Equivariant K3 invariants
In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformal field theory. We consider the invariants calculated by Yau–Zaslow (capturing the Euler characters of the moduli spaces of D2branes on curves of given genus), together with their refinements to carry additional quantum numbers by Katz–Klemm–Vafa (KKV), and Katz–Klemm–Pandharipande (KKP). We show that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently been observed to give rise to mock modular moonshine for a variety of sporadic simple groups that are subgroups of Conway’s group. We also study equivariant versions of these invariants. A K3 sigma model is specified by a choice of 4plane in the K3 Dbrane charge lattice. Symmetries of K3 sigma models are naturally identified with 4plane preserving subgroups of the Conway group, according to the work of Gaberdiel–Hohenegger–Volpato, and one may consider corresponding equivariant refined K3 Gopakumar–Vafa invariants. The same symmetries naturally arise in the auxiliary CFT state space, affording a suggestive alternative view of the same computation. We comment on a lift of this story to the generating function of ellipticmore »
 Authors:

^{[1]};
^{[2]};
^{[3]};
^{[4]}
 Univ. of Amsterdam (Netherlands); Centre National de la Recherche Scientifique (CNRS), Paris (France)
 Emory Univ., Atlanta, GA (United States)
 Harvard Univ., Cambridge, MA (United States)
 Stanford Univ., CA (United States)
 Publication Date:
 Grant/Contract Number:
 AC0276SF00515
 Type:
 Accepted Manuscript
 Journal Name:
 Communications in Number Theory and Physics
 Additional Journal Information:
 Journal Volume: 11; Journal Issue: 1; Journal ID: ISSN 19314523
 Publisher:
 International Press
 Research Org:
 SLAC National Accelerator Lab., Menlo Park, CA (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
 OSTI Identifier:
 1378561
Cheng, Miranda C. N., Duncan, John F. R., Harrison, Sarah M., and Kachru, Shamit. Equivariant K3 invariants. United States: N. p.,
Web. doi:10.4310/CNTP.2017.v11.n1.a2.
Cheng, Miranda C. N., Duncan, John F. R., Harrison, Sarah M., & Kachru, Shamit. Equivariant K3 invariants. United States. doi:10.4310/CNTP.2017.v11.n1.a2.
Cheng, Miranda C. N., Duncan, John F. R., Harrison, Sarah M., and Kachru, Shamit. 2017.
"Equivariant K3 invariants". United States.
doi:10.4310/CNTP.2017.v11.n1.a2. https://www.osti.gov/servlets/purl/1378561.
@article{osti_1378561,
title = {Equivariant K3 invariants},
author = {Cheng, Miranda C. N. and Duncan, John F. R. and Harrison, Sarah M. and Kachru, Shamit},
abstractNote = {In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformal field theory. We consider the invariants calculated by Yau–Zaslow (capturing the Euler characters of the moduli spaces of D2branes on curves of given genus), together with their refinements to carry additional quantum numbers by Katz–Klemm–Vafa (KKV), and Katz–Klemm–Pandharipande (KKP). We show that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently been observed to give rise to mock modular moonshine for a variety of sporadic simple groups that are subgroups of Conway’s group. We also study equivariant versions of these invariants. A K3 sigma model is specified by a choice of 4plane in the K3 Dbrane charge lattice. Symmetries of K3 sigma models are naturally identified with 4plane preserving subgroups of the Conway group, according to the work of Gaberdiel–Hohenegger–Volpato, and one may consider corresponding equivariant refined K3 Gopakumar–Vafa invariants. The same symmetries naturally arise in the auxiliary CFT state space, affording a suggestive alternative view of the same computation. We comment on a lift of this story to the generating function of elliptic genera of symmetric products of K3 surfaces.},
doi = {10.4310/CNTP.2017.v11.n1.a2},
journal = {Communications in Number Theory and Physics},
number = 1,
volume = 11,
place = {United States},
year = {2017},
month = {1}
}