## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 16, Number 3 (2012), 725-804.

### Localization with a surface operator, irregular conformal blocks and open topological string

Hidetoshi Awata, Hiroyuki Fuji, Hiroaki Kanno, Masahide Manabe, and Yasuhiko Yamada

#### Abstract

Following a recent paper by Alday and Tachikawa, we compute the instanton partition function in the presence of the surface operator by the localization formula on the moduli space. For $SU(2)$ theories, we find an exact agreement with conformal field theory (CFT) correlation functions with a degenerate operator insertion, which enables us to work out the decoupling limit of the superconformal theory with four flavors to asymptotically free theories at the level of differential equations for CFT correlation functions (irregular conformal blocks). We also argue that the $K$ theory (or five-dimensional) lift of these computations gives open topological string amplitudes on local Hirzebruch surface and its blow ups, which is regarded as a geometric engineering of the surface operator. By computing the amplitudes in both A and B models we collect convincing evidences of the agreement of the instanton partition function with surface operator and the partition function of open topological string.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 16, Number 3 (2012), 725-804.

**Dates**

First available in Project Euclid: 20 March 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1363792005

**Mathematical Reviews number (MathSciNet)**

MR3024274

**Zentralblatt MATH identifier**

1273.81178

#### Citation

Awata, Hidetoshi; Fuji, Hiroyuki; Kanno, Hiroaki; Manabe, Masahide; Yamada, Yasuhiko. Localization with a surface operator, irregular conformal blocks and open topological string. Adv. Theor. Math. Phys. 16 (2012), no. 3, 725--804. https://projecteuclid.org/euclid.atmp/1363792005