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<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<

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. Braun, Donagi, He, Ovrut, Pantev, ‘06) Phenomenology of heterotic models

String scale: Close to 4d Planck scale 8 6 2 Ms V6 1 Ms V6 17 M = gSM MP = Ms 10 GeV P 2 ; 2 = 2 gs gSM gs ∼ Supersymmetry: Susy model, stabilized hierarchy Mw<

Unification: All SM gauge factors from a single 10d group Unification of gauge coupling constants (at slightly too high scale)

Proton decay: Prevented by large UV scale. Still potentially dangerous model-dependent dim.5 operators

Small extra dimensions: Model has extra dimensions, but of very small size Some further questions

(Quite open, continuous progress in most of these) Yukawa couplings: Present, computable, but no fully compelling mechanism to explain actual values

Supersymmetry breaking: Nature is, at best, susy only above TeV scale. Can we achieve (low-energy) susy breaking?

Cosmological constant: Usual problem with too large quantum vacuum energy, susy does not help after susy breaking

Moduli stabilization: Massless 4d scalars with no potential, vevs specify continuous parameters of model (compactification size, ...) Need to generate moduli potential to fix vevs, make scalars heavy

Many choices: Compactification space, gauge background data,...: large set of vacua, each with different 4d physics (baby version of the “landscape”) Conclusions

Beautiful construction designed to define Quantum Gravity

Strong constraints of self-consistency fix physical properties

Surprisingly enough, leads to gauge interactions & charged fields

Requiring compactification to 4d, leads to chiral matter

Contact with many ideas in BSM

Very remarkable...... and with much progress in the (still many) open questions Very active field. Introduction to String Theory

Angel M. Uranga CERN, Geneva & IFT, Madrid

SILAFAE, Bariloche, Enero 2009 Lecture 2: Beyond perturbation theory Large order behaviour

String theory is finite order by order in perturbation theory The series is not convergent; asymptotic in the string coupling gs (indeed non-analiticity expected from Dyson argument)

Expect non-perturbative dynamics (in fact, string theory contains gauge theory, which has non-perturbative effects e.g. instantons) Strength can be estimated by studying large order behaviour 1/g Using toy models (matrix models), strength e− s ∼ [Shenker ‘90] Suggests role of configurations with action/energy 1/gs 1/g 2 (compare with gauge theory e − YM , instantons, monopoles) ∼ Solitons in string theory

Construct non-perturbative states as field configurations solving the e.o.m. of low energy supergravity effective action (in analogy with field theory monopoles, instantons) Many solutions, even if we restrict to a very (super)symmetric ansatz: BPS states, preserving half of the supersymmetries of vacuum Ex: Dp-branes in type II/I, extended objects with p space dimensions

2 1/2 µ ν 1/2 m m ds = Z(r)− ηµν dx dx + Z(r) dx dx μ 2φ (3 p)/2 x e = Z(r) − 7 p ρ − 7 p (7 p)/2 Z(r) = 1 + 7 p ; ρ − = gsQα" − r − Q H = d(vol) 8 p 8 p (8 p) S − − r − μ = 0,...p ; m = p+1,...,9 xm p even/odd in type IIA/B, p=1,5 in type I Other solitons (NS5-branes, (p,q)-branes,...) not too relevant today BPS states

Supergravity is only a classical low energy approximation of string theory

Why trust the existence and properties of these objects?

BPS property: Certain statements are protected by supersymmetry against inclusion of stringy and quantum corrections [Hull,Townsend ‘94] (analogous to non-renormalization theorems for 4d N=1 superpotential due to susy cancellations) - Existence of state - Dependence of the tension on the parameters (moduli) M p+1 For Dp-branes, the tension (energy per volume) is T s ∼ gs Duality

At finite string coupling, p-brane democracy No object should be considered more fundamental than others Strings become fundamental d.o.f. in the particular limit gs →0 Can other objects appear as fundamental in other regimes? gs →∞

gs →0 gs →∞ Duality: Theories at gs →∞ are equivalent to other perturbative theories, upon a suitable “change of variables” Duality

