GV-spectroscopy for F-theory on genus-one fibrations
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Springer
Received: December 31, Revised: May 27, Accepted: August 5, Published: September 8,
2019 2020 2020 2020
Paul-Konstantin Oehlmanna and Thorsten Schimannekb a
Department of Physics and Astronomy, Uppsala University, Regementsvägen 1, 77120 Uppsala, Sweden b Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
E-mail: [email protected], [email protected] Abstract: We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau d-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber GopakumarVafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polytope. In particular for fibers that generically lead to multiple sections, only multi-sections or that are complete intersections in higher codimension, our technique is vastly more efficient than classical approaches. As an application we study two Higgs chains of six-dimensional supergravities that are engineered by fibrations of codimension two complete intersection fibers. Both chains end on a vacuum with G = Z4 that is engineered by fibrations of bi-quadrics in P3 . We use the detailed knowledge of the structure of the reducible fibers that we obtain from GV-spectroscopy to comment on the corresponding Tate-Shafarevich group. We also show that for all fibers the six-dimensional supergravity anomalies including the discrete anomalies generically cancel. Keywords: F-Theory, M-Theory, Topological Strings ArXiv ePrint: 1912.09493
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)066
JHEP09(2020)066
GV-spectroscopy for F-theory on genus-one fibrations
Contents 1
2 F-theory preliminaries 2.1 Gauge symmetry and matter from reducible fibers 2.2 Discrete symmetries and genus one fibrations
4 4 6
3 Fibrations from families of fibers
9
4 GV-spectroscopy for complete intersection fibers 4.1 Counting fibral curves with Gopakumar-Vafa invariants 4.2 The Batyrev-Borisov construction of CICYs 4.3 Calculating Gopakumar-Vafa invariants with mirror symmetry 4.4 Example: bi-quadric in P3 over B = P2 4.5 Base independent spectra from GV-invariants 4.6 Example: G = SU(3) × SU(2) × Z2
11 11 14 15 19 21 25
5 CICY fibers in the restricted bi-quadric 5.1 nef (0,0): G = Z4 5.2 nef (6,1): G = Z2 × U(1) 5.3 nef (5,1): G = U(1) 5.4 nef (21,0): G = U(1)2 5.5 nef (67,0): G = U(1)3
29 29 31 35 37 42
6 The 6.1 6.2 6.3
46 47 49 51
six- and five-dimensional field theory Higgs transitions among the vacua The structure of the Z4 Tate-Shafarevich group Absence of discrete anomalies
7 Conclusions
52
A Base independent intersection calculus
53
B Example: multiple fibers
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