## Casimir recursion relations for general conformal blocks

## Abstract

We study the structure of series expansions of general spinning conformal blocks. We find that the terms in these expansions are naturally expressed by means of special functions related to matrix elements of Spin(d) representations in Gelfand-Tsetlin basis, of which the Gegenbauer polynomials are a special case. We study the properties of these functions and explain how they can be computed in practice. We show how the Casimir equation in Dolan-Osborn coordinates leads to a simple one-step recursion relation for the coefficients of the series expansion of general spinning conformal block. The form of this recursion relation is determined by 6j symbols of Spin(d – 1). In particular, it can be written down in closed form in d = 3, d = 4, for seed blocks in general dimensions, or in any other situation when the required 6j symbols can be computed. As a result, we work out several explicit examples and briefly discuss how our recursion relation can be used for efficient numerical computation of general conformal blocks.

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics

- Publication Date:

- Research Org.:
- California Inst. of Technology, Pasadena, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1499658

- Grant/Contract Number:
- SC0011632

- Resource Type:
- Accepted Manuscript

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

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal and W Symmetry; Conformal Field Theory

### Citation Formats

```
Kravchuk, Petr. Casimir recursion relations for general conformal blocks. United States: N. p., 2018.
Web. doi:10.1007/jhep02(2018)011.
```

```
Kravchuk, Petr. Casimir recursion relations for general conformal blocks. United States. doi:10.1007/jhep02(2018)011.
```

```
Kravchuk, Petr. Fri .
"Casimir recursion relations for general conformal blocks". United States. doi:10.1007/jhep02(2018)011. https://www.osti.gov/servlets/purl/1499658.
```

```
@article{osti_1499658,
```

title = {Casimir recursion relations for general conformal blocks},

author = {Kravchuk, Petr},

abstractNote = {We study the structure of series expansions of general spinning conformal blocks. We find that the terms in these expansions are naturally expressed by means of special functions related to matrix elements of Spin(d) representations in Gelfand-Tsetlin basis, of which the Gegenbauer polynomials are a special case. We study the properties of these functions and explain how they can be computed in practice. We show how the Casimir equation in Dolan-Osborn coordinates leads to a simple one-step recursion relation for the coefficients of the series expansion of general spinning conformal block. The form of this recursion relation is determined by 6j symbols of Spin(d – 1). In particular, it can be written down in closed form in d = 3, d = 4, for seed blocks in general dimensions, or in any other situation when the required 6j symbols can be computed. As a result, we work out several explicit examples and briefly discuss how our recursion relation can be used for efficient numerical computation of general conformal blocks.},

doi = {10.1007/jhep02(2018)011},

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

number = 2,

volume = 2018,

place = {United States},

year = {2018},

month = {2}

}