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 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(,)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   10

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: E 8 x E 8, 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 spinor 2.The manifolds have SU(3) holonomy 3.Ricci flat Kahler manifolds with c 1 =0 4.They possess nowhere vanishing holomorphic (3,0) form 5.They have two independent Hodge numbers h 1,1 and h 2,1

By embedding the spin connection in the gauge connection, the gauge symmetry is broken as Gauge bosons and gauginos h 2,1 chiral superfields in 27 of E 6 : h 1,1 chiral superfields in 27 of E 6 : Some numbers of E 6 singlets:

Moduli fields We also obtain the following massless fields d=4,N=1 supergravity The dilaton-axion chiral superfield h 2,1 chiral superfields for the complex structure moduli: h 1,1 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 g s N 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 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 T 2 xT 2 xT 2 The D3-brane on a CY singularity and quiver gauge theories: A_i B_i T 1,1 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 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 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 30