nLab higher spin gauge theory

Idea

Generally by a higher spin field theory is meant a quantum field theory that involves fields of spin $\gt 2$ (recalling that a spinor field has spin 1/2, a gauge field has spin 1, a gravitino field has spin 3/2 and the field of gravity has spin 2).

Folklore had it that all higher spin field theories with mass-less higher spin fields are inconsistent due to negative norm states (“ghost”) appearing in their quantization. (This argument underlies the dominance of $\mathcal{N} = 1$ 11-dimensional supergravity, see the introduction to the entry 12-dimensional supergravity for more on this).

But then it was discovered that there is a higher spin gauge theory (Vasiliev 96) which is a kind of higher gauge theory whose field content is an infinite tower of massless fields of ever higher spin.

Ever since the refine folklore says that higher spin theories with a finite number of higher spin field species is inconsistent, but that an infinite tower can fix the problem (…add reference…).

One way that higher spin gauge theories are thought to naturally arise is as the limiting case of string field theory when the string tension is sent to zero (Henneaux-Teitelboim 87, section 2, Gross 88, Sagnotti-Tsulaia 03, Bonelli 03): The excitation spectrum of the string sigma-model contains beyond the massless particles of the effective supergravity theory an infinite tower of massive excitations, of ever higher spin. There are, however, certain limits in which all these masses become negligible to a reference energy scale – the tensionless limit – this is notably so for compactifications on anti de Sitter spaces of small radius. In this limit the string spectrum looks like an infinite collection of massless spinning particles for ever higher spin. Due to their common origin in the string, these share intricate relations among each other, which are argued to be described by higher spin gauge theory. (Notice that at least closed bosonic string field theory is itself already a higher gauge theory, even without sending the string tension to zero, see at closed string field theory – As an ∞-Chern-Simons theory.)

At present there are several classes of higher spin theories that evade the no-go theorems: (1) massless and conformal higher spin theories in three dimensions can be formulated as Chern-Simons theories; (2) conformal higher spin gravity in four dimensions is a higher spin extension of the Weyl gravity; (3) Chiral higher spin gravity is a higher spin extension of self-dual Yang-Mills theory and self-dual Gravity. At the same time the holographic higher spin theories face certain difficulties due non-locality implied by the AdS/CFT duality to vector models

References

General

Original articles:

• Mikhail Vasiliev, Higher-Spin Gauge Theories in Four, Three and Two Dimensions (arXiv:hep-th/9611024)

Reviews and lecture notes:

• Mikhail Vasiliev, Higher Spin Gauge Theories in Various Dimensions, 27th Johns Hopkins Workshop on Current Problems in Particle Theory: Symmetries and Mysteries of M Theory (arXiv:hep-th/0401177)

• Mikhail Vasiliev, Higher spin gauge theories in any dimension, talk at String2004 in Moscow (arXiv:hep-th/0409260)

• R. Argurio, Glenn Barnich, G. Bonelli, M. Grigoriev (eds.) Higher spin gauge theories Solvay Workshops and Symposia (2004) (pdf)

• Xavier Bekaert, S. Cnockaert, Carlo Iazeolla, Mikhail Vasiliev, Nonlinear higher spin theories in various dimensions (arXiv:0503128)

• V.E. Didenko, E.D. Skvortsov, Elements of Vasiliev theory (arXiv:1401.2975)

• Rakibur Rahman, Massimo Taronna, From Higher Spins to Strings: A Primer (arXiv:1512.07932 )

• Pan Kessel, The Very Basics of Higher-Spin Theory (arXiv:1702.03694)

• Ivan Vuković, Higher spin theory (arXiv:1809.02179)

• Dmitry Ponomarev, Basic introduction to higher-spin theories [arXiv:2206.15385]

• Andrea Campoleoni, Stefan Fredenhagen, Higher-spin gauge theories in three spacetime dimensions [arXiv:2403.16567]

Further developments:

• Johan Engquist, Olaf Hohm, Geometry and dynamics of higher-spin frame fields (arXiv:0708.1391)

Concrete higher spin gravities

• Miles Blencowe, A consistent interacting massless higher-spin field theory in $D=2+1$ Classical and quantum gravity, volume 6 no 4 (1998)

• C. N. Pope, P.K. Townsend, Conformal Higher Spin in (2+1)-dimensions, Phys.Lett. B225 (1989) 245-250.

