Gravitational action for a massive Majorana fermion in \(2d\) quantum gravity. (English) Zbl 07821551
Summary: We compute the gravitational action of a free massive Majorana fermion coupled to two-dimensional gravity on compact Riemann surfaces of arbitrary genus. The structure is similar to the case of the massive scalar. The small-mass expansion of the gravitational yields the Liouville action at zeroth order, and we can identify the Mabuchi action at first order. While the massive Majorana action is a conformal deformation of the massless Majorana CFT, we find an action different from the one given by the David-Distler-Kawai (DDK) ansatz.
MSC:
| 81-XX | Quantum theory |
References:
| [1] | A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B103 (1981) 207 [INSPIRE]. |
| [2] | T.L. Curtright and C.B. Thorn, Conformally Invariant Quantization of the Liouville Theory, Phys. Rev. Lett.48 (1982) 1309 [Erratum ibid.48 (1982) 1768] [INSPIRE]. |
| [3] | E. D’Hoker and R. Jackiw, Liouville Field Theory, Phys. Rev. D26 (1982) 3517 [INSPIRE]. |
| [4] | Braaten, E.; Curtright, T.; Thorn, CB, An Exact Operator Solution of the Quantum Liouville Field Theory, Annals Phys., 147, 365 (1983) · Zbl 0538.35071 · doi:10.1016/0003-4916(83)90214-2 |
| [5] | J.-L. Gervais and A. Neveu, Nonstandard Two-dimensional Critical Statistical Models From Liouville Theory, Nucl. Phys. B257 (1985) 59 [INSPIRE]. |
| [6] | F. David, Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge, Mod. Phys. Lett. A3 (1988) 1651 [INSPIRE]. |
| [7] | Distler, J.; Kawai, H., Conformal Field Theory and 2D Quantum Gravity, Nucl. Phys. B, 321, 509 (1989) · doi:10.1016/0550-3213(89)90354-4 |
| [8] | N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl.102 (1990) 319 [INSPIRE]. · Zbl 0790.53059 |
| [9] | Dorn, H.; Otto, HJ, Two and three point functions in Liouville theory, Nucl. Phys. B, 429, 375 (1994) · Zbl 1020.81770 · doi:10.1016/0550-3213(94)00352-1 |
| [10] | A.B. Zamolodchikov and Al.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B477 (1996) 577 [hep-th/9506136] [INSPIRE]. · Zbl 0925.81301 |
| [11] | Harlow, D.; Maltz, J.; Witten, E., Analytic Continuation of Liouville Theory, JHEP, 12, 071 (2011) · Zbl 1306.81287 · doi:10.1007/JHEP12(2011)071 |
| [12] | Bilal, A.; Leduc, L., 2D quantum gravity on compact Riemann surfaces and two-loop partition function: a first principles approach, Nucl. Phys. B, 896, 360 (2015) · Zbl 1331.83022 · doi:10.1016/j.nuclphysb.2015.04.026 |
| [13] | Leduc, L.; Bilal, A., 2D quantum gravity at three loops: a counterterm investigation, Nucl. Phys. B, 903, 226 (2016) · Zbl 1332.83043 · doi:10.1016/j.nuclphysb.2015.12.013 |
| [14] | Bautista, T.; Dabholkar, A.; Erbin, H., Quantum Gravity from Timelike Liouville theory, JHEP, 10, 284 (2019) · Zbl 1427.83018 · doi:10.1007/JHEP10(2019)284 |
| [15] | Bautista, T.; Erbin, H.; Kudrna, M., BRST cohomology of timelike Liouville theory, JHEP, 05, 029 (2020) · Zbl 1437.83077 · doi:10.1007/JHEP05(2020)029 |
| [16] | P. Ginsparg and G. Moore, Lectures on 2D Gravity and 2D String Theory (TASI 1992). |
