Induced Action for Conformal Higher Spins from Worldline Path Integrals
Abstract
:1. Introduction
2. Induced Action for Conformal Higher Spins
2.1. Noether Interaction and Symmetries
2.2. Effective Action and Worldline Path Integral
2.3. The Quadratic Action
3. Discussion and Conclusions
Acknowledgments
Conflicts of Interest
Appendix A. Worldline Phase Space Propagators
Appendix B. Special Functions
Appendix C. Transverse-Traceless Projectors
References
- Kaku, M.; Townsend, P.K.; van Nieuwenhuizen, P. Properties of Conformal Supergravity. Phys. Rev. D 1978, 17, 3179–3187. [Google Scholar] [CrossRef]
- Bergshoeff, E.; de Roo, M.; de Wit, B. Extended Conformal Supergravity. Nucl. Phys. B 1981, 182, 173–204. [Google Scholar] [CrossRef]
- Fradkin, E.S.; Tseytlin, A.A. Conformal Supergravity. Phys. Rep. 1985, 119, 233–362. [Google Scholar] [CrossRef]
- Siegel, W. All Free Conformal Representations in All Dimensions. Int. J. Mod. Phys. A 1989, 4, 2015–2020. [Google Scholar] [CrossRef]
- Fradkin, E.S.; Linetsky, V.Y. Cubic Interaction in Conformal Theory of Integer Higher Spin Fields in Four-dimensional Space-time. Phys. Lett. B 1989, 231, 97–106. [Google Scholar] [CrossRef]
- Tseytlin, A.A. On limits of superstring in AdS5 × S5. Theor. Math. Phys. 2002, 133, 1376–1389. [Google Scholar] [CrossRef]
- Segal, A.Y. Conformal higher spin theory. Nucl. Phys. B 2003, 664, 59–130. [Google Scholar] [CrossRef]
- Shaynkman, O.V.; Tipunin, I.Y.; Vasiliev, M.A. Unfolded form of conformal equations in M dimensions and o(M + 2) modules. Rev. Math. Phys. 2006, 18, 823–886. [Google Scholar] [CrossRef]
- Marnelius, R. Lagrangian conformal higher spin theory. arXiv 2008, arXiv:0805.4686. [Google Scholar]
- Vasiliev, M.A. Bosonic conformal higher-spin fields of any symmetry. Nucl. Phys. B 2010, 829, 176–224. [Google Scholar] [CrossRef]
- Bekaert, X.; Joung, E.; Mourad, J. Effective action in a higher-spin background. J. High Energy Phys. 2011, 2011, 48. [Google Scholar] [CrossRef]
- Bandos, I.A.; de Azcarraga, J.A.; Meliveo, C. Extended supersymmetry in massless conformal higher spin theory. Nucl. Phys. B 2011, 853, 760–776. [Google Scholar] [CrossRef]
- Bekaert, X.; Grigoriev, M. Notes on the ambient approach to boundary values of AdS gauge fields. J. Phys. A 2013, 46, 214008. [Google Scholar] [CrossRef]
- Haehnel, P.; McLoughlin, T. Conformal Higher Spin Theory and Twistor Space Actions. arXiv 2016, arXiv:1604.08209. [Google Scholar]
- Metsaev, R.R. Ordinary-derivative formulation of conformal low spin fields. J. High Energy Phys. 2012, 2012, 64. [Google Scholar] [CrossRef]
- Metsaev, R.R. Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields. J. High Energy Phys. 2012, 2012, 62. [Google Scholar] [CrossRef]
- Nutma, T.; Taronna, M. On conformal higher spin wave operators. J. High Energy Phys. 2014, 2014, 66. [Google Scholar] [CrossRef]
- Grigoriev, M.; Tseytlin, A.A. On conformal higher spins in curved background. J. Phys. A 2017, 50, 125401. [Google Scholar] [CrossRef]
- Beccaria, M.; Tseytlin, A.A. On induced action for conformal higher spins in curved background. Nucl. Phys. B 2017, 919, 359–383. [Google Scholar] [CrossRef]
- Fradkin, E.S.; Tseytlin, A.A. One Loop Beta Function in Conformal Supergravities. Nucl. Phys. B 1982, 203, 157–178. [Google Scholar] [CrossRef]
- Fradkin, E.S.; Tseytlin, A.A. Conformal Anomaly in Weyl Theory and Anomaly Free Superconformal Theories. Phys. Lett. B 1984, 134, 187–193. [Google Scholar] [CrossRef]
