Witten diagrams revisited: the AdS geometry of conformal blocks

Hijano, Eliot; Kraus, Per; Perlmutter, Eric; Snively, River

doi:10.1007/JHEP01(2016)146

Title: Witten diagrams revisited: the AdS geometry of conformal blocks

Abstract

Here, we develop a new method for decomposing blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram", which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we also consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.

Authors:

Hijano, Eliot ^[1]; Kraus, Per ^[1]; Perlmutter, Eric ^[2]; Snively, River ^[1]

Univ. of California, Los Angeles, CA (United States). Dept. of Physics and Astronomy
Princeton Univ., NJ (United States). Dept. of Physics

Publication Date:: Mon Jan 25 00:00:00 EST 2016

Research Org.:: Princeton Univ., NJ (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1327296

Grant/Contract Number:: FG02-91ER40671; NSF PHY11-25915; PHY-1313986

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 1; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT correspondence; conformal and W symmetry

Citation Formats


                    Hijano, Eliot, Kraus, Per, Perlmutter, Eric, and Snively, River. Witten diagrams revisited: the AdS geometry of conformal blocks.  United States: N. p., 2016. 
Web.  doi:10.1007/JHEP01(2016)146.

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                    Hijano, Eliot, Kraus, Per, Perlmutter, Eric, & Snively, River. Witten diagrams revisited: the AdS geometry of conformal blocks.  United States.  https://doi.org/10.1007/JHEP01(2016)146

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                    Hijano, Eliot, Kraus, Per, Perlmutter, Eric, and Snively, River. Mon .  
"Witten diagrams revisited: the AdS geometry of conformal blocks".  United States.  https://doi.org/10.1007/JHEP01(2016)146.  https://www.osti.gov/servlets/purl/1327296.

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@article{osti_1327296,

  title        = {Witten diagrams revisited: the AdS geometry of conformal blocks},

  author       = {Hijano, Eliot and Kraus, Per and Perlmutter, Eric and Snively, River},

  abstractNote = {Here, we develop a new method for decomposing blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram", which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we also consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.},

  doi          = {10.1007/JHEP01(2016)146},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 1,

  volume       = 2016,

  place        = {United States},

  year         = {Mon Jan 25 00:00:00 EST 2016},

  month        = {Mon Jan 25 00:00:00 EST 2016}

}

Copy to clipboard

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Holographic complexity from Crofton’s formula in Lorentzian AdS ₃
journal, December 2019

Huang, Xing; Zhang, Le
International Journal of Modern Physics A, Vol. 34, Issue 35
DOI: 10.1142/s0217751x19502373

A Mellin space approach to the conformal bootstrap
journal, May 2017

Gopakumar, Rajesh; Kaviraj, Apratim; Sen, Kallol
Journal of High Energy Physics, Vol. 2017, Issue 5
DOI: 10.1007/jhep05(2017)027

Quantum Regge trajectories and the Virasoro analytic bootstrap
journal, May 2019

Collier, Scott; Gobeil, Yan; Maxfield, Henry
Journal of High Energy Physics, Vol. 2019, Issue 5
DOI: 10.1007/jhep05(2019)212

Spinning Witten diagrams
journal, June 2017

Sleight, Charlotte; Taronna, Massimo
Journal of High Energy Physics, Vol. 2017, Issue 6
DOI: 10.1007/jhep06(2017)100

Higher-point conformal blocks and entanglement entropy in heavy states
text, January 2016

Banerjee, Pinaki; Datta, Shouvik; Sinha, Ritam
ETH Zurich
DOI: 10.3929/ethz-b-000117244

Charged structure constants from modularity
text, January 2017

Das, Diptarka; Datta, Shouvik; Pal, Sridip
ETH Zurich
DOI: 10.3929/ethz-b-000220012

Charged structure constants from modularity
journal, November 2017

Das, Diptarka; Datta, Shouvik; Pal, Sridip
Journal of High Energy Physics, Vol. 2017, Issue 11
DOI: 10.1007/jhep11(2017)183

Causality Constraints in Conformal Field Theory
text, January 2015

Hartman, Thomas; Jain, Sachin; Kundu, Sandipan
arXiv
DOI: 10.48550/arxiv.1509.00014

