## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Primitive Forms and Related Subjects — Kavli IPMU 2014, K. Hori, C. Li, S. Li and K. Saito, eds. (Tokyo: Mathematical Society of Japan, 2019), 127 - 220

### Notes on the Hemisphere

Kentaro Hori and Mauricio Romo

#### Abstract

In these notes, we provide an introduction to the hemisphere partition function of 2d $(2,2)$ supersymmetric gauge theories, and discuss its relation to the "D-brane central charge" which were studied in superstring theory, in 2d supersymmetric quantum field theory, and in topological string theory. We also discuss relation to "macroscopic loop" in matrix models. They are mostly reviews of the work by the authors, but contains some new results such as the partition function for a rotated supersymmetry as well as the differential equations.

#### Article information

**Dates**

Received: 23 August 2016

Revised: 2 April 2017

First available in Project Euclid:
26 December 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1577379885

**Digital Object Identifier**

doi:10.2969/aspm/08310127

**Subjects**

Primary: 81T60: Supersymmetric field theories

Secondary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14J33: Mirror symmetry [See also 11G42, 53D37] 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 18E30: Derived categories, triangulated categories 32S40: Monodromy; relations with differential equations and D-modules 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33] 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 81T13: Yang-Mills and other gauge theories [See also 53C07, 58E15] 81T30: String and superstring theories; other extended objects (e.g., branes) [See also 83E30] 81T40: Two-dimensional field theories, conformal field theories, etc. 81T45: Topological field theories [See also 57R56, 58Dxx]

**Keywords**

Supersymmetric quantum field theory gauged linear sigma models boundary conditions supersymmetric localization D-brane central charge primitive forms flat structure Gromov-Witten theory

#### Citation

Hori, Kentaro; Romo, Mauricio. Notes on the Hemisphere. Primitive Forms and Related Subjects — Kavli IPMU 2014, 127--220, Mathematical Society of Japan, Tokyo, Japan, 2019. doi:10.2969/aspm/08310127. https://projecteuclid.org/euclid.aspm/1577379885