Modular transformations of tau functions and conformal blocks on the torus

2025/08/19 by Del Monte, Fabrizio, Desiraju, Harini, Gavrylenko, Pavlo
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2508.14030

Abstract

The connection problem for isomonodromic tau functions on the one-punctured torus concerns the ratio between the tau function and its modular transform, associated to dual pants decompositions of the torus. In this paper, we study the modular transformations of the tau function and consequently derive the connection constant. Moreover, through the relation with two-dimensional Conformal Field Theory, we also obtain an exact closed formula for the c=1 Virasoro modular kernel, whose expression was previously unknown, and relate it to the c→∞ (semiclassical) modular kernel and SL₂(ℂ) complex Chern-Simons amplitudes. Finally, we prove that the connection constant and the two, c=1 and c→ ∞, modular kernels are generating functions of canonical transformations on the character variety of the one-punctured torus. Our results are also relevant for the N=2^* gauge theory.

Citations

• Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general q-Painlevé III₃ tau functions
• Proof of Zamolodchikov conjecture for semi-classical conformal blocks on the torus
• Notes on crossing transformations of Virasoro conformal blocks
• Probabilistic conformal blocks for Liouville CFT on the torus
• Tau-functions and monodromy symplectomorphisms
• Circular quiver gauge theories, isomonodromic deformations and W_N\n fermions on the torus
• RG Domain Walls and Hybrid Triangulations
• Chern-Simons Theory and S-duality
• Liouville Correlation Functions from Four-Dimensional Gauge Theories
• From Liouville Theory to the Quantum Geometry of Riemann Surfaces
• Seiberg-Witten Theory and Random Partitions
• Current algebra and Wess-Zumino model in two dimensions

Related

• Twist operator correlators and isomonodromic tau functions from modular Hamiltonians
• The torus one-point block of 2d CFT and null vectors in \mathfraksl₂
• On the Virasoro Crossing Kernels at Rational Central Charge
• Modular Parametrization as Polyakov Path Integral: Cases with CM Elliptic Curves as Target Spaces
• Torus conformal blocks and Casimir equations in the necklace channel