Abstract
We present a systematic construction of bulk solutions that are dual to CFT excited states. The bulk solution is constructed perturbatively in bulk fields. The linearised solution is universal and depends only on the conformal dimension of the primary operator that is associated with the state via the operator-state correspondence, while higher order terms depend on detailed properties of the operator, such as its OPE with itself and generally involve many bulk fields. We illustrate the discussion with the holographic construction of the universal part of the solution for states of two dimensional CFTs, either on $R \times S^1$ or on $R^{1,1}$. We compute the 1-point function both in the CFT and in the bulk, finding exact agreement. We comment on the relation with other reconstruction approaches.
Highlights
Scalar field in a fixed AdS background
The linearised solution is universal and depends only on the conformal dimension of the primary operator that is associated with the state via the operator-state correspondence, while higher order terms depend on detailed properties of the operator, such as its OPE with itself and generally involve many bulk fields
This implies that if we want to work out the linearised bulk solution dual to the state |∆, it suffices to only consider the bulk field that is dual to the operator O∆ in a fixed AdS background
Summary
We setup the problem using the Schwinger-Keldysh formalism. Let us denote by φ(0) the source that couples to O∆. If we compute this path integral for general φ(0)(x) and differentiate w.r.t. φ+(0) and φ−(0), where φ±(0) = φ(0)(0±, 0) and 0± = 0 ± i , and set to zero the sources in the imaginary part of the contour, the resulting expression will be the desired generating functional of in-in correlators in the state |∆. Since 2-point functions in CFT are diagonal the only operator that has a non-zero 1-point function is precisely the operator associated with the excited state This implies that if we want to work out the linearised bulk solution dual to the state |∆ , it suffices to only consider the bulk field that is dual to the operator O∆ in a fixed AdS background.. Where C and Care the normalisations of the 2-point functions in the two cases. The bulk solution dual to this state in global AdS should reproduce (2.6) while the bulk solution in Poincare AdS should yield (2.7)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
