Abstract
In this chapter we will study group actions on manifolds and introduce several standard notions, such as orbits and isotropy groups of group actions. If a Lie group acts on a manifold, then there is an induced infinitesimal action of the Lie algebra, defining so-called fundamental vector fields on the manifold.
An interesting question for Lie group actions on manifolds is under which conditions the quotient space of the action again admits the structure of a smooth manifold. The main result that we prove in this context is Godement’s Theorem, which gives a necessary and sufficient condition that quotient spaces under general equivalence relations are smooth manifolds.
We also finally apply the theory of group actions to construct the exceptional compact simple Lie group G2, that plays an important part in M-theory, a conjectured theory of quantum gravity in 11 dimensions.
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References
Abe, K. et al. (The T2K Collaboration): First combined analysis of neutrino and antineutrino oscillations at T2K. arXiv:1701.00432
Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72, 20–104 (1960)
Adams, J.F.: Vector fields on spheres. Ann. Math. 75, 603–632 (1962)
Argyres, P.C.: An introduction to global supersymmetry. Lecture notes, Cornell University 2001. Available at http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/index.html
Atiyah, M.F.: K-Theory. Notes by D.W. Anderson. W.A. Benjamin, New York/Amsterdam (1967)
Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Phil. Trans. R. Soc. Lond. A 308, 523–615 (1983)
Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3, 3–38 (1964)
Baez, J.C.: The octonions. Bull. Am. Math. Soc. (N.S.) 39, 145–205 (2002); Erratum: Bull. Am. Math. Soc. (N.S.) 42, 213 (2005)
Baez, J., Huerta, J.: The algebra of grand unified theories. Bull. Am. Math. Soc. (N.S.) 47, 483–552 (2010)
Bailin, D., Love, A.: Introduction to Gauge Field Theory. Institute of Physics Publishing, Bristol/Philadelphia (1993)
Ball. P.: Nuclear masses calculated from scratch. Nature, published online 20 November 2008. doi:10.1038/news.2008.1246
Barut, A.O., Raczka, R.: Theory of Group Representations and Applications. Polish Scientific Publishers, Warszawa (1980)
Baum, H.: Spin-Strukturen und Dirac-Operatoren Ă¼ber pseudoriemannschen Mannigfaltigkeiten. Teubner Verlagsgesellschaft, Leipzig (1981)
Baum, H.: Eichfeldtheorie. Springer, Berlin/Heidelberg (2014)
Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin/Heidelberg (2004)
Bleecker, D.: Gauge Theory and Variational Principles. Addison-Wesley Publishing Company, Reading, MA (1981)
Bogolubov, N.N., Logunov, A.A., Todorov, I.T.: Introduction to Axiomatic Quantum Field Theory. W. A. Benjamin, Reading, MA (1975)
Borsanyi, Sz. et al: Ab initio calculation of the neutron-proton mass difference. Science 347(6229), 1452–1455 (2015)
Bott, M.R.: An application of the Morse theory to the topology of Lie-groups, Bull. Soc. Math. France 84, 251–281 (1956)
Bourguignon, J.-P., Hijazi, O., Milhorat, J.-L., Moroianu, A., Moroianu, S.: A Spinorial Approach to Riemannian and Conformal Geometry. European Mathematical Society, ZĂ¼rich (2015)
Brambilla, N. et al.: QCD and strongly coupled gauge theories: challenges and perspectives. Eur. Phys. J. C 74, 2981 (2014)
Branco, G.C., Lavoura, L., Silva, J.P.: CP Violation. Oxford University Press, Oxford (1999)
Bredon, G.E.: Introduction to Compact Transformation Groups. Academic Press, New York/London (1972)
Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Springer, Berlin/Heidelberg/New York (2010)
Bröcker, T., Jänich, K.: EinfĂ¼hrung in die Differentialtopologie. Springer, Berlin/Heidelberg/New York (1990)
Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)
Bryant, R.L.: Submanifolds and special structures on the octonians. J. Differ. Geom. 17, 185–232 (1982)
