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Title: A Borcherds–Kac–Moody Superalgebra with Conway Symmetry

Abstract

We construct here a Borcherds–Kac–Moody (BKM) superalgebra on which the Conway group \(\hbox {Co}_0\) acts faithfully. We show that the BKM algebra is generated by the physical states (BRST cohomology classes) in a chiral superstring theory. Furthermore, we use this construction to produce denominator identities for the chiral partition functions of the Conway module \(V^{s\natural }\), a supersymmetric \(c=12\) chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to \(\hbox {Co}_0\). In particular, these functions satisfy a genus zero property analogous to that of monstrous moonshine. Finally, we suggest how one may promote the denominators to spacetime BPS indices in type II string theory, which might thus furnish a physical explanation of the genus zero property of Conway moonshine.

Authors:
 [1]; ORCiD logo [2];  [3]
  1. McGill Univ., Montreal, QC (Canada)
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States)
  3. Univ. degli Studi di Padova (Italy); Istituto Nazionale di Fisica Nucleare (INFN), Padua (Italy)
Publication Date:
Research Org.:
California Inst. of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science and Engineering Council of Canada; National Science Foundation (NSF)
Contributing Org.:
[Perimeter Institute of Theoretical Physics]
OSTI Identifier:
1597492
Grant/Contract Number:  
[SC0011632; PHY-1607611]
Resource Type:
Accepted Manuscript
Journal Name:
Communications in Mathematical Physics
Additional Journal Information:
[ Journal Volume: 370; Journal Issue: 2]; Journal ID: ISSN 0010-3616
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Harrison, Sarah M., Paquette, Natalie M., and Volpato, Roberto. A Borcherds–Kac–Moody Superalgebra with Conway Symmetry. United States: N. p., 2019. Web. doi:10.1007/s00220-019-03518-0.
Harrison, Sarah M., Paquette, Natalie M., & Volpato, Roberto. A Borcherds–Kac–Moody Superalgebra with Conway Symmetry. United States. doi:10.1007/s00220-019-03518-0.
Harrison, Sarah M., Paquette, Natalie M., and Volpato, Roberto. Tue . "A Borcherds–Kac–Moody Superalgebra with Conway Symmetry". United States. doi:10.1007/s00220-019-03518-0.
@article{osti_1597492,
title = {A Borcherds–Kac–Moody Superalgebra with Conway Symmetry},
author = {Harrison, Sarah M. and Paquette, Natalie M. and Volpato, Roberto},
abstractNote = {We construct here a Borcherds–Kac–Moody (BKM) superalgebra on which the Conway group \(\hbox {Co}_0\) acts faithfully. We show that the BKM algebra is generated by the physical states (BRST cohomology classes) in a chiral superstring theory. Furthermore, we use this construction to produce denominator identities for the chiral partition functions of the Conway module \(V^{s\natural }\), a supersymmetric \(c=12\) chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to \(\hbox {Co}_0\). In particular, these functions satisfy a genus zero property analogous to that of monstrous moonshine. Finally, we suggest how one may promote the denominators to spacetime BPS indices in type II string theory, which might thus furnish a physical explanation of the genus zero property of Conway moonshine.},
doi = {10.1007/s00220-019-03518-0},
journal = {Communications in Mathematical Physics},
number = [2],
volume = [370],
place = {United States},
year = {2019},
month = {7}
}

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Works referenced in this record:

The monster Lie algebra
journal, September 1990


Generalized moonshine, II: Borcherds products
journal, April 2012


Algebras, BPS states, and strings
journal, March 1996


Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants
journal, January 2011


Counting dyons in N = 4 string theory
journal, January 1997


Generalized Kac–Moody algebras, automorphic forms and Conway's group I
journal, April 2004


Automorphic forms with singularities on Grassmannians
journal, May 1998


Second-quantized Mathieu moonshine
journal, January 2014


No more walls! A tale of modularity, symmetry, and wall crossing for 1/4 BPS dyons
journal, May 2017

