The Exact Kähler Potential from Gauge Theory and Mirror Symmetry
The paper "Exact Kähler Potential from Gauge Theory and Mirror Symmetry" by Jaume Gomis and Sungjay Lee addresses the computation of the Kähler potential on the quantum Kähler moduli space of Calabi-Yau manifolds using exact results from two-dimensional N=(2,2) gauge theories. The main contribution of this work is the proof of a conjecture stating that the partition function of these gauge theories on a two-sphere effectively computes the exact Kähler potential. This conjecture had been previously suggested and is shown to align with the computation of worldsheet instantons and Gromov-Witten invariants.
The authors leverage the supersymmetric localization technique to compute the partition function, leading to the intriguing result that the partition function on a squashed two-sphere is independent of the squashing parameter. This result implies that the partition function on a round sphere S2 accurately reflects the inherent dynamics of the associated nonlinear sigma models on Calabi-Yau manifolds. The paper further demonstrates that in the limit of an infinitely squashed sphere, the partition function represents the ground state overlap in the Ramond sector of the infrared conformal field theory — linking it directly to the Kähler potential.
In addition to the gauge theory insights, the paper explores the implications for mirror symmetry, particularly for sigma models on Kähler manifolds. The authors calculate the exact partition function for N=(2,2) Landau-Ginzburg models with arbitrary twisted superpotentials and establish an equivalence of partition functions between abelian gauge theories and their mirror Landau-Ginzburg duals. This equivalence extends to non-abelian gauge theories, rewriting their partition function in terms of mirror Landau-Ginzburg models, providing a compelling evidence for complex mirror symmetry conjectures.
This work has profound implications for both theoretical and computational aspects of supersymmetric field theories and their applications to string theory and algebraic geometry. Practically, the results suggest a new computational tool for Gromov-Witten invariants, offering potentially simpler computations for certain Calabi-Yau manifolds where traditional methods become cumbersome. Theoretically, it bridges aspects of supersymmetry, mirror symmetry, and quantum field theory with the geometry of Calabi-Yau manifolds, reinforcing the interplay between physical and mathematical frameworks.
The research speculates on future developments in the context of dualities in two-dimensional field theories, particularly regarding non-abelian gauge groups and their nontrivial dualities. Moreover, it opens avenues for exploring domain walls and defect operators, potentially leading to a deeper understanding of open Gromov-Witten invariants and wall-crossing phenomena in Calabi-Yau geometries.
In conclusion, this paper enriches the understanding of the Kähler moduli space, providing a novel perspective through the lens of exact gauge theory results and mirror symmetry. It demonstrates the power of localization methods in yielding exact computations within theoretical physics and suggests exciting directions for further research in mathematics and physics.