Operator product expansion for conformal defects

We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.

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“…where k ≤ [d/2]. The operator (22) is required to satisfy the following scaling and transversality relations…”

Section: Tensor Structures For So(d + 1 1) Mixed Symmetry Representamentioning

confidence: 99%

Spinning operators and defects in conformal field theory

Lauria

Meineri

Trevisani

2019

J. High Energ. Phys.

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“…In order to explore the features of these new functions, understand their analytical properties or find useful expansions one could try to follow the same route that was used for four-point blocks, see e.g. [29,30] for some recent work in this direction. It is the central message of this paper, however, that there is another route that gives a much more direct access to defect blocks.…”

Section: Contentsmentioning

confidence: 99%

Calogero-Sutherland approach to defect blocks

Isachenkov

Liendo

Linke

et al. 2018

J. High Energ. Phys.

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Extended objects such as line or surface operators, interfaces or boundaries play an important role in conformal field theory. Here we propose a systematic approach to the relevant conformal blocks which are argued to coincide with the wave functions of an integrable multi-particle Calogero-Sutherland problem. This generalizes a recent observation in [1] and makes extensive mathematical results from the modern theory of multi-variable hypergeometric functions available for studies of conformal defects. Applications range from several new relations with scalar four-point blocks to a Euclidean inversion formula for defect correlators.

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“…Our result based on the analytic continuation is different from the proposal. Motivated by this discrepancy, we present another derivation of the time-like OPE block using a kinematical duality exchanging a pair of time-like separated operators with a space-like codimension-two defect [60,61]. While we are not able to address this issue in full generality, we show that the duality method reduces the time-like configuration to a space-like one, ending up with the surface Witten diagram in two dimensions.…”

Section: Contentsmentioning

confidence: 95%

“…The time-like OPE block was also considered in a different context [58,59] and proposed to have another holographic description known as the surface Witten diagram, which is quite different from our result in section 5.1 in general dimensions. In order to make contact with the surface Witten diagram, we provide an alternative derivation based on the duality between conformal defects [60,61] in section 5.2. Focusing on two-dimensional CFT's, the defect duality exchanges a pair of time-like separated points to a pair of space-like ones.…”

Section: Time-like Ope Blockmentioning

confidence: 99%

“…The key to the derivation of the time-like OPE block in the previous subsection was the analytic continuation of the three-point function with respect to x 2 12 from space-like separation to time-like separation. Here we derive another representation of the time-like OPE block by rewriting the three-point function using the duality between a pair of time-like separated points and a codimension-two spacial defect [60,61]. As a disclaimer, in this subsection we will not be careful about the irrelevant overall normalization constants, as we are only interested in the kinematical structure to compare with a proposed form of the time-like OPE block by [58,59].…”

Section: Dual Defect Picture and Surface Witten Diagram In Two Dimensmentioning

confidence: 99%

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The gravity dual of Lorentzian OPE blocks

Chen

Kobayashi

et al. 2020

J. High Energ. Phys.

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We consider the operator product expansion (OPE) structure of scalar primary operators in a generic Lorentzian CFT and its dual description in a gravitational theory with one extra dimension. The OPE can be decomposed into certain bi-local operators transforming as the irreducible representations under conformal group, called the OPE blocks. We show the OPE block is given by integrating a higher spin field along a geodesic in the Lorentzian AdS space-time when the two operators are space-like separated. When the two operators are time-like separated however, we find the OPE block has a peculiar representation where the dual gravitational theory is not defined on the AdS space-time but on a hyperboloid with an additional time coordinate and Minkowski space-time on its boundary. This differs from the surface Witten diagram proposal for the time-like OPE block, but in two dimensions we reproduce it consistently using a kinematical duality between a pair of time-like separated points and space-like ones.

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Operator product expansion for conformal defects

Cited by 31 publications

References 65 publications

Spinning operators and defects in conformal field theory

Spinning operators and defects in conformal field theory

Calogero-Sutherland approach to defect blocks

The gravity dual of Lorentzian OPE blocks

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