Abstract
We sketch some of the different roles played by Whitham times in connection with averaging, adiabatic invariants, soliton theory, Hamiltonian structures, topological field theory (TFT), Seiberg–Witten (SW) theory, isomonodromy problems, Hitchin systems, WDVV and Picard–Fuchs equations, renormalization, soft supersymmetry breaking, etc.
Similar content being viewed by others
References
Adams, M., Harnad, J. and Previato, E.: Comm.Math.Phys. 117 (1988), 451–500.
Adams, M., Harnad, J. and Hurtubise, J.: Comm.Math.Phys. 134 (1990), 555–585; 155 (1993), 385–413; Lett.Math.Phys. 20 (1990), 299–308.
Adler, M. and van Moerbeke, P.: Comm.Math.Phys. 147 (1992), 25–56.
Álvarez-Gaumé, L., Distler, J., Kounnas, C. and Mariño, M.: Internat.J.Modern Phys.A 11 (1996), 4745–4777.
Álvarez-Gaumé, L. and Mariño, M.: Internat.J.Modern Phys.A 12 (1997), 975–1002.
Álvarez-Gaumé, L. Mariño, M. and Zamora, F.: Internat.J.Modern Phys.A 13 (1998), 403–430; 1847–1880.
Aoyama, S. and Kodama, Y.: Modern Phys.Lett.A 9 (1994), 2481–2492; Comm.Math.Phys. 182 (1996), 185–219.
Argyres, P. and Douglas, M.: Nuclear Phys.B 448 (1995), 93–126.
Arnold, V.: Mathematical Methods of Classical Mechanics, Springer, New York, 1978.
Audin, M.: Spinning, Tops, Cambridge Univ. Press, Cambridge, 1996.
Baez, J. and Muniain, J.: Gauge Fields, Knots, and Gravity,World Scientific, Singapore, 1994.
Bailin, D. and Love, A.: Introduction to Gauge Field Theory, IOP Press, Bristol, 1993.
Bailin, D. and Love, A.: Susy Gauge Field Theory and String Theory, IOP Press, Bristol, 1994.
Beauville, A.: Acta Math. 164 (1990), 211–235.
Belokolos, E., Bobenko, A., Enolskij, V., Its, A. and Matveev, V.: Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, New York, 1994.
Bernard, D.: Nuclear Phys.B 303 (1988), 77–93; 309 (1988), 145–174.
Bertoldi, G. and Matone, M.: hep-th 9712039 and 9712109.
Bloch, A. and Kodama, Y.: SIAM J.Appl.Math. 52 (1992), 909–928.
Bonelli, G. and Matone, M.: Phys.Rev.Lett. 76 (1996), 4107–4110; 77 (1996), 4712–4715; hep-th 9712025.
Braden, H., Marshakov, A., Mironov, A. and Morozov, A.: hep-th 9812078 and 9902205.
Brown, L.: Ann.Phys. 126 (1980), 135–153; (with J. Collins) Ann.Phys. 130 (1980), 215–248.
Calogero, F.: Lett.Nuovo Cimento 13 (1976), 411–417.
Carroll, R. and Chang, J.: solv-int 9612010, Appl.Anal. 64 (1997), 343–378.
Carroll, R. and Kodama, Y.: J.Phys.A 28 (1995), 6373–6387; Proc.Conf.Nonlinear Phys., Gallipoli, 1995, World Scientific, Singapore, 1996, pp. 53–59.
Carroll, R.: J.Nonlinear Sci. 4 (1994), 519–544; Teor.Mat.Fizika 99 (1994), 220–225.
Carroll, R.: hep-th 9712110 and 9802130.
Carroll, R.: Appl.Anal. 65 (1997), 333–352.
Carroll, R.: Appl.Anal. 70 (1998), 127–146.
Carroll, R.: Nuclear Phys.B 502 (1997), 561–593; Lecture Notes in Phys. 502, Springer, New York, 1998, pp. 33–56.
Carroll, R.: Remarks on Whitham dynamics, renormalization, and soft supersymmetry breaking, Invited talk,Workshop Univ. of Edinburgh, Sept. 14–18, 1998; Various roles forWhitham times, Invited talk, AusMS and AMS Joint Meetings, Melbourne, Australia, July 11–16, 1999.
