Abstract
Warped conformal field theory (WCFT) is a two dimensional quantum field theory whose local symmetry algebra consists of a Virasoro algebra and a U(1) Kac-Moody algebra. In this paper, we study correlation functions for primary operators in WCFT. Similar to conformal symmetry, warped conformal symmetry is very constraining. The form of the two and three point functions are determined by the global warped conformal symmetry while the four point functions can be determined up to an arbitrary function of the cross ratio. The warped conformal bootstrap equation are constructed by formulating the notion of crossing symmetry. In the large central charge limit, four point functions can be decomposed into global warped conformal blocks, which can be solved exactly. Furthermore, we revisit the scattering problem in warped AdS spacetime (WAdS), and give a prescription on how to match the bulk result to a WCFT retarded Green’s function. Our result is consistent with the conjectured holographic dualities between WCFT and WAdS.
Highlights
The notion of crossing symmetry as in [3]
Warped conformal field theory (WCFT) is a two dimensional quantum field theory whose local symmetry algebra consists of a Virasoro algebra and a U(1) Kac-Moody algebra
The form of the two and three point functions are determined by the global warped conformal symmetry while the four point functions can be determined up to an arbitrary function of the cross ratio
Summary
The global symmetries are SL(2, R) × U(1), while the local symmetry algebra is a Virasoro algebra plus a U(1) Kac-Moody algebra. A general warped conformal symmetry transformation can be written as x = f (x), y = y + g(x) ,. The warped conformal symmetry are generated by a set of vector fields, Ln = −xn+1∂x, Pn = −ixn∂y. Global symmetry transformations are generated by L±1, L0 and P0, which form a SL(2, R)⊗ U(1) sub-algebra. The commutation relations for the charges form a warped conformal algebra consists of one Virasoro algebra and a Kac-Moody algebra,. The finite transformation properties of the energy momentum tensor and Kac-Moody current are given by,. One can check that Linnv commutate with the Kac-Moody generators, and form a Virasoro algebra with conformal weight c − 1
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