Theory HEP Seminar

K3 surfaces, modular forms, and non-geometric heterotic compactifications

David Morrison

Type IIB string theory has an SL(2, Z) symmetry and a complex scalar field tau valued in the upper half plane, on which SL(2, Z) acts by fractional linear transformations; this naturally suggests building models in which tau is allowed to vary. Although the SL(2, Z)-invariant function j(ô) can reveal some of the structures of these models, for their full construction and study we need SL(2, Z) modular forms, particularly the Eisenstein series E4(ô) and E6(ô) and the corresponding Weierstrass equations. The Weierstrass equations can also be analyzed in algebraic geometry via the theory of elliptic curves. This approach leads to the ''F-theory'' compactifications of type IIB theory.

Similarly, the heterotic string compactified on T2 has a large discrete symmetry group SO(2,18;Z), which acts on the scalars in the theory in a natural way; there have been a number of attempts to construct models in which these scalars are allowed to vary by using SO(2,18;Z)-invariant functions. In our new work, we give (in principle) a more complete construction of these models, using SO(2,18;Z)-modular forms analogous to the Eisenstein series. In practice, we restrict to special cases in which either there are no Wilson lines — and SO(2,2;Z) symmetry — or there is a single Wilson line — and SO(2, 3;Z) symmetry. In those cases, the modular forms can be analyzed in detail and there turns out to be a precise theory of K3 surfaces with prescribed singularities which corresponds to the structure of the modular forms. Using these two approaches — modular forms on the one hand, and the algebraic geometry of the K3 surfaces on the other hand — we can construct non-geometric compactifications of the heterotic theory.

This is a report on two joint projects: one with McOrist and Sethi and the other with Malmendier.