APM  Vol.5 No.6 , May 2015
Super Characteristic Classes and Riemann-Roch Type Formula
Author(s) Tadashi Taniguchi*

The main purpose of this article is to define the super characteristic classes on a super vector bundle over a superspace. As an application, we propose the examples of Riemann-Roch type formula. We also introduce the helicity group and cohomology with respect to coefficient of the helicity group. As an application, we propose the examples of Gauss-Bonnet type formula.

Cite this paper
Taniguchi, T. (2015) Super Characteristic Classes and Riemann-Roch Type Formula. Advances in Pure Mathematics, 5, 353-366. doi: 10.4236/apm.2015.56034.
[1]   Taniguchi, T. (2009) ADHM Construction of Super Yang-Mills Instantons. Journal of Geometry and Physics, 59, 1199-1209.

[2]   Bartocci, C., Bruzzo, U. and Ruipérez, D.H. (1991) The Geometry of Supermanifolds. Mathematics and Its Applications, Volume 71, Kluwer Academic Publishers, Norwell.

[3]   LeBrun, C., Poon, Y.S. and Wells Jr., R.O. (1990) Projective Embedding of Complex Supermanifolds. Communications in Mathematical Physics, 126, 433-452.

[4]   Leites, D.A. (1980) Introduction to the Theory of Supermanifolds. Russian Mathematical Surveys, 35, 1-64.

[5]   Manin, Yu.I. (1997) Gauge Field Theory and Complex Geometry. 2nd Edition, Springer, Berlin.

[6]   Rogers, A. (2007) Supermanifolds Theory and Applications. World Scientific, Singapore City.

[7]   LeBrun, C. and Rothstein, M. (1988) Moduli of Super Riemann Surfaces. Communications in Mathematical Physics, 117, 159-176. http://dx.doi.org/10.1007/BF01228415

[8]   Penkov, I.B. (1983) D-Modules on Supermanifolds. Inventiones Mathematicae, 71, 501-512.

[9]   Bartocci, C. and Bruzzo, U. (1988) Cohomology of the Structure Sheaf of Real and Complex Supermanifolds. Journal of Mathematical Physics, 29, 1789-1794.

[10]   Bott, R. and Tu, L.W. (1982) Differential Forms in Algebraic Topology. Springer, Berlin.

[11]   Bruzzo, U. and Ruipérez, D.H. (1989) Characteristic Classes of Super Vector Bundles. Journal of Mathematical Physics, 30, 1233-1237.

[12]   Hartshorne, R. (1977) Algebraic Geometry. Springer, Berlin.

[13]   Hirzebruch, F. (1966) Topological Methods in Algebraic Geometry. Springer-Verlag, Berlin.

[14]   Lawson Jr., H.B. and Michelsohn, M. (1989) Spin Geometry. Princeton University Press, Princeton.

[15]   Voronov, A.A. and Manin, Y.I. (1990) Elements of Supergeometry. Journal of Mathematical Sciences, 51, 2069-2083.

[16]   Bruzzo, U. and Fucito, F. (2004) Superlocalization Formulas and Supersymmetric Yang-Mills Theories. Nuclear Physics B, 678, 638-655.

[17]   Ninnemann, H. (1992) Deformations of Super Riemann Surfaces. Communications in Mathematical Physics, 150, 267-288.

[18]   Rosly, A.A., Schwarz, A.S. and Voronov, A.A. (1988) Geometry of Superconformal Manifolds. Communications in Mathematical Physics, 119, 129-152.

[19]   Crane, L. and Rabin, J.M. (1988) Super Riemann Surfaces: Uniformization and Teichmüller Theory. Communications in Mathematical Physics, 113, 601-623.

[20]   Manin, Y.I. (1991) Topics in Non-Commutative Geometry. M. B. Porter Lectures at Rice University, Houston.

[21]   Giddings, S.B. and Nelson, P. (1988) The Geometry of Super Riemann Surfaces. Communications in Mathematical Physics, 116, 607-634.

[22]   Giddings, S.B. and Nelson, P. (1988) Line Bundles on Super Riemann Surfaces. Communications in Mathematical Physics, 118, 289-302.

[23]   De Witt, B. (1992) Supermanifolds. 2nd Edition, Cambridge University Press, Cambridge.