## Surface defect indices and 2d-4d BPS states

## Abstract

We conjecture a formula for the Schur index of four-dimensional *N*=2 theories coupled to (2, 2) surface defects in terms of the 2d-4d BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a refined 2d-4d wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the four-dimensional Schur index in terms of infrared BPS particles, with the Cecotti-Vafa formula for limits of the elliptic genus in terms of two-dimensional BPS solitons. We extend our discussion to framed 2d-4d BPS states, and use this to demonstrate a general relationship between surface defect indices and line defect indices. We illustrate our results in the example of *su*(2) super Yang-Mills coupled to the CP ^{1} sigma model defect.

- Authors:

- Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
- Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)

- Publication Date:

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

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

- OSTI Identifier:
- 1502495

- 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: 2017; Journal Issue: 12; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal and W Symmetry; Supersymmetric Gauge Theory

### Citation Formats

```
Córdova, Clay, Gaiotto, Davide, and Shao, Shu-Heng. Surface defect indices and 2d-4d BPS states. United States: N. p., 2017.
Web. doi:10.1007/jhep12(2017)078.
```

```
Córdova, Clay, Gaiotto, Davide, & Shao, Shu-Heng. Surface defect indices and 2d-4d BPS states. United States. doi:10.1007/jhep12(2017)078.
```

```
Córdova, Clay, Gaiotto, Davide, and Shao, Shu-Heng. Thu .
"Surface defect indices and 2d-4d BPS states". United States. doi:10.1007/jhep12(2017)078. https://www.osti.gov/servlets/purl/1502495.
```

```
@article{osti_1502495,
```

title = {Surface defect indices and 2d-4d BPS states},

author = {Córdova, Clay and Gaiotto, Davide and Shao, Shu-Heng},

abstractNote = {We conjecture a formula for the Schur index of four-dimensional N=2 theories coupled to (2, 2) surface defects in terms of the 2d-4d BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a refined 2d-4d wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the four-dimensional Schur index in terms of infrared BPS particles, with the Cecotti-Vafa formula for limits of the elliptic genus in terms of two-dimensional BPS solitons. We extend our discussion to framed 2d-4d BPS states, and use this to demonstrate a general relationship between surface defect indices and line defect indices. We illustrate our results in the example of su(2) super Yang-Mills coupled to the CP1 sigma model defect.},

doi = {10.1007/jhep12(2017)078},

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

number = 12,

volume = 2017,

place = {United States},

year = {2017},

month = {12}

}

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