We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the ``attractor mechanism'' of N=2 supergravity. In IIB string compactification this mechanism singles out certain ``attractor varieties.'' We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N=4 and N=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker's Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled ``Attractors and Arithmetic'' (hep-th/9807056).

Arithmetic and Attractors

We calculate the one-loop corrections to gauge couplings in N = 1 supersymmetric brane world models, which are realized in an type IIA orbifold/orientifold background with several stacks of D6 branes wrapped on 3-cycles with non-vanishing intersections. Contributions arise from both N = 1 and N = 2 open string subsectors. In contrast to what is known from ordinary orbifold theories, N = 1 subsectors do give rise to moduli-dependent one-loop corrections.

Gauge threshold corrections in intersecting brane world models

We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the fivedimensional N =1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N = 2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p, q]-strings as well as M-strings.

Refined stable pair invariants for E-, M- and [ p , q ]-strings

Counting higher genus curves in a Calabi-Yau manifold

We investigate the relevance of Eisenstein series for representing certain G(ℤ)-invariant string theory amplitudes which receive corrections from BPS states only. G(ℤ) may stand for any of the mapping class, T-duality and U-duality groups Sl(d,(ℤ), SO(d,d,(ℤ) or E

 d
 +1(

 d
 +1)((ℤ) respectively. Using G(ℤ)-invariant mass formulae, we construct invariant modular functions on the symmetric space K\G(ℝ) of non-compact type, with K the maximal compact subgroup of G(ℝ), that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincare upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in
one- and g-loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the R

 4 and R

 4

 H

 4

 g
 -4 couplings in toroidal compactifications of M-theory to any dimension D≥ 4 and D≥ 6 respectively.

Eisenstein Series and String Thresholds

We compute certain one-loop corrections to F^4 couplings of the heterotic string compactified on T^2, and show that they can be characterized by holomorphic prepotentials. We then discuss how some of these couplings can be obtained in F-theory, or more precisely from 7-brane geometry in type IIB language. We in particular study theories with E_8 x E_8 and SO(8)^4 gauge symmetry, on certain one-dimensional sub-spaces of the moduli space that correspond to constant IIB coupling. For these theories, the relevant geometry can be mapped to Riemann surfaces. Physically, the computations amount to non-trivial tests of the basic F-theory -- heterotic duality in eight dimensions. Mathematically, they mean to associate holomorphic 5-point couplings of the form (del_t)^5 G = sum[ g_l l^5 q^l/(1-q^l) ] to K3 surfaces. This can be seen as a novel manifestation of the mirror map, acting here between open and closed string sectors.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1998/0002/0005/ATMP-1998-0002-0005-a006.pdf

W. Lerche

Papers

Prepotential, mirror map and $F$-theory on $K3$

Prepotential, Mirror Map and F-Theory on K3