Duke Mathematical Journal

Hodge cohomology of gravitational instantons

Tamás Hausel, Eugenie Hunsicker, and Rafe Mazzeo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the space of $L^2$ harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincaré metrics on $\mathbb{Q}$-rank $1$ ends of locally symmetric spaces and on the complements of smooth divisors in Kähler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The $L^2$ signature formula implied by our result is closely related to the one proved by Dai [25] and more generally by Vaillant [67], and identifies Dai's $\tau$-invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [12] and the forthcoming paper of Cheeger and Dai [17]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about $L^2$ harmonic forms in duality theories in string theory.

Article information

Source
Duke Math. J., Volume 122, Number 3 (2004), 485-548.

Dates
First available in Project Euclid: 22 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082665286

Digital Object Identifier
doi:10.1215/S0012-7094-04-12233-X

Mathematical Reviews number (MathSciNet)
MR2057017

Zentralblatt MATH identifier
1062.58002

Subjects
Primary: 58A14: Hodge theory [See also 14C30, 14Fxx, 32J25, 32S35] 35S35: Topological aspects: intersection cohomology, stratified sets, etc. [See also 32C38, 32S40, 32S60, 58J15]
Secondary: 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE [See also 32C38, 58J15] 35J70: Degenerate elliptic equations

Citation

Hausel, Tamás; Hunsicker, Eugenie; Mazzeo, Rafe. Hodge cohomology of gravitational instantons. Duke Math. J. 122 (2004), no. 3, 485--548. doi:10.1215/S0012-7094-04-12233-X. https://projecteuclid.org/euclid.dmj/1082665286


