Lie algebraic formulation of tt* equations and topological strings

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. I will 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 anti-holomorphic derivatives of tt* and provides these with a finer and algebraic meaning. 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. For d=3, this can be used to provide a purely Lie algebraic formulation of the higher genus topological string theory amplitudes and of the BCOV holomorphic anomaly equations which govern them.