Abstract
We classify the topological terms (in a sense to be made precise) that may appear in a non-linear sigma model based on maps from an arbitrary worldvolume manifold to a homogeneous space G/H (where G is an arbitrary Lie group and H ⊂ G). We derive a new condition for G-invariance of topological terms, which is necessary and sufficient (at least when G is connected), and discuss a variety of examples in quantum mechanics and quantum field theory. In the present work we discuss only terms that may be written in terms of (possibly only locally-defined) differential forms on G/H, leading to an action that is manifestly local. Such terms come in one of two types, with prototypical quantum-mechanical examples given by the Aharonov-Bohm effect and the Dirac monopole. The classification is based on the observation that, for topological terms, the maps from the worldvolume to G/H may be replaced by singular homology cycles on G/H. In a forthcoming paper we apply the results to phenomenological models in which the Higgs boson is composite.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
22 November 2018
1. The words ?exact? and ?closed? should be interchanged in (3.1).
22 November 2018
1. The words ���exact��� and ���closed��� should be interchanged in (3.1).
References
B. Gripaios and D. Sutherland, Quantum mechanics of a generalised rigid body, J. Phys. A 49 (2016) 195201 [arXiv:1504.01406] [INSPIRE].
S. Weinberg, Dynamical approach to current algebra, Phys. Rev. Lett. 18 (1967) 188 [INSPIRE].
D.B. Kaplan, H. Georgi and S. Dimopoulos, Composite Higgs Scalars, Phys. Lett. B 136 (1984) 187 [INSPIRE].
B. Gripaios and D. Sutherland, Quantum Field Theory of Fluids, Phys. Rev. Lett. 114 (2015) 071601 [arXiv:1406.4422] [INSPIRE].
A. Nicolis, R. Penco and R.A. Rosen, Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction, Phys. Rev. D 89 (2014) 045002 [arXiv:1307.0517] [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. II, Phys. Rev. 177 (1969) 2247 [INSPIRE].
E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
E. Witten, Current Algebra, Baryons and Quark Confinement, Nucl. Phys. B 223 (1983) 433 [INSPIRE].
D.T. Son, Low-energy quantum effective action for relativistic superfluids, hep-ph/0204199 [INSPIRE].
S. Dubovsky, L. Hui and A. Nicolis, Effective field theory for hydrodynamics: Wess-Zumino term and anomalies in two spacetime dimensions, Phys. Rev. D 89 (2014) 045016 [arXiv:1107.0732] [INSPIRE].
L.V. Delacrétaz, A. Nicolis, R. Penco and R.A. Rosen, Wess-Zumino Terms for Relativistic Fluids, Superfluids, Solids and Supersolids, Phys. Rev. Lett. 114 (2015) 091601 [arXiv:1403.6509] [INSPIRE].
G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Galileons as Wess-Zumino Terms, JHEP 06 (2012) 004 [arXiv:1203.3191] [INSPIRE].
E. D’Hoker and S. Weinberg, General effective actions, Phys. Rev. D 50 (1994) R6050 [hep-ph/9409402] [INSPIRE].
R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
T.T. Wu and C.N. Yang, Dirac’s Monopole Without Strings: Classical Lagrangian Theory, Phys. Rev. D 14 (1976) 437 [INSPIRE].
O. Alvarez, Topological Quantization and Cohomology, Commun. Math. Phys. 100 (1985) 279 [INSPIRE].
N.S. Manton, A model for the anomalies in gauge field theory, NSF-ITP-83-164 (1983) [INSPIRE].
N.S. Manton, The Schwinger Model and Its Axial Anomaly, Annals Phys. 159 (1985) 220 [INSPIRE].
M.F. Atiyah, Topological quantum field theory, Pub. Math. IH ÉS 68 (1988) 175.
J. Vick, Homology Theory: An Introduction to Algebraic Topology, Graduate Texts in Mathematics, Springer, New York U.S.A. (1994).
D.E. Soper, Classical field theory, Wiley, New York U.S.A. (1976).
A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear σ-models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].
S.R. Coleman, More About the Massive Schwinger Model, Annals Phys. 101 (1976) 239 [INSPIRE].
J.S. Schwinger, Gauge Invariance and Mass. II, Phys. Rev. 128 (1962) 2425 [INSPIRE].
K. Agashe, R. Contino and A. Pomarol, The Minimal composite Higgs model, Nucl. Phys. B 719 (2005) 165 [hep-ph/0412089] [INSPIRE].
B. Gripaios, A. Pomarol, F. Riva and J. Serra, Beyond the Minimal Composite Higgs Model, JHEP 04 (2009) 070 [arXiv:0902.1483] [INSPIRE].
J. Davighi and B. Gripaios, Topological terms in Composite Higgs Models, arXiv:1808.04154 [INSPIRE].
R. Bott and L. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer, New York U.S.A. (1995).
P.A. Horváthy, Prequantisation from path integral viewpoint, in Differential Geometric Methods in Mathematical Physics, H.-D. Doebner, S.I. Andersson and H.R. Petry eds., Springer (1982), pp. 197-206.
D.S. Freed, G.W. Moore and G. Segal, Heisenberg Groups and Noncommutative Fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].
C. Baer and C. Becker, Differential Characters and Geometric Chains, arXiv:1303.6457.
A. Schwarz, Topology for physicists, Grundlehren der mathematischen Wissenschaften, Springer, New York U.S.A. (1994).
C. Chevalley and S. Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Am. Math. Soc. 63 (1948) 85 [INSPIRE].
D.S. Freed, Pions and Generalized Cohomology, J. Diff. Geom. 80 (2008) 45 [hep-th/0607134] [INSPIRE].
B. Gripaios, M. Nardecchia and T. You, On the Structure of Anomalous Composite Higgs Models, Eur. Phys. J. C 77 (2017) 28 [arXiv:1605.09647] [INSPIRE].
P. Forgács and N.S. Manton, Space-Time Symmetries in Gauge Theories, Commun. Math. Phys. 72 (1980) 15 [INSPIRE].
R. Jackiw and N.S. Manton, Symmetries and Conservation Laws in Gauge Theories, Annals Phys. 127 (1980) 257 [INSPIRE].
J. Davighi, B. Gripaios and O. Randal-Williams, Invariant differential characters and quantum field theory, to appear.
D. Finkelstein and J. Rubinstein, Connection between spin, statistics and kinks, J. Math. Phys. 9 (1968) 1762 [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: 1803.07585
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Davighi, J., Gripaios, B. Homological classification of topological terms in sigma models on homogeneous spaces. J. High Energ. Phys. 2018, 155 (2018). https://doi.org/10.1007/JHEP09(2018)155
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2018)155