Hawking from Catalan
text, January 2015

Fitzpatrick, A. Liam; Kaplan, Jared; Walters, Matthew T.
arXiv
DOI: 10.48550/arxiv.1510.00014

Higher-point conformal blocks and entanglement entropy in heavy states
text, January 2016

Banerjee, Pinaki; Datta, Shouvik; Sinha, Ritam
arXiv
DOI: 10.48550/arxiv.1601.06794

A Stereoscopic Look into the Bulk
text, January 2016

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel
arXiv
DOI: 10.48550/arxiv.1604.03110

Conformal Bootstrap in Mellin Space
text, January 2016

Gopakumar, Rajesh; Kaviraj, Apratim; Sen, Kallol
arXiv
DOI: 10.48550/arxiv.1609.00572

Einstein gravity 3-point functions from conformal field theory
text, January 2016

Afkhami-Jeddi, Nima; Hartman, Thomas; Kundu, Sandipan
arXiv
DOI: 10.48550/arxiv.1610.09378

Anatomy of Geodesic Witten Diagrams
text, January 2017

Chen, Heng-Yu; Kuo, En-Jui; Kyono, Hideki
arXiv
DOI: 10.48550/arxiv.1702.08818

Holographic Reconstruction of AdS Exchanges from Crossing Symmetry
text, January 2017

Alday, Luis F.; Bissi, Agnese; Perlmutter, Eric
arXiv
DOI: 10.48550/arxiv.1705.02318

Bulk Phase Shift, CFT Regge Limit and Einstein Gravity
text, January 2017

Kulaxizi, Manuela; Parnachev, Andrei; Zhiboedov, Alexander
arXiv
DOI: 10.48550/arxiv.1705.02934

Witten Diagrams for Torus Conformal Blocks
text, January 2017

Kraus, Per; Maloney, Alexander; Maxfield, Henry
arXiv
DOI: 10.48550/arxiv.1706.00047

From Conformal Blocks to Path Integrals in the Vaidya Geometry
text, January 2017

Anous, Tarek; Hartman, Thomas; Rovai, Antonin
arXiv
DOI: 10.48550/arxiv.1706.02668

Weight Shifting Operators and Conformal Blocks
text, January 2017

Karateev, Denis; Kravchuk, Petr; Simmons-Duffin, David
arXiv
DOI: 10.48550/arxiv.1706.07813

Kinematic space for conical defects
text, January 2017

Cresswell, Jesse C.; Peet, Amanda W.
arXiv
DOI: 10.48550/arxiv.1708.09838

Bands and gaps in Nekrasov partition function
text, January 2017

Gorsky, A.; Milekhin, A.; Sopenko, N.
arXiv
DOI: 10.48550/arxiv.1712.02936

Double-Trace Deformations of Conformal Correlations
text, January 2018

Giombi, Simone; Kirilin, Vladimir; Perlmutter, Eric
arXiv
DOI: 10.48550/arxiv.1801.01477

The Conformal Bootstrap at Finite Temperature
text, January 2018

Iliesiu, Luca; Koloğlu, Murat; Mahajan, Raghu
arXiv
DOI: 10.48550/arxiv.1802.10266

Towards Entanglement of Purification for Conformal Field Theories
text, January 2018

Hirai, Hayato; Tamaoka, Kotaro; Yokoya, Tsuyoshi
arXiv
DOI: 10.48550/arxiv.1803.10539

Fermions in Geodesic Witten Diagrams
text, January 2018

Nishida, Mitsuhiro; Tamaoka, Kotaro
arXiv
DOI: 10.48550/arxiv.1805.00217

A Conformal Collider for Holographic CFTs
text, January 2018

Afkhami-Jeddi, Nima; Kundu, Sandipan; Tajdini, Amirhossein
arXiv
DOI: 10.48550/arxiv.1805.07393

Large $c$ Virasoro Blocks from Monodromy Method beyond Known Limits
text, January 2018

