Gravity, strings, modular and quasimodular forms

P. Marios Petropoulos; Pierre Vanhove

doi:10.5802/ambp.317

Gravity, strings, modular and quasimodular forms
[Gravité, cordes, formes modulaires et quasimodulaires]

P. Marios Petropoulos ¹ ; Pierre Vanhove ²

¹ Centre de Physique Théorique École Polytechnique, CNRS UMR 7644 91128 Palaiseau Cedex France
² IHÉS Le Bois-Marie 91440 Bures-sur-Yvette, France and Institut de Physique Théorique CEA, CNRS URA 2306 91191 Gif-sur-Yvette France

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 2, pp. 379-430.

Résumé
Abstract

Les formes modulaires et quasimodulaires ont joué un rôle important dans la théorie de la gravité et la théorie des cordes. Les séries d’Eisenstein sont apparues de façon systématique dans la détermination des spectres. Les fonctions de partitions sont apparues de façon systématique dans la description des effets non perturbatifs, dans les corrections d’ordre supérieur des espaces de champs scalaires,... Ces dernières apparaissent souvent comme des instantons gravitationnels, c’est-à-dire des solutions particulières des équations d’Einstein. Dans ces notes de cours, nous présentons une classe de telles solutions en dimension quatre, obtenues en exigeant l’autodualité (conforme) et l’homogénéité Bianchi IX. Dans ce cas, un large ensemble de configurations existe qui exhibent d’intéressantes propriétés modulaires. Nous donnons d’autres exemples d’espaces d’Einstein qui bien que n’ayant pas de symétrie Bianchi IX possèdent des caractéristiques similaires. Enfin, nous discutons de l’émergence et du rôle des séries d’Eisenstein dans le cadre des développements perturbatifs de la théorie des champs et des cordes. Nous motivons le besoin d’étudier dans ce cadre de nouvelles structures modulaires.

Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative effects, in higher-order corrections of scalar-field spaces, ...The latter often appear as gravitational instantons i.e. as special solutions of Einstein’s equations. In the present lecture notes we present a class of such solutions in four dimensions, obtained by requiring (conformal) self-duality and Bianchi IX homogeneity. In this case, a vast range of configurations exist, which exhibit interesting modular properties. Examples of other Einstein spaces, without Bianchi IX symmetry, but with similar features are also given. Finally we discuss the emergence and the role of Eisenstein series in the framework of field and string theory perturbative expansions, and motivate the need for unravelling novel modular structures.

MR Zbl

DOI : 10.5802/ambp.317

Affiliations des auteurs :

P. Marios Petropoulos ¹ ; Pierre Vanhove ²

¹ Centre de Physique Théorique École Polytechnique, CNRS UMR 7644 91128 Palaiseau Cedex France
² IHÉS Le Bois-Marie 91440 Bures-sur-Yvette, France and Institut de Physique Théorique CEA, CNRS URA 2306 91191 Gif-sur-Yvette France

@article{AMBP_2012__19_2_379_0,
     author = {P. Marios Petropoulos and Pierre Vanhove},
     title = {Gravity, strings, modular and quasimodular~forms},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {379--430},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {19},
     number = {2},
     year = {2012},
     doi = {10.5802/ambp.317},
     mrnumber = {3025139},
     zbl = {1263.11117},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.317/}
}

TY  - JOUR
AU  - P. Marios Petropoulos
AU  - Pierre Vanhove
TI  - Gravity, strings, modular and quasimodular forms
JO  - Annales mathématiques Blaise Pascal
PY  - 2012
SP  - 379
EP  - 430
VL  - 19
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.317/
DO  - 10.5802/ambp.317
LA  - en
ID  - AMBP_2012__19_2_379_0
ER  -

%0 Journal Article
%A P. Marios Petropoulos
%A Pierre Vanhove
%T Gravity, strings, modular and quasimodular forms
%J Annales mathématiques Blaise Pascal
%D 2012
%P 379-430
%V 19
%N 2
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.317/
%R 10.5802/ambp.317
%G en
%F AMBP_2012__19_2_379_0

P. Marios Petropoulos; Pierre Vanhove. Gravity, strings, modular and quasimodular forms. Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 2, pp. 379-430. doi : 10.5802/ambp.317. https://ambp.centre-mersenne.org/articles/10.5802/ambp.317/

