Abstract
In these lectures we would like to review some of the applications of global methods in geometry and topology to the quantum theory of fields and strings. The use of the theory of characteristic classes, index theory, algebraic geometry and differential topology is becoming common place in recent theoretical physics. It seems likely that these areas of mathematics will soon become part of the mathematical background of theoretical physicists, in the same way that Riemannian geometry, group theory, and complex analysis became part of our mathematical arsenal. These new techniques provide also rather powerful tools to obtain useful qualitative (and sometimes quantitative) information in quantum field theories with gauge and/or gravitational interactions, and in string theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
For a review and references to the original literature, see: L. Alvarez-Gaumé—“An Introduction to Anomalies” HUTP 85/A052, to appear in the Proceedings of the Erice School in Mathematical Physics; G. Velo and A. Wightman, eds. (Plenum Press, 1986 ).
E. Witten—Phys.Lett. 117B (1982) 324; Comm.Math.Phys. 100 (1985) 197.
E. Witten—in Proceedings of the Argonne Symposium on Anomalies Geometry and Topology. W.A. Bardeen and A. White, eds. (World Scientific, 1985 ).
S.L: Adler—Phys.Rev. 177 (1969) 2426, and in “Lectures in Elementary Particles and Quantum Field Theory”, S. Deser et al., eds. (MIT Press, 1970 );
J.S. Bell and R. Jackiw—Nuovo CImento 60A (1969) 47.
See also: R. Jackiw — in “Lectures on Current Algebra and its Application”, (Princeton U. Press, 1972 ).
G. ‘t Hooft — Phys.Rev.Lett. 37 (1976) 8;
C. Callan, R. Dashen and D. Gross — Phys.Lett. 63B (1976) 334;
R. Jackiw and C. Rebbi — Phys.Rev.Lett. 37 (1976) 172.
R. Delbourgo and A. Salam — Phys.Lett. 40B (1972) 381;
T. Eguchi and P. Freund — Phys.Rev.Lett. 37 (1976) 1251.
D.J. Gross and R. Jackiw — Phys.Rev. D6 (1972) 477;
C. Bouchiat, J. Iliopoulos and Ph. Meyer — Phys.Lett. 38B (1972) 519;
H. Georgi and S.L: Glashow — Phys.Rev. D6 (1972) 429.
P. Langacker — Physics Reports 72 (1981) 185.
P.H. Frampton and T.W. Kephart — Phys.Rev.Lett. 50 (1983) 1343, 1347;
P.K. Townsend and G. Sierra — Nucl.Phys. B222 (1983) 493;
B. Zumino, W.Y. Shi and A. Zee — Nucl.Phys. B239 (1984) 477.
L. Alvarez-Gaumé and E. Witten — Nucl.Phys. B234 (1983) 269.
M. Green and J. Schwarz — Phys.Lett. 149B (1984) 117;
M. Green, J. Schwarz and P. West — Nucl.Phys. B254 (1984) 377.
D.J: Gross, J. Harvey, E. Martinec and R. Rohm — Phys.Rev.Lett. 54 (1985) 502;
D.J: Gross, J. Harvey, E. Martinec and R. Rohm — Nucl.Phys. B256 (1985) 253;
D.J: Gross, J. Harvey, E. Martinec and R. Rohm — Nucl.Phys. B267 (1986).
M.F. Atiyah and I.M. Singer — Ann. of Math. 87 (1968) 485, 546;
M.F. Atiyah and I.M. Singer — Ann. of Math. 93 (1971) 1, 119, 139;
M.F. Atiyah and G.B. Segal — Ann. of Math. 87 (1968) 531.
R. Jackiw, C. Nohl and C. Rebbi — in “Particles and Fields”, D. Boch and A. Kamal, eds. (Plenum Press, NY, 1978 );
N.K. Nielsen, H. Roemer and B. Schroer — Nucl.Phys. B136 (1978) 478.
L. Alvarez-Gaumé and P. Ginsparg — Nucl.Phys. B243 (1984) 449.
M.F. Atiyah and I.M. Singer — Proc.Nat.Acad.Sci. U.S.A. 81 (1984) 2597.
S.S. Chern — “Complex Manifolds without Potential Theory” (Van Nostrand, 1967).
See Ref. 1 for a thorough discussion.
For very clear discussions, see: B. Zumino — in “Relativity, Groups and Topology — II”, Proceedings of the Is Houches Summer School, B.S. DeWitt and R. Stora, eds. ( North Holland, 1984 );
R. Stora — in. Stora — in “PRogress in Gauge Theories”, Proceedings of the 1984 Cargèse School, G. ‘t Hooft et al., eds. (Plenum Press, 1984 ).
