Conformal blocks in the Dolan-Osborn coordinates

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Conformal bootstrap

We can use the conformal block decomposition from the previous section to revisit the relations (1.54) expressing the constraints coming from Bose symmetry (in this context also known as crossing symmetry) on the functions G1234(u; v). Symmetry under the exchange 1(x1) $ 3(x3) for example required that v(2+3)=2 G1234(u; v) = u(1+2)=2 G3214(v; u): (1.131) To use a conformal block decomposition, we want to apply the OPE on both sides of this equation simultaneously. On the LHS, this requires that both OPEs (12) and (34) exist, while on the RHS we require the (23) and (14) OPEs. Suppose for now that these conditions are all satised for suitable values of u and v. In that case, Eq. (1.131) can be expanded as a sum over all primaries O in the theory.

Dolan-Osborn kinematics in radial quantization

To exhibit the general structure of conformal blocks in the z; z variables, we propose to use radial quantization, as introduced in section 1.6. Our goal is to express the conformal blocks GO(z; z) in terms of matrix elements of operators on the cylinder R Sd􀀀1, parametrized by the coordinates (t; n). For simplicity, we set the radius R of the cylinder to one. The at-space four-point function with points assigned as in Fig. 2.2 maps to the cylinder matrix element hj(t3; n3)(t2; n2)ji .

Expansion coecients from the Casimir equation

We would like to compute the coecients An;j in (2.18). In principle, this can be done following the radial quantization method to its logical end: imposing the constraints of conformal invariance in the OPE and evaluating the norms of the descendants. The example of scalar exchanged primaries and their rst two descendant levels was considered in [21]. However, it is far more ecient to use the method based on Casimir dierential equations, rst proposed in [16] and reviewed in section 1.8. The idea is that the conformal block satises an eigenvalue equation of the form Du;v G;`(u; v) = c(2) ;` G;`(u; v); c(2) ;` = ( 􀀀 d) + `(` + d 􀀀 2).

Comparison between the z and expansions

We have presented two ways to expand the conformal blocks: the \z-series » (2.25) and the \-series » (2.42). We will now argue that the second expansion is more ecient, in the sense that it converges more rapidly and fewer terms need to be evaluated in order to get a good approximation. This happens because of the better choice of the expansion parameter and the better asymptotic behavior of the series coecients.
Let us start with the expansion parameters. The interesting range for the coordinate is the unit disk jj < 1. The -series will converge absolutely, everywhere in this disk. To prove this, we rst restrict the expansion of the conformal block to the positive real axis grows monotically with N: therefore either it converges or it grows unboundedly (meaning that limN gN() = 1). The latter cannot occur for < 1, because it would mean a physical singularity in the conformal block. We conclude that the expansion (2.55) converges pointwise on [0; 1).

Outlook: potential applications to the conformal bootstrap

In the previous section, we introduced a new way to represent the conformal blocks, by expanding them in the polar coordinates associated with the complex variable . Our interests in the blocks stems from the role they play in the conformal bootstrap program. that the ! 1 blocks from Eq. (2.72) do not satisfy the growth required by the Hardy-Littlewood theorem.
We believe that our new representation will turn out quite useful in this context. Here we will list several ideas, leaving their complete development for the future. We recall that most existing applications of the bootstrap program in d > 3 dimensions follow a scheme rst proposed in [14]. This scheme focuses on a single bootstrap equation, that is obtained by substituting the conformal block expansion (2.2) into the crossing symmetry constraint (2.3) and takes the form: (v 􀀀 u) + X i f2 i [vGi;`i(u; v) 􀀀 (u $ v)] = 0 :

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Table of contents :

Abstract
Resume
Introduction
Citations to published work
Resume substantiel
Conventions and special functions
1 Elements of conformal eld theory 
1.1 Conformal group
1.2 Conformal algebra
1.3 Local operators
1.3.1 Tensor operators
1.3.2 Descendants
1.4 Constraints on correlation functions
1.4.1 Scalar correlators
1.4.2 Spinning correlators
1.4.3 Example: free scalar boson
1.5 Weyl invariance
1.5.1 The free boson revisited
1.6 Radial quantization
1.6.1 State-operator correspondence
1.6.2 Adjoint states and matrix elements
1.6.3 Unitarity constraints on CFTs
1.7 Operator product expansion
1.7.1 Conformal block decomposition
1.7.2 Conformal bootstrap
1.8 Casimir dierential equations
2 Conformal blocks in radial coordinates 
2.1 Conformal blocks in the Dolan-Osborn coordinates
2.1.1 Dolan-Osborn kinematics in radial quantization
2.1.2 Expansion coecients from the Casimir equation
2.1.3 Decoupling of descendants for the leading twist
2.2 Conformal blocks in the coordinate
2.2.1 Comparison between the z and expansions
2.3 Outlook: potential applications to the conformal bootstrap
2.3.1 Inexpensive derivative evaluation for all and `
2.3.2 Truncated bootstrap equation with an error estimate
2.4 Summary
3 Conformal blocks in the diagonal limit 
3.1 Introduction
3.2 Dierential equations on the diagonal
3.3 Frobenius’ method
3.4 Computing conformal blocks and their derivatives eciently
3.5 Summary
4 TCSA for scalar elds d > 2 dimensions 
4.1 Introduction
4.2 Truncated Conformal Space Approach: general setup
4.2.1 A case study for TCSA in d dimensions
4.3 Free scalar in d dimensions
4.3.1 Constructing the Hilbert space
4.3.2 Primaries and descendants
4.3.3 Gram matrix
4.3.4 Null states in integer d
4.3.5 Non-unitarity at fractional d
4.4 TCSA eigenvalue problem
4.4.1 Simple versus generalized eigenvalue problem
4.4.2 Working in the presence of null states
4.4.3 Matrix elements via the OPE method
5 The 2 ow in TCSA 
5.1 Canonical quantization on the cylinder
5.1.1 Casimir energy
5.1.2 Massive states on the cylinder
5.2 Intermezzo: computing observables using conformal perturbation theory
5.2.1 Casimir energy
5.2.2 Excited states
5.3 TCSA setup
5.4 Numerical results
5.4.1 Casimir energy
5.4.2 Massive excitations
5.4.3 Discussion
6 Cuto dependence and renormalization 
6.1 Warm-up: improving 4 theory in at space
6.2 Cuto dependence in TCSA
6.2.1 General remarks
6.2.2 Computation of H via two-point functions
6.2.3 RG improvement
6.2.4 Other treatments of renormalization
6.3 Renormalization for the 2 ow
6.3.1 Renormalization details
6.3.2 Numerical results
7 The Landau-Ginzburg ow 
7.1 Theoretical expectations
7.2 Numerical results
7.3 Non-unitarity and complex energy levels
8 Discussion and outlook 
A Boundedness of the -series coecients 
B Recursion relations for an 
C TCSA computations in practice 
C.1 Constructing the Hilbert space
C.2 OPE matrices
C.2.1 Index-free formalism
C.3 Gram matrix
D Renormalization in TCSA: computations 
D.1 Asymptotics of C(t)
D.2 Mn sequence for 2 2
D.2.1 Renormalization details for the 4 ow
Bibliography

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