Bibliographie
Cité par

[1] M. J. Ablowitz; S. Chakravarty; R. G. Halburd Integrable systems and reductions of the self-dual Yang-Mills equations, J. Math. Phys., Volume 44 (2003) no. 8, pp. 3147-3173 (Integrability, topological solitons and beyond) | DOI | MR | Zbl

[2] M. J. Ablowitz; P. A. Clarkson Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, 149, Cambridge University Press, Cambridge, 1991 | DOI | MR | Zbl

[3] B. S. Acharya; M. O’Loughlin Self-duality in $(D \leq 8)$ -dimensional Euclidean gravity, Phys. Rev. D (3), Volume 55 (1997) no. 8, p. R4521-R4524 | DOI | MR

[4] Sergei Alexandrov Twistor Approach to String Compactifications: a Review (2011) (arXiv:1111.2892v2 [hep-th])

[5] Sergei Alexandrov; Daniel Persson; Boris Pioline On the topology of the hypermultiplet moduli space in type II/CY string vacua, Phys.Rev., Volume D83 (2011), pp. 026001 | DOI

[6] Sergei Alexandrov; Boris Pioline; Frank Saueressig; Stefan Vandoren Linear perturbations of hyperkähler metrics, Lett. Math. Phys., Volume 87 (2009) no. 3, pp. 225-265 | DOI | MR | Zbl

[7] Sergei Alexandrov; Boris Pioline; Frank Saueressig; Stefan Vandoren Linear perturbations of quaternionic metrics, Comm. Math. Phys., Volume 296 (2010) no. 2, pp. 353-403 | DOI | MR | Zbl

[8] Sergei Alexandrov; Boris Pioline; Stefan Vandoren Self-dual Einstein spaces, heavenly metrics, and twistors, J. Math. Phys., Volume 51 (2010) no. 7, pp. 073510, 31 | DOI | MR

[9] Nicola Ambrosetti; Ignatios Antoniadis; Jean-Pierre Derendinger; Pantelis Tziveloglou The Hypermultiplet with Heisenberg Isometry in N=2 Global and Local Supersymmetry, JHEP, Volume 1106 (2011), pp. 139 | DOI | MR

[10] Ignatios Antoniadis; Ruben Minasian; Stefan Theisen; Pierre Vanhove String loop corrections to the universal hypermultiplet, Classical Quantum Gravity, Volume 20 (2003) no. 23, pp. 5079-5102 | DOI | MR | Zbl

[11] M. F. Atiyah; N. J. Hitchin; I. M. Singer Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, Volume 362 (1978) no. 1711, pp. 425-461 | DOI | MR | Zbl

[12] M.F. Atiyah; N.J. Hitchin Low energy scattering of non-Abelian monopoles, Physics Letters A, Volume 107 (1985) no. 1, pp. 21 -25 http://www.sciencedirect.com/science/article/pii/0375960185902385 | DOI | MR | Zbl

[13] M. V. Babich; D. A. Korotkin Self-dual $SU (2)$ -invariant Einstein metrics and modular dependence of theta functions, Lett. Math. Phys., Volume 46 (1998) no. 4, pp. 323-337 | DOI | MR | Zbl

[14] I. Bakas; E. G. Floratos; A. Kehagias Octonionic gravitational instantons, Phys. Lett. B, Volume 445 (1998) no. 1-2, pp. 69-76 | DOI | MR

[15] Ling Bao; Axel Kleinschmidt; Bengt E. W. Nilsson; Daniel Persson; Boris Pioline Instanton corrections to the universal hypermultiplet and automorphic forms on $SU (2, 1)$ , Commun. Number Theory Phys., Volume 4 (2010) no. 1, pp. 187-266 | MR | Zbl

[16] Ling Bao; Axel Kleinschmidt; Bengt E.W. Nilsson; Daniel Persson; Boris Pioline Rigid Calabi-Yau threefolds, Picard Eisenstein series and instantons (2010) (arXiv:1005.4848v1 [hep-th])

[17] A. A. Belavin; A. M. Polyakov; A. S. Schwartz; Yu. S. Tyupkin Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B, Volume 59 (1975) no. 1, pp. 85-87 | DOI | MR

[18] V.A. Belinskii; G.W. Gibbons; D.N. Page; C.N. Pope Asymptotically Euclidean Bianchi IX metrics in quantum gravity, Physics Letters B, Volume 76 (1978) no. 4, pp. 433 -435 http://www.sciencedirect.com/science/article/pii/0370269378908997 | DOI | MR

[19] G. Bossard; P.M. Petropoulos; P. Tripathy Darboux–Halphen system and the action of Geroch group (2012) (Unpublished)