Using BPS-protected quantities, guesswork and consistency, well supported conjectures for the strong coupling behaviour of superstring theories [Witten ‘95] Type IIB self-duality IIB at coupling gs equivalent to dual IIB at coupling gs’=1/gs D1-brane at gs →∞ becomes fundamental with gs’ →0 Actually, Z2 symmetry → full SL(2,Z) of equivalent theories Type I - SO(32) heterotic duality Type I at coupling gs equivalent to heterotic at coupling gs’=1/gs D1-brane at gs →∞ becomes fundamental with gs’ →0 Type IIA, E8 x E8 heterotic and M-theory Strong coupling limit of type IIA related to a new theory Not a string theory, M-theory Also related to strong coupling of E8xE8 heterotic M-theory

k States of k D0-branes in type IIA have mass Mk = gs KK mass formula for a compactification 11d →10d on radius R=gs Propose existence of a new 11d quantum theory, M-theory Compactification on S1 of radius R gives type IIA at coupling gs=R Massless fields are graviton, gravitino, 3-form Effective action is 11d N= 1 supergravity Contains membranes and 5-branes Type IIA and M-theory Type IIA at coupling gs equivalent to M-theory on S1 of radius R=gs 1 E8xE8 heterotic and M-theory S /Z2 Heterotic at coupling gs equivalent M10 E8 to M-theory on S1/Z2 of radius R=gs E8 [Horava,Witten ‘96] The duality web

Perturbative and non-perturbative dualities relate diff. theories Unification of (perturbatively) different theories by duality

Type IIB

Type I orient. T Type IIA S S1 M SO(32) heterotic T S1/Z2 E8xE8 heterotic

Unique answer to the question of quantum gravity! (rather than six) Duality web becomes intricate upon compactification ⇒ Useful in predicting non-perturbative effects Non-perturbative effects from branes

[Strominger ‘95] Type IIB on a CY with a conifold singularity Limit where a 3-cycle shrinks to zero size x2 + y2 + z2 + w2 = ε

ε 0 ε = 0 → ! ε A D3-brane wrapped on the 3-cycle is a 4d particle of mass gs In perturbative theory, generically very massive But becomes massless as New light d.o.f. in effective theory, of non-perturbative origin D-branes

Focus on a particular kind of soliton in type II/I string theories Dp-branes are extended objects, solitons with p spatial dimensions, and propagating in time, whose tension is ~1/gs - At weak coupling gs<<1, behave as part of background - Described as subspaces of 10d space on which open strings end - Equivalently, particles described by oscillation modes of open strings propagate only on volume of Dp-brane ⇒ Brane world

10d spacetime

[Polchinski ’95]

(p+1)dim. volume World-volume dynamics

The zero modes of the soliton are described by the massless oscillation modes of open strings ending on the D-brane Localized on the (p+1) dimensional volume of the Dp-brane * U(1) gauge boson * (9-p) real scalars Φi: Goldstones bosons of the translational symmetries broken in the presence of the D-brane; “vev=position” * Set of fermions: Goldstinos of the 16 supersymmetries of the vacuum broken by the D-brane (other 16 susys still remain) Dynamics is described by a (p+1) dimensional field theory Actually, worldvolume action for the extended Dp-brane!

Ex: Dynamics of scalars Φi(x) describe deformations of the D-brane volume x Coindicent D-branes

N overlapping Dp-branes (stable due to BPS “no-force”) lead to U(N) gauge symmetry on brane

Nx N possible open strings: - “Diagonal” give U(1)N - “Off diagonal” charged under them Enhancement to U(N) N Separation is Higgssing U(N) →U(1) U(2) World-volume effective action

p+1 1 SDp = d x det(G + F ) + gs − ! " 2 + ( Cp+1 + Cp 1 F + Cp 3 tr F + ... ) − ∧ − ∧ !p+1 reduces to SYM at low energies D-branes and black holes

- Microscopic description from open strings can explain properties of the supergravity solution D-brane tension, D-brane charge, ... - Learn about strongly gravitating systems: Consider configurations D-branes, leading to black holes Ex: Compactification to 5d on T4 x S1 (similar 4d examples) D1 – x x x x – x x x x (with momentum P on circle) D5 – x x x x – – – – – 5d S1 T4 Form a 5d black hole with Hawking entropy S=(N1N5 P)1/2 Corresponds to statistical entropy of quantum microstates given by D1-D5 open strings [Strominger, Vafa, ’95] (protected and can be extrapolated to strong gravity regime) Towards understanding quantum properties of black holes and black hole information problem (holography, see later) Near horizon limits

[Maldacena ‘97] Propose complete equivalence of two descriptions of D3-branes - N D3-branes as planes on which open strings end Reduces, at low energies, to 4d N=4 SU(N) super Yang-Mills - N D3-branes as supergravity solution Due to gravitational redshift, low energies correspond to near horizon limit.

