58 research outputs found
Donaldson invariants of CP^1 x CP^1 and Mock Theta Functions
We compute the Moore-Witten regularized u-plane integral on CP^1 x CP^1
directly in a chamber where the elliptic unfolding technique fails to work.
This allows us to determine explicit formulas for its SU(2) and SO(3)-Donaldson
invariants in terms of Mock modular forms.Comment: 20 pages, LaTe
Jacobian elliptic Kummer surfaces and special function identities
We derive formulas for the construction of all inequivalent Jacobian elliptic
fibrations on the Kummer surface of two non-isogeneous elliptic curves from
extremal rational elliptic surfaces by rational base transformations and
quadratic twists. We then show that each such decomposition yields a
description of the Picard-Fuchs system satisfied by the periods of the
holomorphic two-form as either a tensor product of two Gauss' hypergeometric
differential equations, an Appell hypergeometric system, or a GKZ differential
system. As the answer must be independent of the fibration used, identities
relating differential systems are obtained. They include a new identity
relating Appell's hypergeometric system to a product of two Gauss'
hypergeometric differential equations by a cubic transformation.Comment: 20 page
Normal forms for Kummer surfaces
We determine normal forms for the Kummer surfaces associated with abelian
surfaces of polarization of type , , , , and
. Explicit formulas for coordinates and moduli parameters in terms of
Theta functions of genus two are also given. The normal forms in question are
closely connected to the generalized Riemann identities for Theta functions of
Mumford's.Comment: 49 page
Kummer sandwiches and Greene-Plesser construction
In the context of K3 mirror symmetry, the Greene-Plesser orbifolding method
constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in
, starting from a special one-parameter family of K3 varieties
known as the quartic Dwork pencil. We show that certain K3 double covers
obtained from the three-parameter family of quartic Kummer surfaces associated
with a principally polarized abelian surface generalize the relation of the
Dwork pencil and the quartic mirror family. Moreover, for the three-parameter
family we compute a formula for the rational point-count of its generic member
and derive its transformation behavior with respect to -isogenies of the
underlying abelian surface.Comment: 27 pages; minor typos corrected in version
Six line configurations and string dualities
We study the family of K3 surfaces of Picard rank sixteen associated with the
double cover of the projective plane branched along the union of six lines, and
the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special
Nikulin involutions, such that quotienting by the involution and blowing up
recovers the former. We prove that the family of Van Geemen-Sarti partners is a
four-parameter family of K3 surfaces with
lattice polarization. We describe explicit Weierstrass models on both families
using even modular forms on the bounded symmetric domain of type . We also
show that our construction provides a geometric interpretation, called
geometric two-isogeny, for the F-theory/heterotic string duality in eight
dimensions. As a result, we obtain novel F-theory models, dual to non-geometric
heterotic string compactifications in eight dimensions with two non-vanishing
Wilson line parameters.Comment: 42 pages; minor typos corrected in version
The signature of the Seiberg-Witten surface
The Seiberg-Witten family of elliptic curves defines a Jacobian rational
elliptic surface over . We show that for the
-operator along the fiber the logarithm of the regularized
determinant satisfies the
anomaly equation of the one-loop topological string amplitude derived in
Kodaira-Spencer theory. We also show that not only the determinant line bundle
with the Quillen metric but also the -operator itself extends
across the nodal fibers of . The extension introduces current
contributions to the curvature of the determinant line bundle at the points
where the fibration develops nodal fibers. The global anomaly of the
determinant line bundle then determines the signature of which
equals minus the number of hypermultiplets.Comment: 22 page
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