| [17] | Teschner, J., Liouville theory revisited, Class. Quant. Grav., 18, R153 (2001) · Zbl 1022.81047 · doi:10.1088/0264-9381/18/23/201 |
| [18] | Nakayama, Y., Liouville field theory: a decade after the revolution, Int. J. Mod. Phys. A, 19, 2771 (2004) · Zbl 1080.81056 · doi:10.1142/S0217751X04019500 |
| [19] | Al.B. Zamolodchikov and A.B. Zamolodchikov, Lectures on Liouville Theory and Matrix Models, http://qft.itp.ac.ru/ZZ.pdf . · Zbl 0946.81070 |
| [20] | S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE]. · Zbl 1272.81178 |
| [21] | de Lacroix, C.; Erbin, H., A short note on dynamics and degrees of freedom in 2d classical gravity, Gen. Rel. Grav., 52, 9 (2020) · Zbl 1437.83090 · doi:10.1007/s10714-020-2662-7 |
| [22] | Ferrari, F.; Klevtsov, S.; Zelditch, S., Random geometry, quantum gravity and the Kähler potential, Phys. Lett. B, 705, 375 (2011) · doi:10.1016/j.physletb.2011.09.098 |
| [23] | Ferrari, F.; Klevtsov, S.; Zelditch, S., Gravitational Actions in Two Dimensions and the Mabuchi Functional, Nucl. Phys. B, 859, 341 (2012) · Zbl 1246.83074 · doi:10.1016/j.nuclphysb.2012.02.003 |
| [24] | Ferrari, F.; Klevtsov, S., FQHE on curved backgrounds, free fields and large N, JHEP, 12, 086 (2014) · Zbl 1333.83046 · doi:10.1007/JHEP12(2014)086 |
| [25] | Bilal, A.; Ferrari, F.; Klevtsov, S., 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions, Nucl. Phys. B, 880, 203 (2014) · Zbl 1284.83049 · doi:10.1016/j.nuclphysb.2014.01.005 |
| [26] | de Lacroix, C.; Erbin, H.; Svanes, EE, Mabuchi spectrum from the minisuperspace, Phys. Lett. B, 758, 186 (2016) · Zbl 1365.81126 · doi:10.1016/j.physletb.2016.05.013 |
| [27] | C. de Lacroix, H. Erbin and E.E. Svanes, Minisuperspace computation of the Mabuchi spectrum, Class. Quant. Grav.35 (2018) 185011 [arXiv:1704.05855] [INSPIRE]. · Zbl 1409.83058 |
| [28] | Lacoin, H.; Rhodes, R.; Vargas, V., Path integral for quantum Mabuchi K-energy, Duke Math. J., 171, 483 (2022) · Zbl 1513.81114 · doi:10.1215/00127094-2021-0007 |
| [29] | Mabuchi, T., k-energy maps integrating futaki invariants, Tohoku Math. J., 38, 575 (1986) · Zbl 0619.53040 · doi:10.2748/tmj/1178228410 |
| [30] | D.H. Phong and J. Sturm, Lectures on Stability and Constant Scalar Curvature, arXiv:0801.4179. · Zbl 1220.53087 |
| [31] | Bilal, A.; Leduc, L., 2D quantum gravity on compact Riemann surfaces with non-conformal matter, JHEP, 01, 089 (2017) · Zbl 1373.83040 · doi:10.1007/JHEP01(2017)089 |
| [32] | Bilal, A.; de Lacroix, C., 2D gravitational Mabuchi action on Riemann surfaces with boundaries, JHEP, 11, 154 (2017) · Zbl 1383.83097 · doi:10.1007/JHEP11(2017)154 |
| [33] | N.E. Mavromatos and J.L. Miramontes, Regularizing the Functional Integral in 2D Quantum Gravity, Mod. Phys. Lett. A4 (1989) 1847 [INSPIRE]. |
| [34] | E. D’Hoker and P.S. Kurzepa, 2-D Quantum Gravity and Liouville Theory, Mod. Phys. Lett. A5 (1990) 1411 [INSPIRE]. · Zbl 1020.81826 |
| [35] | E. D’Hoker, Equivalence of Liouville theory and 2-D quantum gravity, Mod. Phys. Lett. A6 (1991) 745 [INSPIRE]. · Zbl 1021.81810 |
