Transcript Document

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:
SU (3)  SU (2) U (1)  SU(5)
matter: 3 x ( 10 5 ) + higgs
Further possibilities:
SU(5)  SO(10)  E6
Brief review of particle physics (II)
Additional evidence for simple group:
Supersymmetric unification
Single unified force?
U(1)hyper
GUT group SU(5), SO(10)
a1-1
SU(2)weak a2-1
SU(3)strong a3-1
1 TeV
`Grand Unified Models’
2 1016 GeV
MPl
Can we get this from string theory?
First try: heterotic string theory
’85 Candelas et al.
Below string scale:
E8  E8 Super Yang-Mills in 10d (+ supergravity)
Break symmetry to get 4d SU(5) GUT model
Space-time = R 4  Z
Z = Calabi-Yau three-fold
Bundle V on Z breaks E_8 gauge group to SU(5)
Supersymmetry puts contraints on E_8 bundle V:
F 0, 2  0
Fij g ij  0
Massless fields from KK reduction of E_8 gauginos given by Dolbeault cohomology.
Therefore, find pairs (Z,V) such that:
*
H 0 (Z ,VE8 )  su(3)c  su(2)w  u(1)Y
*
H ( Z ,VE8 )   (10i  5i )  ( H u , H d )
3
1
i 1
*
Yukawas
H 1 (Z ,VE8 )3  C
Non-zero, hierarchical
[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].
Recent years: extend this story to super Yang-Mills in d < 10
10d -- Heterotic
9d –- type I’
Candelas et al, 85
Pantev/MW, to appear.
Donagi/MW, 08
8d –- F-theory
7d –- M-theory
Beasley/Heckman/Vafa, 08
Hayashi et al, 08
Pantev/MW, 09
Main new idea: in d < 10, instead of a bundle V, we need a Higgs bundle
Compactified SYM in lower dimensions: Higgs bundles
* Bundle E with connection
* Adjoint field
A
, interpreted as a map E  E  N
This data has to satisfy first order BPS equations
Hitchin’s equations
Eg. F-theory story:
8d SYM is dimensional reduction of 10d SYM:
A 0,1  A0,1   2,0
8d SYM on compact Kaehler surface S:
E_8 bundle V
Higgs field 
F 0, 2  0,

E  VE8 

VE8  KS
 A  0,
Fij g ij  [, * ]  0
Massless gauginos:
H 0 (S , E  )
Massless chiral fields:
H 1 (S , E  )
Hitchin
Similarly in d=7 and d=9
7d:
9d:
F  [ ,  ]  0 , d A  0 ,
Fzx  J zw Dw
d A*  0
g z z Fz z  iDx  0
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’)
F 0, 2  0,
 A  0,
Fij g ij  [* , ]  0
Standard strategy: first ignore D-term
Constructing solutions (II)
Solution to F-term: use Higgs bundle/spectral cover correspondence
( E , )
Breaking
Spectral sheaf (e-vectors/e-values)
E8  SU (5)
Data:
requires Sl(5,C) Higgs bundle on S
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
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:
y 2  x2  b0 z n   bn1z  bn
n
Consider n lines given by y  x, b0 z   bn1 z  bn  0 varying over S
Defines the `cylinder’ R
We have maps:
Correspondence:
i : R Y,
p: R C
p*i*
H *2 (Y )
H * (C )
i* p*
Summary:
particle physics from strings
compactified SYM
Higgs bundles
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.