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CERN Fundamental Structure of An Overview of

W. Lerche, CERN AcTr, 12/2002 Scale in m Part 1 10-10 Hull of Perturbative string Motivation: the and its Deficiencies as 2d conformal theory 10-14 Nucleus Consistency conditions on string constructions Bosonic and supersymmetric strings in D=26,10 10-15 Compactification to lower 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 equivalences Superstrings 10-34 S-Duality in SUSY gauge theories D- and Stringy Unification of string vacua Tests and Applications General Gravitation "Theoretical experiments": tests and consistency checks Relativity D- approach to QFT Recent developments for files and text write-up see: 1 http://nxth21.cern.ch/~lerche/ 2 Physics of Elementary Deficiencies of the Standard Model

Its structure is quite ad hoc: are there deeper principles ? Carriers elementary ("grand unification" of all matter and ) (Forces) (Gauge Fields) Matter Fields ca 25 free parameters: determined by what ?

Electromagnetism (eg electrons) weak Interactions W,Z- massess Higgs VEVs (Radioactivity) gauge couplings Quarks strong Interactions Gauge hierarchy: (binds Nuclei) There are three why weak scale (100GeV) << Unification scale scale (10^16GeV) "families" of them: Instability of parameters through self-interactions

(: large scale hierarchies problematic)


The "Standard-Model" of perticle physics describes subnuclear between bosons and phenomena with partly stunning accuracy ! (roughly: carriers and matter fields)

improved divergence structure:

- = 0 B F

3 4 Problems of Quantum String-Theory

The Standard-Model of 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:


There are conceptual difficulties with quantum black holes... Excitation spectrum: Expressions of a deeply-rooted problem:

Mass Apparent incompatibility of and

19 "Low physics:" New concepts are necessary....like string theory 10 GeV 1GeV Quarks .... 0 GeV Photon, 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 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 , weighted by

+ + + ......

Variety of field operators in Simple of 2-d D-dimensional space-time field operators

graviton Only one (UV finite) "diagram" at any given in gauge field

Higgs 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

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 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 consistency requires D=26 "central c" is an , 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 , ("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


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


Ghost stress-tensor has

Consistent 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 energy:


2 massive higher fields 1 0 massless Graviton, , Tensorfield -1 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

It has well-defined modular properties, eg:

The vacuum amplitude is indeed modular invariant. ... yield equivalent tori ! ... generate the modular , 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 -- only known consistent theories... The R-sector 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 depends on the "":

From 2d NSR perspective: CFT -> SuperCFT additional fields with c=+11

Critical 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 , 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

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