- Giombi, S.; Klebanov, I.R.; Pufu, S.S.; Safdi, B.R.; Tarnopolsky, G. AdS Description of Induced Higher-Spin Gauge Theory. J. High Energy Phys. 2013, 2013, 16. [Google Scholar] [CrossRef]
- Tseytlin, A.A. On partition function and Weyl anomaly of conformal higher spin fields. Nucl. Phys. B 2013, 877, 598–631. [Google Scholar] [CrossRef]
- Giombi, S.; Klebanov, I.R.; Safdi, B.R. Higher Spin AdSd+1/CFTd at One Loop. Phys. Rev. D 2014, 89, 084004. [Google Scholar] [CrossRef]
- Beccaria, M.; Tseytlin, A.A. Higher spins in AdS5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT. J. High Energy Phys. 2014, 2014, 114. [Google Scholar] [CrossRef]
- Beccaria, M.; Tseytlin, A.A. On higher spin partition functions. J. Phys. A 2015, 48, 275401. [Google Scholar] [CrossRef]
- Vasiliev, M.A. Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions. Phys. Lett. B 1990, 243, 378. [Google Scholar] [CrossRef]
- Vasiliev, M.A. Properties of equations of motion of interacting gauge fields of all spins in (3+1)-dimensions. Class. Quant. Grav. 1991, 8, 1387–1417. [Google Scholar] [CrossRef]
- Vasiliev, M.A. More on equations of motion for interacting massless fields of all spins in (3+1)-dimensions. Phys. Lett. B 1992, 285, 225–234. [Google Scholar] [CrossRef]
- Vasiliev, M.A. Higher spin gauge theories: Star product and AdS space. In The Many Faces of the Superworld; Shifman, M.A., Ed.; World Scientific: Singapore, 2000. [Google Scholar]
- Vasiliev, M.A. Nonlinear equations for symmetric massless higher spin fields in (A)dS(d). Phys. Lett. B 2003, 567, 139–151. [Google Scholar] [CrossRef]
- Bekaert, X.; Cnockaert, S.; Iazeolla, C.; Vasiliev, M.A. Nonlinear higher spin theories in various dimensions. arXiv 2005, arXiv:hep-th/0503128. [Google Scholar]
- Didenko, V.E.; Skvortsov, E.D. Elements of Vasiliev theory. arXiv 2014, arXiv:1401.2975. [Google Scholar]
- Sezgin, E.; Sundell, P. Massless higher spins and holography. Nucl. Phys. B 2002, 644, 303–370, Erratum in 2003, 660, 403. [Google Scholar] [CrossRef]
- Klebanov, I.R.; Polyakov, A.M. AdS dual of the critical O(N) vector model. Phys. Lett. B 2002, 550, 213–219. [Google Scholar] [CrossRef]
- Giombi, S.; Yin, X. Higher Spin Gauge Theory and Holography: The Three-Point Functions. J. High Energy Phys. 2010, 2010, 115. [Google Scholar] [CrossRef]
- Giombi, S.; Yin, X. Higher Spins in AdS and Twistorial Holography. J. High Energy Phys. 2011, 2011, 86. [Google Scholar] [CrossRef]
- Giombi, S.; Yin, X. The Higher Spin/Vector Model Duality. J. Phys. A 2013, 46, 214003. [Google Scholar] [CrossRef]
- Giombi, S. Higher Spin—CFT Duality. In New Frontiers in Fields and Strings, Proceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, 1–26 June 2015; World Scientific: Singapore, 2016; p. 137. [Google Scholar]
- Liu, H.; Tseytlin, A.A. D = 4 superYang-Mills, D = 5 gauged supergravity, and D = 4 conformal supergravity. Nucl. Phys. B 1998, 533, 88–108. [Google Scholar] [CrossRef]
- Boulanger, N.; Sundell, P. An action principle for Vasiliev’s four-dimensional higher-spin gravity. J. Phys. A 2011, 44, 495402. [Google Scholar] [CrossRef]
- Boulanger, N.; Colombo, N.; Sundell, P. A minimal BV action for Vasiliev’s four-dimensional higher spin gravity. J. High Energy Phys. 2012, 2012, 43. [Google Scholar] [CrossRef]
- Boulanger, N.; Sezgin, E.; Sundell, P. 4D Higher Spin Gravity with Dynamical Two-Form as a Frobenius-Chern-Simons Gauge Theory. arXiv 2015, arXiv:1505.04957. [Google Scholar]
- Bonezzi, R.; Boulanger, N.; Sezgin, E.; Sundell, P. An Action for Matter Coupled Higher Spin Gravity in Three Dimensions. J. High Energy Phys. 2016, 2016, 3. [Google Scholar] [CrossRef]
