Realizing Higher-Level Gauge Symmetries in String Theory: New Embeddings for String GUTs

We consider the methods by which higher-level and non-simply laced gauge symmetries can be realized in free-field heterotic string theory. We show that all such realizations have a common underlying feature, namely a dimensional truncation of the charge lattice, and we identify such dimensional truncations with certain irregular embeddings of higher-level and non-simply laced gauge groups within level-one simply-laced gauge groups. This identification allows us to formulate a direct mapping between a given subgroup embedding, and the sorts of GSO constraints that are necessary in order to realize the embedding in string theory. This also allows us to determine a number of useful constraints that generally affect string GUT model-building. For example, most string GUT realizations of higher-level gauge symmetries G_k employ the so-called diagonal embeddings G_k\subset G\times G \times...\times G. We find that there exist interesting alternative embeddings by which such groups can be realized at higher levels, and we derive a complete list of all possibilities for the GUT groups SU(5), SU(6), SO(10), and E_6 at levels k=2,3,4 (and in some cases up to k=7). We find that these new embeddings are always more efficient and require less central charge than the diagonal embeddings which have traditionally been employed. As a byproduct, we also prove that it is impossible to realize SO(10) at levels k>4. This implies, in particular, that free-field heterotic string models can never give a massless 126 representation of SO(10).Comment: 69 pages, LaTeX, 5 figures (Encapsulated PostScript). Revised to match published versio

Broken symmetry and Yang-Mills theory

From its inception in statistical physics to its role in the construction and in the development of the asymmetric Yang-Mills phase in quantum field theory, the notion of spontaneous broken symmetry permeates contemporary physics. This is reviewed with particular emphasis on the conceptual issues.Comment: Latex 30 pages, 9 figures. Typo corrected. Contribution to "Fifty years of Yang Mills theory", editor G. 't Hooft, to be published by World Scientifi

Strings, Loops, Knots and Gauge Fields

The loop representation of quantum gravity has many formal resemblances to a background-free string theory. In fact, its origins lie in attempts to treat the string theory of hadrons as an approximation to QCD, in which the strings represent flux tubes of the gauge field. A heuristic path-integral approach indicates a duality between background-free string theories and generally covariant gauge theories, with the loop transform relating the two. We review progress towards making this duality rigorous in three examples: 2d Yang-Mills theory (which, while not generally covariant, has symmetry under all area-preserving transformations), 3d quantum gravity, and 4d quantum gravity. $SU(N)$ Yang-Mills theory in 2 dimensions has been given a string-theoretic interpretation in the large-$N$ limit by Gross, Taylor, Minahan and Polychronakos, but here we provide an exact string-theoretic interpretation of the theory on $\R\times S^1$ for finite $N$. The string-theoretic interpretation of quantum gravity in 3 dimensions gives rise to conjectures about integrals on the moduli space of flat connections, while in 4 dimensions there may be connections to the theory of 2-tangles.Comment: 34 pages, LaTe

Solving the Hierarchy Problem without Supersymmetry or Extra Dimensions: An Alternative Approach

In this paper, we propose a possible new approach towards solving the gauge hierarchy problem without supersymmetry and without extra spacetime dimensions. This approach relies on the finiteness of string theory and the conjectured stability of certain non-supersymmetric string vacua. One crucial ingredient in this approach is the idea of ``misaligned supersymmetry'', which explains how string theories may be finite even without exhibiting spacetime supersymmetry. This approach towards solving the gauge hierarchy problem is therefore complementary to recent proposals involving both large and small extra spacetime dimensions. This approach may also give a new perspective towards simultaneously solving the cosmological constant problem.Comment: 33 pages, LaTeX, 3 figure