introduction to strings
DESCRIPTION
Introduction to Strings. Yoshihisa Kitazawa KEK Nasu lecture 9/25/06. 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 - PowerPoint PPT PresentationTRANSCRIPT
Why strings?
• We have solved many questions: Standard model of particle physics1. SU(3)xSU(2)xU(1) gauge theory2. 3 generations of quarks and leptons Standard model of cosmology1. Big Bang nucleosynthesis2. Large scale structure formation based on
cold dark matter and inflation
• 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
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
x
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
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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)
• 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
• 4 dimensional models with N=1 SUSY can be obtained from Heterotic string by compactifying extra 6 dimensions into Calabi-Yau manifolds:
1. There exists covaraint constant spinor2. The manifolds have SU(3) holonomy
3. Ricci flat Kahler manifolds with c1=04. They possess nowhere vanishing holom
orphic (3,0) form5. They have two independent Hodge numb
ers 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 fiel
ds
• 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:
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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
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 superstring2. IIA – M theory duality3. type I – Hetero duality• In fact, all string theories are different ma
nifestations of a single theory
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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 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
• 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:
• 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
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 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