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)