Budinich, R., Trautman, A.: The Spinorial Chessboard. Springer, Berlin/Heidelberg (1988)
Bueno, A. et al.: Nucleon decay searches with large liquid Argon TPC detectors at shallow depths: atmospheric neutrinos and cosmogenic backgrounds. JHEP 0704, 041 (2007)
ÄŒap, A., SlovĂ¡k, J.: Parabolic Geometries I: Background and General Theory. American Mathematical Society, Providence, RI (2009)
CERN Press Release: CERN experiments observe particle consistent with long-sought Higgs boson. Available at http://press.cern/press-releases/2012/07/cern-experiments-observe-particle-consistent-long-sought-higgs-boson
Chaichian, M., Nelipa, N.F.: Introduction to Gauge Field Theories. Springer, Berlin/Heidelberg/New York/Tokyo (1984)
Cheng, T.-P., Li, L.-F.: Gauge Theory of Elementary Particle Physics. Oxford University Press, Oxford (1988)
Chevalley, C.: Theory of Lie Groups I. Princeton University Press, Princeton (1946)
Chevalley, C.: The Algebraic Theory of Spinors and Clifford Algebras. Collected Works, Vol. 2. Springer, Berlin/Heidelberg (1997)
Chivukula, R.S.: The origin of mass in QCD. arXiv:hep-ph/0411198
Clay Mathematics Institute: Millenium problems. Yang–Mills and mass gap. Available at http://www.claymath.org/millennium-problems/yang--mills-and-mass-gap
Costello, K.: Renormalization and Effective Field Theory. Mathematical Surveys and Monographs, Vol. 170. American Mathematical Society, Providence, RI (2011)
Darling, R.W.R.: Differential Forms and Connections. Cambridge University Press, Cambridge (1994)
D’Auria, R., Ferrara, S., LledĂ³, M.A., Varadarajan, V.S.: Spinor algebras. J. Geom. Phys. 40, 101–129 (2001)
Derdzinski, A.: Geometry of the Standard Model of Elementary Particles. Springer, Berlin/Heidelberg (1992)
Dissertori, G., Knowles, I., Schmelling, M.: Quantum Chromodynamics. High Energy Experiments and Theory. Oxford University Press, Oxford (2003)
Drexlin, G., Hannen, V., Mertens, S., Weinheimer, C.: Current direct neutrino mass experiments. Adv. High Energy Phys. 2013, Article ID 293986 (2013)
DĂ¼rr, S. et al.: Ab initio determination of light hadron masses. Science 322, 1224–1227 (2008)
Duncan, A.: The Conceptual Framework of Quantum Field Theory. Oxford University Press, Oxford (2013)
Dynkin, E.B.: Semisimple subalgebras of the semisimple Lie algebras. (Russian) Mat. Sbornik 30, 349–462 (1952); English translation: Am. Math. Soc. Transl. Ser. 2 6, 111–244 (1957)
Elliott, C.: Gauge Theoretic Aspects of the Geometric Langlands Correspondence. Ph.D. Thesis, Northwestern University (2016)
Englert, F., Brout R.: Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Lett. 13, 321–323 (1964)
Figueroa-O’Farrill, J.: Majorana Spinors. Lecture Notes. University of Edinburgh (2015)
Flory, M., Helling, R.C., Sluka, C.: How I learned to stop worrying and love QFT. arXiv:1201.2714 [math-ph]
Folland, G.B.: Quantum Field Theory. A Tourist Guide for Mathematicians. American Mathematical Society, Providence, Rhodes Island (2008)
Freed, D.S.: Classical Chern–Simons theory, 1. Adv. Math. 113, 237–303 (1995)
Freed, D.S.: Five Lectures on Supersymmetry. American Mathematical Society, Providence, RI (1999)
Freedman, D.Z., Van Proeyen, A.: Supergravity. Cambridge University Press, Cambridge (2012)
Friedrich, T.: Dirac Operators in Riemannian Geometry. American Mathematical Society, Providence, RI (2000)
Fritzsch, H., Minkowski, P.: Unified interactions of leptons and hadrons. Ann. Phys. 93, 193–266 (1975)
Geiges, H.: An Introduction to Contact Topology. Cambridge University Press, Cambridge (2008)
Georgi, H.: The state of the art – gauge theories. In: Carlson, C.E. (ed.) Particles and Fields – 1974: Proceedings of the Williamsburg Meeting of APS/DPF, pp. 575–582. AIP, New York (1975)
