# $$ \mathcal{N} $$ = 4 superconformal bootstrap of the K _{3} CFT

## Abstract

We study two-dimensional (4; 4) superconformal eld theories of central charge c = 6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap. This is made possible through a surprising relation between the BPS N = 4 superconformal blocks with c = 6 and bosonic Virasoro conformal blocks with c = 28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly from the CFT perspective the signature of a continuous spectrum above a gap at the singular moduli, and fi nd numerically an upper bound on this gap that is saturated by the A1 N = 4 cigar CFT. We also derive an analytic upper bound on the fi rst nonzero eigenvalue of the scalar Laplacian on K3 in the large volume regime, that depends on the K3 moduli data. As two byproducts, we find an exact equivalence between a class of BPS N = 2 superconformal blocks and Virasoro conformal blocks in two dimensions, and an upper boundmore »

- Authors:

- Harvard Univ., Cambridge, MA (United States). Jefferson Physical Lab.
- Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Center for Theoretical Physics

- Publication Date:

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

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

- OSTI Identifier:
- 1389304

- Grant/Contract Number:
- SC0009988; FG02-91ER40654

- 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: 5; 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; Extended Supersymmetry; Field Theories in Lower Dimensions

### Citation Formats

```
Lin, Ying-Hsuan, Shao, Shu-Heng, Simmons-Duffin, David, Wang, Yifan, and Yin, Xi.
```*$ \mathcal{N} $ = 4 superconformal bootstrap of the K3 CFT*. United States: N. p., 2017.
Web. doi:10.1007/JHEP05(2017)126.

```
Lin, Ying-Hsuan, Shao, Shu-Heng, Simmons-Duffin, David, Wang, Yifan, & Yin, Xi.
```*$ \mathcal{N} $ = 4 superconformal bootstrap of the K3 CFT*. United States. doi:10.1007/JHEP05(2017)126.

```
Lin, Ying-Hsuan, Shao, Shu-Heng, Simmons-Duffin, David, Wang, Yifan, and Yin, Xi. Tue .
"$ \mathcal{N} $ = 4 superconformal bootstrap of the K3 CFT". United States.
doi:10.1007/JHEP05(2017)126. https://www.osti.gov/servlets/purl/1389304.
```

```
@article{osti_1389304,
```

title = {$ \mathcal{N} $ = 4 superconformal bootstrap of the K3 CFT},

author = {Lin, Ying-Hsuan and Shao, Shu-Heng and Simmons-Duffin, David and Wang, Yifan and Yin, Xi},

abstractNote = {We study two-dimensional (4; 4) superconformal eld theories of central charge c = 6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap. This is made possible through a surprising relation between the BPS N = 4 superconformal blocks with c = 6 and bosonic Virasoro conformal blocks with c = 28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly from the CFT perspective the signature of a continuous spectrum above a gap at the singular moduli, and fi nd numerically an upper bound on this gap that is saturated by the A1 N = 4 cigar CFT. We also derive an analytic upper bound on the fi rst nonzero eigenvalue of the scalar Laplacian on K3 in the large volume regime, that depends on the K3 moduli data. As two byproducts, we find an exact equivalence between a class of BPS N = 2 superconformal blocks and Virasoro conformal blocks in two dimensions, and an upper bound on the four-point functions of operators of sufficiently low scaling dimension in three and four dimensional CFTs.},

doi = {10.1007/JHEP05(2017)126},

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

number = 5,

volume = 2017,

place = {United States},

year = {Tue May 23 00:00:00 EDT 2017},

month = {Tue May 23 00:00:00 EDT 2017}

}

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