A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves
HTML articles powered by AMS MathViewer
- by I. P. Goulden, J. L. Harer and D. M. Jackson
- Trans. Amer. Math. Soc. 353 (2001), 4405-4427
- DOI: https://doi.org/10.1090/S0002-9947-01-02876-8
- Published electronically: July 9, 2001
- PDF | Request permission
Abstract:
We determine an expression $\xi ^s_g(\gamma )$ for the virtual Euler characteristics of the moduli spaces of $s$-pointed real $(\gamma =1/2$) and complex ($\gamma =1$) algebraic curves. In particular, for the space of real curves of genus $g$ with a fixed point free involution, we find that the Euler characteristic is $(-2)^{s-1}(1-2^{g-1})(g+s-2)!B_g/g!$ where $B_g$ is the $g$th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is $(-1)^{s}(g+s-2)!B_{g+1}/(g+1)(g-1)!$
The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter $\gamma$ that permits specialization of the formula to the real and complex cases. This suggests that $\xi ^s_g(\gamma )$ itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.
References
- D. Bessis, C. Itzykson, and J. B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980), no. 2, 109–157. MR 603127, DOI 10.1016/0196-8858(80)90008-1
- Peter Buser and Mika Seppälä, Symmetric pants decompositions of Riemann surfaces, Duke Math. J. 67 (1992), no. 1, 39–55. MR 1174602, DOI 10.1215/S0012-7094-92-06703-2
- L. Chekhov and A. Zabrodin, A critical matrix model for nonoriented string, Modern Phys. Lett. A 6 (1991), no. 34, 3143–3152. MR 1133827, DOI 10.1142/S0217732391003638
- I. P. Goulden and D. M. Jackson, Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions, Trans. Amer. Math. Soc. 348 (1996), no. 3, 873–892. MR 1325917, DOI 10.1090/S0002-9947-96-01503-6
- I. P. Goulden and D. M. Jackson, Maps in locally orientable surfaces and integrals over real symmetric surfaces, Canad. J. Math. 49 (1997), no. 5, 865–882. MR 1604106, DOI 10.4153/CJM-1997-045-9
- I. P. Goulden and D. M. Jackson, Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials, Canad. J. Math. 48 (1996), no. 3, 569–584. MR 1402328, DOI 10.4153/CJM-1996-029-x
- Phil Hanlon, Jack symmetric functions and some combinatorial properties of Young symmetrizers, J. Combin. Theory Ser. A 47 (1988), no. 1, 37–70. MR 924451, DOI 10.1016/0097-3165(88)90042-8
- John L. Harer, The cohomology of the moduli space of curves, Theory of moduli (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1337, Springer, Berlin, 1988, pp. 138–221. MR 963064, DOI 10.1007/BFb0082808
- John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157–176. MR 830043, DOI 10.1007/BF01388737
- J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), no. 3, 457–485. MR 848681, DOI 10.1007/BF01390325
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- G. ’t Hooft, A planar diagram theory for string interactions, Nuclear Physics B 72 (1974), 461–473.
- C.Itzykson and J.B.Zuber, Combinatorics of mapping class groups and matrix integration, preprint.
- D. M. Jackson, On an integral representation for the genus series for $2$-cell embeddings, Trans. Amer. Math. Soc. 344 (1994), no. 2, 755–772. MR 1236224, DOI 10.1090/S0002-9947-1994-1236224-5
- D. M. Jackson, Counting cycles in permutations by group characters, with an application to a topological problem, Trans. Amer. Math. Soc. 299 (1987), no. 2, 785–801. MR 869231, DOI 10.1090/S0002-9947-1987-0869231-9
- D. M. Jackson and T. I. Visentin, A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus, Trans. Amer. Math. Soc. 322 (1990), no. 1, 343–363. MR 1012517, DOI 10.1090/S0002-9947-1990-1012517-8
- Friedrich Knop and Siddhartha Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9–22. MR 1437493, DOI 10.1007/s002220050134
- Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23. MR 1171758
- Luc Lapointe and Luc Vinet, A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture, Internat. Math. Res. Notices 9 (1995), 419–424. MR 1360620, DOI 10.1155/S1073792895000298
- F. Lesage, V. Pasquier, and D. Serban, Dynamical correlation functions in the Calogero-Sutherland model, Nuclear Phys. B 435 (1995), no. 3, 585–603. MR 1313631, DOI 10.1016/0550-3213(94)00453-L
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
- Andrei Okounkov, Proof of a conjecture of Goulden and Jackson, Canad. J. Math. 49 (1997), no. 5, 883–886. MR 1604110, DOI 10.4153/CJM-1997-046-6
- R. C. Penner, Perturbative series and the moduli space of Riemann surfaces, J. Differential Geom. 27 (1988), no. 1, 35–53. MR 918455
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
- Mika Seppälä, Complex algebraic curves with real moduli, J. Reine Angew. Math. 387 (1988), 209–220. MR 946356, DOI 10.1515/crll.1988.387.209
- M. Seppälä and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties, Math. Z. 201 (1989), no. 2, 151–165. MR 997218, DOI 10.1007/BF01160673
- Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073, DOI 10.1016/0001-8708(89)90015-7
- Kurt Strebel, On quadratic differentials with closed trajectories and second order poles, J. Analyse Math. 19 (1967), 373–382. MR 224808, DOI 10.1007/BF02788726
- Edward Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR 1144529
Bibliographic Information
- I. P. Goulden
- Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
- MR Author ID: 75735
- Email: ipgoulden@math.uwaterloo.ca
- J. L. Harer
- Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708-0320
- MR Author ID: 81320
- Email: harer@math.duke.edu
- D. M. Jackson
- Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
- MR Author ID: 92555
- Email: dmjackson@math.uwaterloo.ca
- Received by editor(s): January 22, 1999
- Received by editor(s) in revised form: April 7, 1999
- Published electronically: July 9, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 4405-4427
- MSC (2000): Primary 58D29, 58C35; Secondary 05C30, 05E05
- DOI: https://doi.org/10.1090/S0002-9947-01-02876-8
- MathSciNet review: 1851176