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Higgs Bundles and String Phenomenology

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

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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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