Download Introduction to Strings

Introduction to Strings
Yoshihisa Kitazawa
KEK
Nasu lecture 9/25/06
Why strings?
•
We have solved many questions:
Standard model of particle physics
1. SU(3)xSU(2)xU(1) gauge theory
2. 3 generations of quarks and leptons
Standard model of cosmology
1. Big Bang nucleosynthesis
2. Large scale structure formation based on
cold dark matter and inflation
We are making progress to solve
important questions
We also find new deep questions
5
• To answer these questions, we need to
understand not only matter but also spacetime at the microscopic level.
• We need to understand all fundamental
interactions including gravity
• String theory is the most promising
approach so far and likely to be in the right
track toward penetrating deeper layers of
space-time and matter
Perturbative strings
• Strings are one dimensionally extended
objects
• There are closed strings and open strings
• Strings sweep two dimensional world
sheets as they propagate
t
xm(s,t)
y
x
•
Polyakov action
•
•
Poincare Invariance in the target space
Conformal invariance with respect to
world sheet metric
Reparametrization invariance with
respect to world sheet metric
•
• Conformal invariance may be spoiled in
general due to quantum anomaly
• The requirement of conformal invariance
(the vanishing of the trace of the energy
momentum tensor) is nothing but classical
equations of motion for strings
• It generalizes Einstein’s equations of
motion
• String perturbation theory is given by
topological expansion of string world sheet
• String theory is free from short distance
divergences if it is modular invariant
t
10
s
• Unlike bosonic string theory, superstring
theories can contain space-time fermions
• The consistent Poincare invariant string
theories exist in 26(bosonic) and
10(superstring) dimensions
• The absence of tachyons (infrared
instability) leads us to 5 superstrings in 10
dimensions:
IIA, IIB, Type I: SO(32),
Hetero: E8 x E8, Hetero: SO(32) x SO(32)
• Closed string consists of left-moving and
right-moving modes, while they are related
in open strings
• Heterotic string is the composite of
superstring(right) and bosonic string(left)
• Type II string consists of superstring
sectors of the opposite (IIA) and the same
chirality (IIB)
• Type I string (unoriented) contains both
the open and closed strings
•
1.
2.
3.
4.
5.
4 dimensional models with N=1 SUSY
can be obtained from Heterotic string by
compactifying extra 6 dimensions into
Calabi-Yau manifolds:
There exists covaraint constant spinor
The manifolds have SU(3) holonomy
Ricci flat Kahler manifolds with c1=0
They possess nowhere vanishing
holomorphic (3,0) form
They have two independent Hodge
numbers h1,1 and h2,1
• By embedding the spin connection in the
gauge connection, the gauge symmetry is
broken as
•
•
•
•
Gauge bosons
and gauginos
h2,1 chiral superfields in 27 of E6:
h1,1 chiral superfields in 27 of E6:
Some numbers of E6 singlets:
Moduli fields
• We also obtain the following massless
fields
• d=4,N=1 supergravity
• The dilaton-axion chiral superfield
• h2,1 chiral superfields for the complex
structure moduli:
• h1,1 chiral superfields for the Kahler
moduli:
15
T-duality
• Closed strings can wind around compact
dimensions (winding modes)
• Momentum modes and winding modes
• The symmetry between them implies the
existence of minimal length
D-branes
• Traditionally free (Neumann) boundary
condition is assumed for open strings
(attached to nothing)
• Conformal invariance allows fixed
(Dirichlet) boundary condition also
(attached to D-brane)
• D-branes restore T-duality for open strings
• D-branes are solitons in string theory
whose tensions scale as the inverse
power of the string coupling
• It is a BPS object which preserves the half
SUSY
• It couples to RR gauge fields to which
fundamental strings do not couple
• D-branes appear as black-brane solutions
in closed string theory
• Supergravity description is good when gsN
is large
• D-brane and black-brane pictures provide
us a dual description (open-closed, weak
vs strong coupling)
•
D-branes (+ orientfold) unify closed
strings and open strings
• They play a crucial role to weak-strong
coupling dualities of string theory:
1. Self duality of IIB superstring
2. IIA – M theory duality
3. type I – Hetero duality
• In fact, all string theories are different
manifestations of a single theory
20
Effective theory for D-branes
• On a Dp-brane, there are p+1 dimensional
gauge fields
• There are also 9-p scalar fields
corresponding to the fluctuations of the Dbrane into orthogonal directions
• U(1) Gauge theory with the maximal
SUSY is realized
• Gauge symmetry is enhanced to U(N)
when N parallel D-branes overlap
• D-branes offer new possibilities for particle
theory model buildings
• They can provide gauge fields and break
SUSY
• Quarks and leptons connect different
branes (bi-fundamental rep.)
• CY + Intersecting D-branes:
D-6 branes in IIA wrapping on T2xT2xT2
• The D3-brane on a CY singularity and
quiver gauge theories:
A_i
B_i
T1,1
Conifold
U(N) x U(N)
Unification of Ideas
• Branes in string theory motivates brane
world scenario
• Critical dimension (10) in string theory
motivates theories based on extra
dimensions
• Large extra dimensions and TeV scale
string
• Warped compactification:
Metric Near D3 brane
• The large hierarchy between the standard
model scale (TeV) and the Planck scale
may be explained by an exponentially
small warp factor
25
• Open-closed string duality suggests a
duality between gauge theory and gravity
• It suggests that strong coupling dynamics
of gauge theory may be investigated by
gravity: AdS/CFT
• It also suggests that gravity may be
formulated as gauge theory or D-brane
inspired matrix models
Space-time and branes
• Moduli fields in CY
compactification may
be fixed by fluxes and
instantons
• (Anti-)Branes may
break SUSY and
provide small positive
cosmological constant
D3
• Brane - Anti-brane systems may cause
inflation
• The Inflaton ( r: the lcation of the brane)
rolls slowly either the potential is flat, or
the warped tension T(r) is small
• Meta-stable branes decay by tachyon
condensation
• D-branes offer microscopic description of
black-holes
• Space-time itself may be formed out of Dbranes
• Formation of fuzzy sphere and higher
dimensional analogs from D0 or D-1
• Matrix models for non-critical strings offer
such an example
Conclusion
• String theory offers us intriguing pictures
of space-time and matter
• It is endowed with numerous stable and
meta-stable vacua
• It offers candidates of new physics to
discover such as SUSY and extradimensions
• Experimental discoveries will be crucial to
its further developments
30