# Open-closed homotopy algebra in mathematical physics

## Abstract

In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A{sub {infinity}} algebras) by closed strings (L{sub {infinity}} algebras)

- Authors:

- Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502 (Japan)
- (United States)

- Publication Date:

- OSTI Identifier:
- 20768729

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 2; Other Information: DOI: 10.1063/1.2171524; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; DEFORMATION; GAUGE INVARIANCE; PERTURBATION THEORY; QUANTIZATION; QUANTUM FIELD THEORY; STRING MODELS; SYMMETRY

### Citation Formats

```
Kajiura, Hiroshige, Stasheff, Jim, and Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395.
```*Open-closed homotopy algebra in mathematical physics*. United States: N. p., 2006.
Web. doi:10.1063/1.2171524.

```
Kajiura, Hiroshige, Stasheff, Jim, & Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395.
```*Open-closed homotopy algebra in mathematical physics*. United States. doi:10.1063/1.2171524.

```
Kajiura, Hiroshige, Stasheff, Jim, and Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395. Wed .
"Open-closed homotopy algebra in mathematical physics". United States.
doi:10.1063/1.2171524.
```

```
@article{osti_20768729,
```

title = {Open-closed homotopy algebra in mathematical physics},

author = {Kajiura, Hiroshige and Stasheff, Jim and Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395},

abstractNote = {In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A{sub {infinity}} algebras) by closed strings (L{sub {infinity}} algebras)},

doi = {10.1063/1.2171524},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 47,

place = {United States},

year = {Wed Feb 15 00:00:00 EST 2006},

month = {Wed Feb 15 00:00:00 EST 2006}

}