| [36] | Ribault, S.; Santachiara, R., Liouville theory with a central charge less than one, JHEP, 08, 109 (2015) · Zbl 1388.81694 · doi:10.1007/JHEP08(2015)109 |
| [37] | R. Rhodes and V. vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity, arXiv:1602.07323 [INSPIRE]. · Zbl 1386.60139 |
| [38] | A. Kupiainen, Constructive Liouville Conformal Field Theory, arXiv:1611.05243 [INSPIRE]. · Zbl 1446.81036 |
| [39] | Schmidhuber, C., Exactly marginal operators and running coupling constants in 2-D gravity, Nucl. Phys. B, 404, 342 (1993) · Zbl 1043.81700 · doi:10.1016/0550-3213(93)90483-6 |
| [40] | Ambjorn, J.; Ghoroku, K., 2-d Quantum gravity coupled to renormalizable matter fields, Int. J. Mod. Phys. A, 9, 5689 (1994) · Zbl 0985.81723 · doi:10.1142/S0217751X94002338 |
| [41] | Bilal, A.; de Lacroix, C.; Erbin, H., Effective gravitational action for 2D massive fermions, JHEP, 11, 165 (2021) · Zbl 1521.83039 · doi:10.1007/JHEP11(2021)165 |
| [42] | H. Erbin, Gravitational Action for a Massive Majorana Fermion in 2d Quantum Gravity - Notes, https://harolderbin.com/files/research/notes/notes_2d_gravity_majorana_fermion.pdf. |
| [43] | Zuber, JB; Itzykson, C., Quantum Field Theory and the Two-Dimensional Ising Model, Phys. Rev. D, 15, 2875 (1977) · doi:10.1103/PhysRevD.15.2875 |
| [44] | Bander, M.; Itzykson, C., Quantum Field Theory Calculation of Two-Dimensional Ising Model Correlation Function, Phys. Rev. D, 15, 463 (1977) · doi:10.1103/PhysRevD.15.463 |
| [45] | P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, 2nd edition, Springer (1999). · Zbl 0869.53052 |
| [46] | M. Namuduri and A. Bilal, Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces, arXiv:2308.05802. |
| [47] | Polyakov, AM, Quantum Geometry of Fermionic Strings, Phys. Lett. B, 103, 211 (1981) · doi:10.1016/0370-2693(81)90744-9 |
| [48] | G. Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford University Press (2009). · Zbl 1434.82004 |
| [49] | R. Blumenhagen, D. Lüst and S. Theisen, Basic Concepts of String Theory, Springer (2014). · Zbl 1262.81001 |
| [50] | C. Itzykson and J.-M. Drouffe, Statistical Field Theory: Volume 2, Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems, Cambridge University Press (1991). |
| [51] | A. Wipf, Introduction to Supersymmetry, https://www.tpi.uni-jena.de/qfphysics/homepage/wipf/lectures/susy/susyhead.pdf, (2016). |
| [52] | D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press (2012). · Zbl 1245.83001 |
| [53] | S. Blau, M. Visser and A. Wipf, Determinants, Dirac Operators, and One Loop Physics, Int. J. Mod. Phys. A4 (1989) 1467 [INSPIRE]. |
| [54] | H. Erbin, Introduction to String Field Theory, http://www.lpthe.jussieu.fr/ erbin/files/reviews/string_theory.pdf, (2019). · Zbl 1457.81002 |
| [55] | D’Hoker, E.; Phong, DH, The Geometry of String Perturbation Theory, Rev. Mod. Phys., 60, 917 (1988) · doi:10.1103/RevModPhys.60.917 |
| [56] | A. Bilal and F. Ferrari, Multi-Loop Zeta Function Regularization and Spectral Cutoff in Curved Spacetime. · Zbl 1284.81220 |