Export citation

References

  • M. Atiyah and N. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton Univ. Press, Princeton, 1987.
  • M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.
  • N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Grundlehren Math. Wiss. 298, Springer, Berlin, 1992.
  • J.-M. Bismut and J. Cheeger, $\eta$-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989), 33–70.
  • R. Bielawski, Betti numbers of $3$-Sasakian quotients of spheres by tori, Bull. London Math. Soc. 29 (1997), 731–736.
  • R. Bielawski and A. Dancer, The geometry and topology of toric hyperkähler manifolds, Comm. Anal. Geom. 8 (2000), 727–760.
  • A. Borel, ed., Intersection Cohomology (Bern, 1983), Progr. Math. 50, Birkhäuser, Boston, 1984.
  • R. Bott and L. V. Tu, Differential Forms in Algebraic Topology, Grad. Texts Math. 82, Springer, New York, 1982.
  • C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of $3$-Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183–220.
  • A. Brandhuber, J. Gomis, S. S. Gubser, and S. Gukov, Gauge theory and large $N$ at new $G_2$ holonomy metrics, Nuclear Phys. B 611 (2001), 179–204.
  • R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), 829–850.
  • G. Carron, $L^2$-cohomology of manifolds with flat ends, Geom. Funct. Anal. 13 (2003), 366–395.
  • E. Cattani, A. Kaplan, and W. Schmid, $L^2$ and intersection cohomologies for a polarizable variation of Hodge structure, Invent. Math. 87 (1987), 217–252.
  • J. M. Charap and M. J. Duff, Space-time topology and a new class of Yang-Mills instanton, Phys. Lett. B 71 (1977), 219–221.
  • J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Natl. Acad. Sci. USA 76 (1979), 2103–2106.
  • –. –. –. –., “On the Hodge theory of Riemannian pseudomanifolds” in Geometry of the Laplace Operator (Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math. 36, Amer. Math. Soc., Providence, 1980, 91–146.
  • J. Cheeger and X. Dai, $L^2$ cohomology of a non-isolated conical singularity and nonmultiplicativity of the signature, in preparation.
  • J. Cheeger, M. Goresky, and R. MacPherson, “$L^2$-cohomology and intersection homology of singular algebraic varieties” in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, 1982, 303–340.
  • S. A. Cherkis and A. Kapustin, $D_k$ gravitational instantons and Nahm equations, Adv. Theor. Math. Phys. 2 (1999), 1287–1306.
  • –. –. –. –., Singular monopoles and gravitational instantons, Comm. Math. Phys. 203 (1999), 713–728.
  • ––––, Hyper-Kähler metrics from periodic monopoles, Phys. Rev. D (3) 65 (2002), no. 084015.
  • Compactifications, Berkeley-Stanford working seminar Spring 2002, homepage of the seminar with lecture notes: http://math.utexas.edu/~hausel/seminars/compactifications/
  • M. Cvetič, G. W. Gibbons, H. Lu, and C. N. Pope, Supersymmetric nonsingular fractional $D2$-branes and NS-NS $2$-branes, Nuclear Phys. B 606 (2001), 18–44.
  • –. –. –. –., Ricci-flat metrics, harmonic forms and brane resolutions, Comm. Math. Phys. 232 (2003), 457–500.
  • X. Dai, Adiabatic limits, nonmultiplicativity of signature, and the Leray spectral sequence, J. Amer. Math. Soc. 4 (1991), 265–321.
  • G. de Rham, Differentiable Manifolds: Forms, Currents, Harmonic Forms, Grundlehren Math. Wiss. 266, Springer, Berlin, 1984.
  • J.-H. Eschenburg and V. Schroeder, Riemannian manifolds with flat ends, Math. Z. 196 (1987), 573–589.
  • G. Etesi and T. Hausel, Geometric construction of new Yang-Mills instantons over Taub-NUT space, Phys. Lett. B 514 (2001), 189–199.
  • –. –. –. –., Geometric interpretation of Schwarzschild instantons, J. Geom. Phys. 37 (2001), 126–136.
  • –. –. –. –., On Yang-Mills instantons over multicentered metrics, Comm. Math. Phys., 235 (2003), 275–288.
  • M. P. Gaffney, A special Stokes's theorem for complete Riemannian manifolds, Ann. of Math. (2) 60 (1954), 140–145.
  • G. W. Gibbons, The Sen conjecture for fundamental monopoles of distinct types, Phys. Lett. B 382 (1996), 53–59.
  • G. W. Gibbons and S. W. Hawking, Gravitational multi-instantons, Phys. Lett. B 78 (1978), 430–432.
  • G. W. Gibbons, P. Rychenkova, and R. Goto, Hyper-Kähler quotient construction of BPS monopole moduli spaces, Comm. Math. Phys. 186 (1997), 581–599.
  • T. Gocho and H. Nakajima, Einstein-Hermitian connections on hyper-Kähler quotients, J. Math. Soc. Japan 44 (1992), 43–51.
  • M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135–162.
  • –. –. –. –., Intersection homology, II, Invent. Math. 72 (1983), 77–129.
  • A. Hassell and A. Vasy, The resolvent for Laplace-type operators on asymptotically conic spaces, Ann. Inst. Fourier (Grenoble) 51 (2001), 1299–1346.
  • T. Hausel, Vanishing of intersection numbers on the moduli space of Higgs bundles, Adv. Theor. Math. Phys. 2 (1998), 1011–1040.
  • T. Hausel and B. Sturmfels, Toric hyperkähler varieties, Doc. Math. 7 (2002), 495–534.
  • T. Hausel and E. Swartz, Intersection form of toric hyperkähler varieties, preprint.
  • S. W. Hawking, Gravitational instantons, Phys. Lett. A 60 (1977), 81–83.
  • N. J. Hitchin, “A new family of Einstein metrics” in Manifolds and Geometry (Pisa, Italy, 1993), Sympos. Math. 36, Cambridge Univ. Press, Cambridge, 1996, 190–222.
  • –. –. –. –., $L^2$-cohomology of hyperkähler quotients, Comm. Math. Phys. 211 (2000), 153–165.
  • N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535–589.
  • E. Hunsicker, $L^2$ harmonic forms for a class of complete Kähler metrics, Michigan Math. J. 50 (2002), 339–349.
  • C. Kim, K. Lee, and P. Yi, Discrete light cone quantization of fivebranes, large $N$ screening, and $L^2$ harmonic forms on Calabi manifolds, Phys. Rev. D (3) 65 (2002), no. 065024.
  • K. Kodaira, Harmonic fields in Riemannian manifolds (generalized potential theory), Ann. of Math. (2) 50 (1949), 587–665.
  • P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665–683.
  • –. –. –. –., A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), 685–697.
  • P. B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), 263–307.
  • R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), 309–339.
  • –. –. –. –., Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations 16 (1991), 1616–1664.
  • R. Mazzeo and R. B. Melrose, “Pseudodifferential operators on manifolds with fibred boundaries” in Mikio Sato: A Great Japanese Mathematician of the Twentieth Century, Asian J. Math. 2, Int. Press, Cambridge, Mass., 1998, 833–866.
  • R. Mazzeo and R. S. Phillips, Hodge theory on hyperbolic manifolds, Duke Math. J. 60 (1990), 509–559.
  • R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem, Res. Notes Math. 4, A. K. Peters, Wellesly, Mass., 1993.
  • –. –. –. –., “Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces” in Spectral and Scattering Theory (Sanda, Japan, 1992), Lecture Notes in Pure and Appl. Math. 161, Dekker, New York, 1994, 85–130.
  • ––––, Geometric Scattering Theory, Stanford Lectures, Cambridge Univ. Press, Cambridge, 1995.
  • W. Müller, Manifolds with cusps of rank one, Lecture Notes in Math. 1244, Springer, New York, 1987.
  • A. Nair, Weighted cohomology of arithmetic groups, Ann. of Math. (2) 150 (1999), 1–31.
  • P. J. Ruback, The notion of Kaluza-Klein monopoles, Comm. Math. Phys. 107 (1986), 93–102.
  • L. Saper and M. Stern, $L^2$-cohomology of arithmetic varieties, Ann. of Math. (2) 132 (1990), 1–69.
  • A. Sen, Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and $\SL (2,\mathbb Z)$ invariance in string theory, Phys. Lett. B 329 (1994), 217–221.
  • –. –. –. –., Dynamics of multiple Kaluza-Klein monopoles in M-theory and string theory, Adv. Theor. Math. Phys. 1 (1997), 115–126.
  • M. B. Stenzel, Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math. 80 (1993), 151–163.
  • C. Vafa and E. Witten, A strong coupling test of $S$-duality, Nuclear Phys. B 431 (1994), 3–77.
  • B. Vaillant, Index and spectral theory for manifolds with generalized fibred cusps, Ph.D. dissertation, Bonner Math. Schriften 344, Univ. Bonn, Mathematisches Institut, Bonn, 2001.
  • S. Zucker, $L_2$ cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982/83), 169–218.
  • –. –. –. –., On the reductive Borel-Serre compactification: $L^p$ cohomology of arithmetic groups (for large $p$), Amer. J. Math. 123 (2001), 951–984.