- Bonezzi, R.; Boulanger, N.; Sezgin, E.; Sundell, P. Frobenius–Chern–Simons gauge theory. J. Phys. A 2017, 50, 055401. [Google Scholar] [CrossRef]
- Bekaert, X.; Erdmenger, J.; Ponomarev, D.; Sleight, C. Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange. J. High Energy Phys. 2015, 2015, 170. [Google Scholar] [CrossRef]
- Bekaert, X.; Erdmenger, J.; Ponomarev, D.; Sleight, C. Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory. J. High Energy Phys. 2015, 2015, 149. [Google Scholar] [CrossRef]
- Sleight, C.; Taronna, M. Higher Spin Interactions from Conformal Field Theory: The Complete Cubic Couplings. Phys. Rev. Lett. 2016, 116, 181602. [Google Scholar] [CrossRef] [PubMed]
- Gopakumar, R. From free fields to AdS. Phys. Rev. D 2004, 70, 025009. [Google Scholar] [CrossRef]
- Gopakumar, R. From free fields to AdS. 2. Phys. Rev. D 2004, 70, 025010. [Google Scholar] [CrossRef]
- Gopakumar, R. From free fields to AdS: III. Phys. Rev. D 2005, 72, 066008. [Google Scholar] [CrossRef]
- Maldacena, J.M. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 1999, 38, 1113–1133. [Google Scholar] [CrossRef] [Green Version]
- Gubser, S.S.; Klebanov, I.R.; Polyakov, A.M. Gauge theory correlators from noncritical string theory. Phys. Lett. B 1998, 428, 105–114. [Google Scholar] [CrossRef]
- Witten, E. Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 1998, 2, 253–291. [Google Scholar] [CrossRef]
- Eastwood, M.G. Higher symmetries of the Laplacian. Ann. Math. 2005, 161, 1645–1665. [Google Scholar] [CrossRef]
- Balasubramanian, V.; Gimon, E.G.; Minic, D.; Rahmfeld, J. Four-dimensional conformal supergravity from AdS space. Phys. Rev. D 2001, 63, 104009. [Google Scholar] [CrossRef]
- Compere, G.; Marolf, D. Setting the boundary free in AdS/CFT. Class. Quant. Grav. 2008, 25, 195014. [Google Scholar] [CrossRef]
- Schubert, C. Perturbative quantum field theory in the string inspired formalism. Phys. Rept. 2001, 355, 73–234. [Google Scholar] [CrossRef]
- Bern, Z.; Kosower, D.A. The Computation of loop amplitudes in gauge theories. Nucl. Phys. B 1992, 379, 451–561. [Google Scholar] [CrossRef]
- Strassler, M.J. Field theory without Feynman diagrams: One loop effective actions. Nucl. Phys. B 1992, 385, 145–184. [Google Scholar] [CrossRef]
- Bastianelli, F.; van Nieuwenhuizen, P. Trace anomalies from quantum mechanics. Nucl. Phys. B 1993, 389, 53–80. [Google Scholar] [CrossRef]
- D’Hoker, E.; Gagne, D.G. Worldline path integrals for fermions with general couplings. Nucl. Phys. B 1996, 467, 297–312. [Google Scholar] [CrossRef]
- Reuter, M.; Schmidt, M.G.; Schubert, C. Constant external fields in gauge theory and the spin 0, 1/2, 1 path integrals. Ann. Phys. 1997, 259, 313–365. [Google Scholar] [CrossRef]
- Bastianelli, F.; Zirotti, A. Worldline formalism in a gravitational background. Nucl. Phys. B 2002, 642, 372–388. [Google Scholar] [CrossRef]
- Bastianelli, F.; Corradini, O.; Zirotti, A. Dimensional regularization for N=1 supersymmetric sigma models and the worldline formalism. Phys. Rev. D 2003, 67, 104009. [Google Scholar] [CrossRef]
- Bastianelli, F.; Benincasa, P.; Giombi, S. Worldline approach to vector and antisymmetric tensor fields. J. High Energy Phys. 2005, 2005, 10. [Google Scholar] [CrossRef]
- Dai, P.; Huang, Y.T.; Siegel, W. Worldgraph Approach to Yang-Mills Amplitudes from N=2 Spinning Particle. J. High Energy Phys. 2008, 2008, 27. [Google Scholar] [CrossRef]
