Bootstrapping the half-BPS line defect

110

We derive a Lorentzian OPE inversion formula for the principal series of sl(2, R). Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric four-point functions and makes crossing symmetry manifest. In particular, inverting a single conformal block in the crossed channel returns the coefficient function of the crossing-symmetric sum of Witten exchange diagrams in AdS, including the direct-channel exchange. The inversion kernel exhibits poles at the double-trace scaling dimensions, whose contributions must cancel out in a generic solution to crossing. In this way the inversion formula leads to a derivation of the Polyakov bootstrap for sl(2, R). The residues of the inversion kernel at the double-trace dimensions give rise to analytic bootstrap functionals discussed in recent literature, thus providing an alternative explanation for their existence. We also use the formula to give a general proof that the coefficient function of the principal series is meromorphic in the entire complex plane with poles only at the expected locations.

Section: Discussion and Open Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A crossing-symmetric OPE inversion formula

Mazáč

2019

110

“…Also the bootstrap programme, both in its numerical and analytical incarnation, has been applied to higher dimensional superconformal field theories, see [22][23][24]. Significant progress has been made by studying four-point blocks of half-BPS operators [25][26][27][28][29][30][31][32][33][34][35], or superprimary components of correlators for operators that are not half-BPS [36][37][38][39]. For correlation functions of BPS operators, the conformal blocks are similar to those of the ordinary bosonic conformal symmetry and hence they are well known.…”

Section: Contentsmentioning

confidence: 99%

“…There are many potentially interesting applications, in particular to line defects in superconformal field theories, see e.g. [35,[64][65][66]. A Conformal partial waves and blocks for sl(2|1)…”

Section: Applications To 4-dimensional N = 1 Theoriesmentioning

confidence: 99%

Superconformal blocks: general theory

Burić

Schomerus

Sobko

2020

In this work we launch a systematic theory of superconformal blocks for four-point functions of arbitrary supermultiplets. Our results apply to a large class of superconformal field theories including 4-dimensional models with any number N of supersymmetries. The central new ingredient is a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solutions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact. We illustrate the general theory at the example of d = 1 dimensional theories with N = 2 supersymmetry for which we recover known superblocks. The paper concludes with an outlook to 4-dimensional blocks with N = 1 supersymmetry.

“…τ . It is convenient to parametrize these polynomials in terms of eigenfunctions Y k 12 ,k 34 OPEs S k 1 × S k 2 and S k 3 × S k 4 in (2.2), we can expand A k 12 ,k34 …”

mentioning

confidence: 99%

The M-theory archipelago

Agmon

Chester

Pufu

2020

105

We combine supersymmetric localization results and the numerical conformal bootstrap technique to study the 3d maximally supersymmetric (N = 8) CFT on N coincident M2branes (the U (N ) k × U (N ) −k ABJM theory at Chern-Simons level k = 1). In particular, we perform a mixed correlator bootstrap study of the superconformal primaries of the stress tensor multiplet and of the next possible lowest-dimension half-BPS multiplet that is allowed by 3d N = 8 superconformal symmetry. Of all known 3d N = 8 SCFTs, the k = 1 ABJM theory is the only one that contains both types of multiplets in its operator spectrum. By imposing the values of the short OPE coefficients that can be computed exactly using supersymmetric localization, we are able to derive precise islands in the space of semi-short OPE coefficients for an infinite number of such coefficients. We find that these islands decrease in size with increasing N . More generally, we also analyze 3d N = 8 SCFT that contain both aforementioned multiplets in their operator spectra without inputing any additional information that is specific to ABJM theory. For such theories, we compute upper and lower bounds on the semi-short OPE coefficients as well as upper bounds on the scaling dimension of the lowest unprotected scalar operator. These latter bounds are more constraining than the analogous bounds previously derived from a single correlator bootstrap of the stress tensor multiplet. This leads us to conjecture that the U (N ) 2 × U (N + 1) −2 ABJ theory, and not the k = 1 ABJM theory, saturates the single correlator bounds. July 2019correlator system that is the primary focus of this paper has three advantages:1. In principle, the free multiplet S 1 is allowed to appear in the mixed OPE S 2 × S 3 , so 3 Lower bounds can be computed for the OPE coefficients of the operators that have been assumed to be relevant, since there is now a gap between them and the continuum of irrelevant operators. Islands in the space of these OPE coefficients are reported in [14,17].4 Note that for 4d SCFTs with N ≥ 3, all the protected operators that appear in the four-point function of half-BPS operator are already fixed by the 2d chiral algebra [36], so the bootstrap studies [37-39] could not compute any upper bounds on OPE coefficients. 5 Percent error as computed, for instance, in (4.13). 6 Here, Λ is a parameter counting the number of derivatives in the functionals used for numerical bootstrap. See [41] for the precise definition. 9 Our blocks are normalized as G k12,k34 ∆, (r, η) ∼ r ∆ P (η) as r → 0, where P (η) is a Legendre polynomial.