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Introduction to String Theory

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Introduction to String Theory Introduction to String Theory Angel M. Uranga CERN, Geneva & IFT, Madrid SILAFAE, Bariloche, Enero 2009 Plan of the lectures Lecture 1: Perturbative string theory (80’s) Lecture 2: Beyond perturbation theory (90’s) Lecture 3: Model building with D-branes (00’s) Lecture 1: Perturbative string theory Motivation The question of Quantum Gravity (at the perturbative level) Perturbative quantum interactions of spin-2 particle (graviton) coupled to energy-momentum tensor Analogy: Fermi General Relativity G - Vertex G G G - Coupling GF GN -1 19 -1 - Cutoff LF ≈ (100 GeV) LF ≈ (10 GeV) e ν G - UV compl. W G ? ν e G Whatever the proposed UV completion, take it seriously! Motivation String theory is a UV completion of gravity, which implements modifications at a high scale Ms (around/below Mp) In addition, the theory is very rich - incorporates gauge interactions and charged fermions ⇒ models of particle physics - automatically accommodates ideas for new physics: extra dimensions, supersymmetry, unficiation, ... - Has applications in many fields of theoretical physics BSM phenomenology, heavy ion physics, cosmology, condensed matter,... Certainly worth studying! What is string theory? Elementary particles are not pointlike, but small one-dimensional objects (how small? think roughly Ls =1/Mp, quantify later) zoom or Different states of oscillation correspond to different particles (i.e. different particles are just different states of oscillation of a unique kind of string ⇒ Ultimate unification! ) Example etc etc 0 αµ 0 αµαν 0 | ! | ! | ! What is string theory? Spectrum: massless modes plus infinite tower of string resonances step of order Ms Only massless (compared to Ms) modes are accesible at E<<Ms What are these particles? Closed string massless sector ⇒ Graviton! contains a spin-2 particle Open string massless sector ⇒ Gauge bosons! contains spin-1,1/2, 0 particles matter! (Quantum) Theory of gravity and gauge interactions String interactions Generalize Feynman diagrams of QFT: worldsheet - Propagators - Vertices 2 2 gs gs gs Two consequences: 1. Theories of open strings must contain closed strings 11. Only one fundamental vertex, which is 3-leg, e.g. the 4-leg splits as = Perturbative expansion 2 Interaction strength: gs for 3-open vertex, gs for 3-closed vertex Build Feynman diagrams by glueing vertices: Riemann surfaces with g holes (and possibly h boundaries) 2+2g+h Each diagram weigthed by (gs)− String theory version of ... and many others Vertices delocalized in a distance 1/Ms: regularizes loop amplitudes Finiteness of string perturbation theory order by order (Perturbative series is asymptotic: non-perturbative effects) Finiteness and conformal invariance UV of string theory differs from QFT, and is free of divergences field theory string theory E M p E Ms ~ s UV E M ~ s E > M s = 8 IR in dual channel A crucial property underlying this behaviour: conformal invariance (invariance under rescaling of small patches on the worldsheet) Bosonic strings - Bosonic string theory: dynamis is described in terms of functions Xµ(σ, t) specifying the spacetime position X µ of each worldsheet point (σ,t) (σ, t) with μ=0,...,D-1 labeling directions in D-dimensions flat space - Invariance under reparametrizations of the worldsheet ⇒ Some combinations of functions can be gauged away Physical oscillations are only those transverse to the worldsheet Denote these (D-2) functions by Xi(σ, t) Bosonic strings - Dynamics is encoded in an action (Nambu-Goto, Polyakov actions) which essentially reduces to S[X] = M 2 dσ dt ( ∂ Xi ∂ Xi ∂ Xi ∂ Xi ) s t t − σ σ ! Equation of motion is a wave equation ∂ 2Xi ∂ 2Xi = 0 t − σ Free oscillation of the string in the D-2 transverse directions - Interestingly, a 2d quantum field theory of D-2 free massless bosons S[X] d2ξ ∂ Xi ∂αXi ∼ α Physics on the 2d worldsheet ! Conformally invariant (classically, see later for quantum) (OBS: interacting 2d theories describe curved spacetimes) Extra dimensions Critical dimension - Conformal invariance of 2d action is anomalous at quantum level The anomaly would spoil good UV properties of string theory - As is familiar from other contexts, can cancel anomaly by choosing a suitable “matter” content of the 2d theory The field content of our 2d field theory are (D-2) massless bosons The anomaly cancels for D=26 - This fixes the dimension of spacetime on which a quantum string can propagate consistently! (critical dimension for superstrings is D=10 see later; reduction to 4d will require “compactification” see later) Spacetime spectrum General solution of wave equation: Standing waves as superposition of left- and right-moving waves i i i i 2πi n(t+σ) i 2πi n(t σ) X (σ, t) = x + p t + αn e + α˜n e− − n=0 n=0 !! !! Oscillation states correspond to particles in spacetime with mass 2 M ˜ Lowest states are 2 = N + N 2 Ms − lightest particles Light modes 0 T tachyon! ☹ | ! µ ν α α˜ 0 Gµν , Bµν , φ graviton, 2-form, dilaton | ! φ ☺ Dilaton vev fixes coupling g s = e − : No external parameters! ☺ Invariance under general coordinate transformations in 26d! Effective theory for light modes is GR coupled to extra fields! ☹ No spacetime fermions! Superstrings - Define new, more interesting, consistent string theories - Change the 2d field theory on worldsheet by adding extra fields Propose to define this “extension” by using a symmetry principle Supersymmetry on the worldsheet! - Define new theories by considering dynamics of 2d free bosonic and fermionic fields, roughly Xi(σ, t) , ψi(σ, t) - Cancellation of conformal anomaly now requires D=10 OBS: Worldsheet fermions and worldsheet susy do not necessarily imply spacetime fermions and spacetime susy ... But it does happen in the most familiar superstring theories (but not in some less familiar string theories, like type 0) The five superstrings Different superstring theories from above ingredients - Type IIA and type IIB Theories of closed strings. Have N=2 susy in 10d (2 gravitinos) Worldsheet theory: 10 fields X, ψ in both left- and right-movers Massless fields: graviton, gravitinos, dilaton, dilatinos, and p-forms - Type I Theory of non-oriented closed and open strings. Has N=1 susy in 10d Worldsheet theory related to type IIB. Closed sector: graviton, gravitino, dilaton, dilatino and p-forms Open sector: SO(32) gauge bosons and gauginos - E8 x E8 and SO(32) heterotics Theories of closed strings. Have N=1 susy in 10d (1 gravitino) Worldsheet theory: 10 right-moving fields X, ψ, 26 left-moving fields X Spectrum: graviton, gravitino, dilaton, dilatino and p-forms gauge bosons of E8xE8 or SO(32), and gauginos The five superstrings Different superstring theories from above ingredients Kind of Worldsheet Spacetime ZOO Massless particles strings theory susy 10 X and ψ 10d N=2 Graviton, gravitinos, dilaton, Type IIA Closed for L and R (non-chiral) dilatinos, odd p-forms 10 X and ψ 10d N=2 Graviton, gravitinos, dilaton, Type IIB Closed for L and R (chiral) dilatinos, even p-forms Open and Graviton, gravitino, dilaton, 10 X and ψ Type I closed, 10d N=1 dilatino, SO(32) gauge bosons for L and R unoriented and gauginos Graviton, gravitino, dilaton, E8xE8 10 X and ψ for R Closed 10d N=1 dilatino, E8xE8 gauge bosons heterotic 26 X forL and gauginos Graviton, gravitino, dilaton, SO(32) 10 X and ψ for R Closed 10d N=1 dilatino, SO(32) gauge bosons heterotic 26 X forL and gauginos Compactification - Although UV theory is 10d, need 4d physics at low energies - Consider the theory propagating on spacetime M4 x X6 with X6 a compact 6d manifold with typical size L - Strings on small curved spacetimes is in general rather difficult but can use effective theory approximation if size is 1/L<<Ms X } int Ls }Xint }M4 }M4 1/L<< Ms 1/L ~ Ms - Recover 4d physics at E << 1/L << Ms At low energies, not enough resolution to detect additional dims. Compactification Kaluza-Klein compactification: - 5d → 4d toy model of massless scalar ϕ ( x 0 , . , x 4 ) on M4 x S1 Fourier-expand on periodic x4 0 4 2πikx4/L 0 3 ϕ(x , . , x ) = e ϕk(x , . , x ) k Z !∈ Traded one 5d field for infinite KK tower of 4d fields, of mass 2πk M = k L At low E << 1/L only the massless state (zero mode) is observable 4d physics: Effective theory for one massless 4d scalar field 4d effective action obtained by integrating over → factor of L - Generalization to 10d → 4d for all fields in string theory (tools of KK compactification in supergravity) Calabi-Yau compactification - Interested on compactifications preserving supersymmetry Theoretically convenient, possibly phenomenologically relevant - Calabi-Yau spaces are 6d geometries with the property that there is a zero mode in the KK compactification of the 10d gravitino ⇒ 4d massless gravitinos, i.e. 4d supersymmetry (mathematically, property of SU(3) holonomy implies existence of a globally defined covariantly constant spinor) Ex. of CY: the 2-torus T2 in 2d, the 6-torus T6 in 6d - Parameters of compactification space are sizes of 2- and 3-cycles 2d and 3d “holes” inside the 6d geometry Generalization of parameter L in circle compactification Correspond to vevs of scalar moduli fields in 4d theory Kahler moduli and complex structure moduli CY compactification of heterotic strings [Candelas, Horowitz, Strominger, Witten, ‘85] a - E8 x E8 heterotic string on 6d CY space Gi j A i }X6 - In addition to non-trivial metric, need to introduce non-trivial internal magnetic M4 fields for some of the gauge bosons } e.g. SU(3) magnetic fields in E8 ⊃ E6 x SU(3) 4d physics*: E6 GUT + hidden sector - CY ⇒ 4d N=1 supersymmetry - Gauge group broken to subgroup commuting with bcknd: E6 x E8 - KK reduction of 10d gauginos leads to 4d chiral fermions in 27’s - Replication of GUT families, given by topological number χ/2 * Generalizations lead to models much closer to (MS)SM (e.g.
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