[20] F. Bourliot; J. Estes; P. M. Petropoulos; Ph Spindel $G 3$ -homogeneous gravitational instantons, Classical Quantum Gravity, Volume 27 (2010) no. 10, pp. 105007, 17 | DOI | MR | Zbl

[21] F. Bourliot; J. Estes; P. M. Petropoulos; Ph. Spindel Gravitational instantons, self-duality, and geometric flows, Phys. Rev. D, Volume 81 (2010) no. 10, pp. 104001, 5 | DOI | MR

[22] D. Brecher; M. J. Perry Ricci-flat branes, Nuclear Phys. B, Volume 566 (2000) no. 1-2, pp. 151-172 | DOI | MR | Zbl

[23] D. J. Broadhurst; D. Kreimer Association of multiple zeta values with positive knots via Feynman diagrams up to $9$ loops, Phys. Lett. B, Volume 393 (1997) no. 3-4, pp. 403-412 | DOI | MR | Zbl

[24] D. J. Broadhurst; D. Kreimer Feynman diagrams as a weight system: four-loop test of a four-term relation, Phys. Lett. B, Volume 426 (1998) no. 3-4, pp. 339-346 | DOI | MR | Zbl

[25] Francis Brown On the decomposition of motivic multiple zeta values (2011) (arXiv:1102.1310v2 [math.NT])

[26] Francis C.S. Brown On the periods of some Feynman integrals (2010) (arXiv:0910.0114v2 [math.AG])

[27] M. Cahen; R. Debever; L. Defrise A complex vectorial formalism in general relativity, J. Math. Mech., Volume 16 (1967), pp. 761-785 | MR | Zbl

[28] D. M. J. Calderbank; H. Pedersen Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 3, pp. 921-963 | DOI | Numdam | MR | Zbl

[29] David M. J. Calderbank; Henrik Pedersen Selfdual Einstein metrics with torus symmetry, J. Differential Geom., Volume 60 (2002) no. 3, pp. 485-521 http://projecteuclid.org/getRecord?id=euclid.jdg/1090351125 | MR | Zbl

[30] J. Chazy Sur les équations différentielles dont l’intégrale générale possède une coupure essentielle mobile., C.R. Acad. Sc. Paris, Volume 150 (1910), pp. 456-458

[31] J. Chazy Sur les équations différentielles du troisième ordre et d’ordre supérieur dont líntégrale générale a ses points critiques fixes., Acta Math., Volume 34 (1911), pp. 317-385 | DOI | MR

[32] Bennett Chow; Dan Knopf The Ricci flow: an introduction, Mathematical Surveys and Monographs, 110, American Mathematical Society, Providence, RI, 2004 | MR | Zbl

[33] S. Coleman Aspects of Symmetry, Cambridge University Press, 1985 | Zbl

[34] M. Cvetič; G. W. Gibbons; H. Lü; C. N. Pope Cohomogeneity one manifolds of Spin(7) and $G_{2}$ holonomy, Phys. Rev. D (3), Volume 65 (2002) no. 10, pp. 106004, 29 | DOI | MR | Zbl

[35] Mirjam Cvetic; G.W. Gibbons; H. Lu; C.N. Pope Bianchi IX selfdual Einstein metrics and singular G(2) manifolds, Class.Quant.Grav., Volume 20 (2003), pp. 4239-4268 | DOI | MR | Zbl

[36] Gaston Darboux Mémoire sur la théorie des coordonnées curvilignes, et des systèmes orthogonaux, Ann. Sci. École Norm. Sup. (2), Volume 7 (1878), p. 101-150, 227–260, 275–348 | Numdam

[37] P. Deligne Multizêtas [d’après Francis Brown] (Janvier 2012) (Séminaire Bourbaki)

[38] Eric D’Hoker; D. H. Phong The box graph in superstring theory, Nuclear Phys. B, Volume 440 (1995) no. 1-2, pp. 24-94 | DOI | MR | Zbl

[39] Tohru Eguchi; Peter B. Gilkey; Andrew J. Hanson Gravitation, gauge theories and differential geometry, Phys. Rep., Volume 66 (1980) no. 6, pp. 213-393 | DOI | MR

[40] Tohru Eguchi; Andrew J. Hanson Gravitational instantons, Gen. Relativity Gravitation, Volume 11 (1979) no. 5, pp. 315-320 | DOI | MR

