Communications in Mathematical Physics Ricci-Flat Metrics , Harmonic Forms and Brane Resolutions

We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of Sn+1. We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic selfdual (p, q)-forms in the middle dimension p + q = (n + 1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p, p)-forms areL2-normalisable, while for (p, q)-forms the degree of divergence grows with |p−q|. We also construct a set of Ricci-flat metrics whose level surfaces are U(1) bundles over a product of N Einstein-Kähler manifolds, and we construct examples of harmonic forms there.As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2, 1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1, 2) and (2, 1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions.

[1]  G. Gibbons,et al.  Supersymmetric Non-singular Fractional D2-branes and NS-NS 2-branes , 2001, hep-th/0101096.

[2]  K. Becker A Note on compactifications on spin(7) - holonomy manifolds , 2000, hep-th/0011114.

[3]  M. Frau,et al.  Fractional D-branes and their gauge duals , 2000, hep-th/0011077.

[4]  L. Girardello,et al.  A note on regular type 0 solutions and confining gauge theories , 2000, hep-th/0011041.

[5]  M. Cvetič,et al.  Brane resolution through transgression , 2000, hep-th/0011023.

[6]  K. Becker,et al.  Compactifying M-theory to four dimensions , 2000, hep-th/0010282.

[7]  A. Tseytlin,et al.  3-branes on resolved conifold , 2000, hep-th/0010088.

[8]  J. Polchinski,et al.  Supersymmetric three-form flux perturbations on AdS 5 , 2000, hep-th/0009211.

[9]  I. Klebanov,et al.  Supergravity and a confining gauge theory: Duality cascades and χSB-resolution of naked singularities , 2000, hep-th/0007191.

[10]  C. Pope,et al.  Kaluza-Klein consistency, Killing vectors and Kähler spaces , 2000, hep-th/0005172.

[11]  M. Cvetič,et al.  Consistent sphere reductions and universality of the Coulomb branch in the domain-wall/QFT correspondence , 2000, hep-th/0004201.

[12]  G. Shiu,et al.  Warped compactifications in M and F theory , 2000, hep-th/0004103.

[13]  Tamás Hausel,et al.  Geometric interpretation of Schwarzschild instantons , 2000, Journal of Geometry and Physics.

[14]  I. Klebanov,et al.  Gravity Duals of Supersymmetric SU(N) x SU(N+M) Gauge Theories , 2000, hep-th/0002159.

[15]  M. Wang,et al.  The cohomogeneity one Einstein equations from the Hamiltonian viewpoint , 2000 .

[16]  S. Gubser,et al.  Symmetric potentials of gauged supergravities in diverse dimensions and Coulomb branch of gauge theories , 1999, hep-th/9909121.

[17]  K. Sfetsos,et al.  States and curves of five-dimensional gauged supergravity , 1999, hep-th/9909041.

[18]  K. Sfetsos,et al.  Non-standard compactifications with mass gaps and Newton's law , 1999, hep-th/9908116.

[19]  S. Gubser,et al.  Continuous distributions of D3-branes and gauged supergravity , 1999, hep-th/9906194.

[20]  S. Trivedi,et al.  The Coulomb branch of gauge theory from rotating branes , 1998, hep-th/9811120.

[21]  Mckenzie Y. Wang,et al.  Kähler-Einstein metrics of cohomogeneity one , 1998 .

[22]  S. Hawking,et al.  Bulk charges in eleven dimensions , 1997, hep-th/9711042.

[23]  M. Duff,et al.  The octonionic membrane , 1997, hep-th/9706124.

[24]  G. Segal,et al.  The cohomology of the space of magnetic monopoles , 1996 .

[25]  Matthew B. Stenzel Ricci-flat metrics on the complexification of a compact rank one symmetric space , 1993 .

[26]  Xenia de la Ossa,et al.  Comments on Conifolds , 1990 .

[27]  R. Bryant,et al.  On the construction of some complete metrics with exceptional holonomy , 1989 .

[28]  D. Page,et al.  Inhomogeneous Einstein metrics on complex line bundles , 1987 .

[29]  D. Page,et al.  Stability analysis of compactifications of D = 11 supergravity with SU(3) × SU(2) × U(1) symmetry☆ , 1984 .

[30]  J. Schwarz Covariant field equations of chiral N = 2 D = 10 supergravity , 1983 .

[31]  G. Gibbons,et al.  The Positive Action conjecture and asymptotically Euclidean metrics in quantum gravity , 1979 .

[32]  A. Hanson,et al.  Asymptotically flat self-dual solutions to euclidean gravity , 1978 .

[33]  G. Gibbons,et al.  Einstein matrices on S3, R3 and R4 bundles , 1990 .