Asymptotic One-Point Functions in Gauge-String Duality with Defects

Abstract

We take the first step in extending the integrability approach to one-point functions in AdS/dCFT to higher loop orders. More precisely, we argue that the formula encoding all tree-level one-point functions of SU(2) operators in the defect version of N=4 supersymmetric Yang-Mills theory, dual to the D5-D3 probe-brane system with flux, has a natural asymptotic generalization to higher loop orders. The asymptotic formula correctly encodes the information about the one-loop correction to the one-point functions of nonprotected operators once dressed by a simple flux-dependent factor, as we demonstrate by an explicit computation involving a novel object denoted as an amputated matrix product state. Furthermore, when applied to the Berenstein-Maldacena-Nastase vacuum state, the asymptotic formula gives a result for the one-point function which in a certain double-scaling limit agrees with that obtained in the dual string theory up to wrapping order.

Circular Wilson loops in defect $$ \mathcal{N} $$ = 4 SYM: phase transitions, double-scaling limits and OPE expansions

“…(4.20). We see that our result non-trivially matches with the one-point functions derived in [10][11][12] if…”

“…There are a number of different directions that can be explored in order to improve the present investigations. First of all one should compute independently the Wilson coefficient for the Konishi in the circular BPS Wilson loop OPE: this would represent a non-trivial check of the result obtained for one-point functions in [10][11][12] or would enlighten potential subtleties in our OPE description.…”

“…(4.20)) should be recovered from the onepoint functions of the Konishi operator K(x) and of the combination of chiral primaries Y a (θ)O a (x), from the one-loop anomalous dimension of K(x) and from the Wilson coefficient c 2 . Fortunately in a beautiful series of papers [10][11][12] the NBI group has studied the one-point functions of scalar operators in the defect theory, obtaining explicit result both at tree and one-loop level through perturbative computations and at all-order applying integrability techniques. We take advantage of their efforts and we adapt their results to our relevant operators (see Appendix F for the full details)…”

Spin chain overlaps and the twisted Yangian

“…A challenge is of course in the first place to take the present calculations beyond the scalar section of N = 4 SYM which is not closed at higher loop orders. An argument in favour of integrability of the defect systems at higher loop orders is that an exact expression for one-loop one-point functions of SU (2) sub-sector of the D3-D5 probe-brane set-up has been found and a possible asymptotic form of these presented [26].…”

“…The dCFT versions of N = 4 SYM theory with non-vanishing vevs and their dual string theory configurations. The discussion of the integrability properties of the corresponding matrix product states can be found in [13,14] and the closed expression for the overlap formula for the D3-D5 case appears in [9][10][11]13] for tree level and in [26] for one-loop, see also [27,28].…”