[41] Tohru Eguchi; Andrew J. Hanson Selfdual Solutions to Euclidean Gravity, Annals Phys., Volume 120 (1979), pp. 82-106 | DOI | MR | Zbl

[42] S. Ferrara; S. Sabharwal Quaternionic Manifolds for Type II Superstring Vacua of Calabi-Yau Spaces, Nucl.Phys., Volume B332 (1990), pp. 317 | DOI | MR

[43] E. G. Floratos; A. Kehagias Eight-dimensional self-dual spaces, Phys. Lett. B, Volume 427 (1998) no. 3-4, pp. 283-290 | DOI | MR

[44] Daniel Harry Friedan Nonlinear Models in Two + Epsilon Dimensions, Annals Phys., Volume 163 (1985), pp. 318 | DOI | MR | Zbl

[45] Herbert Gangl; Masanobu Kaneko; Don Zagier Double zeta values and modular forms, Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, pp. 71-106 | DOI | MR | Zbl

[46] Robert Geroch A method for generating solutions of Einstein’s equations, J. Mathematical Phys., Volume 12 (1971), pp. 918-924 | DOI | MR | Zbl

[47] G. W. Gibbons; S. W. Hawking Classification of gravitational instanton symmetries, Comm. Math. Phys., Volume 66 (1979) no. 3, pp. 291-310 http://projecteuclid.org/getRecord?id=euclid.cmp/1103905051 | DOI | MR

[48] G. W. Gibbons; N. S. Manton Classical and quantum dynamics of BPS monopoles, Nuclear Phys. B, Volume 274 (1986) no. 1, pp. 183-224 | DOI | MR

[49] G. W. Gibbons; C. N. Pope $C P^{2}$ as a gravitational instanton, Comm. Math. Phys., Volume 61 (1978) no. 3, pp. 239-248 | DOI | MR | Zbl

[50] G. W. Gibbons; C. N. Pope The positive action conjecture and asymptotically Euclidean metrics in quantum gravity, Comm. Math. Phys., Volume 66 (1979) no. 3, pp. 267-290 http://projecteuclid.org/getRecord?id=euclid.cmp/1103905050 | DOI | MR

[51] G.W. Gibbons; S.W. Hawking Gravitational multi-instantons, Physics Letters B, Volume 78 (1978) no. 4, pp. 430 -432 http://www.sciencedirect.com/science/article/pii/0370269378904781 | DOI

[52] A.S. Goncharov Hodge correlators (2010) (arXiv:0803.0297v2 [math.AG]) | Zbl

[53] Michael B. Green; Stephen D. Miller; Jorge G. Russo; Pierre Vanhove Eisenstein series for higher-rank groups and string theory amplitudes, Commun.Num.Theor.Phys., Volume 4 (2010), pp. 551-596 | MR | Zbl

[54] Michael B. Green; Jorge G. Russo; Pierre Vanhove Low energy expansion of the four-particle genus-one amplitude in type II superstring theory, J. High Energy Phys. (2008) no. 2, pp. 020, 56 | DOI | MR

[55] Michael B. Green; John H. Schwarz; Edward Witten Superstring theory. Vol. 1, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987 (Introduction) | MR | Zbl

[56] Michael B. Green; John H. Schwarz; Edward Witten Superstring theory. Vol. 2., Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987 (Loop amplitudes, anomalies and phenomenology) | MR | Zbl

[57] Michael B. Green; Pierre Vanhove The Low-energy expansion of the one loop type II superstring amplitude, Phys.Rev., Volume D61 (2000), pp. 104011 | DOI | MR

[58] G. H. Halphen Sur certains systéme d’équations différetielles, C. R. Acad. Sci Paris, Volume 92 (1881), pp. 1404-1407

[59] G. H. Halphen Sur une systéme d’équations différetielles, C. R. Acad. Sci Paris, Volume 92 (1881), pp. 1101-1103

[60] Richard S. Hamilton Three-manifolds with positive Ricci curvature, J. Differential Geom., Volume 17 (1982) no. 2, pp. 255-306 http://projecteuclid.org/getRecord?id=euclid.jdg/1214436922 | MR | Zbl

[61] N. J. Hitchin Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom., Volume 42 (1995) no. 1, pp. 30-112 http://projecteuclid.org/getRecord?id=euclid.jdg/1214457032 | MR | Zbl

[62] G. ’t Hooft Symmetry Breaking through Bell-Jackiw Anomalies, Phys. Rev. Lett., Volume 37 (1976), pp. 8-11 http://link.aps.org/doi/10.1103/PhysRevLett.37.8 | DOI

