Abstract

Consistent coupling of effective field theories with a quantum theory of gravity appears to require bounds on the rank of the gauge group and the amount of matter. We consider landscapes of field theories subject to such to boundedness constraints. We argue that appropriately “coarse-grained” aspects of the randomly chosen field theory in such landscapes, such as the fraction of gauge groups with ranks in a given range, can be statistically predictable. To illustrate our point we show how the uniform measures on simple classes of N=1 quiver gauge theories localize in the vicinity of theories with certain typical structures. Generically, this approach would predict a high energy theory with very many gauge factors, with the high rank factors largely decoupled from the low rank factors if we require asymptotic freedom for the latter.

Highlights

  • Consistent coupling of effective field theories with a quantum theory of gravity appears to require bounds on the the rank of the gauge group and the amount of matter

  • Asymptotic freedom is less compelling than anomaly cancellation, as the set of random quiver theories may well represent a set of low-energy effective field theories obtained e.g. in string theory

  • One could imagine weighing field theories by the dimension or even size of the cohomology of their respective moduli spaces, having the close connection between quiver gauge theories and D-brane moduli spaces in mind. As another simple example of how dynamics can affect the measure, if we suppose that dynamical effects can give the matter fields any expectation value, generically all the gauge groups will be broken to U(1) and analysis of the distribution of gauge factors is moot

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Summary

The set of field theories

A natural, large class of field theories to consider is the set of quiver gauge theories. We will restrict attention to N = 1 supersymmetric quiver gauge theories where the gauge group is a product of unitary groups, L. Gauge theories of quiver type are ubiquitous in string theory, and this is the main motivation to restrict attention to this class. A typical setup to engineer quiver N = 1 theories is to consider D6-branes wrapping 3-cyles inside a Calabi-Yau manifold in type IIA string theory, in which case the number of bifundamentals is related to the intersection number of the 3-cycles. We can engineer quiver gauge theories with SO and Sp gauge factors, but we will postpone a study of these theories to another occasion. Before looking at some concrete examples, we are first going to make some general remarks on possible further restrictions on the set of gauge theories, on the choice of measure on the space of theories, and the kinds of properties we might predict

Interesting classes of quiver gauge theories
Averages and typicality
Choice of measure
Theories without matter: coarse graining and typicality
The canonical ensemble breaks down
Microcanonical analysis
Thinking about the general quiver
Implementing anomaly cancellation
Dealing with discrete quiver symmetries: an example
Towards dynamics
General quiver with unequal gauge groups
Conclusion
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