Counting RG flows
Abstract
Here, interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different "topological sectors" for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts  from counting RG walls to AdS/CFT correspondence  will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott's version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and "topological obstructions" for the strong form of the Ctheorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong Ctheorem happens and predicts "phase transitions" along the RG flow when the topological obstruction is nontrivial. Along the way, we also find new conformal manifolds in wellknown 4d CFT's and point out connections withmore »
 Authors:

 California Inst. of Technology (CalTech), Pasadena, CA (United States); MaxPlanckInstitut fur Mathematik, Bonn (Germany)
 Publication Date:
 Research Org.:
 California Institute of Technology (CalTech), Pasadena, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), High Energy Physics (HEP)
 OSTI Identifier:
 1327329
 Grant/Contract Number:
 SC0011632
 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 10298479
 Publisher:
 Springer Berlin
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; supersymmetric gauge theory; renormalization group
Citation Formats
Gukov, Sergei. Counting RG flows. United States: N. p., 2016.
Web. doi:10.1007/JHEP01(2016)020.
Gukov, Sergei. Counting RG flows. United States. doi:10.1007/JHEP01(2016)020.
Gukov, Sergei. Tue .
"Counting RG flows". United States. doi:10.1007/JHEP01(2016)020. https://www.osti.gov/servlets/purl/1327329.
@article{osti_1327329,
title = {Counting RG flows},
author = {Gukov, Sergei},
abstractNote = {Here, interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different "topological sectors" for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts  from counting RG walls to AdS/CFT correspondence  will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott's version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and "topological obstructions" for the strong form of the Ctheorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong Ctheorem happens and predicts "phase transitions" along the RG flow when the topological obstruction is nontrivial. Along the way, we also find new conformal manifolds in wellknown 4d CFT's and point out connections with the superconformal index and classifying spaces of global symmetry groups.},
doi = {10.1007/JHEP01(2016)020},
journal = {Journal of High Energy Physics (Online)},
number = 1,
volume = 2016,
place = {United States},
year = {2016},
month = {1}
}
Web of Science
Works referenced in this record:
The renormalization group and the ε expansion
journal, August 1974
 Wilson, K.
 Physics Reports, Vol. 12, Issue 2
Gradient properties of the renormalisation group equations in multicomponent systems
journal, July 1975
 Wallace, D. J.; Zia, R. K. P.
 Annals of Physics, Vol. 92, Issue 1
Is there a ctheorem in four dimensions?
journal, December 1988
 Cardy, John L.
 Physics Letters B, Vol. 215, Issue 4
Evidence for the strongest version of the 4d atheorem via amaximization along RG flows
journal, December 2004
 Barnes, Edwin; Intriligator, Ken; Wecht, Brian
 Nuclear Physics B, Vol. 702, Issue 12
On renormalization group flows in four dimensions
journal, December 2011
 Komargodski, Zohar; Schwimmer, Adam
 Journal of High Energy Physics, Vol. 2011, Issue 12
Conductance fluctuations in disordered superconductors with broken timereversal symmetry near two dimensions
journal, October 2007
 Ryu, S.; Furusaki, A.; Ludwig, A. W. W.
 Nuclear Physics B, Vol. 780, Issue 3
Exactly marginal deformations and global symmetries
journal, June 2010
 Green, Daniel; Komargodski, Zohar; Seiberg, Nathan
 Journal of High Energy Physics, Vol. 2010, Issue 6
A note on the holographic beta and C functions
journal, May 2000
 Anselmi, D.; Girardello, L.; Porrati, M.
 Physics Letters B, Vol. 481, Issue 24
Nondegenerate Critical Manifolds
journal, September 1954
 Bott, Raoul
 The Annals of Mathematics, Vol. 60, Issue 2
Naturalness versus supersymmetric nonrenormalization theorems
journal, December 1993
 Seiberg, Nathan
 Physics Letters B, Vol. 318, Issue 3
Supersymmetry and Morse theory
journal, January 1982
 Witten, Edward
 Journal of Differential Geometry, Vol. 17, Issue 4
The YangMills Equations over Riemann Surfaces
journal, March 1983
 Atiyah, M. F.; Bott, R.
