Taylor's research program in recent years has focused on developing a global understanding of the space of supersymmetric string vacua and supergravity theories. In theories with many space-time dimensions, supersymmetry strongly constrains the set of possible theories, so that the set of quantum consistent theories of gravity with supersymmetry matches closely with the set of possible string theory vacua. Studying these higher-dimensional theories has given tools for the global characterization of the set of theories. Taylor and collaborators are now applying analogous methods in four dimensions to identify generic features in the large class of string theory vacua described through "F-theory". This work has also given rise to new insights and progress on the long-standing mathematical problem of classifying Calabi-Yau manifolds.
Generic features in 4D F-theory vacua: Most attempts to construct string theory vacua that reproduce the observed standard model of particle physics involve combining a complicated set of ingredients that require a high degree of fine-tuning. The geometry of F-theory vacua, however, suggests that in this approach to string theory a certain set of gauge groups and matter representations arise naturally in generic classes of vacua. The "non-Higgsable clusters" that can arise in 4D F-theory compactifications were analyzed in Morrison and Taylor. While there are many open theoretical questions regarding how the underlying geometric structure of these models corresponds to observed low-energy physics, the connection between these structures and observed physics is rather suggestive. The possible gauge structures that can arise in generic F-theory geometries include the nonabelian part of the standard model gauge group, SU(3) x SU(2), as found by Grassi, Halverson, Shaneson, and Taylor, but cannot for example include the grand unification groups SU(5) or SO(10). There are also strong constraints on dark matter structures that can arise in this framework.
Classifying elliptic Calabi-Yau threefolds: Calabi-Yau manifolds are a special type of geometric space that play an important role in supersymmetric compactifications of string theory. Since they were first used in string theory almost 30 years ago, physicists and mathematicians have studied these geometric spaces quite intensively. Nonetheless, basic questions about the classification of Calabi-Yau threefolds (three complex dimensions = six real dimensions) are still unanswered. In particular, it is not known if there are a finite or infinite number of topologically distinct Calabi-Yau threefolds. Recent work of Morrison and Taylor identified a set of "non-Higgsable clusters" of Kodaira singularities that appear in generic Weierstrass models over a given complex surface forming the base of an elliptic Calabi-Yau threefold. Combined with the mathematical minimal model program for surfaces and work of Grassi, these clusters can be used to systematically bound and analyze the finite set of elliptic Calabi-Yau threefolds, as further described in these papers by Taylor and collaborators Morrison, Martini, and Johnson.
Further work in these directions is currently underway.Ten-dimensional supergravity and string theory
Supersymmetry and quantum consistency constraints very strongly limit the set of possible consistent theories of gravity that are possible in ten dimensions. With the minimal amount of supersymmetry, gravity in ten dimensions can be coupled to massless gauge fields associated with a gauge group G. Anomaly cancellation conditions found by Green and Schwarz can only be satisfied when G is one of the groups E_8 x E_8, SO(32), U(1)^496, or U(1)^248 x E_8. Adams, DeWolfe and Taylor showed in 2010 that the latter two theories cannot be simultaneously consistent with supergravity and anomaly cancellation. This shows that the only consistent spectra for a 10D gravity theory with N = 1 supersymmetry are those realized by string theory.
6D supergravity and string vacua
In six dimensions, low-energy consistency constraints including anomaly cancellation place strong restrictions on the set of possible N = 1 supergravity theories. It was conjectured by Kumar and Taylor that the set of consistent spectra for such supergravity theories corresponds precisely to those that can be realized by string theory constructions. In a series of works by Taylor with collaborators Kumar and Morrison, it was shown that the constraints on the low-energy supergravity theory correspond closely with the structure imposed by the geometrical F-theory approach to string compactification. While "string universality" for 6D supergravity theories is not yet proven, further progress on this program has been made in these papers. The picture that is emerging suggests that all 6D N = 1 supergravity theories live on a single connected moduli space, with different branches (corresponding to elliptic fibrations over different base surfaces in the F-theory picture) connected through tensionless string transitions at superconformal fixed points. Thus, in 6D there may only be essentially one consistent theory of N = 1 supergravity, which is that described by F-theory.
Four-dimensional physics from F-theory
While in six dimensions, F-theory seems to describe almost all possible quantum consistent supergravity theories, in four dimensions the story is much more complicated. In 4D, N = 1 supergravity vacua consist of an enormous number of disjoint classes of vacua, separated by potential barriers, and F-theory seems to only describe one piece of the full "landscape" of string vacua. Despite this increased complexity, Taylor's extensive study of supersymmetric F-theory vacua in six dimensions has provided a rich set of tools and insight that apply surprisingly well in the much more complicated context of 4D compactifications. Some initial steps towards developing for four dimensions some analogous structures to those used to characterize supergravity theories and F-theory vacua in six dimensions were taken by Grimm and Taylor. In particular, in this work the Green-Schwarz like couplings of the 4D theory were used to give a general description of heterotic/F-theory duality at the topological level. This was then used by Anderson and Taylor to give a systematic description of all heterotic/F-theory duals where the F-theory geometry has the shape of a P^1 bundle over a complex Kaehler base surface. The set of possible "non-Higgsable clusters" for F-theory compactifications to four dimensions was analyzed by Morrison and Taylor, and the role of these clusters in constructing the standard model gauge group was considered by Grassi, Halverson, Shaneson, and Taylor, as described in more detail in the research highlights section above.
Classifying Calabi-Yau threefolds and fourfolds
Some of Taylor and collaborator's recent work on classifying elliptic Calabi-Yau threefolds is described above in research highlights. Current projects include a parallel analysis of elliptic Calabi-Yau fourfolds.
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