Algebraic structure of \(t\,t{*}\) equations for Calabi-Yau sigma models (Q2362796)

The \(tt^*\) equations were introduced in [Nucl. Phys., B 367, No. 2, 359--461 (1991; Zbl 1136.81403)] by \textit{S. Cecotti} and \textit{C. Vafa} to describe the variation of the ground states of \(2\)D, \(\mathcal{N}=2\) quantum field theories. When the theories are superconformal they can be realized as non-linear sigma-models with Calabi-Yau targets to whose special geometry \(tt^*\) equations are equivalent, and into whose cohomology the chiral rings of the theories translate. On a subset of the chiral ring the \(tt^*\) equations describe the variation of Hodge structure. The purpose of this paper is to give a holomorphic/algebraic formulation of the non-holomorphic content of the \(tt^*\) connection restricted to the conformal deformation directions for Calabi-Yau targets of dimensions 1, 2 and 3. The idea is to provide more rigorous approach to these equations, make explicit their symmetries and connections to some integrable systems (known in special cases), and to Nekrasov's deformation corrections to string amplitudes in geometrically engineered \(4\)D, \(\mathcal{N}=2\) gauge theories. For the mirror quintic, {\S. Yamaguchi} and \textit{S. Yau} [``Topological string partition functions as polynomials''. J. High Energy Phys. 2004, No. 7, Paper No. 047 (2004)] showed that the non-holomorphic content is captured by finitely many functions closed under derivatives, for the elliptic curve the corresponding functions are the classical quasi-modular forms. The author develops an extension of these differential rings to the moduli of lattice polarized \(K3\) manifolds, and gives explicitly the constraints on the Kähler metric imposed by the flatness of the \(tt^*\) equations for the elliptic curve, which transfers to lattice polarized \(K3\) manifolds. The generators of the rings parametrize the moduli \(T\) of the Calabi-Yau enhanced by differential forms that respect the Hodge filtration and the constant pairing, which has 3 dimensions instead of the usual 1 (it was introduced by Movasati for the elliptic curve). The constant pairing is symplectic for \(d=1,3\) and symmetric for \(d=2\), and \(T\) corresponds to choosing the elements of the chiral ring that lie in the conformal deformation bundle, which in its turn corresponds to a subset of Ramond ground states of the theory with the spectral flow. Linear combinations of vector fields along the generators on \(T\) support a Lie algebra structure, which replaces the non-holomorphic derivatives of the \(tt^*\) equations and gives them an algebraic expression. In examples, e.g. the mirror quartic in \(\mathbb{P}^3\), coordinates on \(T\) are shown to correspond to quasi-modular forms and their generalizations.