Georgi, H., Glashow, S.L.: Unity of all elementary-particle forces. Phys. Rev. Lett. 32, 438–441 (1974)
Georgi, H.M., Glashow, S.L., Machacek, M.E., Nanopoulos, D.V.: Higgs Bosons from two-gluon annihilation in proton-proton collisions. Phys. Rev. Lett. 40 692 (1978)
Georgi, H., Quinn, H.R., Weinberg, S.: Hierarchy of interactions in unified gauge theories. Phys. Rev. Lett. 33, 451–454 (1974)
Giunti, C., Kim, C.W.: Fundamentals of Neutrino Physics and Astrophysics. Oxford University Press, Oxford (2007)
Glashow, S.L.: Trinification of all elementary particle forces. In: 5th Workshop on Grand Unification, Providence, RI, April 12–14, 1984
Glashow, S.L., Iliopoulos, J., Maiani, L.: Weak interactions with lepton-hadron symmetry. Phys. Rev. D 2, 1285–1292 (1970)
Gleason, A.M.: Groups without small subgroups. Ann. Math. 56, 193–212 (1952)
Grimus, W., Rebelo, M.N.: Automorphisms in gauge theories and the definition of CP and P. Phys. Rep. 281, 239–308 (1997)
Guralnik, G.S., Hagen, C.R., Kibble, T.W.B.: Global conservation laws and massless particles. Phys. Rev. Lett. 13, 585–587 (1964)
GĂ¼rsey, F., Ramond, P., Sikivie, P.: A universal gauge theory model based on E6. Phys. Lett. B 60, 177–180 (1976)
Haag, R.: Local Quantum Physics. Fields, Particles, Algebras. Springer, Berlin/ Heidelberg/New York (1996)
Hall, B.C.: Lie Groups, Lie Algebras and Representations. An Elementary Introduction. Springer, Cham Heidelberg/New York/Dordrecht/London (2016)
Halzen, F., Martin, A.D.: Quarks and Leptons. An Introductory Course in Modern Particle Physics. Wiley, New York/Chichester/Brisbane/Toronto/Singapore (1984)
Hartanto, A., Handoko L.T.: Grand unified theory based on the SU(6) symmetry. Phys. Rev. D 71, 095013 (2005)
Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
Hatcher, A.: Vector bundles and K-theory. Version 2.1, May 2009
Heeck, J.: Interpretation of lepton flavor violation. Phys. Rev. D 95, 015022 (2017)
Higgs, P.W.: Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 13, 508–509 (1964)
Hilgert, J., Neeb, K.-H.: Structure and Geometry of Lie Groups. Springer, New York/ Dordrecht/Heidelberg/London (2012)
Hirsch, M.W.: Differential Topology. Springer, New York/Berlin/Heidelberg (1997)
Hoddeson, L., Brown, L., Riordan, M., Dresden, M. (ed.): The Rise of the Standard Model: Particle Physics in the 1960s and 1970s. Cambridge University Press, Cambridge (1997)
Hollowood, T.J.: Renormalization Group and Fixed Points in Quantum Field Theory. Springer, Heidelberg/New York/Dordrecht/London (2013)
Husemoller, D.: Fibre Bundles. Springer, New York (1994)
Klaczynski, L.: Haag’s Theorem in renormalisable quantum field theory. Ph.D. Thesis, Humboldt Universität zu Berlin (2015)
Knapp, A.W.: Lie Groups Beyond an Introduction. Birkhäuser, Boston/Basel/Berlin (2002)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Vol. I. Interscience Publishers, New York/London (1963)
Kounnas, C., Masiero, A., Nanopoulos, D.V., Olive, K.A.: Grand Unification with and Without Supersymmetry and Cosmological Implications. World Scientific, Singapore (1984)
Lancaster, T., Blundell, S. J.: Quantum Field Theory for the Gifted Amateur. Oxford University Press, Oxford (2014)
Langacker, P.: Grand unified theories and proton decay. Phys. Rep. 72, 185–385 (1981)
Lawson, H.B. Jr., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton, NJ (1989)
Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York/Heidelberg/Dordrecht/ London (2013)
Leigh, R.G., Strassler, M.J.: Duality of Sp(2N c ) and SO(N c ) supersymmetric gauge theories with adjoint matter. Phys. Lett. B 356, 492–499 (1995)
Martin, S.P.: A supersymmetry primer. arXiv:hep-ph/9709356
Mayer, M.E.: Review: David D. Bleecker, Gauge theory and variational principles. Bull. Am. Math. Soc. (N.S.) 9, 83–92 (1983)
Meinrenken, E.: Clifford Algebras and Lie Theory. Springer, Berlin/Heidelberg (2013)
