the quotient space perspective, it would not dependent on the angular position.