Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula

Matone, Marco; Volpato, Roberto

doi:10.1090/S0002-9939-2012-11526-6

Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula
HTML articles powered by AMS MathViewer

by Marco Matone and Roberto Volpato PDF

Proc. Amer. Math. Soc. 141 (2013), 2575-2587 Request permission

Abstract:

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for $g\geq 4$, a new class of vector-valued modular forms, defined on the Teichmüller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus $4$ is proportional to the square root of the products of Thetanullwerte $\chi _{68}$, which is a proof of the recently rediscovered Klein “amazing formula”. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, $\chi _{68}$ and the theta series corresponding to the even unimodular lattices $E_8\oplus E_8$ and $D_{16}^+$. We also find, for $g=4$, a functional relation between the singular component of the theta divisor and the Riemann period matrix.

References

Marco Matone and Roberto Volpato, Higher genus superstring amplitudes from the geometry of moduli space, Nuclear Phys. B 732 (2006), no. 1-2, 321–340. MR 2192117, DOI 10.1016/j.nuclphysb.2005.10.036
Marco Matone and Roberto Volpato, Superstring measure and non-renormalization of the three-point amplitude, Nuclear Phys. B 806 (2009), no. 3, 735–747. MR 2467587, DOI 10.1016/j.nuclphysb.2008.08.011
Marco Matone and Roberto Volpato, Getting superstring amplitudes by degenerating Riemann surfaces, Nuclear Phys. B 839 (2010), no. 1-2, 21–51. MR 2670813, DOI 10.1016/j.nuclphysb.2010.05.020
Marco Matone and Roberto Volpato, Linear relations among holomorphic quadratic differentials and induced Siegel’s metric on $\scr M_g$, J. Math. Phys. 52 (2011), no. 10, 102305, 13. MR 2894590, DOI 10.1063/1.3653550
M. Matone, Extending the Belavin-Knizhnik “wonderful formula” by the characterization of the Jacobian, JHEP 1210 (2012), 175.
John Fay, Kernel functions, analytic torsion, and moduli spaces, Mem. Amer. Math. Soc. 96 (1992), no. 464, vi+123. MR 1146600, DOI 10.1090/memo/0464
John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
M. Matone and R. Volpato, Determinantal characterization of canonical curves and combinatorial theta identities, Math. Ann. DOI:10.1007/s00208-012-0787-z
David Mumford, Stability of projective varieties, Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272
A. A. Belavin and V. G. Knizhnik, Algebraic geometry and the geometry of quantum strings, Phys. Lett. B 168 (1986), no. 3, 201–206. MR 830618, DOI 10.1016/0370-2693(86)90963-9
Loriano Bonora, Adrian Lugo, Marco Matone, and Jorge Russo, A global operator formalism on higher genus Riemann surfaces: $b$-$c$ systems, Comm. Math. Phys. 123 (1989), no. 2, 329–352. MR 1002042
J.-B. Bost and T. Jolicœur, A holomorphy property and the critical dimension in string theory from an index theorem, Phys. Lett. B 174 (1986), no. 3, 273–276. MR 848539, DOI 10.1016/0370-2693(86)91097-X
Jean-Michel Bismut and Daniel S. Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles, Comm. Math. Phys. 106 (1986), no. 1, 159–176. MR 853982
Luis Alvarez-Gaumé, Jean-Benoît Bost, Gregory Moore, Philip Nelson, and Cumrun Vafa, Bosonization on higher genus Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 3, 503–552. MR 908551
D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177. MR 383463, DOI 10.2307/1970909
A. A. Beĭlinson and Yu. I. Manin, The Mumford form and the Polyakov measure in string theory, Comm. Math. Phys. 107 (1986), no. 3, 359–376. MR 866195
Erik Verlinde and Herman Verlinde, Chiral bosonization, determinants and the string partition function, Nuclear Phys. B 288 (1987), no. 2, 357–396. MR 896667, DOI 10.1016/0550-3213(87)90219-7
Gerard van der Geer, Siegel modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 181–245. MR 2409679, DOI 10.1007/978-3-540-74119-0_{3}
M. Matone and R. Volpato, The singular locus of the theta divisor and quadrics through a canonical curve, arXiv:0710.2124 [math.AG].
N. I. Shepherd-Barron, Thomae’s formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves, arXiv:0802.3014 [math.AG].
A. Belavin, V. Knizhnik, A. Morozov, and A. Perelomov, Two- and three-loop amplitudes in the bosonic string theory, Phys. Lett. B 177 (1986), no. 3-4, 324–328. MR 860363, DOI 10.1016/0370-2693(86)90761-6
A. Morozov, Explicit formulae for one-, two-, three- and four-loop string amplitudes, Phys. Lett. B 184 (1987), no. 2-3, 171–176. MR 879231, DOI 10.1016/0370-2693(87)90563-6
E. D’Hoker and D. H. Phong, Two-loop superstrings. IV. The cosmological constant and modular forms, Nuclear Phys. B 639 (2002), no. 1-2, 129–181. MR 1925297, DOI 10.1016/S0550-3213(02)00516-3
Takashi Ichikawa, On Teichmüller modular forms, Math. Ann. 299 (1994), no. 4, 731–740. MR 1286895, DOI 10.1007/BF01459809
Takashi Ichikawa, Teichmüller modular forms of degree $3$, Amer. J. Math. 117 (1995), no. 4, 1057–1061. MR 1342841, DOI 10.2307/2374959
Eric D’Hoker and D. H. Phong, Asyzygies, modular forms, and the superstring measure. II, Nuclear Phys. B 710 (2005), no. 1-2, 83–116. MR 2126049, DOI 10.1016/j.nuclphysb.2004.12.020
Jun-ichi Igusa, Schottky’s invariant and quadratic forms, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 352–362. MR 661078
Jun-ichi Igusa, On the irreducibility of Schottky’s divisor, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 531–545 (1982). MR 656035
Shigeaki Tsuyumine, Thetanullwerte on a moduli space of curves and hyperelliptic loci, Math. Z. 207 (1991), no. 4, 539–568. MR 1119956, DOI 10.1007/BF02571407
Gilles Lachaud, Christophe Ritzenthaler, and Alexey Zykin, Jacobians among abelian threefolds: a formula of Klein and a question of Serre, Math. Res. Lett. 17 (2010), no. 2, 323–333. MR 2644379, DOI 10.4310/MRL.2010.v17.n2.a11
Felix Klein, Zur Theorie der Abel’schen Functionen, Math. Ann. 36 (1890), no. 1, 1–83 (German). MR 1510611, DOI 10.1007/BF01199432
G. Salmon, Traité de géométrie analytique à trois dimensions. Troisième partie. Ouvrage traduit de l’anglais sur la quatrième édition, Paris, 1892.
A. A. Belavin and V. G. Knizhnik, Complex geometry and the theory of quantum strings, Zh. Èksper. Teoret. Fiz. 91 (1986), no. 2, 364–390 (Russian); English transl., Soviet Phys. JETP 64 (1986), no. 2, 214–228 (1987). MR 888370
Juan Mateos Guilarte and José M. Muñoz Porras, Four-loop vacuum amplitudes for the bosonic string, Proc. Roy. Soc. London Ser. A 451 (1995), no. 1942, 319–329. MR 1362484, DOI 10.1098/rspa.1995.0127