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- W4300912898 abstract "The $tt^*$ equations define a flat connection on the moduli spaces of $2d, mathcal{N}=2$ quantum field theories. For conformal theories with $c=3d$, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the $tt^*$ equations in the cases $d=1,2,3$ is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of $tt^*$ and provides these with a finer and algebraic meaning. For sigma models into lattice polarized $K3$ manifolds, the differential ring of special functions on the moduli space is constructed, extending known structures for $d=1$ and 3. The generators of the differential rings of special functions are given by quasi-modular forms for $d=1$ and their generalizations in $d=2,3$. Some explicit examples are worked out including the case of the mirror of the quartic in $CP^3$, where due to further algebraic constraints, the differential ring coincides with quasi modular forms." @default.
- W4300912898 created "2022-10-04" @default.
- W4300912898 creator A5083833304 @default.
- W4300912898 date "2014-12-10" @default.
- W4300912898 modified "2023-10-05" @default.
- W4300912898 title "Algebraic structure of $tt^*$ equations for Calabi-Yau sigma models" @default.
- W4300912898 doi "https://doi.org/10.48550/arxiv.1412.3454" @default.
- W4300912898 hasPublicationYear "2014" @default.
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