# On the positive geometry of conformal field theory

## Abstract

It has long been clear that the conformal bootstrap is associated with a rich geometry. In this paper we undertake a systematic exploration of this geometric structure as an object of study in its own right. We study conformal blocks for the minimal SL(2 *, R*) symmetry present in conformal field theories in all dimensions. Unitarity demands that the Taylor coefficients of the four-point function lie inside a polytope **U** determined by the operator spectrum, while crossing demands they lie on a plane **X**. The conformal bootstrap is then geometrically interpreted as demanding a non-empty intersection of **U** $$\cap$$ **X**. We find that the conformal blocks enjoy a surprising positive determinant property. This implies that **U** is an example of a famous polytope — the cyclic polytope. The face structure of cyclic polytopes is completely understood. This lets us fully characterize the intersection **U** $$\cap$$ **X** by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory.

- Authors:

- Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
- National Taiwan Univ., Taipei (Taiwan); National Tsing Hua Univ., Hsinchu (Taiwan)

- Publication Date:

- Research Org.:
- Inst. for Advanced Study, Princeton, NJ (United States)

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

- OSTI Identifier:
- 1596091

- Grant/Contract Number:
- SC0009988

- 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: 2019; Journal Issue: 6; 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; Conformal and W Symmetry

### Citation Formats

```
Arkani-Hamed, Nima, Huang, Yu-tin, and Shao, Shu-Heng. On the positive geometry of conformal field theory. United States: N. p., 2019.
Web. doi:10.1007/JHEP06(2019)124.
```

```
Arkani-Hamed, Nima, Huang, Yu-tin, & Shao, Shu-Heng. On the positive geometry of conformal field theory. United States. doi:10.1007/JHEP06(2019)124.
```

```
Arkani-Hamed, Nima, Huang, Yu-tin, and Shao, Shu-Heng. Tue .
"On the positive geometry of conformal field theory". United States. doi:10.1007/JHEP06(2019)124. https://www.osti.gov/servlets/purl/1596091.
```

```
@article{osti_1596091,
```

title = {On the positive geometry of conformal field theory},

author = {Arkani-Hamed, Nima and Huang, Yu-tin and Shao, Shu-Heng},

abstractNote = {It has long been clear that the conformal bootstrap is associated with a rich geometry. In this paper we undertake a systematic exploration of this geometric structure as an object of study in its own right. We study conformal blocks for the minimal SL(2, R) symmetry present in conformal field theories in all dimensions. Unitarity demands that the Taylor coefficients of the four-point function lie inside a polytope U determined by the operator spectrum, while crossing demands they lie on a plane X. The conformal bootstrap is then geometrically interpreted as demanding a non-empty intersection of U $\cap$ X. We find that the conformal blocks enjoy a surprising positive determinant property. This implies that U is an example of a famous polytope — the cyclic polytope. The face structure of cyclic polytopes is completely understood. This lets us fully characterize the intersection U $\cap$ X by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory.},

doi = {10.1007/JHEP06(2019)124},

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

number = 6,

volume = 2019,

place = {United States},

year = {2019},

month = {6}

}

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