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The geometry of gauged linear sigma model correlation functions. (English) Zbl 1395.81157

Summary: Applying advances in exact computations of supersymmetric gauge theories, we study the structure of correlation functions in two-dimensional \(\mathcal{N} = (2, 2)\) abelian and non-abelian gauge theories. We determine universal relations among correlation functions, which yield differential equations governing the dependence of the gauge theory ground state on the Fayet-Iliopoulos parameters of the gauge theory. For gauge theories with a non-trivial infrared \(\mathcal{N} = (2, 2)\) superconformal fixed point, these differential equations become the Picard-Fuchs operators governing the moduli-dependent vacuum ground state in a Hilbert space interpretation. For gauge theories with geometric target spaces, a quadratic expression in the Givental \(I\)-function generates the analyzed correlators. This gives a geometric interpretation for the correlators, their relations, and the differential equations. For classes of Calabi-Yau target spaces, such as threefolds with up to two Kähler moduli and fourfolds with a single Kähler modulus, we give general and universally applicable expressions for Picard-Fuchs operators in terms of correlators. We illustrate our results with representative examples of two-dimensional \(\mathcal{N} = (2, 2)\) gauge theories.

MSC:

81T10 Model quantum field theories
81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
35P05 General topics in linear spectral theory for PDEs
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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