Kusuki, Yuya
arXiv
DOI: 10.48550/arxiv.1806.04352

Holographic relations for OPE blocks in excited states
text, January 2018

Cresswell, Jesse C.; Jardine, Ian T.; Peet, Amanda W.
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DOI: 10.48550/arxiv.1809.09107

An Analytic Approach to BCFT$_d$
text, January 2018

Mazáč, Dalimil; Rastelli, Leonardo; Zhou, Xinan
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DOI: 10.48550/arxiv.1812.09314

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text, January 2019

Parikh, Sarthak
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DOI: 10.48550/arxiv.1901.01267

Structure Constants from Modularity in Warped CFT
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Song, Wei; Xu, Jianfei
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DOI: 10.48550/arxiv.1903.01346

Dynamical Constraints on RG Flows and Cosmology
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Albayrak, Soner; Kharel, Savan
arXiv
DOI: 10.48550/arxiv.1908.01835

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Title: Witten diagrams revisited: the AdS geometry of conformal blocks

Abstract

Citation Formats

Manifestly conformal covariant operator-product expansion journal, December 1971

Covariant expansion of the conformal four -point function journal, December 1972

Conformal four point functions and the operator product expansion journal, April 2001

Conformal partial waves and the operator product expansion journal, February 2004

Spinning conformal blocks journal, November 2011

Bounding scalar operator dimensions in 4 D CFT journal, December 2008

The Large-N Limit of Superconformal Field Theories and Supergravity journal, January 1999

Gauge theory correlators from non-critical string theory journal, May 1998

Anti de Sitter space and holography journal, January 1998

Comments on 4-point functions in the CFT/AdS correspondence journal, April 1999

General scalar exchange in AdSd+1 journal, June 1999

Graviton exchange and complete four-point functions in the AdS/CFT correspondence journal, November 1999

The operator product expansion of SYM and the 4-point functions of supergravity journal, November 2000

Analyticity of AdS scalar exchange graphs in the crossed channel journal, April 2000

Aspects of the conformal operator product expansion in $\mathrm{AdS}$/$\mathrm{CFT}$ correspondence journal, January 2000

Writing CFT correlation functions as AdS scattering amplitudes journal, March 2011

Towards Feynman rules for Mellin amplitudes in AdS/CFT journal, October 2011

A natural language for AdS/CFT correlators journal, November 2011

Unitarity and the holographic S-Matrix journal, October 2012

Factorization of Mellin amplitudes journal, October 2015

Holographic entanglement entropy from 2d CFT: heavy states and local quenches journal, February 2015

Virasoro conformal blocks and thermality from classical background fields journal, November 2015

Universality of long-distance AdS physics from the CFT bootstrap journal, August 2014

Worldline approach to semi-classical conformal blocks journal, July 2015

Conformal symmetry in two-dimensional space: Recursion representation of conformal block journal, October 1987

Conformal symmetry in two dimensions: An explicit recurrence formula for the conformal partial wave amplitude journal, September 1984

Bootstrapping the O(N ) vector models journal, June 2014

Holography from conformal field theory journal, October 2009

Universal spectrum of 2d conformal field theory in the large c limit journal, September 2014

Correlation functions and massive Kaluza–Klein modes in the AdS/CFT correspondence journal, August 2003

Gauge boson exchange in AdSd+1 journal, April 1999

Graviton and gauge boson propagators in AdS+1 journal, November 1999

AdS/CFT four-point functions: How to succeed at z-integrals without really trying journal, November 1999

Non-perturbative effects in AdS 5 × S 5 string theory and d = 4 SUSY Yang-Mills journal, May 1998

String corrections to four point functions in the AdS/CFT correspondence journal, January 1999

The large Nc limit of four-point functions in N = 4 super-Yang-Mills theory from anti-de Sitter supergravity journal, August 1999

Bonus symmetries of N = 4 super-Yang-Mills correlation functions via AdS duality journal, July 1999

Near-extremal correlators and vanishing supergravity couplings in AdS/CFT journal, November 2000

Operator product expansion of the lowest weight CPOs in =4 at strong coupling journal, October 2000

Partial non-renormalisation of the stress-tensor four-point function in N=4 SYM and AdS/CFT journal, July 2001

Conjectures for large N superconformal chiral primary four point functions journal, August 2006

Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions journal, April 2007

Generalized Recursion Relations for Correlators in the Gauge-Gravity Correspondence journal, February 2011

Spinning AdS propagators journal, September 2014

On Feynman rules for Mellin amplitudes in AdS/CFT journal, May 2012

Four point function of N = 4 $$ \mathcal{N}=4 $$ stress-tensor multiplet at strong coupling journal, April 2015

Projectors, shadows, and conformal blocks journal, April 2014

Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces journal, December 1984

Conformal correlators of mixed-symmetry tensors journal, February 2015

The analytic bootstrap and AdS superhorizon locality journal, December 2013

Convexity and liberation at large spin journal, November 2013

Higher-spin correlators journal, October 2013

Analytic bootstrap at large spin journal, November 2015

Universal anomalous dimensions at large spin and large twist journal, July 2015

Conformal blocks in the large D limit journal, August 2013

Generalized additivity in unitary conformal field theories journal, October 2015

Properties of Partial-Wave Amplitudes in Conformal Invariant Field Theories text, January 1975

Holography from Conformal Field Theory text, January 2009

More Holography from Conformal Field Theory text, January 2010

Effective Conformal Theory and the Flat-Space Limit of AdS text, January 2010

Emergent Spacetime and Holographic CFTs text, January 2011

A Natural Language for AdS/CFT Correlators text, January 2011

Spinning Conformal Blocks text, January 2011

Analyticity and the Holographic S-Matrix text, January 2011

Projectors, Shadows, and Conformal Blocks text, January 2012

Conformal Regge theory text, January 2012

Conformal Blocks in the Large D Limit text, January 2013

Bootstrapping the O(N) Vector Models text, January 2013

Universality of Long-Distance AdS Physics from the CFT Bootstrap text, January 2014

Factorization of Mellin amplitudes preprint, January 2014

Lessons from crossing symmetry at large N preprint, January 2014

Conformal correlators of mixed-symmetry tensors text, January 2014

Generalized Additivity in Unitary Conformal Field Theories text, January 2014

Worldline approach to semi-classical conformal blocks preprint, January 2015

Virasoro Conformal Blocks and Thermality from Classical Background Fields text, January 2015

Universal anomalous dimensions at large spin and large twist text, January 2015

Classical conformal blocks via AdS/CFT correspondence text, January 2015

Manifestly conformal covariant operator-product expansion
journal, December 1971

Covariant expansion of the conformal four -point function
journal, December 1972

Conformal four point functions and the operator product expansion
journal, April 2001

Conformal partial waves and the operator product expansion
journal, February 2004

Spinning conformal blocks
journal, November 2011

Bounding scalar operator dimensions in 4 D CFT
journal, December 2008

The Large-N Limit of Superconformal Field Theories and Supergravity
journal, January 1999

Gauge theory correlators from non-critical string theory
journal, May 1998

Anti de Sitter space and holography
journal, January 1998

Comments on 4-point functions in the CFT/AdS correspondence
journal, April 1999

General scalar exchange in AdSd+1
journal, June 1999

Graviton exchange and complete four-point functions in the AdS/CFT correspondence
journal, November 1999

The operator product expansion of SYM and the 4-point functions of supergravity
journal, November 2000

Analyticity of AdS scalar exchange graphs in the crossed channel
journal, April 2000

Aspects of the conformal operator product expansion in $\mathrm{AdS}$/$\mathrm{CFT}$ correspondence
journal, January 2000

Writing CFT correlation functions as AdS scattering amplitudes
journal, March 2011

Towards Feynman rules for Mellin amplitudes in AdS/CFT
journal, October 2011

A natural language for AdS/CFT correlators
journal, November 2011

Unitarity and the holographic S-Matrix
journal, October 2012

Factorization of Mellin amplitudes
journal, October 2015

Holographic entanglement entropy from 2d CFT: heavy states and local quenches
journal, February 2015

Virasoro conformal blocks and thermality from classical background fields
journal, November 2015

Universality of long-distance AdS physics from the CFT bootstrap
journal, August 2014