[63] James Isenberg; Martin Jackson Ricci flow of locally homogeneous geometries on closed manifolds, J. Differential Geom., Volume 35 (1992) no. 3, pp. 723-741 http://projecteuclid.org/getRecord?id=euclid.jdg/1214448265 | MR | Zbl

[64] Evgeny Ivanov; Galliano Valent Harmonic space construction of the quaternionic Taub-NUT metric, Classical Quantum Gravity, Volume 16 (1999) no. 3, pp. 1039-1056 | DOI | MR | Zbl

[65] John David Jackson Classical electrodynamics, John Wiley & Sons Inc., New York, 1975 | MR | Zbl

[66] Michio Jimbo; Tetsuji Miwa Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D, Volume 2 (1981) no. 3, pp. 407-448 | DOI | MR | Zbl

[67] Edward Kasner Geometrical Theorems on Einstein’s Cosmological Equations, Amer. J. Math., Volume 43 (1921) no. 4, pp. 217-221 | DOI | MR

[68] Neal Koblitz Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, 97, Springer-Verlag, New York, 1993 | MR | Zbl

[69] H. A. Kramers; G. H. Wannier Statistics of the two-dimensional ferromagnet. I, Phys. Rev. (2), Volume 60 (1941), pp. 252-262 | DOI | MR | Zbl

[70] Robert P. Langlands On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin, 1976 | MR | Zbl

[71] C. R. LeBrun $ℋ$ -space with a cosmological constant, Proc. Roy. Soc. London Ser. A, Volume 380 (1982) no. 1778, pp. 171-185 | DOI | MR | Zbl

[72] D. Lorenz Gravitational instanton solutions for Bianchi types I–IX, Acta Phys.Polon., Volume B14 (1983), pp. 791-805 | MR

[73] Dieter Lorenz-Petzold Gravitational instanton solutions, Progr. Theoret. Phys., Volume 81 (1989) no. 1, pp. 17-22 | DOI | MR

[74] Andrzej J. Maciejewski; Jean-Marie Strelcyn On the algebraic non-integrability of the Halphen system, Phys. Lett. A, Volume 201 (1995) no. 2-3, pp. 161-166 | DOI | MR | Zbl

[75] N. S. Manton A remark on the scattering of BPS monopoles, Phys. Lett. B, Volume 110 (1982) no. 1, pp. 54-56 | DOI | MR | Zbl

[76] R. Maszczyk; L. J. Mason; N. M. J. Woodhouse Self-dual Bianchi metrics and the Painlevé transcendents, Classical Quantum Gravity, Volume 11 (1994) no. 1, pp. 65-71 http://stacks.iop.org/0264-9381/11/65 | DOI | MR | Zbl

[77] John Milnor Curvatures of left invariant metrics on Lie groups, Advances in Math., Volume 21 (1976) no. 3, pp. 293-329 | DOI | MR | Zbl

[78] C. Mœglin; J.-L. Waldspurger Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, 113, Cambridge University Press, Cambridge, 1995 (Une paraphrase de l’Écriture [A paraphrase of Scripture]) | DOI | MR | Zbl

[79] C. Montonen; D. Olive Magnetic monopoles as gauge particles?, Physics Letters B, Volume 72 (1977) no. 1, pp. 117 -120 http://www.sciencedirect.com/science/article/pii/0370269377900764 | DOI

[80] E. Newman; L. Tamburino; T. Unti Empty-space generalization of the Schwarzschild metric, J. Mathematical Phys., Volume 4 (1963), pp. 915-923 | DOI | MR | Zbl

[81] Lars Onsager Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev., Volume 65 (1944), pp. 117-149 http://link.aps.org/doi/10.1103/PhysRev.65.117 | DOI | MR | Zbl

[82] A.R. Osborne; T.L. Burch Internal Solitons in the Andaman Sea, Science, Volume 208 (1980), pp. 451-460 | DOI

[83] H. Pedersen Eguchi-Hanson metrics with cosmological constant, Classical Quantum Gravity, Volume 2 (1985) no. 4, pp. 579-587 http://stacks.iop.org/0264-9381/2/579 | DOI | MR | Zbl

[84] H. Pedersen Einstein metrics, spinning top motions and monopoles, Math. Ann., Volume 274 (1986) no. 1, pp. 35-59 | DOI | MR | Zbl

