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. * * * *