Reduces to type IIB string theory on AdS5 x S5 2 r2 µ ν R2 2 2 2 F = N ds = R2 (ηµν dx dx ) + r2 dr + R dΩ5 , S5 5 ! AdS/CFT

[Maldacena ‘97] Propose complete equivalence of two descriptions of D3-branes

4d N=4 SU(N) SYM IIB on AdS5 x S5

Seem two very different theories! “Strong-weak duality” due to mapping of parameters R4 2 recall α! Ms− 2 = 4πgsN α! ! Some kinematical checks: Symmetries - SU(4) R-symmetry of N=4 is SO(6) isometry of S5 - SO(4,2) conformal symmetry of N=4 SYM is isometry of AdS5 with very non-trivial implications radial direction in AdS5 identified with energy scale in 4d SYM Large N limit

Fits well with ‘t Hooft picture of large N limit of gauge theories - Double line notation for adjoints

a) b) = = - Feynman diagrams become Riemann surfaces by “filling the holes”

- Large N expansion at fixed λ=g2N is an expansion in the genus of the 2 Riemann surface O(1/N )

AdS/CFT relates large N limit of SYM, at large ‘t Hooft coupling, with weakly coupled strings/supergravity on AdS5 O(1/N) Holography

[Gubser, Klebanov,Polyakov; Witten ‘98]

4d N=4 SU(N) SYM IIB on AdS5 x S5

Holographic relation between the 4d field theory and the 5d gravitational theory (Holography: describe gravity in D dims by non-gravitational theory in D-1 dims) AdS5 has a (conformal) boundary which is 4d Minkowski space Values of supergrativy fields at boundary act as sources for the 4d field theory on the boundary Can use classical supergravity (good approximation at large R) to describe properties of strongly coupled (large gsN) gauge theory Ex.correlators Towards hot QCD

Use AdS/CFT for N=4 SYM and other gauge theories as toy model for QCD in strongly coupled regimes

Successful application to hydrodynamical properties of hot QCD in heavy ion collisions (at RHIC, Brookhaven, and soon at LHC, CERN)

(More than) reasonable quantitative agreement in some observables: viscosity to entropy ratio 1 η/s = 4π Computations for other phenomena: jet quenching, drag force, ... Other applications

Many new interesting connections of very general interest - Application to QCD spectrum, AdS/QCD - Hydrodynamics and gravity Navier-Stokes equation ⇔ Einsteins equations - Starting study of non-relativistic limits and applications to condensed matter physics: gravity dual of superconductivity or other non-trivial phenomena - Hydrodynamics of fluids of trapped cold atoms - ...

Very surprising applications, far from original quantum gravity motivations Conclusions

Interesting steps into non-perturbative structure of theory

Role of other extended objects and of dualities

Although mainly supersymmetric, some more general lessons

The AdS/CFT correspondence as an example of holography

Many new application areas of string theory: Brane worlds, QCD, condensed matter,...

Very surprising applications, far from original quantum gravity motivations... Still, expectation to provide a theory of elementary particles very much alive, see tomorrow! Introduction to String Theory

Angel M. Uranga CERN, Geneva & IFT, Madrid

SILAFAE, Bariloche, Enero 2009 Lecture 3: Model building with D-branes Type II orientifolds

CY compactifications of type II have 4d N=2 supersymmetry, and do not contain non-abelian gauge symmetries

Adding D-branes can break down to 4d N=1 and yield non-abelian gauge symmetries (and eventually charged chiral fermions)

Models contain open and closed string sectors (usually unoriented) Related to CY compactifications of type I Referred to as type II orientifolds or D-brane models

10d spacetime Realize brane world scenario: Gravity(closed strings) propagate on 10d spacetime Gauge intractions (open strings) propagate on lower-dim. branes (p+1)dim. volume The brane-world idea

[(Antoniadis), Arkani-Hamed, Dimopoulos, Dvali] Different volume dependence of gravitational and gauge couplings

Gravity propagates over the whole 6d extra dimensions 8 2 Ms V6 MP = 2 gs Gauge ints. propagate on Dp-brane: 4d space + p-3 extra dimensions p 3 1 Ms − VΠ 2 = gSM gs Hence 11 p 2 2 Ms − V MP gSM = ⊥ gs

