Abstract
We consider the (twisted) warped Virasoro group Diff(S1)⋉C∞(S1) in the presence of its three cocycles. We compute the Kirillov-Kostant-Souriau symplectic 2-form on coadjoint orbits. We then construct the Euclidean action of the ‘warped Schwarzian theory’ associated to the orbit with SL(2,ℝ)×U(1) stabilizer as the effective theory of the reparametrization over the base circle and evaluate the corresponding one-loop-exact path integral. We further discuss thermodynamics of the wSch theory in comparison with the complex SYK model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467][INSPIRE] .
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. 126B (1983) 41 [INSPIRE].
R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev. X 5 (2015) 041025 [arXiv:1506.05111][INSPIRE] .
R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849][INSPIRE] .
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818][INSPIRE] .
D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [arXiv:1703.04612][INSPIRE] .
E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].
A. Alekseev and S.L. Shatashvili, Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity, Nucl. Phys. B 323 (1989) 719 [INSPIRE].
I. Bakas, Conformal Invariance, the KdV Equation and Coadjoint Orbits of the Virasoro Algebra, Nucl. Phys. B 302 (1988) 189 [INSPIRE].
B. Rai and V.G.J. Rodgers, From Coadjoint Orbits to Scale Invariant WZNW Type Actions and 2-D Quantum Gravity Action, Nucl. Phys. B 341 (1990) 119 [INSPIRE].
A. Kirillov, Elements of the theory of representations, Springer, Berlin, Germany, (1976).
B. Kostant, Quantization and unitary representations, Lecture Notes in Math. 170 (1970).
J.-M. Souriau, Structure des syst̀emes dynamiques, Dunod, Paris, France (1970).
P.B. Wiegmann, Multivalued Functionals and Geometrical Approach for Quantization of Relativistic Particles and Strings, Nucl. Phys. B 323 (1989) 311 [INSPIRE].
J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263][INSPIRE] .
G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, JHEP 11 (2017) 046 [arXiv:1702.04266][INSPIRE] .
V. Ovsienko, Large coadjoint representation of virasoro-type lie algebras and differential operators on tensor-densities, math-ph/0602009.
G. Barnich, H. A. González and P. Salgado-ReboLledó, Geometric actions for three-dimensional gravity, Class. Quant. Grav. 35 (2018) 014003 [arXiv:1707.08887][INSPIRE] .
M. Vergne, Representations of lie groups and the orbit method, in Emmy Noether in Bryn Mawr, Springer New York, U.S.A., (1983), pp. 59–101.
R.F. Penna and C. Zukowski, Kinematic space and the orbit method, JHEP 07 (2019) 045 [arXiv:1812.02176][INSPIRE] .
B. Oblak, BMS Particles in Three Dimensions, Ph.D. thesis, Brussels U., 2016. arXiv:1610.08526. 10.1007/978-3-319-61878-4 [INSPIRE].
G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: I. Induced representations, JHEP 06 (2014) 129 [arXiv:1403.5803] [INSPIRE].
G.W. Delius, P. van Nieuwenhuizen and V.G.J. Rodgers, The Method of Coadjoint Orbits: An Algorithm for the Construction of Invariant Actions, Int. J. Mod. Phys. A 5 (1990) 3943 [INSPIRE].
H. Afshar, S. Detournay, D. Grumiller and B. Oblak, Near-Horizon Geometry and Warped Conformal Symmetry, JHEP 03 (2016) 187 [arXiv:1512.08233][INSPIRE] .
D.M. Hofman and A. Strominger, Chiral Scale and Conformal Invariance in 2D Quantum Field Theory, Phys. Rev. Lett. 107 (2011) 161601 [arXiv:1107.2917][INSPIRE] .
S. Detournay, T. Hartman and D.M. Hofman, Warped Conformal Field Theory, Phys. Rev. D 86 (2012) 124018 [arXiv:1210.0539][INSPIRE] .
C. Roger and J. Unterberger, The Schrödinger-Virasoro Lie group and algebra: From geometry to representation theory, Annales Henri Poincaŕe 7 (2006) 1477 [math-ph/0601050] [INSPIRE].
E. Arbarello, C. De Concini, V.G. Kac and C. Procesi, Moduli spaces of curves and representation theory, Commun. Math. Phys. 117 (1988) 1.
Y. Billig, Representations of the twisted heisenberg-virasoro algebra at level zero, Can. Math. Bull. 46 (2003) 529.
P. Chaturvedi, Y. Gu, W. Song and B. Yu, A note on the complex SYK model and warped CFTs, JHEP 12 (2018) 101 [arXiv:1808.08062][INSPIRE] .
G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP 03 (2015) 033 [arXiv:1502.00010] [INSPIRE].
I. Marshall, A lie algebraic setting for miura maps related to an energy dependent linear problem, Commun. Math. Phys. 133 (1990) 509.
V. Ovsienko and C. Roger, Extensions of Virasoro group and Virasoro algebra by modules of tensor-densities on S1 , hep-th/9409067 [INSPIRE].
J. Unterberger and C. Roger, The Schrödinger-Virasoro Algebra, Springer, Berlin, Germany, (2012).
W. Song and J. Xu, Correlation Functions of Warped CFT, JHEP 04 (2018) 067 [arXiv:1706.07621][INSPIRE] .
L. Apolo and W. Song, Bootstrapping holographic warped CFTs or: how I learned to stop worrying and tolerate negative norms, JHEP 07 (2018) 112 [arXiv:1804.10525][INSPIRE] .
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408][INSPIRE] .
[Addendum: Phys. Rev.D95 069904(2017)] W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [Addendum ibid. D 95 (2017) 069904] [arXiv:1610.08917] [INSPIRE].
D.J. Gross and V. Rosenhaus, A Generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093 [arXiv:1610.01569][INSPIRE] .
R. Bhattacharya, D.P. Jatkar and A. Kundu, Chaotic Correlation Functions with Complex Fermions, arXiv:1810.13217 [INSPIRE].
K. Bulycheva, A note on the SYK model with complex fermions, JHEP 12 (2017) 069 [arXiv:1706.07411][INSPIRE] .
J. Yoon, SYK Models and SYK-like Tensor Models with Global Symmetry, JHEP 10 (2017) 183 [arXiv:1707.01740][INSPIRE] .
P. Narayan and J. Yoon, Supersymmetric SYK Model with Global Symmetry, JHEP 08 (2018) 159 [arXiv:1712.02647][INSPIRE] .
D. Grumiller, R. McNees, J. Salzer, C. Valcárcel and D. Vassilevich, Menagerie of AdS2 boundary conditions, JHEP 10 (2017) 203 [arXiv:1708.08471][INSPIRE] .
H.A. González, D. Grumiller and J. Salzer, Towards a bulk description of higher spin SYK, JHEP 05 (2018) 083 [arXiv:1802.01562][INSPIRE] .
J. Liu and Y. Zhou, Note on global symmetry and SYK model, JHEP 05 (2019) 099 [arXiv:1901.05666][INSPIRE] .
T.G. Mertens and G.J. Turiaci, Defects in Jackiw-Teitelboim Quantum Gravity, JHEP 08 (2019) 127 [arXiv:1904.05228][INSPIRE].
H. Afshar, H.A. González, D. Grumiller and D. Vassilevich, Flat space holography and complex SYK, arXiv:1911.05739 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1908.08089
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Afshar, H.R. Warped Schwarzian theory. J. High Energ. Phys. 2020, 126 (2020). https://doi.org/10.1007/JHEP02(2020)126
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2020)126