  • Paquette, Natalie M.; Volpato, Roberto; Zimet, Max
  • Journal of High Energy Physics, Vol. 2017, Issue 5
  • DOI: 10.1007/JHEP05(2017)047

Conway subgroup symmetric compactifications of heterotic string
journal, July 2018

  • Harvey, Jeffrey A.; Moore, Gregory W.
  • Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 35
  • DOI: 10.1088/1751-8121/aac9d1

Automorphic forms onO s +2,2(R) and infinite products
journal, December 1995

  • Borcherds, Richard E.
  • Inventiones Mathematicae, Vol. 120, Issue 1
  • DOI: 10.1007/BF01241126

BRST cohomology of the super-Virasoro Algebras
journal, June 1989

  • Lian, Bong H.; Zuckerman, Gregg J.
  • Communications In Mathematical Physics, Vol. 125, Issue 2
  • DOI: 10.1007/BF01217910

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
  • DOI: 10.1186/2197-9847-1-3

Super-moonshine for Conway's largest sporadic group
journal, August 2007


On the Algebras of BPS States
journal, October 1998

  • Harvey, Jeffrey A.; Moore, Gregory
  • Communications in Mathematical Physics, Vol. 197, Issue 3
  • DOI: 10.1007/s002200050461

Vertex Algebras, Lie Algebras, and Superstrings
journal, February 1998


Derived equivalences of K3 surfaces and twined elliptic genera
journal, February 2016

  • Duncan, John F. R.; Mack-Crane, Sander
  • Research in the Mathematical Sciences, Vol. 3, Issue 1
  • DOI: 10.1186/s40687-015-0050-9

Symmetries of K3 sigma models
journal, January 2012

  • Gaberdiel, Matthias R.; Hohenegger, Stefan; Volpato, Roberto
  • Communications in Number Theory and Physics, Vol. 6, Issue 1
  • DOI: 10.4310/CNTP.2012.v6.n1.a1

The Fake Monster Superalgebra
journal, May 2000


Umbral moonshine
journal, January 2014

  • Cheng, Miranda C. N.; Duncan, John F. R.; Harvey, Jeffrey A.
  • Communications in Number Theory and Physics, Vol. 8, Issue 2
  • DOI: 10.4310/CNTP.2014.v8.n2.a1

3D string theory and Umbral moonshine
journal, September 2017

  • Kachru, Shamit; Paquette, Natalie M.; Volpato, Roberto
  • Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 40
  • DOI: 10.1088/1751-8121/aa6e07

K3 surfaces, $\mathcal{N}=4$ dyons and the Mathieu group $M_{24}$
journal, January 2010


Dyon spectrum in generic Script N = 4 supersymmetric Bbb Z N orbifolds
journal, January 2007


The Moonshine Module for Conway’S Group
journal, June 2015

  • Duncan, John F. R.; Mack-Crane, Sander
  • Forum of Mathematics, Sigma, Vol. 3
  • DOI: 10.1017/fms.2015.7

Equivariant K3 invariants
journal, January 2017

  • Cheng, Miranda C. N.; Duncan, John F. R.; Harrison, Sarah M.
  • Communications in Number Theory and Physics, Vol. 11, Issue 1
  • DOI: 10.4310/CNTP.2017.v11.n1.a2

Super vertex algebras, meromorphic Jacobi forms and umbral moonshine
journal, December 2018


Fricke S-duality in CHL models
journal, December 2015

  • Persson, Daniel; Volpato, Roberto
  • Journal of High Energy Physics, Vol. 2015, Issue 12
  • DOI: 10.1007/JHEP12(2015)156

Rademacher sums, moonshine and gravity
journal, January 2011

  • Duncan, John F. R.; Frenkel, Igor B.
  • Communications in Number Theory and Physics, Vol. 5, Issue 4
  • DOI: 10.4310/CNTP.2011.v5.n4.a4

Umbral Moonshine and K3 Surfaces
journal, June 2015

  • Cheng, Miranda C. N.; Harrison, Sarah
  • Communications in Mathematical Physics, Vol. 339, Issue 1
  • DOI: 10.1007/s00220-015-2398-5