• E. S. Fradkin, V. Ya. Linetsky, a Superconformal Theory of Massless Higher Spin Fields in D=2+1 (web)

• Maxim Grigoriev, Karapet Mkrtchyan, Evgeny Skvortsov, Matter-free higher spin gravities in 3D: Partially-massless fields and general structure (arXiv:2005.05931)

• Maxim Grigoriev, Iva Lovrekovic, Evgeny Skvortsov, New Conformal Higher Spin Gravities in $3d$ (arXiv:1909.13305)

• Arkady A. Tseytlin, On limits of superstring in AdS(5) x S5 (arXiv:0201112)

• Arkady Y. Segal, Conformal higher spin theory (arXiv:0207212)

• Dmitry Ponomarev, E.D. Skvortsov, Light-Front Higher-Spin Theories in Flat Space (arXiv:1609.04655)

Relation to string theory

The idea that higher spin gauge theory appears as the limiting case of string field theory where the string tension vanishes goes back to

• Marc Henneaux, Claudio Teitelboim, section 2 of First And Second Quantized Point Particles Of Any Spin, in Claudio Teitelboim, Jorge Zanelli (eds.) Santiago 1987, Proceedings, Quantum mechanics of fundamental systems 2, pp. 113-152. Plenum Press.

• David Gross, High-Energy Symmetries Of String Theory, Phys. Rev. Lett. 60 (1988) 1229.

and is further developed for instance in

• Augusto Sagnotti, M. Tsulaia, On higher spins and the tensionless limit of String Theory, Nucl. Phys. B682:83-116, 2004 (arXiv:hep-th/0311257)

• G. Bonelli, On the Tensionless Limit of Bosonic Strings, Infinite Symmetries and Higher Spins, Nucl. Phys. B669 (2003) 159-172 (arXiv:hep-th/0305155)

• Augusto Sagnotti, M. Taronna, String Lessons for Higher-Spin Interactions, Nucl. Phys. B842:299-361,2011 (arXiv:1006.5242)

• Augusto Sagnotti, Notes on Strings and Higher Spins (arXiv:1112.4285)

And conversely:

• Rakibur Rahman, Massimo Taronna, From Higher Spins to Strings: A Primer in Stefan Fredenhagen (ed.) Introduction to Higher Spin Theory (arXiv:1512.07932)

• Matthias Gaberdiel, Rajesh Gopakumar, String Theory as a Higher Spin Theory, J. High Energ. Phys. (2016) 2016: 85 (arXiv:1512.07237, doi:10.1007/JHEP09(2016)085)

• Simon Caron-Huot, Zohar Komargodski, Amit Sever, Alexander Zhiboedov, Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude, JHEP10(2017)026 (arXiv:1607.04253)

• Amit Sever, Alexander Zhiboedov, On Fine Structure of Strings: The Universal Correction to the Veneziano Amplitude, JHEP06(2018)054 (arXiv:1707.05270)

Relation with double field theory

Discussion of the appearance of spin-2 Fierz-Pauli fields in double field theory:

• Chen-Te Ma, Franco Pezzella, More Stringy Effects in Target Space from Double Field Theory, JHEP 08 (2020) 113 (arXiv:1909.00411)

Relation to other systems

Relation to Kac-Moody algebras is discussed in

• Marc Henneaux, Axel Kleinschmidt, Hermann Nicolai, Higher spin gauge fields and extended Kac-Moody symmetries (arXiv:1110.4460)

Expression in terms of AKSZ sigma-models is discussed in

• K.B. Alkalaev, Maxim Grigoriev, E.D. Skvortsov, Uniformizing higher-spin equations (arXiv:1409.6507)

Higher spin Chern-Simons theory

Chern-Simons theory for higher spin fields is discussed in

• Miles Blencowe, A consistent interacting massless higher-spin field theory in $D=2+1$ Classical and quantum gravity, volume 6 no 4 (1998)

• E. S. Fradkin, V. Ya. Linetsky, a Superconformal Theory of Massless Higher Spin Fields in D=2+1 (web)

• Johan Engquist, Olaf Hohm, Higher-spin Chern-Simons theories in odd dimensions (arXiv:0705.3714)

Relation to holography

We list references that discuss the relation of higher spin gauge theory to the AdS/CFT correspondence.

• Simone Giombi, Xi Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions (arXiv:0912.3462)

• Simone Giombi, Xi Yin, Higher Spins in AdS and Twistorial Holography (arXiv:1004.3736)

• Simone Giombi, TASI Lectures on the Higher Spin - CFT duality (arXiv:1607.02967)

• Charlotte Sleight, Lectures on Metric-like Methods in Higher Spin Holography (arXiv:1701.08360)