 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 308, Issue 1505
Stochastic Differential Equations on TwoDimensional Theory Space and Morse Theory
journal, April 1989
 Das, Sumit R.; Mandal, Gautam; Wadia, Spenta R.
 Modern Physics Letters A, Vol. 04, Issue 08
cTheorem and the topology of 2D QFTs
journal, September 1988
 Vafa, Cumrun
 Physics Letters B, Vol. 212, Issue 1
Domain walls, black holes, and supersymmetric quantum mechanics
journal, May 2001
 Behrndt, Klaus; Gukov, Sergei; Shmakova, Marina
 Nuclear Physics B, Vol. 601, Issue 12
On Conformal Deformations
journal, September 2002
 Kol, Barak
 Journal of High Energy Physics, Vol. 2002, Issue 09
On dangerous irrelevant operators
journal, May 1982
 Amit, Daniel J.; Peliti, Luca
 Annals of Physics, Vol. 140, Issue 2
A comment on duality in N = 1 supersymmetric nonabelian gauge theories
journal, May 1995
 Kutasov, D.
 Physics Letters B, Vol. 351, Issue 13
On duality in supersymmetric YangMills theory
journal, July 1995
 Kutasov, D.; Schwimmer, A.
 Physics Letters B, Vol. 354, Issue 34
New RG fixed points and duality in supersymmetric SP(Nc) and SO(Nc) gauge theories
journal, August 1995
 Intriligator, K.
 Nuclear Physics B, Vol. 448, Issue 12
Chiral rings, singularity theory and electricmagnetic duality
journal, January 1996
 Kutasov, D.; Schwimmer, A.; Seiberg, N.
 Nuclear Physics B, Vol. 459, Issue 3
Accidental symmetries and N = 1 duality in supersymmetric gauge theory
journal, July 1997
 Leigh, Robert G.; Strassler, Matthew J.
 Nuclear Physics B, Vol. 496, Issue 12
Δa curiosities in some 4d susy RG flows
journal, November 2012
 Amariti, Antonio; Intriligator, Kenneth
 Journal of High Energy Physics, Vol. 2012, Issue 11
Defects and bulk perturbations of boundary LandauGinzburg orbifolds
journal, April 2008
 Brunner, Ilka; Roggenkamp, Daniel
 Journal of High Energy Physics, Vol. 2008, Issue 04
Accidents in (0,2) LandauGinzburg theories
journal, December 2014
 Bertolini, Marco; Melnikov, Ilarion V.; Plesser, M. Ronen
 Journal of High Energy Physics, Vol. 2014, Issue 12
Phases of N = 2 theories in two dimensions
journal, August 1993
 Witten, Edward
 Nuclear Physics B, Vol. 403, Issue 12
RG flow in N = 1 discrete series
journal, April 1989
 Kastor, D. A.; Martinec, E. J.; Shenker, S. H.
 Nuclear Physics B, Vol. 316, Issue 3
Algebraic geometry and effective lagrangians
journal, February 1989
 Martinec, Emil J.
 Physics Letters B, Vol. 217, Issue 4
Catastrophes and the classification of conformal theories
journal, February 1989
 Vafa, Cumrun; Warner, Nicholas
 Physics Letters B, Vol. 218, Issue 1
Electricmagnetic duality, monopole condensation, and confinement in N=2 supersymmetric YangMills theory
journal, September 1994
 Seiberg, N.; Witten, E.
 Nuclear Physics B, Vol. 426, Issue 1
Electricmagnetic duality in supersymmetric nonAbelian gauge theories
journal, February 1995
 Seiberg, N.
 Nuclear Physics B, Vol. 435, Issue 12
Renormalization group flows from holography: Supersymmetry and a $c$theorem
journal, January 1999
 Freedman, D. Z.; Gubser, S. S.; Pilch, K.