| [57] | A. Dettki, I. Sachs and A. Wipf, Generalized gauged Thirring model on curved space-times, hep-th/9308067 [INSPIRE]. |
| [58] | Steiner, J., A geometrical mass and its extremal properties for metrics on s^2, Duke Math. J., 129, 63 (2005) · Zbl 1144.53055 · doi:10.1215/S0012-7094-04-12913-6 |
| [59] | P. Doyle and J. Steiner, Blowing bubbles on the torus, arXiv:1710.09865. |
| [60] | K. Okikiolu, Extremals for Logarithmic Hardy-Littlewood-Sobolev inequalities on compact manifolds, math/0603717. · Zbl 1140.58003 |
| [61] | Okikiolu, K., A negative mass theorem for surfaces of positive genus, Commun. Math. Phys., 290, 1025 (2009) · Zbl 1184.53046 · doi:10.1007/s00220-008-0722-z |
| [62] | Morpurgo, C., The logarithmic hardy-littlewood-sobolev inequality and extremals of zeta functions on s_n, Geom. Funct. Anal., 6, 146 (1996) · Zbl 0852.58079 · doi:10.1007/BF02246771 |
| [63] | Van Proeyen, A., Tools for supersymmetry, Ann. U. Craiova Phys., 9, 1 (1999) |
| [64] | G.W. Moore, Gravitational phase transitions and the Sine-Gordon model, hep-th/9203061 [INSPIRE]. |
| [65] | Eguchi, T., c = 1 Liouville theory perturbed by the black hole mass operator, Phys. Lett. B, 316, 74 (1993) · Zbl 0973.81526 · doi:10.1016/0370-2693(93)90660-A |
| [66] | Hsu, E.; Kutasov, D., The Gravitational Sine-Gordon model, Nucl. Phys. B, 396, 693 (1993) · doi:10.1016/0550-3213(93)90668-F |
| [67] | C. Schmidhuber, RG flow in 2-d field theory coupled to gravity, in the proceedings of the 28th International Symposium on Particle Theory, Wendisch-Rietz, Germany, August 30 - September 03 (1994) [hep-th/9412051] [INSPIRE]. |
| [68] | M. Reuter, Weyl invariant quantization of Liouville field theory, in the proceedings of the 3rd International Conference on Renormalization Group (RG 96), Dubna, Russian Federation, August 26-31 (1996) [hep-th/9612158] [INSPIRE]. |
| [69] | Reuter, M.; Wetterich, C., Quantum Liouville field theory as solution of a flow equation, Nucl. Phys. B, 506, 483 (1997) · Zbl 0925.81330 · doi:10.1016/S0550-3213(97)00447-1 |
| [70] | B. Carneiro da Cunha and E.J. Martinec, Closed string tachyon condensation and world sheet inflation, Phys. Rev. D68 (2003) 063502 [hep-th/0303087] [INSPIRE]. |
| [71] | Doyon, B.; Fonseca, P., Ising field theory on a Pseudosphere, J. Stat. Mech., 0407, P07002 (2004) · Zbl 1076.82008 |
| [72] | Al.B. Zamolodchikov, Scaling Lee-Yang model on a sphere. 1. Partition function, JHEP07 (2002) 029 [hep-th/0109078] [INSPIRE]. |
| [73] | Al.B. Zamolodchikov, Perturbed conformal field theory on fluctuating sphere, in the proceedings of the 1st Balkan Workshop on Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model: Perspectives of Balkans Collaboration, Vrnjacka Banja, Serbia, August 29 - September 02 (2003) [hep-th/0508044] [INSPIRE]. |
| [74] | Y. Ishimoto and Al.B. Zamolodchikov, Massive Majorana fermion coupled to 2D gravity and random lattice Ising model, Theor. Math. Phys.147 (2006) 755 [INSPIRE]. · Zbl 1177.81093 |
| [75] | Martinec, EJ; Moore, WE, Modeling Quantum Gravity Effects in Inflation, JHEP, 07, 053 (2014) · doi:10.1007/JHEP07(2014)053 |
