SENIOR THESIS PROPOSAL
**Supersymmetric Three-Cycles in String Theory**

**Ian Weiner**

Advisor: Weiqing Gu

For my thesis I will work with Prof. Gu on a problem of great importance in theoretical physics, particularly in String theory. We will try to find a so-called "supersymmetric three-cycle of G_{2}holonomy in R^{6}x S^{1}".The problem was first suggested by Edward Witten to Prof. Gu, and is discussed in his paper "Branes and the Dynamics of QCD" (section 4.2) which appeared in Nuclear Physics B (hep-th/9706109). Witten suggests that a three cycle as mentioned above could represent a BPS saturated domain wall in string theory. He does not, however, prove the existence of such a cycle which obeys the appropriate boundary conditions, and has no concrete examples of such cycles.

Prof. Gu's previous work in the field and experience with the method of calibration will prove useful in identifying the cycle we seek. One of Prof. Gu's main research areas is the application of her previous work in calibration to identify associative, coassociative, and Cayley cycles in String theory (which physicists refer to as BPS states). The supersymmetric three-cycles we seek correspond to associative cycles in a Riemannian manifold R

^{6}x S^{1}.My thesis will involve an important part of the solution of the problem posed by Edward Witten. Prof. Gu and I plan to fully solve the problem sometime in the future, perhaps by the end of this summer, if our work goes well.

Although there is much background material on this thesis topic that I must learn before I can fully understand the project at hand, I have gained some experience this summer working with Prof. Gu on a project using Quaternion numbers, whose extension to eight dimensions, Cayley numbers, are integral to the project.

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*Last modified: September 2000