Milnor, J.: On manifolds homeomorphic to the 7-sphere. Ann. Math. 64, 399–405 (1956)
Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman and Company, New York (1973)
Mohapatra, R.N.: Unification and Supersymmetry. The Frontiers of Quark-Lepton Physics. Springer, New York/Berlin/Heidelberg (2003)
Montgomery, D., Zippin, L.: Small subgroups of finite-dimensional groups. Ann. Math. 56, 213–241 (1952)
Moore, J.D.: Lectures on Seiberg–Witten Invariants. Springer, Berlin/Heidelberg/New York (2001)
Morgan, J.W.: The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-Manifolds. Princeton University Press, Princeton, NJ (1996)
Mosel, U.: Fields, Symmetries, and Quarks. Springer, Berlin/Heidelberg (1999)
Naber, G.L.: Topology, Geometry and Gauge Fields. Foundations. Springer, New York (2011)
Naber, G.L.: Topology, Geometry and Gauge Fields. Interactions. Springer, New York (2011)
Nakahara, M.: Geometry, Topology and Physics, 2nd edn. IOP Publishing Ltd, Bristol/ Philadelphia (2003)
O’Raifeartaigh, L.: Group Structure of Gauge Theories. Cambridge University Press, Cambridge (1986)
Patrignani, C. et al. (Particle Data Group): 2016 Review of particle physics. Chin. Phys. C 40, 100001 (2016). http://www-pdg.lbl.gov/
Patrignani, C. et al. (Particle Data Group): 2016 Review of particle physics. Particle listings. Chin. Phys. C 40, 100001 (2016). http://www-pdg.lbl.gov/
Patrignani, C. et al. (Particle Data Group): 2016 Review of particle physics. Particle listings. Neutrino mixing. Chin. Phys. C 40, 100001 (2016). http://www-pdg.lbl.gov/
Patrignani, C. et al. (Particle Data Group): 2016 Review of particle physics. Reviews, tables, and plots. 1. Physical constants. Chin. Phys. C 40, 100001 (2016). http://www-pdg.lbl.gov/
Patrignani, C. et al. (Particle Data Group): 2016 Review of particle physics. Reviews, tables, and plots. 10. Electroweak model and constraints on new physics. Chin. Phys. C 40, 100001 (2016). http://www-pdg.lbl.gov/
Patrignani, C. et al. (Particle Data Group): 2016 Review of particle physics. Reviews, tables, and plots. 12. The CKM quark-mixing matrix. Chin. Phys. C 40, 100001 (2016). http://www-pdg.lbl.gov/
Patrignani, C. et al. (Particle Data Group): 2016 Review of particle physics. Reviews, tables, and plots. 16. Grand unified theories. Chin. Phys. C 40, 100001 (2016). http://www-pdg.lbl.gov/
Pich, A.: The Standard Model of electroweak interactions. arXiv:1201.0537 [hep-ph]
Quigg, C.: Gauge Theories of the Strong, Weak, and Electromagnetic Interactions. Westview Press, Boulder, Colorado (1997)
Robinson, M.: Symmetry and the Standard Model. Mathematics and Particle Physics. Springer, New York/Dordrecht/Heidelberg/London (2011)
Roe, J.: Elliptic Operators, Topology and Asymptotic Methods. Longman Scientific & Technical, Harlow (1988)
Roman, P.: Introduction to Quantum Field Theory. Wiley, New York/London/Sydney/Toronto (1969)
Royal Swedish Academy of Sciences: The official web site of the nobel prize. https://www.nobelprize.org/nobel_prizes/physics/
Royal Swedish Academy of Sciences: Asymptotic freedom and quantum chromodynamics: the key to the understanding of the strong nuclear forces. Advanced information on the Nobel Prize in Physics, 5 October 2004. https://www.nobelprize.org/nobel_prizes/physics/laureates/2004/advanced.html
Royal Swedish Academy of Sciences: Class of Physics. Broken symmetries. Scientific Background on the Nobel Prize in Physics 2008. https://www.nobelprize.org/nobel_prizes/physics/laureates/2008/advanced.html
Royal Swedish Academy of Sciences: Class of Physics. The BEH-mechanism, interactions with short range forces and scalar particles. Scientific Background on the Nobel Prize in Physics 2013. https://www.nobelprize.org/nobel_prizes/physics/laureates/2013/advanced.html
