# Boundary terms and three-point functions: an AdS/CFT puzzle resolved

## Abstract

N=8 superconformal field theories, such as the ABJM theory at Chern-Simons level k = 1 or 2, contain 35 scalar operators O _{IJ} with Δ = 1 in the 35 v representation of SO(8). The 3-point correlation function of these operators is non-vanishing, and indeed can be calculated non-perturbatively in the field theory. But its AdS _{4} gravity dual, obtained from gauged N=8 supergravity, has no cubic A ^{3} couplings in its Lagrangian, where A ^{IJ} is the bulk dual of OIJ. So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetry to the AdS _{4} boundary requires the introduction of a finite A ^{3} counterterm that does provide a perfect match to the 3-point correlator. Boundary supersymmetry also requires infinite counterterms which agree with the method of holographic renormalization. The generating functional of correlation functions of the Δ = 1 operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.

- Authors:

- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States); Stanford Univ., CA (United States)
- Univ. of Southern California, Los Angeles, CA (United States)
- Princeton Univ., NJ (United States). Joseph Henry Lab. of Physics

- Publication Date:

- Research Org.:
- Univ. of Southern California, Los Angeles, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1426908

- Grant/Contract Number:
- SC0011687

- Resource Type:
- Journal Article: Accepted Manuscript

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

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 6; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Freedman, Daniel Z., Pilch, Krzysztof, Pufu, Silviu S., and Warner, Nicholas P..
```*Boundary terms and three-point functions: an AdS/CFT puzzle resolved*. United States: N. p., 2017.
Web. doi:10.1007/JHEP06(2017)053.

```
Freedman, Daniel Z., Pilch, Krzysztof, Pufu, Silviu S., & Warner, Nicholas P..
```*Boundary terms and three-point functions: an AdS/CFT puzzle resolved*. United States. doi:10.1007/JHEP06(2017)053.

```
Freedman, Daniel Z., Pilch, Krzysztof, Pufu, Silviu S., and Warner, Nicholas P.. Mon .
"Boundary terms and three-point functions: an AdS/CFT puzzle resolved". United States.
doi:10.1007/JHEP06(2017)053. https://www.osti.gov/servlets/purl/1426908.
```

```
@article{osti_1426908,
```

title = {Boundary terms and three-point functions: an AdS/CFT puzzle resolved},

author = {Freedman, Daniel Z. and Pilch, Krzysztof and Pufu, Silviu S. and Warner, Nicholas P.},

abstractNote = {N=8 superconformal field theories, such as the ABJM theory at Chern-Simons level k = 1 or 2, contain 35 scalar operators OIJ with Δ = 1 in the 35 v representation of SO(8). The 3-point correlation function of these operators is non-vanishing, and indeed can be calculated non-perturbatively in the field theory. But its AdS4 gravity dual, obtained from gauged N=8 supergravity, has no cubic A 3 couplings in its Lagrangian, where A IJ is the bulk dual of OIJ. So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetry to the AdS4 boundary requires the introduction of a finite A 3 counterterm that does provide a perfect match to the 3-point correlator. Boundary supersymmetry also requires infinite counterterms which agree with the method of holographic renormalization. The generating functional of correlation functions of the Δ = 1 operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.},

doi = {10.1007/JHEP06(2017)053},

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

number = 6,

volume = 2017,

place = {United States},

year = {Mon Jun 12 00:00:00 EDT 2017},

month = {Mon Jun 12 00:00:00 EDT 2017}

}