Search for a holographic dual to AdS₃ × S³ × S³ × S¹

The problem of finding a holographic dual to string theory on AdS₃ × S³ × S³ × S¹ is examined in depth. This background supports a large N = 4 superconformal symmetry. While in some respects similar to the familiar small N = 4 systems on AdS₃ × S³ × K3 and AdS₃ × S³ × T ⁴, there are important qualitative differences. Using an analogue of the elliptic genus for large N = 4 theories we rule out all extant proposals - in their simplest form - for a holographic duality to supergravity at generic values of the background fluxes. Modifications of these extant proposals and other possible duals are discussed.