Abstract
Conformal field theories with (0,4) worldsheet supersymmetry and K3 target can be used to compactify the E8xE8 heterotic string to six dimensions in a supersymmetric manner. The data specifying such a model includes an appropriate configuration of 24 gauge instantons in the E8xE8 gauge group to satisfy the constraints of anomaly cancellation. In this note, we compute twining genera - elliptic genera with appropriate insertions of discrete symmetry generators in the trace - for (0,4) theories with various instanton embeddings. We do this by constructing linear sigma models which flow to the desired conformal field theories, and using the techniques of localization. We present several examples of such twining genera which are consistent with a moonshine relating these (0,4) models to the finite simple sporadic group M24.
Highlights
Vanishing first Chern class c1 = 0, in the simplest case), the left-moving fermions in the sigma model couple to the gauge connections on these bundles, and cL = 4 + r1 + r2
We used the recently derived localization formula for the elliptic genus of (0, 2) supersymmetric rank one two-dimensional gauge theories [3] to compute twining genera of (0,4) gauged linear sigma models with K3 target. We did this for a variety of discrete symmetries in (0, 4) models with four different sets of instanton numbers (n(1), n(2))
We found that the simple discrete symmetries give twining genera which are consistent with those of M24 elements of the same order, with the trace in the elliptic genus taken over the M24 module conjectured to exist in [6] and constructed in [11]
Summary
We will write down (0,4) linear sigma models by working in (0,2) superspace and using vector bundles constructed as the cohomology of an exact sequence, as in [2]. We will be considering (0,2) gauge theories with U(1) gauge group, so we need to discuss the (0,2) gauge multiplet. It consists of a pair of superfields V, A whose expansion, in Wess-Zumino gauge, is given by. A gauge invariant kinetic term for a charged chiral multiplet Φ with charge Q is SΦ = d2z(∂z − Qa)φ(∂z + Qa)φ + (∂z − Qa)φ(∂z + Qa)φ +2ψ(∂z + Qa)ψ + Q(αψφ − αφφ) − QDφφ ,. While a gauge invariant kinetic term for a charged Fermi multiplet Λ of charge Q is. D2z dθ−ΛF (Φ) + h.c. Here, F needs to be chosen to be a homogeneous polynomial of the appropriate degree in the charged field Φ so that (2.12) is gauge invariant
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