## Modular bootstrap revisited

## Abstract

We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how “extremal spectra” which maximize the degeneracy at the gap can be determined numerically.

- Authors:

- Harvard Univ., Cambridge, MA (United States). Jefferson Physical Lab.

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1507580

- Grant/Contract Number:
- FG02-91ER40654

- 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: 9; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Black Holes; Conformal and W Symmetry; Conformal Field Theory

### Citation Formats

```
Collier, Scott, Lin, Ying -Hsuan, and Yin, Xi. Modular bootstrap revisited. United States: N. p., 2018.
Web. doi:10.1007/jhep09(2018)061.
```

```
Collier, Scott, Lin, Ying -Hsuan, & Yin, Xi. Modular bootstrap revisited. United States. doi:10.1007/jhep09(2018)061.
```

```
Collier, Scott, Lin, Ying -Hsuan, and Yin, Xi. Wed .
"Modular bootstrap revisited". United States. doi:10.1007/jhep09(2018)061. https://www.osti.gov/servlets/purl/1507580.
```

```
@article{osti_1507580,
```

title = {Modular bootstrap revisited},

author = {Collier, Scott and Lin, Ying -Hsuan and Yin, Xi},

abstractNote = {We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how “extremal spectra” which maximize the degeneracy at the gap can be determined numerically.},

doi = {10.1007/jhep09(2018)061},

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

number = 9,

volume = 2018,

place = {United States},

year = {2018},

month = {9}

}