# Light-cone modular bootstrap and pure gravity

## Abstract

We explore the large spin spectrum in two-dimensional conformal field theories with a finite twist gap, using the modular bootstrap in the light-cone limit. By recursively solving the modular crossing equations associated with different $PSL(2,\mathbb{Z})$ elements, we identify the universal contribution to the density of large spin states from the vacuum in the dual channel. Our result takes the form of a sum over $PSL(2,\mathbb{Z})$ elements, whose leading term generalizes the usual Cardy formula to a wider regime. Rather curiously, the contribution to the density of states from the vacuum becomes negative in a specific limit, which can be canceled by that from a nonvacuum Virasoro primary whose twist is no bigger than $\frac{c-1}{16}$. This suggests a new upper bound of $\frac{c-1}{16}$ on the twist gap in any $c>1$ compact, unitary conformal field theory with a vacuum, which would in particular imply that pure ${\mathrm{AdS}}_{3}$ gravity does not exist. We confirm this negative density of states in the pure gravity partition function by Maloney, Witten, and Keller. We generalize our discussion to theories with $\mathcal{N}=(1,1)$ supersymmetry and find similar results

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP); Simons Foundation; National Science Foundation (NSF)

- OSTI Identifier:
- 1564551

- Alternate Identifier(s):
- OSTI ID: 1600526

- Grant/Contract Number:
- SC0011632; PHY-1606531; PHY-1620059; PHY-1607611

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 6; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Benjamin, Nathan, Ooguri, Hirosi, Shao, Shu-Heng, and Wang, Yifan. Light-cone modular bootstrap and pure gravity. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.066029.
```

```
Benjamin, Nathan, Ooguri, Hirosi, Shao, Shu-Heng, & Wang, Yifan. Light-cone modular bootstrap and pure gravity. United States. doi:10.1103/PhysRevD.100.066029.
```

```
Benjamin, Nathan, Ooguri, Hirosi, Shao, Shu-Heng, and Wang, Yifan. Sun .
"Light-cone modular bootstrap and pure gravity". United States. doi:10.1103/PhysRevD.100.066029.
```

```
@article{osti_1564551,
```

title = {Light-cone modular bootstrap and pure gravity},

author = {Benjamin, Nathan and Ooguri, Hirosi and Shao, Shu-Heng and Wang, Yifan},

abstractNote = {We explore the large spin spectrum in two-dimensional conformal field theories with a finite twist gap, using the modular bootstrap in the light-cone limit. By recursively solving the modular crossing equations associated with different PSL(2,Z) elements, we identify the universal contribution to the density of large spin states from the vacuum in the dual channel. Our result takes the form of a sum over PSL(2,Z) elements, whose leading term generalizes the usual Cardy formula to a wider regime. Rather curiously, the contribution to the density of states from the vacuum becomes negative in a specific limit, which can be canceled by that from a nonvacuum Virasoro primary whose twist is no bigger than c-116. This suggests a new upper bound of c-116 on the twist gap in any c>1 compact, unitary conformal field theory with a vacuum, which would in particular imply that pure AdS3 gravity does not exist. We confirm this negative density of states in the pure gravity partition function by Maloney, Witten, and Keller. We generalize our discussion to theories with N=(1,1) supersymmetry and find similar results},

doi = {10.1103/PhysRevD.100.066029},

journal = {Physical Review D},

number = 6,

volume = 100,

place = {United States},

year = {2019},

month = {9}

}

DOI: 10.1103/PhysRevD.100.066029

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