Automorphic function on the moduli space and analytic torsion

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We have proved the fact that analytic torsion of a K3 surface with anti-symplectic involution (corrected by that of the fixed curve), regarded as a function on the moduli space, is given by an automorphic form in the wider sense on the Hermitian domain of type IV . In particular, we have shown that analytic torsion of an Enriques surface coincides with Borcherds Φ-function, and that analogous result is valid for various 2-elementary lattices. As a result, we found that analytic torsion of certain K3 surfaces related, with Del Pezzo surfaces coincides with the denominator function of some generalized Kac-Moody algebras . These algebras correspond to the odd unimodular hyperbolic lattices with Weyl vector. Together with a series of generalized Kac-Moody algebras corresponding to even unimodular hyper-bolic lattices with Weyl vector, our algebras seem to form an interesting series of generalized Kac-Moody algebras whose denominator function is given by the automorphic form characterizing the discriminant locus. The geometric meaning of these algebras is far from understanding .