Worldline approach to semi-classical conformal blocks
journal, July 2015

Conformal symmetry in two-dimensional space: Recursion representation of conformal block
journal, October 1987

Conformal symmetry in two dimensions: An explicit recurrence formula for the conformal partial wave amplitude
journal, September 1984

Bootstrapping the O(N ) vector models
journal, June 2014

Holography from conformal field theory
journal, October 2009

Universal spectrum of 2d conformal field theory in the large c limit
journal, September 2014

Correlation functions and massive Kaluza–Klein modes in the AdS/CFT correspondence
journal, August 2003

Gauge boson exchange in AdSd+1
journal, April 1999

Graviton and gauge boson propagators in AdS+1
journal, November 1999

AdS/CFT four-point functions: How to succeed at z-integrals without really trying
journal, November 1999

Non-perturbative effects in AdS ₅ × S ⁵ string theory and d = 4 SUSY Yang-Mills
journal, May 1998

String corrections to four point functions in the AdS/CFT correspondence
journal, January 1999

The large Nc limit of four-point functions in N = 4 super-Yang-Mills theory from anti-de Sitter supergravity
journal, August 1999

Bonus symmetries of N = 4 super-Yang-Mills correlation functions via AdS duality
journal, July 1999

Near-extremal correlators and vanishing supergravity couplings in AdS/CFT
journal, November 2000

Operator product expansion of the lowest weight CPOs in =4 at strong coupling
journal, October 2000

Partial non-renormalisation of the stress-tensor four-point function in N=4 SYM and AdS/CFT
journal, July 2001

Conjectures for large N superconformal chiral primary four point functions
journal, August 2006

Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions
journal, April 2007

Generalized Recursion Relations for Correlators in the Gauge-Gravity Correspondence
journal, February 2011

Spinning AdS propagators
journal, September 2014

On Feynman rules for Mellin amplitudes in AdS/CFT
journal, May 2012

Four point function of N = 4 $$ \mathcal{N}=4 $$ stress-tensor multiplet at strong coupling
journal, April 2015

Projectors, shadows, and conformal blocks
journal, April 2014

Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces
journal, December 1984

Conformal correlators of mixed-symmetry tensors
journal, February 2015

The analytic bootstrap and AdS superhorizon locality
journal, December 2013

Convexity and liberation at large spin
journal, November 2013

Higher-spin correlators
journal, October 2013

Analytic bootstrap at large spin
journal, November 2015

Universal anomalous dimensions at large spin and large twist
journal, July 2015

Conformal blocks in the large D limit
journal, August 2013

Generalized additivity in unitary conformal field theories
journal, October 2015

Properties of Partial-Wave Amplitudes in Conformal Invariant Field Theories
text, January 1975

Holography from Conformal Field Theory
text, January 2009

More Holography from Conformal Field Theory
text, January 2010

Effective Conformal Theory and the Flat-Space Limit of AdS
text, January 2010

Emergent Spacetime and Holographic CFTs
text, January 2011

A Natural Language for AdS/CFT Correlators
text, January 2011

Spinning Conformal Blocks
text, January 2011

Analyticity and the Holographic S-Matrix
text, January 2011

Projectors, Shadows, and Conformal Blocks
text, January 2012

Conformal Regge theory
text, January 2012

Conformal Blocks in the Large D Limit
text, January 2013

Bootstrapping the O(N) Vector Models
text, January 2013

Universality of Long-Distance AdS Physics from the CFT Bootstrap
text, January 2014

Factorization of Mellin amplitudes
preprint, January 2014

Lessons from crossing symmetry at large N
preprint, January 2014

Conformal correlators of mixed-symmetry tensors
text, January 2014

Generalized Additivity in Unitary Conformal Field Theories
text, January 2014

Worldline approach to semi-classical conformal blocks
preprint, January 2015

Virasoro Conformal Blocks and Thermality from Classical Background Fields
text, January 2015

Universal anomalous dimensions at large spin and large twist
text, January 2015

Classical conformal blocks via AdS/CFT correspondence
text, January 2015

Deconstructing Conformal Blocks in 4D CFT
preprint, January 2015

Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions
text, January 2006

Gauge Theory Correlators from Non-Critical String Theory
text, January 1998