# Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine

## Abstract

We demonstrate that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi-Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skew-holomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This proposes a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.

- Authors:

- Univ. of Amsterdam, Amsterdam (Netherlands)
- Emory Univ., Atlanta, GA (United States)
- Harvard Univ., Cambridge, MA (United States); McGill Univ., Montreal, QC (Canada)
- Univ. of Chicago, Chicago, IL (United States)
- Stanford Univ., Palo Alto, CA (United States). Stanford Inst. for Theoretical Physics

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1505229

- Grant/Contract Number:
- SC0007870

- 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: 7; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; Black Holes in String Theory; Discrete Symmetries; M-Theory; Superstrings and Heterotic Strings

### Citation Formats

```
Cheng, Miranda C. N., Duncan, John F. R., Harrison, Sarah M., Harvey, Jeffrey A., Kachru, Shamit, and Rayhaun, Brandon C. Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine. United States: N. p., 2018.
Web. doi:10.1007/jhep07(2018)130.
```

```
Cheng, Miranda C. N., Duncan, John F. R., Harrison, Sarah M., Harvey, Jeffrey A., Kachru, Shamit, & Rayhaun, Brandon C. Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine. United States. doi:10.1007/jhep07(2018)130.
```

```
Cheng, Miranda C. N., Duncan, John F. R., Harrison, Sarah M., Harvey, Jeffrey A., Kachru, Shamit, and Rayhaun, Brandon C. Thu .
"Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine". United States. doi:10.1007/jhep07(2018)130. https://www.osti.gov/servlets/purl/1505229.
```

```
@article{osti_1505229,
```

title = {Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine},

author = {Cheng, Miranda C. N. and Duncan, John F. R. and Harrison, Sarah M. and Harvey, Jeffrey A. and Kachru, Shamit and Rayhaun, Brandon C.},

abstractNote = {We demonstrate that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi-Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skew-holomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This proposes a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.},

doi = {10.1007/jhep07(2018)130},

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

number = 7,

volume = 2018,

place = {United States},

year = {2018},

month = {7}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Heegner cycles, modular forms and jacobi forms

journal, January 1991

- Skoruppa, Nils-Peter
- Journal de Théorie des Nombres de Bordeaux, Vol. 3, Issue 1

##
Naturality in conformal field theory

journal, January 1989

- Moore, Gregory; Seiberg, Nathan
- Nuclear Physics B, Vol. 313, Issue 1

##
Calabi–Yau Black Holes and (0,4) Sigma Models

journal, February 2000

- Minasian, Ruben; Moore, Gregory; Tsimpis, Dimitrios
- Communications in Mathematical Physics, Vol. 209, Issue 2

##
A strong coupling test of S-duality

journal, December 1994

- Vafa, Cumrun; Witten, Edward
- Nuclear Physics B, Vol. 431, Issue 1-2

##
Jacobi forms and a certain space of modular forms

journal, February 1988

- Skoruppa, Nils-Peter; Zagier, Don
- Inventiones Mathematicae, Vol. 94, Issue 1

##
The Betti numbers of the Hilbert scheme of points on a smooth projective surface

journal, March 1990

- Göttsche, Lothar
- Mathematische Annalen, Vol. 286, Issue 1-3

##
Finite simple groups which projectively embed in an exceptional Lie group are classified!

journal, January 1999

- Griess Jr., Robert L.; Ryba Jr., A. J. E.
- Bulletin of the American Mathematical Society, Vol. 36, Issue 01

##
Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes

journal, August 2015

- Alim, Murad; Haghighat, Babak; Hecht, Michael
- Communications in Mathematical Physics, Vol. 339, Issue 3

##
Split states, entropy enigmas, holes and halos

journal, November 2011

- Denef, Frederik; Moore, Gregory W.
- Journal of High Energy Physics, Vol. 2011, Issue 11

##
Notes on the K3 Surface and the Mathieu Group *M* _{24}

journal, March 2011

- Eguchi, Tohru; Ooguri, Hirosi; Tachikawa, Yuji
- Experimental Mathematics, Vol. 20, Issue 1

##
Umbral moonshine and the Niemeier lattices

journal, June 2014

- Cheng, Miranda CN; Duncan, John FR; Harvey, Jeffrey A.
- Research in the Mathematical Sciences, Vol. 1, Issue 1

##
Integral Symmetric Bilinear Forms and some of Their Applications

journal, February 1980

- Nikulin, V. V.
- Mathematics of the USSR-Izvestiya, Vol. 14, Issue 1

##
E-strings and N = 4 topological Yang-Mills theories

journal, September 1998

- Minahan, J. A.; Nemeschansky, D.; Vafa, C.
- Nuclear Physics B, Vol. 527, Issue 3

##
Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms

journal, December 1990

- Skoruppa, Nils-Peter
- Inventiones Mathematicae, Vol. 102, Issue 1

##
Nonrenormalization of the superstring tension

journal, July 1989

- Dabholkar, Atish; Harvey, Jeffrey A.
- Physical Review Letters, Vol. 63, Issue 5

##
The M5-brane elliptic genus: modularity and BPS states

journal, August 2007

- Gaiotto, Davide; Strominger, Andrew; Yin, Xi
- Journal of High Energy Physics, Vol. 2007, Issue 08

##
The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface

journal, May 1995

- Yoshioka, K?ta
- Mathematische Annalen, Vol. 302, Issue 1

##
Weight one Jacobi forms and umbral moonshine

journal, February 2018

- Cheng, Miranda C. N.; Duncan, John F. R.; Harvey, Jeffrey A.
- Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 10

##
Traces of singular moduli and moonshine for the Thompson group

journal, January 2016

- Harvey, Jeffrey A.; Rayhaun, Brandon C.
- Communications in Number Theory and Physics, Vol. 10, Issue 1

##
A modern fareytail

journal, January 2010

- Manschot, Jan; Moore, Gregory W.
- Communications in Number Theory and Physics, Vol. 4, Issue 1

##
Black holes and Hurwitz class numbers

journal, October 2017

- Kachru, Shamit; Tripathy, Arnav
- International Journal of Modern Physics D, Vol. 26, Issue 12