Carroll, R.: In: Proc.NEEDS Workshop, Los Alamos, 1994, World Scientific, Singapore, 1995, pp. 24–33.
Carroll, R.: Appl.Anal. 49 (1993), 1–31; 56 (1995), 147–164.
Carroll, R.: Nonlinear Anal. 30 (1997), 187–198.
Carroll, R.: In: Proc.ISSAC Conf., June 1997, Kluwer Acad. Publ., to appear.
Carroll, R.: Topics in Soliton Theory, North-Holland, Amsterdam, 1991.
Carroll, R.: Phys.Lett.A 234 (1997), 171–180.
Danilov, V. and Shokurov, V.: Algebraic Curves, Algebraic Manifolds, and Schemes, Springer, New York, 1998.
Davis, K.: hep-th 9308039.
Dijkgraaf, R., Verlinde, E. and Verlinde, H.: Nuclear Phys.B 348 (1991), 435–456; 352 (1991), 59–86.
Dijkgraaf, R. and Witten, E.: Nuclear Phys.B 342 (1990), 486–522.
Dobrokhotov, S. and Maslov, V.: J.Soviet Math. 16 (1981), 1433–1487.
Dolan, B.: hep-th 9406061.
Dolan, B.: hep-th 9702156 and 9710161.
Dolan, B. and Lewis, A.: hep-th 9904119.
Dolan, B.: Internat.J.Modern Phys.A 9 (1994), 1261–1286.
Dolan, B.: hep-th 9307023, 9307024, and 9403070; cond-mat 9412031.
Dolan, B.: hep-th 9511175; Internat.J.Modern Phys.A 12 (1997), 2413–2424.
Donagi, R.: alg-geom 9705010.
Donagi, R. and Witten, E.: Nuclear Phys.B 460 (1996), 299–334.
Donagi, R.: alg-geom 9505009, Math.Sci.Res.Inst.Publ. 28 (1995), 65–86.
Donagi, R. and Markman, E.: Lecture Notes in Math. 1620, Springer, New York, 1996, pp. 1–119.
Dubrovin, B.: Russian Math.Surveys 36 (1981), 11–92.
Dubrovin, B.: Lecture Notes in Math. 1620, Springer, New York, 1996, pp. 120–348; Nuclear Phys.B 379 (1992), 627–689; Comm.Math.Phys. 145 (1992), 195–207; Integrable Systems, Birkhäuser, 1993, pp. 313–359.
Dubrovin, B.: hep-th 9206037 and 9303152.
Dubrovin, B.: math.AG/9807034.
Dubrovin, B., Krichever, I. and Novikov, S.: Math.Phys.Rev. 3 (1982), 1–150.
Dubrovin, B. and Novikov, S.: Russian Math.Surveys 44 (1989), 35–124; Math.Phys.Rev. 9 (1991), 3–136.
Dubrovin, B. and Zhang, Y.: hep-th 9712232.
Dubrovin, B.: hep-th 9206037.
Earle, C. and Eells, J.: J.Differential Geom. 3 (1969), 19–43.
Edelstein, J., Mariño, M. and Mas, J.: hep-th 9805172.
Edelstein, J. and Mas, J.: hep-th 9901006 and 9902161.
Edelstein, J., Gómez-Reino, M. and Mas, J.: hep-th 9904087.
Eguchi, T. and Yang, S.: Modern Phys.Lett.A 11 (1996), 131–138.
Eguchi, T., Yamada, Y. and Yang, S.: Modern Phys.Lett.A 8 (1993), 1627–1637.
Eguchi, T., Hori, K. and Yang, S.: hep-th 9503017.
Eguchi, T. and Yang, S.: hep-th 9612086.
Enriques, B. and Rubtsov, V.: Math.Phys.Lett. 3 (1996), 343–357.
Etinghof, P. and Kirillov, A.: Duke Math.J. 74 (1994), 585–614.
Falceto, F. and Gawedzky, K.: hep-th 9502161; Comm.Math.Phys. 183 (1997), 267–290.
Fay, J.: Theta Functions on Riemann Surfaces, Lecture Notes in Math. 352, Springer, New York, 1973.
Felder, G.: hep-th 9609153.
Felder, G. and Wieczerkowski, C.: hep-th 9411004.
