Abstract
We develop an analytic approach to Boundary Conformal Field Theory (BCFT), focussing on the two-point function of a general pair of scalar primary operators. The resulting crossing equation can be thought of as a vector equation in an infinite-dimensional space \( \mathcal{V} \) of analytic functions of a single complex variable. We argue that in a unitary theory, functions in \( \mathcal{V} \) satisfy a boundedness condition in the Regge limit. We identify a useful basis for \( \mathcal{V} \), consisting of bulk and boundary conformal blocks with scaling dimensions which appear in OPEs of the mean field theory correlator. Our main achievement is an explicit expression for the action of the dual basis (the basis of linear functionals on \( \mathcal{V} \)) on an arbitrary conformal block. The practical merit of our basis is that it trivializes the study of perturbations around mean field theory. Our results are equivalent to a BCFT version of the Polyakov bootstrap. Our derivation of the expressions for the functionals relies on the identification of the Polyakov blocks with (suitably improved) boundary and bulk Witten exchange diagrams in AdSd+1. We also provide another conceptual perspective on the Polyakov block expansion and the associated functionals, by deriving a new Lorentzian OPE inversion formula for BCFT.
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References
J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys.B 240 (1984) 514 [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys.135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
W.-H. Hsiao and D.T. Son, Duality and universal transport in mixed-dimension electrodynamics, Phys. Rev.B 96 (2017) 075127 [arXiv:1705.01102] [INSPIRE].
C.P. Herzog, K.-W. Huang, I. Shamir and J. Virrueta, Superconformal Models for Graphene and Boundary Central Charges, JHEP09 (2018) 161 [arXiv:1807.01700] [INSPIRE].
A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP06 (2001) 063 [hep-th/0105132] [INSPIRE].
O. Aharony, O. DeWolfe, D.Z. Freedman and A. Karch, Defect conformal field theory and locally localized gravity, JHEP07 (2003) 030 [hep-th/0303249] [INSPIRE].
N. Andrei et al., Boundary and Defect CFT: Open Problems and Applications, 2018, arXiv:1810.05697 [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
A. Gadde, Conformal constraints on defects, arXiv:1602.06354 [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N} \) = 4 SYM with defects, JHEP01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
M. Hogervorst, Crossing Kernels for Boundary and Crosscap CFTs, arXiv:1703.08159 [INSPIRE].
L. Rastelli and X. Zhou, The Mellin Formalism for Boundary CFTd , JHEP10 (2017) 146 [arXiv:1705.05362] [INSPIRE].
A. Karch and Y. Sato, Boundary Holographic Witten Diagrams, JHEP09 (2017) 121 [arXiv:1708.01328] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Radial coordinates for defect CFTs, JHEP11 (2018) 148 [arXiv:1712.07668] [INSPIRE].
M. Lemos, P. Liendo, M. Meineri and S. Sarkar, Universality at large transverse spin in defect CFT, JHEP09 (2018) 091 [arXiv:1712.08185] [INSPIRE].
P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect, JHEP10 (2018) 077 [arXiv:1806.01862] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal field theory, JHEP08 (2019) 066 [arXiv:1807.02522] [INSPIRE].
A. Bissi, T. Hansen and A. Söderberg, Analytic Bootstrap for Boundary CFT, JHEP01 (2019) 010 [arXiv:1808.08155] [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys.B 324 (1989) 581 [INSPIRE].
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett.B 259 (1991) 274 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
F. Gliozzi, Truncatable bootstrap equations in algebraic form and critical surface exponents, JHEP10 (2016) 037 [arXiv:1605.04175] [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett.111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFTd, JHEP07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
D. Mazáč, A Crossing-Symmetric OPE Inversion Formula, JHEP06 (2019) 082 [arXiv:1812.02254] [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP11 (2007) 019 [arXiv:0708.0672] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett.119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz.66 (1974) 23 [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal Bootstrap in Mellin Space, Phys. Rev. Lett.118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
R. Gopakumar and A. Sinha, On the Polyakov-Mellin bootstrap, JHEP12 (2018) 040 [arXiv:1809.10975] [INSPIRE].
C. Behan, Conformal manifolds: ODEs from OPEs, JHEP03 (2018) 127 [arXiv:1709.03967] [INSPIRE].
A. Karch and Y. Sato, Conformal Manifolds with Boundaries or Defects, JHEP07 (2018) 156 [arXiv:1805.10427] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, Long-range critical exponents near the short-range crossover, Phys. Rev. Lett.118 (2017) 241601 [arXiv:1703.03430] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, A scaling theory for the long-range to short-range crossover and an infrared duality, J. Phys.A 50 (2017) 354002 [arXiv:1703.05325] [INSPIRE].
L. Di Pietro, D. Gaiotto, E. Lauria and J. Wu, 3d Abelian Gauge Theories at the Boundary, JHEP05 (2019) 091 [arXiv:1902.09567] [INSPIRE].
A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of D-branes, Nucl. Phys.B 545 (1999) 233 [hep-th/9811237] [INSPIRE].
A. Kaviraj and M.F. Paulos, The Functional Bootstrap for Boundary CFT, arXiv:1812.04034 [INSPIRE].
X. Zhou, Polyakov blocks and functionals for BCFTd , Bootstrap 2018, Caltech, 17 July 2018.
D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys.B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
O. DeWolfe, D.Z. Freedman and H. Ooguri, Holography and defect conformal field theories, Phys. Rev.D 66 (2002) 025009 [hep-th/0111135] [INSPIRE].
P. Dey, K. Ghosh and A. Sinha, Simplifying large spin bootstrap in Mellin space, JHEP01 (2018) 152 [arXiv:1709.06110] [INSPIRE].
A. Karch and L. Randall, Localized gravity in string theory, Phys. Rev. Lett.87 (2001) 061601 [hep-th/0105108] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
X. Zhou, Recursion Relations in Witten Diagrams and Conformal Partial Waves, JHEP05 (2019) 006 [arXiv:1812.01006] [INSPIRE].
M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP11 (2017) 193 [arXiv:1702.08471] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev.D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
V. Goncalves and G. Itsios, A note on defect Mellin amplitudes, arXiv:1803.06721 [INSPIRE].
E. D’Hoker, D.Z. Freedman and L. Rastelli, AdS/CFT four point functions: How to succeed at z integrals without really trying, Nucl. Phys.B 562 (1999) 395 [hep-th/9905049] [INSPIRE].
E. D’Hoker and D.Z. Freedman, General scalar exchange in AdS(d+1), Nucl. Phys.B 550 (1999) 261 [hep-th/9811257] [INSPIRE].
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Mazáč, D., Rastelli, L. & Zhou, X. An analytic approach to BCFTd. J. High Energ. Phys. 2019, 4 (2019). https://doi.org/10.1007/JHEP12(2019)004
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DOI: https://doi.org/10.1007/JHEP12(2019)004