- Bastianelli, F.; Bonezzi, R. One-loop quantum gravity from a worldline viewpoint. J. High Energy Phys. 2013, 2013, 16. [Google Scholar] [CrossRef]
- Bastianelli, F.; Bonezzi, R.; Corradini, O.; Latini, E. Particles with non abelian charges. J. High Energy Phys. 2013, 2013, 98. [Google Scholar] [CrossRef]
- Bastianelli, F.; van Nieuwenhuizen, P. Path Integrals and Anomalies in Curved Space; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Gershun, V.D.; Tkach, V.I. Classical And Quantum Dynamics Of Particles With Arbitrary Spin. J. Exp. Theor. Phys. Lett. 1979, 29, 288–291. [Google Scholar]
- Henneaux, M.; Teitelboim, C. First and second quantized point particles of any spin. In Quantum Mechanics of Fundamental Systems 2; Springer: Berlin, Germany, 1989; pp. 113–152. [Google Scholar]
- Howe, P.S.; Penati, S.; Pernici, M.; Townsend, P.K. Wave Equations for Arbitrary Spin From Quantization of the Extended Supersymmetric Spinning Particle. Phys. Lett. B 1988, 215, 555–558. [Google Scholar] [CrossRef]
- Kuzenko, S.M.; Yarevskaya, Z.V. Conformal invariance, N extended supersymmetry and massless spinning particles in anti-de Sitter space. Mod. Phys. Lett. A 1996, 11, 1653–1664. [Google Scholar] [CrossRef]
- Bastianelli, F.; Corradini, O.; Latini, E. Spinning particles and higher spin fields on (A)dS backgrounds. J. High Energy Phys. 2008, 2008, 54. [Google Scholar] [CrossRef]
- Bastianelli, F.; Corradini, O.; Waldron, A. Detours and Paths: BRST Complexes and Worldline Formalism. J. High Energy Phys. 2009, 2009, 17. [Google Scholar] [CrossRef]
- Bastianelli, F.; Bonezzi, R.; Corradini, O.; Latini, E. Effective action for higher spin fields on (A)dS backgrounds. J. High Energy Phys. 2012, 2012, 113. [Google Scholar] [CrossRef]
- Segal, A.Y. Point particle in general background fields versus gauge theories of traceless symmetric tensors. Int. J. Mod. Phys. A 2003, 18, 4999–5021. [Google Scholar] [CrossRef]
- Bonezzi, R.; Corradini, O.; Franchino Vinas, S.A.; Pisani, P.A.G. Worldline approach to noncommutative field theory. J. Phys. A 2012, 45, 405401. [Google Scholar] [CrossRef]
- Ahmadiniaz, N.; Corradini, O.; D’Ascanio, D.; Estrada-Jiménez, S.; Pisani, P. Noncommutative U(1) gauge theory from a worldline perspective. J. High Energy Phys. 2015, 2015, 69. [Google Scholar] [CrossRef]
- Craigie, N.S.; Dobrev, V.K.; Todorov, I.T. Conformally Covariant Composite Operators in Quantum Chromodynamics. Ann. Phys. 1985, 159, 411–444. [Google Scholar] [CrossRef]
- Berends, F.A.; Burgers, G.J.H.; van Dam, H. Explicit Construction of Conserved Currents for Massless Fields of Arbitrary Spin. Nucl. Phys. B 1986, 271, 429–441. [Google Scholar] [CrossRef]
- Bekaert, X.; Joung, E.; Mourad, J. On higher spin interactions with matter. J. High Energy Phys. 2009, 2009, 126. [Google Scholar] [CrossRef]
- Weyl, H. Quantum mechanics and group theory. Z. Phys. 1927, 46, 1. [Google Scholar] [CrossRef]
- Wigner, E.P. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 1932, 40, 749–759. [Google Scholar] [CrossRef]
- Moyal, J.E. Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc. 1949, 45, 99–124. [Google Scholar] [CrossRef]
- Sato, M.A. Operator Ordering and Perturbation Expansion in the Path Integration Formalism. Prog. Theor. Phys. 1977, 58, 1262–1270. [Google Scholar] [CrossRef]
- Joung, E.; Nakach, S.; Tseytlin, A.A. Scalar scattering via conformal higher spin exchange. J. High Energy Phys. 2016, 2016, 125. [Google Scholar] [CrossRef]
- Beccaria, M.; Nakach, S.; Tseytlin, A.A. On triviality of S-matrix in conformal higher spin theory. J. High Energy Phys. 2016, 2016, 34. [Google Scholar] [CrossRef]