Flaschka, H., Forest, M. and McLaughlin, D.: Comm.Pure Appl.Math. 33 (1980), 739–784.
Flaschka, H. and Newell, A.: Comm.Math.Phys. 76 (1980), 65–116; Physica D 3 (1981), 203–221.
Forster, O.: Lectures on Riemann Surfaces, Springer, New York, 1981.
Fré, P. and Soriani, P.: The N = 2 Wonderland, World Scientific, Singapore, 1995.
Fucito, F., Gamba, A., Martellini, M. and Ragnisco, O.: Internat.J.Modern Phys.B 6 (1992), 2123–2147.
Gawedzki, K. and Tran-Ngoc-Bich, P.: hep-th 9710025 and 9803101.
Gibbons, J. and Kodama, Y.: Singular Limits of Dispersive Waves, Plenum, New York, 1994, pp. 61–66.
Giradello, L. and Grisaru, M.: Nuclear Phys.B 194 (1982), 65–76.
Gorsky, A., Krichever, I., Marshakov, A., Mironov, A. and Morozov, A.: Phys.Lett.B 355 (1995), 466–474.
Gorsky, A., Marshakov, A., Mironov, A. and Morozov, A.: hep-th 9802007, Nuclear Phys.B 527 (1998), 690–716.
Gorsky, A.: hep-th 9612238.
Gorsky, A., Gukov, S. and Mironov, A.: hep-th 9707120 and 9710239.
Gorsky, A., Nekrasov, N. and Rubtsov, V.: hep-th 9901089.
Gorsky, A. and Mironov, A.: hep-th 9902030.
Gorsky, A. and Marshakov, A.: Phys.Lett.B 375 (1996), 127–134.
Gorsky, A.: hep-th 9812250.
Gorsky, A.: Phys.Lett.B 410 (1997), 22–26.
Griffiths, P.: Amer.J.Math. 107 (1985), 1445–1483.
Gukov, S.: hep-th 9709138.
Gunning, R.: Lectures on Riemann Surfaces, Princeton Univ. Press, Princeton, 1966; Lectures of Vector Bundles over Riemann Surfaces, Princeton Univ. Press, Princeton, 1967.
Hanany, A., Oz, Y. and Plesser, M.: hep-th 9401030.
Harnad, J.: Comm.Math.Phys. 166 (1994), 337–365.
Harnad, J. and Wisse, M.: Fields Inst.Commun. 7 (1996), 155–169.
Harnad, J. and Its, A.: solv-int 9706002.
Harnad, J.: hep-th 9406078.
Hatfield, B.: Quantum Field Theory of Point Particles and Strings, Addison-Wesley, Reading, MA, 1992.
Hitchin, N.: Duke Math.J. 54 (1987), 91–114.
Hitchin, N.: J.Differential Geom. 42 (1995), 52–134.
Hitchin, N.: Proc.London Math.Soc. 55 (1987), 59–126.
Hitchin, N.: Lectures on Riemann Surfaces, World Scientific, Singapore, 1989, pp. 99–118.
Hitchin, N.: Integrable Systems, Oxford Univ. Press, Oxford, 1999, pp. 1–52.
d'Hoker, E. and Phong, D.: hep-th 9701055, 9709053, 9804124, 9804125, 9804126, 9903002, and 9903068.
d'Hoker, E., Krichever, I. and Phong, D.: hep-th 9609041, 9609145; Nuclear Phys.B 494 (1997), 89–104.
Hurtubise, J.: Duke Math.J. 83 (1996), 19–50.
Hyun, S. and Park, J. S.: hep-th 9409009, 9503036, 9503201, and 9508162.
Imayoshi, Y. and Taniguchi, M.: An Introduction to Teichmüller Space, Springer, New York, 1992.
Inozemtsev, V.: Lett.Math.Phys. 17 (1989), 11–17.
Isidro, J.: hep-th 9805051.
Ito, K. and Yang, S.: hep-th 9712018 and 9803126; Phys.Lett.B 366 (1996), 165–173 and 415 (1997), 45–53.
Itoyama, H. and Morozov, A.: hep-th 9511126, 9512161, and 9601168; Nuclear Phys.B 477 (1996), 855–877 and 491 (1997), 529–573.
Ivanov, D.: hep-th 9610207.