L.D. Faddeev — Phys.Lett. 145B (1984) 81;
L.D. Faddeev and S. Shatashvili — Teor. and Math.Phys. 60 (1984) 206;
B. Zumino — Nucl.Phys. B253 (1985) 477;
R. Jackiw — MIT Preprint CTP 1298 (198), to appear in Comm.Nucl. Part.Phys.;
P. Nelson and L. Alvarez-gaumé — Comm.Math.Phys. 99 (1985) 103.
G. ‘t Hooft — in “Recent Progress in gauge Theories”, G. ‘t Hooft et al., eds. (Plenum Press, N.Y., 1980 ).
W.A. Bardeen and B. Zumino — Nucl.Phys. b244 (1984) 241.
A.N. Redlich — Phys.Rev.Lett. 52 (1984) 1. See also Ref. 10.
L. Alvarez-Gaumé, S. Della Pietra and G. Moore — Ann.Phys. 163 (1985) 288.
A. Niemi and G. Semenoff — Physics Reports 136 (1986) 100.
M.F. Atiyah, V.I. Patodi and J.M. Singer — Proc.Camb.Phil.Soc. 77 (1975) 43;
M.F. Atiyah, V.I. Patodi and J.M. Singer — Proc.Camb.Phil.Soc. 78 (1975) 405;
M.F. Atiyah, V.I. Patodi and J.M. Singer — Proc.Camb.Phil.Soc. 79 (1976) 71.
J. Schwarz — Physics Reports 89 (1982) 223; and Proceedings of the Santa Barbara Workshop on Superstrings, M. Green and D. Gross, eds. ( World Scientific, 1986 ).
F. Gliozzi, J. Scherk and D. Olive — Nucl.Phys. B122 (1977) 253.
See Ref. 3.
J. Birman — “Braids, Links and Mapping Class Groups”, Lecture Notes in Mathematics (Princeton University Press, 1974 ).
R. Stong — “Lectures on Cobordism Theory”, Lecture Notes in Mathematics (Princeton University Press, 1968 ).
L. Alvarez-Gaumé, P. Ginsparg, G. Moore and C. Vafa — HUTP 86/A013; L. Dixon and J. Harvey — Princeton Preprint
B.A. Schellekens and N. Warner — CERN Preprints TH. 4464 and 4465 (1986).
A.M. Polyakov — Phys.Lett. 103B (1981) 207, 211;
D. Friedan — in. Friedan — in “Recent Advances in Field Theory and Statistical Mechanics”, J.B. Zuber and R. Stora, eds. (Elsevier, 1984 );
O. Alvarez — Nucl.Phys. B216 (1983) 125;
J. Polchinski — Comm.Math.Phys. 104 (1986) 37;
G. Moore and P. Nelson — Nucl.Phys. B266 (1986) 58;
E. D’Hoker and D. Phong — Nucl.Phys. B269 (1986) 206.
H. Harris — in “Proceedings of the International Congress of Mathematicians”, Warszawa, Poland, C. Olech and Z. Ciesielski, eds. ( Elsevier, Amsterdam, 1984 ).
A. Belavin and V.I. Knizhnik — “Algebraic Geometry and the Geometry of Quantum Strings”, Landau Institute Preprint (to appear in Nucl.Phys.);
R. Catenacci, M. Cornalba, M. Martellini and C. Reina -. Reina — “Algebraic Geometry and Path Integrals for Closed Strings”, Pavia Preprint; J.B. Bost and T. Jolicoeur — Saclay PhT/86–28 (1986).
L. Alvarez-Gaumé, G. Moore and C. Vafa — Comm.Math.Phys. 106 (1986) 40;
J.B. Bost and P. Nelson — Phys.Rev.Lett. 57 (1986) 795.
D. Friedan, E. Martinec and S. Shenker — Nucl.Phys. B271 (1986) 93.
L. Alvarez-Gaumé, J.B. Bost, G. Moore, P. Nelson and C. Vafa — HUTP 86/A036 (1986) (to appear in Phys.Lett. B) and in preparation.
D. Quillen — Funct.Anal.Appl. 19 (1986) 31.
G. Faltings — Ann. of Math. 119 (1984) 387.
H. Farkas and J. Kra — “Riemann Surfaces”, Springer (1980).
E. D’Hoker and D. Phong — Comm.Math.Phys. 104 (1986) 537.
D. Mumford — “Tata Lectures on Theta”, Birkhauser (1983).
G. Gunning — “Introduction to Riemann Surfaces”, Princeton Lecture Notes in Mathematics (Princeton University Press, 1966 ).
Yu. Manin — JETP Lett. 43 (1986) 204.
D. Friedan and S. Shenker — Chicago Preprints EFI-86–18A, B (1986).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Plenum Press, New York
About this chapter
Cite this chapter
Alvarez-Gaumé, L. (1990). Anomalies, Strings and Algebraic Geometry. In: Zichichi, A. (eds) The Superworld I. The Subnuclear Series, vol 24. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-1318-2_4
Download citation
DOI: https://doi.org/10.1007/978-1-4684-1318-2_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4684-1320-5
Online ISBN: 978-1-4684-1318-2
eBook Packages: Springer Book Archive