## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, J.-P. Bourguignon, M. Kotani, Y. Maeda and N. Tose, eds. (Tokyo: Mathematical Society of Japan, 2009), 51 - 67

### Instanton counting and the chiral ring relations in supersymmetric gauge theories

#### Abstract

We compute topological one-point functions of the chiral operator $\mathrm{Tr}\ \varphi^k$ in the maximally confining phase of $U(N)$ supersymmetric gauge theory. These chiral one-point functions are of particular interest from gauge/string theory correspondence, since they are related to the equivariant Gromov–Witten theory of $\mathbf{P}^1$. By considering the power sums of Jucys–Murphy elements in the class algebra of the symmetric group we can derive a combinatorial identity that leads the relations among chiral one-point functions. Using the operator formalism of free fermions, we also compute the vacuum expectation value of the loop operator $\langle \mathrm{Tr}\ e^{it\varphi}\rangle$ which gives the generating function of the one-point functions.

#### Article information

**Dates**

Received: 27 November 2008

First available in Project Euclid:
28 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543447901

**Digital Object Identifier**

doi:10.2969/aspm/05510051

**Mathematical Reviews number (MathSciNet)**

MR2463490

**Zentralblatt MATH identifier**

1206.05101

**Subjects**

Primary: 05E10: Combinatorial aspects of representation theory [See also 20C30] 81T45: Topological field theories [See also 57R56, 58Dxx] 81T60: Supersymmetric field theories

**Keywords**

Instanton supersymmetric gauge theory

#### Citation

Kanno, Hiroaki. Instanton counting and the chiral ring relations in supersymmetric gauge theories. Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, 51--67, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05510051. https://projecteuclid.org/euclid.aspm/1543447901