Volume 16, issue 3 (2012)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 29
Issue 5, 2251–2782
Issue 4, 1693–2250
Issue 3, 1115–1691
Issue 2, 549–1114
Issue 1, 1–548

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
$\mathrm{SO}(3)$–Donaldson invariants of $\mathbb{C}\mathrm{P}^2$ and mock theta functions

Andreas Malmendier and Ken Ono

Geometry & Topology 16 (2012) 1767–1833
Bibliography
1 L Álvarez-Gaumé, M Mariño, F Zamora, Softly broken $N=2$ QCD with massive quark hypermultiplets, I, Internat. J. Modern Phys. A 13 (1998) 403 MR1621192
2 G E Andrews, Mordell integrals and Ramanujan's “lost” notebook, from: "Analytic number theory (Philadelphia, PA, 1980)" (editor M I Knopp), Lecture Notes in Math. 899, Springer (1981) 10 MR654518
3 M F Atiyah, The logarithm of the Dedekind $\eta$–function, Math. Ann. 278 (1987) 335 MR909232
4 M F Atiyah, I M Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. U.S.A. 81 (1984) 2597 MR742394
5 K Bringmann, K Ono, The $f(q)$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006) 243 MR2231957
6 K Bringmann, K Ono, Dyson's ranks and Maass forms, Ann. of Math. 171 (2010) 419 MR2630043
7 K Bringmann, K Ono, R C Rhoades, Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008) 1085 MR2425181
8 J H Bruinier, Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, Lecture Notes in Math. 1780, Springer (2002) MR1903920
9 J H Bruinier, J Funke, On two geometric theta lifts, Duke Math. J. 125 (2004) 45 MR2097357
10 J H Bruinier, K Ono, Heegner divisors, $L$–functions and harmonic weak Maass forms, Ann. of Math. 172 (2010) 2135 MR2726107
11 J H Bruinier, G van der Geer, G Harder, D Zagier, The 1-2-3 of modular forms, (K Ranestad, editor), Universitext, Springer (2008) MR2385372
12 J H Bruinier, T Yang, Faltings heights of CM cycles and derivatives of $L$–functions, Invent. Math. 177 (2009) 631 MR2534103
13 Y Choie, M H Lee, Rankin–Cohen brackets on pseudodifferential operators, J. Math. Anal. Appl. 326 (2007) 882 MR2280950
14 S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Math. Monogr., Oxford Science Publ., The Clarendon Press, Oxford Univ. Press (1990) MR1079726
15 F J Dyson, A walk through Ramanujan's garden, from: "Ramanujan revisited (Urbana-Champaign, IL, 1987)" (editors G E Andrews, R A Askey, B C Berndt, K G Ramanathan, R A Rankin), Academic Press (1988) 7 MR938957
16 G Ellingsrud, L Göttsche, Wall-crossing formulas, the Bott residue formula and the Donaldson invariants of rational surfaces, Quart. J. Math. Oxford Ser. 49 (1998) 307 MR1645556
17 D S Freed, K K Uhlenbeck, Instantons and four-manifolds, MSRI Publ. 1, Springer (1991) MR1081321
18 A L Gorodenzev, Top Chern classes of universal bundles on the complex projective plane and correlation functions of asymptotically free SYM $N=2$ QFT, J. Math. Sci. $($New York$)$ 106 (2001) 3240 MR1878046
19 L Göttsche, Modular forms and Donaldson invariants for $4$–manifolds with $b_{+}=1$, J. Amer. Math. Soc. 9 (1996) 827 MR1362873
20 L Göttsche, Donaldson invariants in algebraic geometry, from: "School on Algebraic Geometry (Trieste, 1999)" (editor L Göttsche), ICTP Lect. Notes 1, Abdus Salam Int. Cent. Theoret. Phys. (2000) 101 MR1795862
21 L Göttsche, H Nakajima, K Yoshioka, Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008) 343 MR2472477
22 L Göttsche, D Zagier, Jacobi forms and the structure of Donaldson invariants for $4$–manifolds with $b_+=1$, Selecta Math. 4 (1998) 69 MR1623706
23 F Hirzebruch, D Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976) 57 MR0453649
24 A A Klyachko, Moduli of vector bundles and numbers of classes, Funktsional. Anal. i Prilozhen. 25 (1991) 81 MR1113131
25 D Kotschick, $\mathrm{SO}(3)$–invariants for $4$–manifolds with $b^ +_2=1$, Proc. London Math. Soc. 63 (1991) 426 MR1114516
26 D Kotschick, P Lisca, Instanton invariants of $\mathbf C\mathrm{P}^2$ via topology, Math. Ann. 303 (1995) 345 MR1348804
