Mapping 6D \( \mathcal{N} = 1 \) supergravities to F-theory

Journal of High Energy Physics, Feb 2010

We develop a systematic framework for realizing general anomaly-free chiral 6D supergravity theories in F-theory. We focus on 6D (1, 0) models with one tensor multiplet whose gauge group is a product of simple factors (modulo a finite abelian group) with matter in arbitrary representations. Such theories can be decomposed into blocks associated with the simple factors in the gauge group; each block depends only on the group factor and the matter charged under it. All 6D chiral supergravity models can be constructed by gluing such blocks together in accordance with constraints from anomalies. Associating a geometric structure to each block gives a dictionary for translating a supergravity model into a set of topological data for an F-theory construction. We construct the dictionary of F-theory divisors explicitly for some simple gauge group factors and associated matter representations. Using these building blocks we analyze a variety of models. We identify some 6D supergravity models which do not map to integral F-theory divisors, possibly indicating quantum inconsistency of these 6D theories.

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Mapping 6D \( \mathcal{N} = 1 \) supergravities to F-theory

Vijay Kumar 1 David R. Morrison 0 Washington Taylor 1 0 Departments of Mathematics and Physics, University of California , Santa Barbara, CA 93106, U.S.A 1 Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology , Cambridge, MA 02139, U.S.A We develop a systematic framework for realizing general anomaly-free chiral 6D supergravity theories in F-theory. We focus on 6D (1, 0) models with one tensor multiplet whose gauge group is a product of simple factors (modulo a finite abelian group) with matter in arbitrary representations. Such theories can be decomposed into blocks associated with the simple factors in the gauge group; each block depends only on the group factor and the matter charged under it. All 6D chiral supergravity models can be constructed by gluing such blocks together in accordance with constraints from anomalies. Associating a geometric structure to each block gives a dictionary for translating a supergravity model into a set of topological data for an F-theory construction. We construct the dictionary of F-theory divisors explicitly for some simple gauge group factors and associated matter representations. Using these building blocks we analyze a variety of models. We identify some 6D supergravity models which do not map to integral F-theory divisors, possibly indicating quantum inconsistency of these 6D theories. 1 Introduction 2 3 4 Anomaly-free (1, 0) supergravity models in 6D 2.1 Review of anomaly conditions 2.2 Finite number of models 2.3 Classification of SU(N ) models F-theory realizations of SU(N ) product models 3.1 Review of 6D F-theory constructions 3.2 Mapping SU(N ) models into F-theory More representations and groups 4.1 Other representations of SU(N ) 4.1.1 Adjoint representation 4.1.2 3-index antisymmetric representation 4.1.3 Symmetric representation 4.1.4 4-index antisymmetric representations 4.1.5 Larger representations 4.2 SU(2) and SU(3) 4.3 Tri-fundamental representation of SU(M ) SU(N ) SU(P ) 4.4 SO(N ) 4.5 Exceptional groups 4.6 Non-simply laced groups 5.1 Weierstrass models on Hirzebruch surfaces 5.2 SU(N ) 5.2.1 F2 5.2.2 F1 5.3 E6 5.4 E7 6 Some exceptional cases 7 Conclusions 1 String theory appears to provide a framework in which gravity can be consistently coupled to many different low-energy field theories in different dimensions. The problem of understanding precisely which low-energy gravity theories admit a UV completion, and which can be realized in string theory, is a longstanding challenge. Many different string constructions exist, which have been shown to give a variety of low-energy theories through compactifications of perturbative string theory or M/F-theory. In four space-time dimensions, while there are many string constructions, giving a rich variety of field theory models coupled to gravity, there is no general understanding as yet of which gravity theories admit a UV completion and which do not. In six dimensions, however, we may be closer to developing a systematic understanding of the set of allowed low-energy theories and their UV completions through string theory. For chiral (1, 0) supersymmetric theories in six dimensions, cancellation of gravitational, gauge, and mixed anomalies give extremely strong constraints on the set of possible consistent models [1]. In [2], it was shown that (with restrictions to nonabelian gauge group structure and one tensor multiplet) the number of possible distinct combinations of gauge groups and matter representations appearing in such models is finite. In [3], it was conjectured that all consistent models of this type have realizations in string theory. The goal of this paper is to connect the set of allowed chiral 6D supergravity theories to their string realizations by developing a systematic approach to realizing these theories in F-theory. In a general 6D supergravity theory, the gauge group can be decomposed into a product of simple factors modulo a finite abelian group (G = (G1 Gk)/) [In this paper we ignore U(1) factors]. In [2] it was shown that when there is one tensor multiplet, the anomaly cancellation conditions in 6D independently constrain each nonabelian factor Gi in the gauge group, along with the associated matter representations, into a finite number of distinct building blocks. Each building block makes a contribution to the overall gravitational anomaly nh nv = 244, where nh, nv respectively are the numbers of hyper and vector multiplets in the theory. An arbitrary model can be constructed by combining these building blocks to saturate the gravitational anomaly (with neutral hypermultiplets added as needed). The basic idea of the approach we take in this paper is to construct a dictionary between these building blocks of anomaly-free 6D theories and geometric structures in F-theory. F-theory [4] is a framework for constructing type IIB string vacua where the axio-dilaton varies over the internal space. The nonper (...truncated)


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Vijay Kumar, David R. Morrison, Washington Taylor. Mapping 6D \( \mathcal{N} = 1 \) supergravities to F-theory, Journal of High Energy Physics, 2010, pp. 99, Volume 2010, Issue 2, DOI: 10.1007/JHEP02(2010)099