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Higgs Bundles and String Phenomenology

Recap of Standard Model, Grand Unified Models, and String Theory linking to Higgs Bundles in String Phenomenology, with emphasis on Symmetry Breaking and Solution Strategies for BPS Equations.

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Higgs Bundles and String Phenomenology

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

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

  3. Brief review of particle physics (II) Additional evidence for simple group: Supersymmetric unification Single unified force? GUT group SU(5), SO(10) U(1)hyper a1-1 SU(2)weak a2-1 SU(3)strong a3-1 1 TeV 2 1016 GeV MPl `Grand Unified Models’

  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. Therefore, find pairs (Z,V) such that: * * * Yukawas Non-zero, hierarchical

  6. [Aside: Anno 2011: Landscape Bousso/Polchinski ‘00 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 On the other hand, justifies naturalness: dim’less parameters order one unless extra structure End aside].

  7. Recent years: extend this story to super Yang-Mills in d < 10 10d -- Heterotic Candelas et al, 85 9d –- type I’ Pantev/MW, to appear. Donagi/MW, 08 Beasley/Heckman/Vafa, 08 8d –- F-theory Hayashi et al, 08 7d –- M-theory Pantev/MW, 09 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 F , , 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 Construct K_S-twisted Higgs bundle on complex surface S (eg. S = del Pezzo) Hitchin’s equations split into: , , a complex equation (`F-term’) * a moment map (`D-term’) * Standard strategy: first ignore D-term

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

  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 strings compactified SYM Higgs bundles

  15. Some questions for mathematicians: Construct solutions of complex part of Hitchin type equations in odd dimensions * Comments: T-duality (Pantev/MW) 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? * 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.

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