Algebraic structure of tt* equations for Calabi-Yau sigma models

The tt∗ equations define a flat connection on the moduli spaces of 2d,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 CP3, where due to further algebraic constraints, the differential ring coincides with quasi modular forms.