Iwasaki, K.: J.Fac.Sci.Univ.Tokyo Sect.1A 38 (1991), 431–531; Pacific J.Math. 155 (1992), 319–340.
Kapustin, A.: Nuclear Phys.B 534 (1998), 531–545; Adv.Theoret.Math.Phys. 2 (1998), 571–591.
Kharchev, S., Marshakov, A., Mironov, A. and Morozov, A.: Modern Phys.Lett.A 8 (1993), 1047–1061 (hep-th 9208046); Nuclear Phys.B 397 (1993), 339–378; Internat J.Modern Phys.A 10 (1995), 2015–2051.
Kharchev, S., Marshakov, A., Mironov, A., Morozov, A. and Zabrodin, A.: Nuclear Phys.B 380 (1992), 181–240; hep-th 9111037.
Knizhnik, V. and Zamolodchikov, A. B.: Nuclear Phys.B 247 (1984), 83–103.
Kobayashi, S.: Differential Geometry of Complex Vector Bundles, Princeton Univ. Press, Princeton, 1987.
Kodama, Y. and Gibbons, J.: Proc.Fourth Workshop on Nonlinear and Turbulent Processes in Physics, World Scientific, Singapore, 1990, pp. 166–180.
Kodaira, K.: Complex Manifolds and Deformation of Complex Structure, Springer, New York, 1986.
Konopelchenko, B.: Introduction to Multidimensional Integrable Equations, Plenum, New York, 1992; Solitons in Multidimensions, World Scientific, Singapore, 1993.
Korotkin, D. and Samtleben, J.: Internat J.Modern Phys.A, 1–2 (1997), 2013–2033.
Kresch, A.: alg-geom 9703015.
Krichever, I.: Funktsional Anal.i Prilozhen. 22 (1988), 200–213.
Krichever, I. and Phong, D.: J.Differential Geom. 45 (1997), 349–389.
Krichever, I.: Comm.Pure Appl.Math. 47 (1994), 437–475; Acta Appl.Math. 39 (1995), 93–125; Comm.Math.Phys. 143 (1992), 415–429.
Krichever, I. and Phong, D.: hep-th 9708170.
Kuroki, G. and Takebe, T.: q-alg 9612033 and math.QA 9809157.
Lässig, M.: Nuclear Phys.B 334 (1990), 652–668.
Latorre, J. and Lütken, C.: hep-th 9711150.
Lebedev, O. and Manin, Yu.: Phys.Lett.A 7 (1979), 164.
Levin, A. and Olshanetsky, M.: alg-geom 9706010 and hep-th 9709207.
Levin, A. and Olshanetsky, M.: Comm.Math.Phys. 188 (1997), 449–466.
Levin, A. and Olshanetsky, M.: math-ph 9904023.
Losev, A., Nekrasov, N. and Shatashvili, S.: hep-th 9711108.
Losev, A.: J.Experiment.Theoret.Phys. 65 (1997), 374–379; hep-th 9801179.
Losev, A.: hep-th 9211090.
Losev, A. and Polyubin, I.: hep-th 9305079.
Lübke, M. and Teleman, A.: The Kobayashi-Hitchin Correspondence, World Scientific, Singapore, 1995.
Lykken, J.: hep-th 9612114.
Manin, Yu. and Merkulov, S.: alg-geom 9702014.
Manin, Yu.: math.QA 9801006.
Mariño, M. and Moore, G.: hep-th 9712062, 9802185, and 9804104.
Mariño, M. and Zamora, F.: Nuclear Phys.B 533 (1998), 373–405.
Mariño, M.: hep-th 9905053.
Markman, E.: Compositio Math. 93 (1994), 255–290.
Marshakov, A., Mironov, A. and Morozov, A.: hep-th 9607109 and 9710123; Modern Phys.Lett.A 12 (1997), 773–787.
Marshakov, A.: hep-th 9709001.
Marshakov, A.: Seiberg-Witten Theory and Integrable Systems, World Scientific, Singapore, 1999.
Marshakov, A. and Mironov, A.: hep-th 9809196.
Martinec, E.: hep-th 9510204.
Martinec, E. and Warner, N.: hep-th 9509161.
Matone, M.: Phys.Lett.B 357 (1995), 342–348; Phys.Rev.D 53 (1996), 7354–7358; Phys. Rev.Lett 78 (1997), 1412–1415.