27 D Kotschick, J W Morgan, $\mathrm{SO}(3)$–invariants for $4$–manifolds with $b^ +_2=1$. II, J. Differential Geom. 39 (1994) 433 MR1267898
28 P B Kronheimer, T S Mrowka, Embedded surfaces and the structure of Donaldson's polynomial invariants, J. Differential Geom. 41 (1995) 573 MR1338483
29 J M F Labastida, C Lozano, Duality in twisted $\mathscr{N}=4$ supersymmetric gauge theories in four dimensions, Nuclear Phys. B 537 (1999) 203 MR1659299
30 T G Leness, Degeneracy loci of families of Dirac operators, Trans. Amer. Math. Soc. 364 (2012) 5995 MR2946940
31 A Losev, N Nekrasov, S Shatashvili, Issues in topological gauge theory, Nuclear Phys. B 534 (1998) 549 MR1663467
32 A Malmendier, |Expressions for the generating function of the Donaldson invariants for $\mathbb{C}\mathrm{P}^2$, PhD thesis, Massachusetts Insitute of Technology (2007)
33 A Malmendier, The signature of the Seiberg–Witten surface, from: "Surveys in differential geometry. Volume XV. Perspectives in mathematics and physics" (editors T Mrowka, S T Yau), Surv. Differ. Geom. 15, Int. Press (2011) 255 MR2815730
34 M Mariño, G Moore, Integrating over the Coulomb branch in $\mathscr{N}=2$ gauge theory, Nuclear Phys. B Proc. Suppl. 68 (1998) 336 MR1641983
35 R Miranda, Persson's list of singular fibers for a rational elliptic surface, Math. Z. 205 (1990) 191 MR1076128
36 R Miranda, An overview of algebraic surfaces, from: "Algebraic geometry (Ankara, 1995)" (editor S Sertöz), Lecture Notes in Pure and Appl. Math. 193, Dekker (1997) 157 MR1483329
37 G Moore, E Witten, Integration over the $u$–plane in Donaldson theory, Adv. Theor. Math. Phys. 1 (1997) 298 MR1605636
38 W Nahm, On electric-magnetic duality, Nuclear Phys. B Proc. Suppl. 58 (1997) 91 MR1486333
39 K Oguiso, T Shioda, The Mordell–Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul. 40 (1991) 83 MR1104782
40 K Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, from: "Current developments in mathematics, 2008" (editors D Jerison, B Mazur, T Mrowka, W Schmid, R P Stanley, S T Yau), Int. Press (2009) 347 MR2555930
41 U Persson, Configurations of Kodaira fibers on rational elliptic surfaces, Math. Z. 205 (1990) 1 MR1069483
42 N Seiberg, E Witten, Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang–Mills theory, Nuclear Phys. B 426 (1994) 19 MR1293681
43 N Seiberg, E Witten, Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric QCD, Nuclear Phys. B 431 (1994) 484 MR1306869
44 Y Shimizu, Seiberg–Witten integrable systems and periods of rational elliptic surfaces, from: "Primes and knots" (editors T Kohno, M Morishita), Contemp. Math. 416, Amer. Math. Soc. (2006) 237 MR2276144
45 G Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973) 440 MR0332663
46 T Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972) 20 MR0429918
47 P F Stiller, Elliptic curves over function fields and the Picard number, Amer. J. Math. 102 (1980) 565 MR584462
48 P F Stiller, Monodromy and invariants of elliptic surfaces, Pacific J. Math. 92 (1981) 433 MR618076
49 S A Strømme, Ample divisors on fine moduli spaces on the projective plane, Math. Z. 187 (1984) 405 MR757480
50 C Vafa, E Witten, A strong coupling test of $S$–duality, Nuclear Phys. B 431 (1994) 3 MR1305096
51 E Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988) 353 MR953828
52 E Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769 MR1306021
53 E Witten, On $S$–duality in abelian gauge theory, Selecta Math. 1 (1995) 383 MR1354602
54 K Yoshioka, The Betti numbers of the moduli space of stable sheaves of rank $2$ on $\mathbf P^2$, J. Reine Angew. Math. 453 (1994) 193 MR1285785
55 D Zagier, Ramanujan's mock theta functions and their applications (after Zwegers and Ono–Bringmann), from: "Séminaire Bourbaki. Vol. 2007/2008", Astérisque 326, Soc. Math. France (2009) 143 MR2605321
56 S P Zwegers, Mock theta functions, PhD thesis, Universiteit Utrecht (2002)