The $(5,5)$ String

We study an $N=(2,1)$ heterotic string with right-movers on R2,2 and left-movers of signature $(5,5)$, obtained by adjoining the lattice II3,3. Unlike the standard $N=(2,1)$ strings it requires no null current, and its target space remains four-dimensional. All of its states, an infinite tower with sixteen supercharges, are massless. For the neutral bosons, the Kutasov-Martinec determinant action is dual to a cubic action for a self-dual metric coupled to self-dual Maxwell fields, and we complete it with fermions using a single superfield. For the charged states we give evidence that the couplings are the structure constants of a Lie superalgebra formed by the physical states of the left-movers. We are thus led to propose that the classical target theory is a self-dual gauge theory for this superalgebra: every state of the string is a gauge field, and gravity is part of the gauge algebra. This would be a purely cubic field theory of all the modes of a closed string.