McDuff, D. and Salamon, D.: Introduction to Symplectic Topology, Oxford Univ. Press, Oxford, 1995.
McLaughlin, D.: Physica D 3 (1981), 335–343.
Miranda, R.: Algebraic Curves and Riemann Surfaces, Amer. Math. Soc., Providence, RI, 1995.
Mironov, A.: Nuclear Phys.B, Supp. 61A (1998), 177–185.
Mironov, A. and Morozov, A.: hep-th 9712177.
Mironov, A.: hep-th 9903088.
Moore, G. and Witten, E.: hep-th 9709193.
Morozov, A.: hep-th 9711194.
Morozov, A.: hep-th 9303139 and 9502091.
Morozov, A.: hep-th 9810031 and 9903087.
Morozov, A.: Soviet Phys.Uspekhi 35 (1992), 671–714; 62 (1994), 1–55; hep-th 9502091.
Morrow, J. and Kodaira, K.: Complex Manifolds, Holt, Rinehart & Winston, New York, 1971.
Nakatsu, T. and Takasaki, K.: Modern Phys.Lett.A 11 (1996), 157–168.
Nakatsu, T.: Modern Phys.Lett.A 9 (1994), 3313–3324.
Nekrasov, N.: Comm.Math.Phys. 180 (1996), 587–603.
Novikov, S., Manakov, S., Pitaevskij, L. and Zakharov, V.: Theory of Solitons, Plenum, New York, 1984.
O'Connor, D. and Stephens, C.: hep-th 9304095, 9310086, and 9310198.
Ohta, J.: J.Math.Phys. 40 (1999), 1891–1900.
Okamoto, K.: Funkcial Ekvac. 14 (1971), 137–152; J.Fac.Sci.Univ.Tokyo Sect.1A 24 (1977), 357–371.
Olshanetsky, M.: hep-th 9901019.
Olshanetsky, M.: hep-th 9510143.
Peskin, M. and Schroeder, D.: An Introduction to QFT, Addison-Wesley, Reading, MA, 1995.
LePotier, J.: Lectures on Vector Bundles, Cambridge Univ. Press, Cambridge, 1997.
Reshetikhin, N.: Comm.Math.Phys. 26 (1992), 167–172.
Ritz, A.: hep-th 9710112.
Seiberg, N. and Witten, E.: Nuclear Phys.B 426 (1994), 19–52.
Seiberg, N.: Phys.Lett.B 206 (1988), 75–80; 318 (1993), 469–475.
Simpson, C.: Publ.Math.IHES 75 (1992), 5–95.
Sklyanin, E. and Takebe, T.: q-alg 9601028 and solv-int 9807008.
Sonoda, H.: Nuclear Phys.B 352 (1991), 585–600 and 601–615; hep-th 9306119.
Stephens, C.: hep-th 9611062.
Taimanov, I.: alg-geom 9609016.
Takasaki, K. and Takebe, T.: Internat.J.Modern Phys.A Supp. (1992), 889–922; Rev.Math.Phys. 7 (1995), 743–808.
Takasaki, K.: hep-th 9803217 and 9901120.
Takasaki, K.: hep-th 9403190.
Takasaki, K. and Takebe, T.: hep-th 9301070.
Takasaki, K. and Nakatsu, T.: hep-th 9603069.
Takasaki, K.: hep-th 9705162; Lett.Math.Phys. 43 (1988), 123–135.
Takasaki, K.: solv-int 9704004.
Takasaki, K.: hep-th 9711058.
Takebe, T.: hep-th 9210086.
Vanhaecke, P.: Lecture Notes in Math. 1638, Springer, New York, 1996.
Vereschagin, V.: hep-th 9605092.
Whitham, G.: Linear and Nonlinear Waves, Wiley, New York, 1974.
Witten, E.: hep-th 9703166.
Witten, E.: Nuclear Phys.B 340 (1990), 281–332.
Yoneya, T.: Comm.Math.Phys. 144 (1992), 623–639.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Carroll, R. Various Aspects of Whitham Times. Acta Applicandae Mathematicae 60, 225–316 (2000). https://doi.org/10.1023/A:1006483905847
Issue Date:
DOI: https://doi.org/10.1023/A:1006483905847