Introduction%20to%20Strings

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Introduction to Strings

• Yoshihisa Kitazawa
• KEK
• Nasu lecture 9/25/06

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Why strings?

• We have solved many questions
• Standard model of particle physics
• SU(3)xSU(2)xU(1) gauge theory
• 3 generations of quarks and leptons
• Standard model of cosmology
• Big Bang nucleosynthesis
• Large scale structure formation based on cold
  dark matter and inflation

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We are making progress to solve important
questions
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We also find new deep questions
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• To answer these questions, we need to understand
  not only matter but also space-time 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

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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
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• Polyakov action
• Poincare Invariance in the target space
• Conformal invariance with respect to world sheet
  metric
• Reparametrization invariance with respect to
  world sheet metric

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• 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 Einsteins equations of motion

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• 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
s
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• 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)

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• 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

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• 4 dimensional models with N1 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 c10
• They possess nowhere vanishing holomorphic (3,0)
  form
• They have two independent Hodge numbers h1,1 and
  h2,1

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• 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

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Moduli fields

• We also obtain the following massless fields
• d4,N1 supergravity
• The dilaton-axion chiral superfield
• h2,1 chiral superfields for the complex structure
  moduli
• h1,1 chiral superfields for the Kahler moduli

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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

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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

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• 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

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• 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)

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• D-branes ( orientfold) unify closed strings and
  open strings
• They play a crucial role to weak-strong coupling
  dualities of string theory
• Self duality of IIB superstring
• IIA M theory duality
• type I Hetero duality
• In fact, all string theories are different
  manifestations of a single theory

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Effective theory for D-branes

• On a Dp-brane, there are p1 dimensional gauge
  fields
• There are also 9-p scalar fields corresponding to
  the fluctuations of the D-brane 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

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• 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.)

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• 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)
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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

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• Warped compactification
• The large hierarchy between the standard model
  scale (TeV) and the Planck scale may be explained
  by an exponentially small warp factor

Metric Near D3 brane
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• 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

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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
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• 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

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• Meta-stable branes decay by tachyon condensation
• D-branes offer microscopic description of
  black-holes
• Space-time itself may be formed out of D-branes
• Formation of fuzzy sphere and higher dimensional
  analogs from D0 or D-1
• Matrix models for non-critical strings offer such
  an example

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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 extra-dimensions
• Experimental discoveries will be crucial to its
  further developments

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