# An algebraic approach to the analytic bootstrap

## Abstract

We develop an algebraic approach to the analytic bootstrap in CFTs. By acting with the Casimir operator on the crossing equation we map the problem of doing large spin sums to any desired order to the problem of solving a set of recursion relations. We compute corrections to the anomalous dimension of large spin operators due to the exchange of a primary and its descendants in the crossed channel and show that this leads to a Borel-summable expansion. Here, we analyse higher order corrections to the microscopic CFT data in the direct channel and its matching to infinite towers of operators in the crossed channel. We apply this method to the critical O(N ) model. At large N we reproduce the first few terms in the large spin expansion of the known two-loop anomalous dimensions of higher spin currents in the traceless symmetric representation of O(N ) and make further predictions. At small N we present the results for the truncated large spin expansion series of anomalous dimensions of higher spin currents.

- Authors:

- Univ. of Oxford (United Kingdom). Mathematical Inst.
- Harvard Univ., Cambridge, MA (United States). Center for the Fundamental Laws of Nature

- Publication Date:

- Research Org.:
- Harvard Univ., Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); European Research Council (ERC)

- OSTI Identifier:
- 1368372

- Grant/Contract Number:
- SC0007870; 306260

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 4; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Conformal Field Theory; AdS-CFT Correspondence

### Citation Formats

```
Alday, Luis F., and Zhiboedov, Alexander.
```*An algebraic approach to the analytic bootstrap*. United States: N. p., 2017.
Web. doi:10.1007/JHEP04(2017)157.

```
Alday, Luis F., & Zhiboedov, Alexander.
```*An algebraic approach to the analytic bootstrap*. United States. doi:10.1007/JHEP04(2017)157.

```
Alday, Luis F., and Zhiboedov, Alexander. Thu .
"An algebraic approach to the analytic bootstrap". United States.
doi:10.1007/JHEP04(2017)157. https://www.osti.gov/servlets/purl/1368372.
```

```
@article{osti_1368372,
```

title = {An algebraic approach to the analytic bootstrap},

author = {Alday, Luis F. and Zhiboedov, Alexander},

abstractNote = {We develop an algebraic approach to the analytic bootstrap in CFTs. By acting with the Casimir operator on the crossing equation we map the problem of doing large spin sums to any desired order to the problem of solving a set of recursion relations. We compute corrections to the anomalous dimension of large spin operators due to the exchange of a primary and its descendants in the crossed channel and show that this leads to a Borel-summable expansion. Here, we analyse higher order corrections to the microscopic CFT data in the direct channel and its matching to infinite towers of operators in the crossed channel. We apply this method to the critical O(N ) model. At large N we reproduce the first few terms in the large spin expansion of the known two-loop anomalous dimensions of higher spin currents in the traceless symmetric representation of O(N ) and make further predictions. At small N we present the results for the truncated large spin expansion series of anomalous dimensions of higher spin currents.},

doi = {10.1007/JHEP04(2017)157},

journal = {Journal of High Energy Physics (Online)},

number = 4,

volume = 2017,

place = {United States},

year = {Thu Apr 27 00:00:00 EDT 2017},

month = {Thu Apr 27 00:00:00 EDT 2017}

}

*Citation information provided by*

Web of Science

Web of Science