String-Theory
CERN Fundamental Structure of Matter An Overview of String Theory
W. Lerche, CERN AcTr, 12/2002 Scale in m Part 1 10-10 Hull of electrons Perturbative string theories Motivation: the Standard Model and its Deficiencies String theory as 2d conformal field theory 10-14 Nucleus Consistency conditions on string constructions Protons Bosonic and supersymmetric strings in D=26,10 Neutrons 10-15 Compactification to lower dimensions Particle Quarks Physics T-Duality, minimal length scales <10-18 Supersymmetry, geometry and zero modes exper bound Parameter spaces, geometrization of coupling constants Stringy predictions ? Non-perturbative string dualities Non-perturbative quantum equivalences Superstrings 10-34 S-Duality in SUSY gauge theories D-branes and Stringy Geometry Unification of string vacua Tests and Applications General Gravitation "Theoretical experiments": tests and consistency checks Relativity D-brane approach to QFT Recent developments for files and text write-up see: 1 http://nxth21.cern.ch/~lerche/ 2 Physics of Elementary Particles Deficiencies of the Standard Model
Its structure is quite ad hoc: are there deeper principles ? Interactions Carriers elementary ("grand unification" of all matter and forces) (Forces) (Gauge Fields) Matter Fields ca 25 free parameters: determined by what ?
Electromagnetism Photon Leptons (eg electrons) weak Interactions W,Z-Bosons massess Higgs VEVs (Radioactivity) gauge couplings Quarks strong Interactions Gauge hierarchy: Gluons (binds Nuclei) There are three why weak scale (100GeV) << Unification scale scale (10^16GeV) "families" of them: Instability of parameters through self-interactions
(Renormalization: large scale hierarchies problematic)
The "Standard-Model" of perticle physics describes subnuclear Symmetry between bosons and fermions phenomena with partly stunning accuracy ! (roughly: force carriers and matter fields)
improved divergence structure:
- = 0 B F
3 4 Problems of Quantum Gravity String-Theory
The Standard-Model of Particle Physics is not complete: small (10-34m), one-dimensional objects: gravity is not included !
The usual quantum field theoretical formulation of gravity does not work, due to incurable divergences (is "not renormalizable"):
scale determined by gravitational coupling:
Graviton
There are conceptual difficulties with quantum black holes... Excitation spectrum: Expressions of a deeply-rooted problem:
Mass Apparent incompatibility of Quantum Mechanics and general Relativity
19 "Low energy physics:" New concepts are necessary....like string theory 10 GeV 1GeV Quarks .... 0 GeV Photon, Graviton As a "by-product", it also provides the grand unification of all particles and their interactions ! "elementary" Particles of the Standard Model = (almost) massless zero modes
5 6 String Theory as 2d Field Theory The String "Miracle"
Perturbative effective action in D-dimensional space-time: Strings trace out two-dimensional "world-sheets" : 2d action (eg. free)
+ + ...
general relativity, small string corrections gauge theory etc infinitely many predictions Perturbative string theory = 2d field theory on Riemann surfaces Building blocks: can be eg. free 2d bosons ‘‘Loop expansion’’ = sum over 2d topologies, weighted by
+ + + ......
Variety of field operators in Simple combinatorics of 2-d D-dimensional space-time field operators
graviton Only one (UV finite) "diagram" at any given order in perturbation theory gauge field
Higgs boson Discrete reparametriztions of have no analog in particle theory; important to make sense eg. of graviton scattering. ‘‘Feynman rules’’ are substantially different from particle theory. Intrinsic unification of particles + interactions ! In particular, gravity is automatically built in. String theory, even in perturbation theory, is more than just having infinitely many particles plus a cutoff !