[85] Henrik Pedersen; Yat Sun Poon Hyper-Kähler metrics and a generalization of the Bogomolny equations, Comm. Math. Phys., Volume 117 (1988) no. 4, pp. 569-580 http://projecteuclid.org/getRecord?id=euclid.cmp/1104161817 | DOI | MR | Zbl

[86] Henrik Pedersen; Yat Sun Poon Kähler surfaces with zero scalar curvature, Classical Quantum Gravity, Volume 7 (1990) no. 10, pp. 1707-1719 http://stacks.iop.org/0264-9381/7/1707 | DOI | MR | Zbl

[87] Grisha Perelman The Entropy formula for the Ricci flow and its geometric applications (2002) (arXiv:math/0211159v1 [math.DG]) | Zbl

[88] Grisha Perelman Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003) (arXiv:math/0307245v1 [math.DG]) | Zbl

[89] Grisha Perelman Ricci flow with surgery on three-manifolds (2003) (arXiv:math/0303109v1 [math.DG]) | Zbl

[90] P.M. Petropoulos; V. Pozzoli; K. Siampos Self-dual gravitational instantons and geometric flows of all Bianchi types (2011) (arXiv:1108.0003v2 [hep-th]) | MR | Zbl

[91] Michael P. Ryan; Lawrence C. Shepley Homogeneous relativistic cosmologies, Princeton University Press, Princeton, N.J., 1975 (Princeton Series in Physics) | MR

[92] O. Schlotterer; S. Stieberger Motivic Multiple Zeta Values and Superstring Amplitudes (2012) (arXiv:1205.1516v1 [hep-th])

[93] Peter Scott The geometries of $3$ -manifolds, Bull. London Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[94] N. Seiberg; E. Witten Electric-magnetic duality, monopole condensation, and confinement in $N = 2$ supersymmetric Yang-Mills theory, Nuclear Physics B, Volume 426 (1994) no. 1, pp. 19 -52 http://www.sciencedirect.com/science/article/pii/0550321394901244 (Erratum [95]) | DOI | MR | Zbl

[95] N. Seiberg; E. Witten Erratum, Nuclear Physics B, Volume 430 (1994) no. 2, pp. 485 -486 http://www.sciencedirect.com/science/article/pii/0550321394004498 | DOI | MR | Zbl

[96] Jean-Pierre Serre Cours d’arithmétique, Presses Universitaires de France, Paris, 1977 (Deuxième édition revue et corrigée, Le Mathématicien, No. 2) | MR | Zbl

[97] K. Sfetsos T-duality and RG-flows (18-22 September 2006) (ERG2006, Lefkada, Greece, unpublished.)

[98] Philippe Spindel Gravity before supergravity, Supersymmetry (Bonn, 1984) (NATO Adv. Sci. Inst. Ser. B Phys.), Volume 125, Plenum, New York, 1985, pp. 455-533 | MR

[99] L. A. Takhtajan A simple example of modular forms as tau-functions for integrable equations, Teoret. Mat. Fiz., Volume 93 (1992) no. 2, pp. 330-341 | DOI | MR | Zbl

[100] Audrey Terras Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985 | MR | Zbl

[101] W. Thurston The Geometry and Topology of Three-Manifolds (1978 – 1981) (Princeton lecture notes)

[102] K. P. Tod A comment on: “Kähler surfaces with zero scalar curvature” [Classical Quantum Gravity 7 (1990), no. 10, 1707–1719; MR1075860 (91i:53057)] by H. Pedersen and Y. S. Poon, Classical Quantum Gravity, Volume 8 (1991) no. 5, pp. 1049-1051 http://stacks.iop.org/0264-9381/8/1049 | DOI | MR | Zbl

[103] K. P. Tod Self-dual Einstein metrics from the Painlevé VI equation, Phys. Lett. A, Volume 190 (1994) no. 3-4, pp. 221-224 | DOI | MR | Zbl

[104] R. S. Ward Self-dual space-times with cosmological constant, Comm. Math. Phys., Volume 78 (1980/81) no. 1, pp. 1-17 http://projecteuclid.org/getRecord?id=euclid.cmp/1103908499 | DOI | MR | Zbl

[105] R. S. Ward Integrable and solvable systems, and relations among them, Philos. Trans. Roy. Soc. London Ser. A, Volume 315 (1985) no. 1533, pp. 451-457 (With discussion, New developments in the theory and application of solitons) | DOI | MR | Zbl

Cité par Sources :