## Advances in Theoretical and Mathematical Physics

### Direct integration for general $\Omega$ backgrounds

#### Abstract

We extend the direct integration method of the holomorphic anomalyequations to general $\Omega$ backgrounds $\epsilon_1\neq -\epsilon_2$for pure SU(2) $N=2$ Super-Yang-Mills theory and topological stringtheory on non-compact Calabi-Yau threefolds. We find that anextension of the holomorphic anomaly equation, modularity and boundary conditions provided by the perturbative terms as well as by the gap condition at the conifold are sufficient to solve the generalized theory in the above cases. In particular, we use the method to solve the topological string for the general $\Omega$ backgrounds on non-compact toric Calabi-Yau spaces. The conifold boundary condition follows from that the $N=2$ Schwinger-loop calculation with Bogomol'nyi-Prasad-Sommerfield (BPS) states coupled to a self-dual and an anti-self-dual field strength. We calculate such BPS states also for the large base decompactification limit of Calabi-Yau spaces with regular $K3$ fibrations and half $K3$s embedded in Calabi-Yau backgrounds.

#### Article information

Source
Adv. Theor. Math. Phys., Volume 16, Number 3 (2012), 805-849.

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1363792006

Mathematical Reviews number (MathSciNet)
MR3024275

Zentralblatt MATH identifier
1276.81098

#### Citation

Huang, Min-xin; Klemm, Albrecht. Direct integration for general $\Omega$ backgrounds. Adv. Theor. Math. Phys. 16 (2012), no. 3, 805--849. https://projecteuclid.org/euclid.atmp/1363792006