7 8 Consistency Conditions Conformal Field Theory
The 2d theory cannot be arbitrary but is highly constrained Is a tool set by which one can relatively easily do explicit computations, in terms of simple building blocks
Stress-Energy Tensor T(z) = generator of conformal transformations 2d properties D-dimensional properties
... satisfies an operator product algebra: Conformal invariance implies gauge and general coordinate invariance in spacetime consistency requires D=26 "central charge c" is an anomaly, an obstruction to a good transformation behavior of T(z)
Modular invariance strongly constrains spectrum For D free 2d scalars: absence of UV divergences and anomalies each contributes 1, so
consistency allows only few possible spectra
2d Supersymmetry space-time fermions For theories with more symmetries, ("Superstrings") like supersymmetry or gauge symmetries, one has does not imply that D-dimensional a corresponding generalization of this current algebra space-time theory is supersymmetric !
consistency requires D=10
9 10 The Bosonic String Consistency at 1-loop level
Polyakov-action:
reduces in conformal gauge (2d metric g->1) to a CFT of D free scalar fields Particles: (QFT partition Gauge-fixing of 2d reparametrization symmetries: function) Faddeev-Popov ghosts b,c
Strings:
Ghost stress-tensor has central charge
Consistent quantization requires total central charge to vanish: modular parameter of torus = complexified proper time
Bosonic string exists only in 26 dimensions !
Quantization also implies a shift of vacuum energy:
m2
2 massive higher spin fields 1 0 massless Graviton, Dilaton, Tensorfield -1 Tachyon No stable vacuum !
11 12 Modular transformations Modular Invariance of Partition Function
Torus is defined by identifying sides of parallelogram:
For bosonic string:
oscillator excitations
zero modes 0 1 The "modular" parameter determines its shape
Global reparametrizations: where (Dedekind function)
is the (inverse) oscillator partition function of a scalar field
It has well-defined modular properties, eg:
The vacuum amplitude is indeed modular invariant. ... yield equivalent tori ! ... generate the modular group, PSL(2,Z): This "global consistency condition" has no analog in particle QFT; and is responsible for many stringy features...
Physical amplitudes must be invariant under such modular transformations !
13 14 Vacuum amplitude at 1-loop UV and IR Divergences
In string theory, we need to integrate precisely once over To obtain vacuum amplitude, still need to integrate; try: all inequivalent shapes of the torus. These are described by the "fundamental region F":
t IR limit projection on R=0 t = proper time F t IR limit 1
UV limit S s -1/2 0 1/2 Every point outside F is equivalent (via a modular UV limit: divergence from symmetry transformation) to a point inside F; s integrating over it would be overcounting -1/2 0 1/2
As t=0 is not integrated over, there is no UV divergence possible
However, this is all-too-naive particle field theory thinking..... (NB: is more than a cutoff...) However, there is an IR divergence for large t in string theory,
has a definite geometrical meaning, namely it is the modular parameter of a torus ! Pole is due to Tachyon state (vacuum instability)
Superstrings !
15 16 Superstrings Spin Structures
2d fermions can have non-trivial boundary conditions: 2d supersymmetry --> space-time fermions typically, but not necessarily supersymmetric in space-time !
Typically have no tachyons -- only known consistent theories... The R-sector leads to space-time fermions, the NS-sector to bosons ! Each cycle can be either periodic (P) or anti-periodic (A) Two formulations: "Green-Schwarz": covariant, but difficult to quantize
"Neveu-Schwarz-Ramond": easier to quantize, but not manifestly covariant The partition function of a fermion depends on the "spin structure":
From 2d NSR perspective: CFT -> SuperCFT additional ghost fields with c=+11
Critical dimension for superstrings is D=10 These "theta"-functions have well-defined modular properties .
Modular invariance of the partition function requires particular, Main novel technical ingredient: consistent choices for the boundary conditions, and these determine the physical spectrum boundary conditions of the 2d fermions Modular invariance strongly constrains the possible physical spectra
.... chiral anomalies always cancel 17 18 Modular Invariant Partition Functions Supersymmetric String Theories in D=10
How to construct modular invariant sums over spin structures ?
By combining superstring (S) and bosonic string (B) building blocks, one can construct five types of string theories in D=10:
Systematic procedure: map this to certain lattice sums ! Modular invariance self-duality of lattice, Combination Name Gauge group easy classification of all possibilities
The result is various possibilities in D=10 : 1) Holomorphic and anti-holomorphic sectors separately invariant
"Type IIA"
"Type IIB" These theories have one dimensionful parameter, "heterotic" the string tension , besides the coupling . (two kinds, related to uniqueness of self-dual lattices: E8xE8 or SO(32) gauge symm.) They have very different spectra (c.f., gauge groups) ! 2) Non-trivial correlation of spin structures This gives various non-supersymmetric theories, only one of which is tachyon-free
In addition, there is one more "open" supersymmetric string
19 20