Royal Swedish Academy of Sciences: Class of Physics. Neutrino oscillations. Scientific Background on the Nobel Prize in Physics 2015. https://www.nobelprize.org/nobel_prizes/physics/laureates/2015/advanced.html
Rudolph, G., Schmidt, M.: Differential Geometry and Mathematical Physics. Part I. Manifolds, Lie Groups and Hamiltonian Systems. Springer Netherlands, Dordrecht (2013)
Rudolph, G., Schmidt, M.: Differential Geometry and Mathematical Physics. Part II. Fibre Bundles, Topology and Gauge Fields. Springer Netherlands, Dordrecht (2017)
Ryder, L.H.: Quantum Field Theory. Cambridge University Press, Cambridge (1996)
Schwartz, M.D.: Quantum Field Theory and the Standard Model. Cambridge University Press, Cambridge (2014)
Seiberg, N.: Five dimensional SUSY field theories, non-trivial fixed points and string dynamics. Phys. Lett. B 388, 753–760 (1996)
Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19–52 (1994). Erratum: Nucl. Phys. B 430, 485–486 (1994)
Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994)
Sepanski, M.R.: Compact Lie Groups. Springer Science+Business Media LLC, New York (2007)
Serre, J.-P.: Lie Algebras and Lie Groups. 1964 Lectures given at Harvard University. Springer, Berlin/Heidelberg (1992)
Slansky, R.: Group theory for unified model building. Phys. Rep. 79, 1–128 (1981)
Srednicki, M.: Quantum Field Theory. Cambridge University Press, Cambridge (2007)
Steenrod, N.: The Topology of Fibre Bundles. Princeton University Press, Princeton (1951)
Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. Princeton University Press, Princeton, NJ (2000)
Tao, T.: Hilbert’s Fifth Problem and Related Topics. Graduate Studies in Mathematics, Vol. 153. American Mathematical Society, Providence, RI (2014)
Taubes, C.H.: Differential Geometry. Bundles, Connections, Metrics and Curvature. Oxford University Press, Oxford (2011)
Thomson, M.: Modern Particle Physics. Cambridge University Press, Cambridge (2013)
Vafa, C., Zwiebach, B.: N = 1 dualities of SO and USp gauge theories and T-duality of string theory. Nucl. Phys. B 506, 143–156 (1997)
van den Ban, E.P.: Notes on quotients and group actions. Fall 2006. Universiteit Utrecht
Van Proeyen, A.: Tools for supersymmetry. arXiv:hep-th/9910030
van Vulpen, I.: The Standard Model Higgs boson. Part of the Lecture Particle Physics II, University of Amsterdam Particle Physics Master 2013–2014
Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Springer, New York (2010)
Weinberg, S.: The Quantum Theory of Fields, Vol. I. Foundations. Cambridge University Press, Cambridge (1995)
Weinberg, S.: The Quantum Theory of Fields, Vol. II. Modern Applications. Cambridge University Press, Cambridge (1996)
Weinberg, S.: The Quantum Theory of Fields, Vol. III. Supersymmetry. Cambridge University Press, Cambridge (2005)
Wess, J., Bagger, J.: Supersymmetry and Supergravity, 2nd edn. Princeton University Press, Princeton, NJ (1992)
Wilczek, F.: Decays of heavy vector mesons into Higgs particles. Phys. Rev. Lett. 39, 1304 (1977)
Witten, E.: Quest for unification. arXiv:hep-ph/0207124
Witten, E.: Chiral ring of Sp(N) and SO(N) supersymmetric gauge theory in four dimensions. Chin. Ann. Math. 24, 403 (2003)
Witten, E.: Newton lecture 2010: String theory and the universe. Available at http://www.iop.org/resources/videos/lectures/page_44292.html Cited 20 Nov 2016
Yamamoto, K.: SU(7) Grand Unified Theory. Ph.D. thesis, Kyoto University (1981)
Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)
Ziller, W.: Lie Groups. Representation theory and symmetric spaces. Lecture Notes, University of Pennsylvania, Fall 2010. Available at https://www.math.upenn.edu/~wziller/
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Hamilton, M.J.D. (2017). Chapter 3 Group Actions. In: Mathematical Gauge Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-68439-0_3
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