| [76] | Dorn, H.; Otto, HJ, Analysis of all dimensionful parameters relevant to gravitational dressing of conformal theories, Phys. Lett. B, 280, 204 (1992) · doi:10.1016/0370-2693(92)90056-A |
| [77] | Kawai, H.; Kitazawa, Y.; Ninomiya, M., Ultraviolet stable fixed point and scaling relations in (2+epsilon)-dimensional quantum gravity, Nucl. Phys. B, 404, 684 (1993) · Zbl 1009.81524 · doi:10.1016/0550-3213(93)90594-F |
| [78] | J. Polchinski, String Theory: Volume 1, An Introduction to the Bosonic String, Cambridge University Press (2005). · Zbl 1075.81054 |
| [79] | T. Bautista and A. Dabholkar, Quantum Cosmology Near Two Dimensions, Phys. Rev. D94 (2016) 044017 [arXiv:1511.07450] [INSPIRE]. |
| [80] | Iorio, A.; O’Raifeartaigh, L.; Sachs, I.; Wiesendanger, C., Weyl gauging and conformal invariance, Nucl. Phys. B, 495, 433 (1997) · Zbl 0935.83026 · doi:10.1016/S0550-3213(97)00190-9 |
| [81] | Klebanov, IR; Kogan, II; Polyakov, AM, Gravitational dressing of renormalization group, Phys. Rev. Lett., 71, 3243 (1993) · Zbl 0972.81603 · doi:10.1103/PhysRevLett.71.3243 |
| [82] | Tanii, Y.; Kojima, S-I; Sakai, N., Physical scaling and renormalization group in two-dimensional gravity, Phys. Lett. B, 322, 59 (1994) · doi:10.1016/0370-2693(94)90491-X |
| [83] | Dorn, H., On gravitational dressing of renormalization group beta functions beyond lowest order of perturbation theory, Phys. Lett. B, 343, 81 (1995) · doi:10.1016/0370-2693(94)01450-Q |
| [84] | Schmidhuber, C., RG flow on random surfaces with handles and closed string field theory, Nucl. Phys. B, 453, 156 (1995) · Zbl 0925.81109 · doi:10.1016/0550-3213(95)00436-V |
| [85] | Dorn, H., On scheme dependence of gravitational dressing of renormalization group functions, Nucl. Phys. B Proc. Suppl., 49, 81 (1996) · Zbl 0957.81560 · doi:10.1016/0920-5632(96)00320-9 |
| [86] | Kutasov, D., Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B, 220, 153 (1989) · doi:10.1016/0370-2693(89)90028-2 |
| [87] | H. Sonoda, Connection on the theory space, in the proceedings of the International Conference on Strings 93, Berkeley, U.S.A., May 24-29 (1993) [hep-th/9306119] [INSPIRE]. · Zbl 0844.58092 |
| [88] | Ranganathan, K., Nearby CFTs in the operator formalism: the role of a connection, Nucl. Phys. B, 408, 180 (1993) · Zbl 1030.81521 · doi:10.1016/0550-3213(93)90136-D |
| [89] | Ranganathan, K.; Sonoda, H.; Zwiebach, B., Connections on the state space over conformal field theories, Nucl. Phys. B, 414, 405 (1994) · Zbl 1007.81543 · doi:10.1016/0550-3213(94)90436-7 |
| [90] | Carneiro da Cunha, B., Crumpled Wires and Liouville Field Theory, EPL, 88, 31001 (2009) · doi:10.1209/0295-5075/88/31001 |
| [91] | Dorn, H.; Otto, HJ, On Scaling Dimensions of Vertex Operators in Conformally Gauged 2-D Quantum Gravity, Phys. Lett. B, 232, 327 (1989) · doi:10.1016/0370-2693(89)90752-1 |
| [92] | P.H. Ginsparg, Matrix models of 2-d gravity, hep-th/9112013 [INSPIRE]. · Zbl 0985.82500 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