The umbral moonshine module for the unique unimodular Niemeier root system
journal, January 2017


Self-dual vertex operator superalgebras and superconformal field theory
journal, December 2017

  • Creutzig, Thomas; Duncan, John F. R.; Riedler, Wolfgang
  • Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 3
  • DOI: 10.1088/1751-8121/aa9af5

Generalized Kac-Moody algebras
journal, June 1988


K3 string theory, lattices and moonshine
journal, July 2018

  • Cheng, Miranda C. N.; Harrison, Sarah M.; Volpato, Roberto
  • Research in the Mathematical Sciences, Vol. 5, Issue 3
  • DOI: 10.1007/s40687-018-0150-4

Monstrous BPS-algebras and the superstring origin of moonshine
journal, January 2016

  • Paquette, Natalie M.; Persson, Daniel; Volpato, Roberto
  • Communications in Number Theory and Physics, Vol. 10, Issue 3
  • DOI: 10.4310/CNTP.2016.v10.n3.a2

Twisting the Fake Monster Superalgebra
journal, December 2001


K3 Elliptic Genus and an Umbral Moonshine Module
journal, March 2019

  • Anagiannis, Vassilis; Cheng, Miranda C. N.; Harrison, Sarah M.
  • Communications in Mathematical Physics, Vol. 366, Issue 2
  • DOI: 10.1007/s00220-019-03314-w

Some Numerology between the Fischer-Griess Monster and the Elliptic Modular Function
journal, October 1979

  • Thompson, J. G.
  • Bulletin of the London Mathematical Society, Vol. 11, Issue 3
  • DOI: 10.1112/blms/11.3.352

Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra
journal, April 1998


Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model
journal, June 1972


Notes on the K3 Surface and the Mathieu Group M 24
journal, March 2011


Mock modular Mathieu moonshine modules
journal, July 2015

  • Cheng, Miranda C. N.; Dong, Xi; Duncan, John F. R.
  • Research in the Mathematical Sciences, Vol. 2, Issue 1
  • DOI: 10.1186/s40687-015-0034-9

Heterotic sigma models on T 8 and the Borcherds automorphic form Φ12
journal, October 2017

  • Harrison, Sarah M.; Kachru, Shamit; Paquette, Natalie M.
  • Journal of High Energy Physics, Vol. 2017, Issue 10
  • DOI: 10.1007/JHEP10(2017)121

Modular equations and the genus zero property of moonshine functions
journal, August 1997


Lie algebra homology and the Macdonald-Kac formulas
journal, February 1976

  • Garland, Howard; Lepowsky, James
  • Inventiones Mathematicae, Vol. 34, Issue 1
  • DOI: 10.1007/BF01418970

BPS algebras, genus zero and the heterotic Monster
journal, September 2017

  • Paquette, Natalie M.; Persson, Daniel; Volpato, Roberto
  • Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 41
  • DOI: 10.1088/1751-8121/aa8443

On Rademacher sums, the largest Mathieu group and the holographic modularity of moonshine
journal, January 2012

  • Cheng, Miranda C. N.; Duncan, John F. R.
  • Communications in Number Theory and Physics, Vol. 6, Issue 3
  • DOI: 10.4310/CNTP.2012.v6.n3.a4

Monstrous Moonshine
journal, October 1979

  • Conway, J. H.; Norton, S. P.
  • Bulletin of the London Mathematical Society, Vol. 11, Issue 3
  • DOI: 10.1112/blms/11.3.308

Quiver W-algebras
journal, March 2018


Monstrous moonshine and monstrous Lie superalgebras
journal, December 1992

  • Borcherds, Richard E.
  • Inventiones Mathematicae, Vol. 109, Issue 1
  • DOI: 10.1007/BF01232032

A Characterization of Generalized Kac-Moody Algebras
journal, June 1995


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
  • DOI: 10.1088/1751-8121/aaa819

Finite Groups and Modular Functions
journal, October 1979

  • Thompson, J. G.
  • Bulletin of the London Mathematical Society, Vol. 11, Issue 3
  • DOI: 10.1112/blms/11.3.347