## The 3d stress-tensor bootstrap

## Abstract

Here, we study the conformal bootstrap for 4-point functions of stress tensors in parity-preserving 3d CFTs. To set up the bootstrap equations, we analyze the constraints of conformal symmetry, permutation symmetry, and conservation on the stress-tensor 4-point function and identify a non-redundant set of crossing equations. Studying these equations numerically using semidefinite optimization, we compute bounds on the central charge as a function of the independent coefficient in the stress-tensor 3-point function. With no additional assumptions, these bounds numerically reproduce the conformal collider bounds and give a general lower bound on the central charge. We also study the effect of gaps in the scalar, spin-2, and spin-4 spectra on the central charge bound. We find general upper bounds on these gaps as well as tighter restrictions on the stress-tensor 3-point function coefficients for theories with moderate gaps. When the gap for the leading scalar or spin-2 operator is sufficiently large to exclude large N theories, we also obtain upper bounds on the central charge, thus finding compact allowed regions. Finally, assuming the known low-lying spectrum and central charge of the critical 3d Ising model, we determine its stress-tensor 3-point function and derive a bound on its leading parity-odd scalar.

- Authors:

- Univ. of Kentucky, Lexington, KY (United States). Dept. of Physics and Astronomy; Skolkovo Inst. of Science and Technology, Moscow (Russian Federation). Skolkovo Innovation Center
- Univ. of California, Berkeley, CA (United States). Berkeley Center for Theoretical Physics and Dept. of Physics ; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Theoretical Physics Group
- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics
- Yale Univ., New Haven, CT (United States). Dept. of Physics
- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics; Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Scientific User Facilities Division; National Science Foundation (NSF); Simons Foundation

- OSTI Identifier:
- 1461289

- Alternate Identifier(s):
- OSTI ID: 1493252

- Grant/Contract Number:
- SC0009988; SC0011632; AC02-05CH11231; PHY-1720374; 488657; PHY-1350180; 488651

- 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 Field Theory; AdS-CFT Correspondence; Conformal and W Symmetry; Field Theories in Higher Dimensions

### Citation Formats

```
Dymarsky, Anatoly, Kos, Filip, Kravchuk, Petr, Poland, David, and Simmons-Duffin, David. The 3d stress-tensor bootstrap. United States: N. p., 2018.
Web. doi:10.1007/JHEP02(2018)164.
```

```
Dymarsky, Anatoly, Kos, Filip, Kravchuk, Petr, Poland, David, & Simmons-Duffin, David. The 3d stress-tensor bootstrap. United States. doi:10.1007/JHEP02(2018)164.
```

```
Dymarsky, Anatoly, Kos, Filip, Kravchuk, Petr, Poland, David, and Simmons-Duffin, David. Tue .
"The 3d stress-tensor bootstrap". United States. doi:10.1007/JHEP02(2018)164. https://www.osti.gov/servlets/purl/1461289.
```

```
@article{osti_1461289,
```

title = {The 3d stress-tensor bootstrap},

author = {Dymarsky, Anatoly and Kos, Filip and Kravchuk, Petr and Poland, David and Simmons-Duffin, David},

abstractNote = {Here, we study the conformal bootstrap for 4-point functions of stress tensors in parity-preserving 3d CFTs. To set up the bootstrap equations, we analyze the constraints of conformal symmetry, permutation symmetry, and conservation on the stress-tensor 4-point function and identify a non-redundant set of crossing equations. Studying these equations numerically using semidefinite optimization, we compute bounds on the central charge as a function of the independent coefficient in the stress-tensor 3-point function. With no additional assumptions, these bounds numerically reproduce the conformal collider bounds and give a general lower bound on the central charge. We also study the effect of gaps in the scalar, spin-2, and spin-4 spectra on the central charge bound. We find general upper bounds on these gaps as well as tighter restrictions on the stress-tensor 3-point function coefficients for theories with moderate gaps. When the gap for the leading scalar or spin-2 operator is sufficiently large to exclude large N theories, we also obtain upper bounds on the central charge, thus finding compact allowed regions. Finally, assuming the known low-lying spectrum and central charge of the critical 3d Ising model, we determine its stress-tensor 3-point function and derive a bound on its leading parity-odd scalar.},

doi = {10.1007/JHEP02(2018)164},

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

number = 2,

volume = 2018,

place = {United States},

year = {2018},

month = {2}

}

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