Heterotic string theory

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I propose writing a dissertation in string theory. More specifically, I will examine the consistency of heterotic string vacua. Both the standard model of particle physics and general relativitity are highly successful theories in their respective domains but if we are to have a consistent quantum theory that encompasses both of them it seems inevitable that we are led to string theory. While string theory may well be incomplete, at the time of writing, there does not seem to be any viable alternatives.Within string theory, there are a class of vacua, the heterotic vacua, that seem closest to the observed four dimensional world of particle physics. Despite the fact that the existence or these vacua has been knwon since early in the "string revolution" of 1985-1986, very little is known in detail about such vacua. These vacua we first proposed using semi-classical arguments and relatively little progress has been made until recently either with respect to finding explicit examples of heterotic vacua that might be realistic or even whether these vacua really exist at the quantum level. A significant issue here is that certain Yukawa couplings in the theory have to vanish if the perturbative heteroticvacuum is to exist as a true vacuum of the theory. I propose writing a thesis in this area. The main question is whether heterotic vacua really exist. If they do, then an importantquestion is to find such a vacuum that describes the real world. The Yukawa couplings referred to above are, in reality, couplings between the parameters of the (semi-classical) vacua. An important first step is therefore to correctly identify these parameters. Workremains to be done here but a significant step has been taken in recent papers. I will try to extend the geometrical methods found in these papers.My first research topic will be to check the classical Yukawa couplings associated with these parameters and (hopefully) verify the fact that the couplings are between true parameters of the vacuum ensures that associated couplings vanish. A second problem will be to check that the quantum corrected couplings also vanish. This is more difficult since the quantum corrections to the couplings result from instanton contributions and there are known examples where the individual instanton contributions to a given coupling are non-zero but the sum over all instanton contributions leads to a vanishing result.[4]. At present the cancellations are mysterious and my aim would be to understand whenand why the vacua are the vacua.The impact of this work will in the first instance, be among physicists and mathematicians. However, since what is sought is to make contact between string theory and the observed world of particle physics, it is hard to overstate the importance of this aim.This project falls within the EPSRC Mathematical Physics research area