I am interested in the unification of the fundamental forces of physics, i.e. of those described by the standard model of strong, weak and electromagnetic interactions, and of gravity as described by Einstein’s general relativity. The path to this unification appears to be through supersymmetry, supergravity and superstring theory. I have contributed to these subjects in the past and continue to work on them. Occasionally these studies require the use of techniques of advanced algebra and differential geometry, and my work has involved, and will continue to involve, the development of new mathematical tools. The study of noncommutative differential geometry is an example.
In some recent papers my collaborators and I have extended previous work done in collaboration with M.K. Gaillard, and given a general theory of self-duality for nonlinear Lagrangians with several abelian gauge fields. By using Lagrange multiplier auxiliary fields we were also able to give the supersymmetric form of those Lagrangians in four space-time dimensions. There are various directions in which this work can be further extended. In a recent paper Gibbons and Hashimoto study a different model with several abelian gauge fields in presence of the gravitational field and show that the energy momentum tensor vanishes for self-dual gauge fields. This is interesting because solutions of Einstein’s gravitational equations are not modified when self-dual fields are taken as sources and one can study various coupled solitonic configurations. We have already verified that our model has the same property of vanishing energy momentum tensor. This suggests the possibility that one could prove it directly as a consequence of the self-duality of the theory. Other lines of development are the study of Born Infeld theory on noncommutative tori and the study of nonabelian Born-Infeld theory. Interesting papers on these topics already exist, by E. Verlinde and collaborators and by Tseytlin, but these subjects are very rich ones and more work seems appropriate (see also below).
I also plan to extend the work I have done with various collaborators on noncommutative geometry and in particular the work on Yang-Mills on noncommutative Tori done with D. Brace and B. Morariu. We have some ideas on how to use the Seiberg-Witten map to construct noncommutative Yang-Mills theories for an arbitrary Lie group.
C.-S. Chu, P.-M. Ho, and B. Zumino, “The quantum 2-sphere as a complex quantum manifold,” Zeits f. Physik C70, 339 (1996).
C.-S. Chu, P.-M. Ho, and B. Zumino, “The braided quantum 2-sphere,” Mod. Phys. Letts. A11, 307 (1996).
C.-S. Chu, P.-M. Ho, and B. Zumino, “Geometry of the quantum complex projective space CPq(N),” Zeits f. Physik C72, 163 (1996).
C.-S. Chu, P.-M. Ho, and B. Zumino, “Some complex quantum manifolds and their geometry,” in Quantum Fields and Quantum Space Time. Proceedings of a NATO Advanced Study Institute, Cargèse, France G.‘t Hooft, A. Jaffe, G. Mack, P. K. Mitter, et al. (Eds.), pp. 281-322, New York: Plenum Press (1996).
C.-S. Chu, P.-M. Ho, and B. Zumino, “Non-Abelian anomalies and effective actions for a homogeneous space,” Nucl. Phys. B475, 484 (1996).
B. Zumino, “Supersymmetric sigma-models in two-dimensions,” LBNL-41392, in Duality and Supersymmetric Theories, D. Olive and P. C. West (Eds.), Cambridge U. Press (1999).
D. Brace, B. Morariu, and B. Zumino, “T-duality and Ramond-Ramond backgrounds in the matrix model,” Nucl. Phys. B549, 181 (1999).
D. Brace, B. Morariu, and B. Zumino, “Dualities of the matrix model from T-duality of the Type II string,” Nucl. Phys. B545, 192 (1999).
B. Morariu and B. Zumino, “Super Yang-Mills on the noncommutative torus,” LBNL-42104, UCB-PTH-98/38, hep-th/9807198, in Relativity, Particle Physics, and Cosmology, R. E. Allen (Ed.), World Scientific (1999).
M. K. Gaillard and B. Zumino, “Self-duality in non-linear electromagnetism,” in Supersymmetry and Quantum Field Theory, J. Wess and V.P Akulov (Eds.), Lecture Notes in Physics, 509, 121, Springer-Verlag (1998).
M. K. Gaillard and B. Zumino, “Nonlinear electromagnetic self-duality and Legendre transformations,” in Duality and Supersymmetric Theories, D. Olive and P. West (Eds.), Cambridge University Press (1999).
D. Brace, B. Morariu, and B. Zumino, “Duality invariant Born-Infeld theory,” in The Many Faces of the Superworld, M. Shifman (Ed.), World Scientific (2000).
P. Aschieri, D. Brace, B. Morariu, and B. Zumino, “Nonlinear self-duality in even dimensions,” Nucl. Phys. B574, 551 (2000)
P. Aschieri, D. Brace, B. Morariu, and B. Zumino, “Proof of a symmetrized trace conjecture,” Nucl. Phys. B588, 521 (2000).
B. L. Cerchiai and B. Zumino, “Properties of perturbative solutions of unilateral matrix equations,” hep-th/0009013, Lett. Math. Phys. 54, 33 (2000).
B. L. Cerchiai and B. Zumino, “Some remarks on unilateral matrix equations,” hep-th/0105065, Modern Physics Letters A16, 191 (2001).
D. Brace, B. Cerchiai, A. Pasqua, U. Varadarajan, and B. Zumino, “A cohomological approach to the non-abelian Seiberg-Witten map,” hep-th/0105192, JHEP 06, 047 (2001).