 Advances in Theoretical and Mathematical Physics, Vol. 3, Issue 2
Domain Walls and Flow Equations in Supergravity
journal, May 2001
 Behrndt, Klaus
 Fortschritte der Physik, Vol. 49, Issue 46
Btype defects in LandauGinzburg models
journal, August 2007
 Brunner, Ilka; Roggenkamp, Daniel
 Journal of High Energy Physics, Vol. 2007, Issue 08
A new supersymmetric index
journal, November 1992
 Cecotti, Sergio; Fendley, Paul; Intriligator, Ken
 Nuclear Physics B, Vol. 386, Issue 2
Scattering and thermodynamics of fractionallycharged supersymmetric solitons
journal, March 1992
 Fendley, P.; Intriligator, K.
 Nuclear Physics B, Vol. 372, Issue 3
On classification ofN=2 supersymmetric theories
journal, December 1993
 Cecotti, Sergio; Vafa, Cumrun
 Communications in Mathematical Physics, Vol. 158, Issue 3
Walls, lines, and spectral dualities in 3d gauge theories
journal, May 2014
 Gadde, Abhijit; Gukov, Sergei; Putrov, Pavel
 Journal of High Energy Physics, Vol. 2014, Issue 5
Matter coupled AdS3 supergravities and their black strings
journal, May 2000
 Deger, N. S.; Kaya, A.; Sezgin, E.
 Nuclear Physics B, Vol. 573, Issue 12
Chern–Simons supergravity plus matter near the boundary of AdS3
journal, June 2001
 Deger, N. S.; Kaya, A.; Sezgin, E.
 Nuclear Physics B, Vol. 604, Issue 12
Gauged locally supersymmetric D=3 nonlinear sigma models
journal, November 2003
 de Wit, Bernard; Herger, Ivan; Samtleben, Henning
 Nuclear Physics B, Vol. 671
Renormalization Group Flows from D = 3, N = 2 Matter Coupled Gauged Supergravities
journal, November 2002
 Deger, Nihat Sadik
 Journal of High Energy Physics, Vol. 2002, Issue 11
Gauge theory correlators from noncritical string theory
journal, May 1998
 Gubser, S. S.; Klebanov, I. R.; Polyakov, A. M.
 Physics Letters B, Vol. 428, Issue 12
Anti de Sitter space and holography
journal, January 1998
 Witten, Edward
 Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2
The exact superconformal Rsymmetry extremizes Z
journal, May 2012
 Jafferis, Daniel L.
 Journal of High Energy Physics, Vol. 2012, Issue 5
Towards the Ftheorem: $ \mathcal{N} = 2 $ field theories on the threesphere
journal, June 2011
 Jafferis, Daniel L.; Klebanov, Igor R.; Pufu, Silviu S.
 Journal of High Energy Physics, Vol. 2011, Issue 6
Towards a derivation of holographic entanglement entropy
journal, May 2011
 Casini, Horacio; Huerta, Marina; Myers, Robert C.
 Journal of High Energy Physics, Vol. 2011, Issue 5
Aspects of N = 2 supersymmetric gauge theories in three dimensions
journal, August 1997
 Aharony, O.; Hanany, A.; Intriligator, K.
 Nuclear Physics B, Vol. 499, Issue 12
On mirror symmetry in three dimensional Abelian gauge theories
journal, April 1999
 Kapustin, Anton; Strassler, Matthew J.
 Journal of High Energy Physics, Vol. 1999, Issue 04
Indices for superconformal field theories in 3, 5 and 6 dimensions
journal, February 2008
 Bhattacharya, Jyotirmoy; Bhattacharyya, Sayantani; Minwalla, Shiraz
 Journal of High Energy Physics, Vol. 2008, Issue 02
Extremal $\CN=(2,2)$ 2D conformal field theories and constraints of modularity
journal, January 2008
 Gaberdiel, Matthias R.; Gukov, Sergei; Keller, Christoph A.
 Communications in Number Theory and Physics, Vol. 2, Issue 4
Superconformal indices of threedimensional theories related by mirror symmetry
journal, June 2011
 Krattenthaler, C.; Spiridonov, V. P.; Vartanov, G. S.
