Introduction to Strings

Yoshihisa Kitazawa

KEK

Nasu lecture 9/25/06

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

We are making progress to solve important questions

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

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

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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