1. | We will discuss only bosonic totally symmetric fields. In arbitrary even dimensions one has to add a power of the laplacian. |
2. | |
3. | In arbitrary even dimension d the conformal weight of is . Given an nth order vertex with fields of spin , the number of derivatives is fixed to . |
4. | In four dimensions the Weyl anomaly contains only two relevant structures: the Euler density, whose coefficient is usually named a, and the square of the Weyl tensor, whose coefficient is . |
5. | |
6. | Direct matching of free gauge theory correlators with AdS Witten diagrams has been investigated in [49,50,51] in order to exploit open-closed string duality. In particular, in [49] one-loop open string diagrams in the field theory limit (hence worldline loops) were shown to reproduce tree level diagrams in AdS by direct change of variables in the moduli space. |
7. | In the standard AdS/CFT context [52,53,54] the boundary values of bulk fields are fixed, non dynamical sources for CFT correlators. From a pure boundary perspective, however, one can see the coupling as a Noether coupling that gauges the infinite dimensional symmetry algebra [31,55] generated by the charges associated to the currents . Moreover, even in the AdS/CFT context one can give different, Neumann type, boundary conditions to bulk fields, allowing them to fluctuate on the boundary [22,56,57]. |
8. | |
9. | Indices denoted with the same letter and groups of indices are intended as symmetrized with strength one, e.g., . |
10. | The logarithmic divergence is present only in even dimensions, that is the only case we will treat here. |
11. | The induced action contains vertices with arbitrary powers of higher spin fields but, due to the absence of dimensionful parameters, the number of derivatives is bounded by the number of fields and sum of the spins involved. |
12. | The quantity is gauge invariant on the circle; hence it constitutes a modulus to be integrated over after gauge fixing. |
13. | See Appendix A for details. |
14. | This can be seen by just changing variables in integrals of periodic and translation invariant functions. However, a more precise justification comes from the gauge fixing procedure on the circle: The einbein possesses, on topology, a Killing vector that is not fixed by the gauge and that generates global translations around the circle. A natural way to fix the leftover global symmetry is then to fix the position of one vertex on the circle, e.g., by setting , as it is customary in String Theory. |
15. | The field independent cannot contribute to the logarithmic divergence and neither can the linear in . |
16. | In the variables one can exchange x with for free. |
17. | For details see the original derivation [11]. |
18. | See Appendix C for the explicit form of the projectors. |
© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Bonezzi, R. Induced Action for Conformal Higher Spins from Worldline Path Integrals. Universe 2017, 3, 64. https://doi.org/10.3390/universe3030064
Bonezzi R. Induced Action for Conformal Higher Spins from Worldline Path Integrals. Universe. 2017; 3(3):64. https://doi.org/10.3390/universe3030064
Chicago/Turabian StyleBonezzi, Roberto. 2017. "Induced Action for Conformal Higher Spins from Worldline Path Integrals" Universe 3, no. 3: 64. https://doi.org/10.3390/universe3030064
APA StyleBonezzi, R. (2017). Induced Action for Conformal Higher Spins from Worldline Path Integrals. Universe, 3(3), 64. https://doi.org/10.3390/universe3030064