 Journal of High Energy Physics, Vol. 2011, Issue 6
Exactly marginal operators and duality in four dimensional N = 1 supersymmetric gauge theory
journal, July 1995
 Leigh, Robert G.; Strassler, Matthew J.
 Nuclear Physics B, Vol. 447, Issue 1
Central charges and U(1) _{ R } symmetries in Script N = 1 super YangMills
journal, November 2003
 Kutasov, David; Parnachev, Andrei; Sahakyan, David A.
 Journal of High Energy Physics, Vol. 2003, Issue 11
RG fixed points and flows in SQCD with adjoints
journal, January 2004
 Intriligator, Ken; Wecht, Brian
 Nuclear Physics B, Vol. 677, Issue 12
The $ \mathcal{N} $ = 1 superconformal index for class $ \mathcal{S} $ fixed points
journal, April 2014
 Beem, Christopher; Gadde, Abhijit
 Journal of High Energy Physics, Vol. 2014, Issue 4
New phenomena in SU(3) supersymmetric gauge theory
journal, August 1995
 Argyres, Philip C.; Douglas, Michael R.
 Nuclear Physics B, Vol. 448, Issue 12
Central charges and RG flow of stronglycoupled $ \mathcal{N}=2 $ theory
journal, March 2013
 Xie, Dan; Zhao, Peng
 Journal of High Energy Physics, Vol. 2013, Issue 3
Works referencing / citing this record:
Properties of the new N$$ \mathcal{N} $$ = 1 AdS4 vacuum of maximal supergravity
journal, January 2020
 Bobev, Nikolay; Fischbacher, Thomas; Pilch, Krzysztof
 Journal of High Energy Physics, Vol. 2020, Issue 1
Geometric constraints on the space of N $$ \mathcal{N} $$ = 2 SCFTs. Part I: physical constraints on relevant deformations
journal, February 2018
 Argyres, Philip; Lotito, Matteo; Lü, Yongchao
 Journal of High Energy Physics, Vol. 2018, Issue 2
3D supergravity from wrapped M5branes
journal, March 2016
 Karndumri, Parinya; Ó Colgáin, Eoin
 Journal of High Energy Physics, Vol. 2016, Issue 3
Multiplets of superconformal symmetry in diverse dimensions
journal, March 2019
 Córdova, Clay; Dumitrescu, Thomas T.; Intriligator, Kenneth
 Journal of High Energy Physics, Vol. 2019, Issue 3
Holographic RG flows for fourdimensional N = 2 $$ \mathcal{N}=2 $$ SCFTs
journal, June 2018
 Bobev, Nikolay; Cassani, Davide; Triendl, Hagen
 Journal of High Energy Physics, Vol. 2018, Issue 6
N $$ \mathcal{N} $$ =1 Lagrangians for generalized ArgyresDouglas theories
journal, October 2017
 Agarwal, Prarit; Sciarappa, Antonio; Song, Jaewon
 Journal of High Energy Physics, Vol. 2017, Issue 10
Anyonic Chains, Topological Defects, and Conformal Field Theory
journal, October 2017
 Buican, Matthew; Gromov, Andrey
 Communications in Mathematical Physics, Vol. 356, Issue 3
${\mathrm{SQED}}_{3}$ and ${\mathrm{SQCD}}_{3}$ : Phase transitions and integrability
journal, September 2019
 Santilli, Leonardo; Tierz, Miguel
 Physical Review D, Vol. 100, Issue 6
Nonperturbative Renormalization of Operators in NearConformal Systems Using Gradient Flows
journal, November 2018
 Carosso, Andrea; Hasenfratz, Anna; Neil, Ethan T.
 Physical Review Letters, Vol. 121, Issue 20
Landscape of Simple Superconformal Field Theories in 4D
journal, March 2019
 Maruyoshi, Kazunobu; Nardoni, Emily; Song, Jaewon
 Physical Review Letters, Vol. 122, Issue 12
Entanglement entropy: Holography and renormalization group
journal, September 2018
 Nishioka, Tatsuma
 Reviews of Modern Physics, Vol. 90, Issue 3