A crossing-symmetric OPE inversion formula

Self Cite

155

We clarify the relationships between different approaches to the conformal bootstrap. A central role is played by the so-called extremal functionals. They are linear functionals acting on the crossing equation which are directly responsible for the optimal bounds of the numerical bootstrap. We explain in detail that the extremal functionals probe the Regge limit. We construct two complete sets of extremal functionals for the crossing equation specialized to z =z, associated to the generalized free boson and fermion theories. These functionals lead to non-perturbative sum rules on the CFT data which automatically incorporate Regge boundedness of physical correlators. The sum rules imply universal properties of the OPE at large ∆ in every unitary solution of SL(2) crossing. In particular, we prove an upper and lower bound on a weighted sum of OPE coefficients present between consecutive generalized free field dimensions. The lower bound implies the φ×φ OPE must contain at least one primary in the interval [2∆ φ + 2n, 2∆ φ + 2n + 4] for all sufficiently large integer n. The functionals directly compute the OPE decomposition of crossingsymmetrized Witten exchange diagrams in AdS 2 . Therefore, they provide a derivation of the Polyakov bootstrap for SL(2), in particular fixing the so-called contact-term ambiguity. We also use the resulting sum rules to bootstrap several Witten diagrams in AdS 2 up to two loops.

Section: Jhep02(2019)163mentioning

confidence: 67%

Section: Jhep02(2019)163mentioning

confidence: 99%

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The analytic functional bootstrap. Part II. Natural bases for the crossing equation

Mazáč

Paulos

2019

Self Cite

155

“…This is one of the infinitely many equivalent choices of λ, see [60] for more details. encode the information of the complete set of functionals for CFT 1 (see [61,62] for earlier related work, also [63]). A basis of analytic functionals is given by α n and β m , labelled by integers n = 0, 1, 2, .…”

Section: Analytic Functionals In Cftmentioning

confidence: 99%

Recursion relations in Witten diagrams and conformal partial waves

Zhou

2019

We revisit the problem of performing conformal block decomposition of exchange Witten diagrams in the crossed channel. Using properties of conformal blocks and Witten diagrams, we discover infinitely many linear relations among the crossed channel decomposition coefficients. These relations allow us to formulate a recursive algorithm that solves the decomposition coefficients in terms of certain seed coefficients. In one dimensional CFTs, the seed coefficient is the decomposition coefficient of the double-trace operator with the lowest conformal dimension. In higher dimensions, the seed coefficients are the coefficients of the double-trace operators with the minimal conformal twist. We also discuss the conformal block decomposition of a generic contact Witten diagram with any number of derivatives. As a byproduct of our analysis, we obtain a similar recursive algorithm for decomposing conformal partial waves in the crossed channel.

“…Apart from their intrinsic interest, 1d CFTs are also a useful laboratory in which bootstrap ideas can be explored. Recent work includes exact functionals that allow to extract the spectrum analytically [24][25][26], inversion formulas [27,28] (see also [29,30] for the closely related case of BCFT), and intriguing positivity properties [31].…”

Section: Introductionmentioning

confidence: 99%

Bootstrapping line defects in $$ \mathcal{N} $$ = 2 theories

Liendo

2020

We study half-BPS line defects in N = 2 superconformal theories using the bootstrap approach. We concentrate on local excitations constrained to the defect, which means the system is a 1d defect CFT with osp(4 * |2) symmetry. In order to study correlation functions we construct a suitable superspace, and then use the Casimir approach to calculate a collection of new superconformal blocks. Special emphasis is given to the displacement operator, which controls deformations orthogonal to the defect and is always present in a defect CFT. After setting up the crossing equations we proceed with a numerical and analytical bootstrap analysis. We obtain numerical bounds on the CFT data and compare them to known solutions. We also present an analytic perturbative solution to the crossing equations, and argue that this solution captures line defects in N = 2 gauge theories at strong coupling.