Download presentation

Presentation is loading. Please wait.

Published byJames Blankley Modified over 2 years ago

1
Higgs Bundles and String Phenomenology M. Wijnholt, LMU Munich String-Math Philadelphia, June 7 2011

2
Brief review of some particle physics Gauge group: SU(3) x SU(2) x U(1) Standard Model: matter: 3 x (Q,U,D,L,E) + higgs Fits nicely into simple group: matter: 3 x ( ) + higgs Further possibilities:

3
`Grand Unified Models’ Additional evidence for simple group: Supersymmetric unification 1 TeV 2 10 16 GeV 3 -1 2 -1 1 -1 M Pl SU(3) strong SU(2) weak U(1) hyper Single unified force? GUT group SU(5), SO(10) Brief review of particle physics (II)

4
Can we get this from string theory? First try: heterotic string theory ’85 Candelas et al. Below string scale: Super Yang-Mills in 10d (+ supergravity) Break symmetry to get 4d SU(5) GUT model Space-time = Z = Calabi-Yau three-fold Bundle V on Z breaks E_8 gauge group to SU(5)

5
Supersymmetry puts contraints on E_8 bundle V: Massless fields from KK reduction of E_8 gauginos given by Dolbeault cohomology. Yukawas Therefore, find pairs (Z,V) such that: Non-zero, hierarchical * * *

6
Anno 2011: Landscape Denef/Douglas ’04 : # vacua ~ Lambda^betti Lambda = tadpole cut-off, betti = rank of flux lattice Vacua = classical, SUSY field configurations with fixed Kaehler moduli, stabilized complex moduli Donagi/MW ’09: betti ~ 10^3 just in visible sector of heterotic/local F-theory models # vacua ~ 10^1000 in visible sector of heterotic/F models These numbers are so astronomical that it is pointless to `find’ the SM [Aside: End aside]. On the other hand, justifies naturalness: dim’less parameters order one unless extra structure Bousso/Polchinski ‘00

7
10d -- Heterotic 9d –- type I’ 8d –- F-theory 7d –- M-theory Recent years: extend this story to super Yang-Mills in d < 10 Donagi/MW, 08 Beasley/Heckman/Vafa, 08 Pantev/MW, 09 Candelas et al, 85 Pantev/MW, to appear. Hayashi et al, 08 Main new idea: in d < 10, instead of a bundle V, we need a Higgs bundle

8
Compactified SYM in lower dimensions: Higgs bundles * Bundle E with connection * Adjoint field, interpreted as a map This data has to satisfy first order BPS equations Hitchin’s equations

9
Eg. F-theory story: 8d SYM is dimensional reduction of 10d SYM: 8d SYM on compact Kaehler surface S: E_8 bundle V Higgs field Hitchin Massless gauginos: Massless chiral fields:,,

10
,, Similarly in d=7 and d=9 7d: 9d:

11
Constructing solutions Focus on best-understood case: 8d SYM/F-theory Hitchin’s equations split into: a complex equation (`F-term’) a moment map (`D-term’) Construct K_S-twisted Higgs bundle on complex surface S (eg. S = del Pezzo),, Standard strategy: first ignore D-term * *

12
Constructing solutions (II) Solution to F-term: use Higgs bundle/spectral cover correspondence Solution to D-term: use Uhlenbeck-Yau HE metric existsHiggs bundle/spectral sheaf is poly-stable Spectral sheaf (e-vectors/e-values) Breakingrequires Sl(5,C) Higgs bundle on S Spectral cover C given by degree 5 polynomial Spectral line bundle in Pic(C) Data: (If only it were this easy for d=7 and d=9)

13
Embedding in string theory Requires Higgs bundle/ALE-fibration correspondence For simplicity consider Sl(n,C) Higgs bundle on S ALE-fibration Y over S: Consider n lines given by varying over S Defines the `cylinder’ R We have maps: Correspondence:

14
Summary: particle physics from stringscompactified SYM Higgs bundles

15
Some questions for mathematicians: Construct solutions of complex part of Hitchin type equations in odd dimensions Analogue of Uhlenbeck-Yau: Existence of hermitian metric solving moment map equation? Correct notion of stability for A-branes? Conceptual: what classifies first order deformations in ALE fibration picture? Comments: T-duality (Pantev/MW) Relation between 5d Higgs bundles and Kapustin-Orlov type coisotropic branes? Our equations are naturally non-abelian, but even in the abelian case they do not seem to coincide. * * * *

Similar presentations

OK

1 Supersymmetry Yasuhiro Okada (KEK) January 14, 2005, at KEK.

1 Supersymmetry Yasuhiro Okada (KEK) January 14, 2005, at KEK.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on ram and rom memory Ppt on credit policy meaning Ppt on chemical and magnetic effect of electric current Ppt on three types of computer languages Ppt on eddy current inspection Ppt on global warming in india free download Download ppt on civil disobedience movement martin Ppt on varactor diodes Project ppt on product launch Ppt on indian politics quoted