BMN Correspondence at Finite J/N

Employing the string bit formalism of hep-th/0209215, we identify the basis transformation that relates BMN operators in N = 4 gauge theory to string states in the dual string field theory at finite g2 = J 2/N . In this basis, the supercharge truncates at linear order in g2, and the mixing amplitude between 1 and 2-string states precisely matches with the (corrected) answer of hep-th/0206073 for the 3-string amplitude in light-cone string field theory. Supersymmetry then predicts the order g2 2 contact term in the string bit Hamiltonian. The resulting leading order mass renormalization of string states agrees with the recently computed shift in conformal dimension of BMN operators in the gauge theory. Introduction and Philosophy The BMN correspondence [1] equates type IIB string theory on a plane wave background with a certain limit of N = 4 gauge theory at large R-charge J , where N is taken to infinity while the quantities λ = g2 YMN J2 , g2 = J2 N (1) are held fixed. The proposal is based on a natural identification between the basis of string theory states and the basis of gauge theory operators, and between the light-cone string Hamiltonian P and the generator ∆ of conformal transformation in the gauge theory via1 2 μ P = ∆− J . (2) BMN argued, and it was subsequently confirmed to all orders in λ [2, 3], that this identification holds at the level of free string theory (g2=0). This beautiful proposal equates two operators which act on completely different spaces: the light-cone Hamiltonian P acts on the Hilbert space of string field theory, and allows for the splitting and joining of strings, while H ≡ ∆ − J acts on the operators of the field theory, and in general mixes single-trace operators with doubleand higher-trace operators. Light-cone string field theory in the plane wave background has been constructed in [4, 5]. On the field theory side, a number of impressive papers [7–14] have pushed the calculations to higher order in g2 with the aim of showing that (2) continues to hold, thereby providing an equality between a perturbative, interacting string theory and perturbative N = 4 gauge theory. It is clear, however, that at finite g2 the natural identification between single string states and single trace operators breaks down. For example, 1-string states are orthogonal to 2-string states for all g2, but single-trace operators and double-trace operators are not. This raises the question how to formulate the BMN correspondence in the interacting string theory. In order to prove that two operators in (2) are equal, it is sufficient to prove that they have the same eigenvalues. If they do, then there is guaranteed to exist a unitary transformation between the spaces on which the two operators act. A basis independent formulation of the BMN correspondence, therefore, is that the interacting string field theory Hamiltonian 2 μP − and the gauge theory operator H must have the same eigenvalues. While this is the minimum that we are allowed to expect from the BMN correspondence, we can hope to do better. Light-cone string field theory, as formulated in [4–6], comes with a natural choice of basis: this string basis (of single and multiple strings) is neither the BMN basis (of single and multiple traces) nor the basis of eigenstates of the light-cone Hamiltonian. But how do we identify the string basis in the gauge theory? The parameter μ can be introduced by performing a boost and serves merely as a bookkeeping device.