arXiv subjects

Andreas Malmendier

Publications and source records attributed to Andreas Malmendier.

41 records · Page 3Linked to original sources

arXiv2012

Moonshine and Donaldson invariants of CP^2

Eguchi, Ooguri, and Tachikawa recently conjectured a new moonshine phenomenon. They conjecture that the coefficients of a certain mock modular form H(tau), which arises from the K3 surface elliptic genus, are sums of dimensions of irreducible representations of the Mathieu group M24. We prove that H(tau) surprisingly also plays a significant role in the theory of Donaldson invariants. We prove that the Moore-Witten u-plane integrals for H(tau) are the SO(3)-Donaldson invariants of CP^2. This result then implies a moonshine phenomenon where these invariants conjecturally are expressions in the dimensions of the irreducible representations of M24. Indeed, we obtain an explicit expression for the Donaldson invariant generating function Z(p,S) in terms of the derivatives of H(tau).

math.DG↗

arXiv2012

Kummer surfaces associated with Seiberg-Witten curves

By carrying out a rational transformation on the base curve $\mathbb{CP}^1$ of the Seiberg-Witten curve for $\mathcal{N}=2$ supersymmetric pure $\mathrm{SU}(2)$-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure $\mathrm{SU}(2)$-gauge theory to the one for $\mathrm{SU}(2)$-gauge theory with $N_f=2$ massless hypermultiplets extends to define a Nikulin involution on each K3 surface in the family. We show that the desingularization of the quotient of the K3 surface by the involution is isomorphic to a Kummer surface of the Jacobian variety of a curve of genus two. We then derive a relation between the Yukawa coupling associated with the elliptic K3 surface and the Yukawa coupling of pure $\mathrm{SU}(2)$-gauge theory.

math.AG↗

arXiv2011

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.

math.DG↗

arXiv2011

The signature of the Seiberg-Witten surface

The Seiberg-Witten family of elliptic curves defines a Jacobian rational elliptic surface $\Z$ over $\mathbb{C}\mathrm{P}^1$. We show that for the $\bar{\partial}$-operator along the fiber the logarithm of the regularized determinant $-1/2 \log \det' (\bar\partial^* \bar\partial)$ 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 $\bar{\partial}$-operator itself extends across the nodal fibers of $\mathrm{Z}$. 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 $\mathrm{Z}$ which equals minus the number of hypermultiplets.

math.DG↗

arXiv2003

The eigenvalue equation on the Eguchi-Hanson space

We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their asymptotic scattering phases with the help of the Liouville-Green approximation (WKB). Furthermore, for specific discrete eigenvalues obtained by a continued T-fraction we construct the solution by the Frobenius method and determine its scattering phase by a monodromy computation.

math.DG↗