Abstract
Abstract We propose a correspondence between vertex operator superalgebras and families of sigma models in which the two structures are related by symmetry properties and a certain reflection procedure. The existence of such a correspondence is motivated by previous work on ${\cal N}=(4,4)$ supersymmetric non-linear sigma models on K3 surfaces, and on a vertex operator superalgebra with Conway group symmetry. Here we present an example of the correspondence for ${\cal N}=(4,4)$ supersymmetric non-linear sigma models on four-tori, and compare it to the K3 case.
Highlights
The relation between sporadic finite simple groups and symmetries of K3 surfaces and K3 sigma models has attracted a lot of attention since the pioneering work of [1] and [2]
Apart from the Mathieu groups featured in [1,2], symmetries of N = (4, 4) supersymmetric non-linear sigma models on K3 surfaces have been related to other groups, including the sporadic simple Conway groups [17,18,19], and the groups of umbral moonshine [20, 21]
Of special interest is the fact that many of the twined elliptic genera of sigma models on K3 surfaces can be reproduced by the vertex operator superalgebra (VOSA) V s, which has played a prominent role in Conway moonshine [19,22,23]. (Here and in the remainder of this work we use sigma model as a shorthand for N = (4, 4) supersymmetric non-linear sigma model.)
Summary
The relation between sporadic finite simple groups and symmetries of K3 surfaces and K3 sigma models has attracted a lot of attention since the pioneering work of [1] and [2]. In §C we explain how all but one of the twined K3 elliptic genera may be recovered from V s if we allow non-Conway group symmetries (which is to say symmetries that do not preserve supersymmetry), or Conway group symmetries that are not of the expected order This novel chiral/non-chiral connection between V s and K3 sigma models has been made precise at a special (orbifold) point in the moduli space, where V s can be retrieved as the image of the corresponding K3 theory under reflection: a procedure explored in [19] for the specific case of V s and later formerly investigated in more generality by Taormina–Wendland in [24]. We review the relationship between V f [22] and V s [19, 23], explain a sense in which the Conway group arises naturally as a group of automorphisms of V s , and explain why they are the same as far as twinings of the K3 elliptic genus are concerned
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