Phases of 5d SCFTs from M-/F-theory on non-flat fibrations
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Springer
Received: February 7, 2019 Accepted: May 14, 2019 Published: May 28, 2019
Fabio Apruzzi,a Ling Linb and Christoph Mayrhofer a
Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United Kingdom b Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We initiate the systematic investigation of non-flat resolutions of non-minimal singularities in elliptically fibered Calabi-Yau threefolds. Compactification of M-theory on these geometries provides an alternative approach to studying phases of five-dimensional superconformal field theories (5d SCFTs). We argue that such resolutions capture nontrivial holonomies in the circle reduction of the 6d conformal matter theory that is the F-theory interpretation of the singular fibration. As these holonomies become mass deformations in the 5d theory, non-flat resolutions furnish a novel method in the attempt to classify 5d SCFTs through 6d SCFTs on a circle. A particularly pleasant aspect of this proposal is the explicit embedding of the 5d SCFT’s enhanced flavor group inside that of the parent 6d SCFT, which can be read off from the geometry. We demonstrate these features in toric examples which realize 5d theories up to rank four. Keywords: Conformal Field Models in String Theory, F-Theory, M-Theory, String Duality ArXiv ePrint: 1811.12400
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)187
JHEP05(2019)187
Phases of 5d SCFTs from M-/F-theory on non-flat fibrations
Contents 1
2 5d supersymmetric gauge theories 2.1 5d fixed points 2.2 su(2) gauge theory with matter 2.3 5d gauge theories and fixed points from geometry: brief summary
5 8 8 9
3 SCFTs in M-/F-theory duality 3.1 F-theory on elliptic threefolds with non-minimal singularities 3.2 Circle reduction of conformal matter theories and fiber resolutions
11 11 12
4 Non-flat resolutions of non-minimal singularities 4.1 Global symmetry from the global resolution 4.2 Non-flat resolutions from toric constructions 4.3 Explicit example: E8 -top over F10
14 15 17 21
5 5d rank 1 theories from 6d E-string 5.1 Gauge enhancement at weak coupling 5.2 Global symmetries at weak and strong coupling
23 24 26
6 Higher rank theories from non-flat fibers 6.1 5d rank 2 theories from 6d (E8 , SU(2)) conformal matter 6.2 5d rank 4 theories from 6d (E8 , SU(3)) conformal matter
30 30 36
7 Summary and outlook
39
A Weierstrass model of the E8 top on F10
42
B Geometric computation of Chern-Simons levels for rank four example
43
C Fiber structure in non-flat resolutions of the E-string
44
1
Introduction
One of the remarkable achievements of string theory is to provide evidence for the existence of higher dimensional super(symmetric-)conformal field theories (SCFTs). Such theories are always strongly coupled and lack a canonical Lagrangian description, thus making them difficult to approach using
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