Can have large Ms and small transverse volume...... or low Ms and large transverse volume: low string scale models e.g. choose Ms ~ TeV and no hierarchy problem Chirality

Isolated D-branes in smooth geometries cannot lead to chiral gauge theories

X 6 R 6

D3 D3 4d N=4

Two setups for SM model building - D-branes at singularities [Aldazabal, Ibanez, Quevedo, AU, ‘00; Verlinde, Wijnholt, ‘05] - Intersecting D-branes [Blumenhagen, Gorlich, Kors, Lust; Aldazabal, Franco, Ibanez, Rabadan, AU; ‘00] (or magnetized branes [Bachas ‘95; Angelantonj, Antoniadis, Dudas, Sagnotti, ‘00]) Focus on intersecting D-branes Related to others by string dualities Intersecting D6-branes

[Berkooz, Douglas, Leigh,’96] Consider type IIA string theory with two stacks of D6-branes (i.e. 7d subspaces) intersecting over a 4d subspace of their volumes

D6 2 D6 1 ! 2 ! 3 ! 1

2 2 2 M 4 R R R

Three sectors of open strings

- D61-D61: U(N1) on 7d plane 1 - D62-D62: U(N2) on 7d plane 2

- D61-D62: 4d chiral fermion in (N1,N2) on 4d intersection (possibly also light scalars in same representation) Chirality is a consequence of the geometry of the intersection e.g. two D5’s intersecting over 4d leads to non-chiral fermions Scalars at intersections and Higgs mechanism

In addition, the D61-D62 sector contains 4d scalars, which are generically massive, and potentially light 1 M 2 = ( θ θ θ ) M 2 2 1 ± 2 ± 3 s Could be tachyonic, massless or massive Possible instability corresponds to possible brane recombination

D6 D6 2 1 Higgs effect e.g. SU(2) x U(1)→ U(1)

Nice geometric interpretation in terms of volume minimization [Douglas, ’01] Type II orientifolds on CY

These configurations arise in !2 X6 type IIA orientifolds with D6-branes !1 wrapped on 3-cycles on the CY !3

M4

For instance, in toroidal CY !"#!"% !"#(% !"#!"% !"#'%

!"#$%

!"#"%

&' & ' &' Replication of fermions from multiple number of intersections Iab Direct SM constructions

[Ibáñez, Marchesano, Rabadán; Cremades, Ibáñez, Marchesano;’01]

Introduce four stacks of D6’s a,b,c,d with U(3) SU(2) U(1) U(1) a × b × c × d b- Left c- Right Iab = 3 QL gluon → a- Baryonic U(3) Q Iac = 3, Iac! = 3 UR, DR L U , D − → R R Idb = 3 L → W Idc = 3, Idc! = 3 ER, νR − − → d- Leptonic U(1) L

Spectrum of SM LL ER U(2) U(1) Hypercharge S R 1 1 1 Y = Q Q Q 6 a − 2 c − 2 d other U(1)’s massive, remain as global symmetries Many explicit models realizing this construction Direct SM constructions

[Ibáñez, Marchesano, Rabadán; Cremades, Ibáñez, Marchesano;’01]

Introduce four stacks of D6’s a,b,c,d with U(3) SU(2) U(1) U(1) a × b × c × d I = 3 QL ab → Iac = 3, Iac = 3 U , D − ! → R R Idb = 3 L I = 3→, I = 3 E , ν dc − dc! − → R R Spectrum of SM

Hypercharge S 1 1 1 Y = Q Q Q 6 a − 2 c − 2 d other U(1)’s massive, remain as global symmetries Many explicit models realizing this construction GUTs

Can also construct models of SU(5) Grand Unification (No SO(10): requires families in spinor representation)

- 5 overlapping D-branes 5 - Intersection with image gives 10 SU(5) - Intersection with extra D-brane 10 gives 5

Need to break GUT to SM

Yukawa couplings - 10.5.5 is allowed - 10.10.5 is forbidden by perturbatively exact U(1) symmetry Need non-perturbative effects The type IIB picture: D7-branes

Similar pictures can be constructed using intersecting D7- branes in type IIB theory Each stack of D7branes wraps a 4-cycle in the CY: Gauge group Pairs of 4-cycles intersect over 2-cycles: Matter 2-cyles intersect at points: Yukawa couplings F-theory GUTS

[Beasley, Heckman, Vafa ‘07] F-theory: Generalization of type IIB with D7-branes to configuration including other non-perturbative 7-branes

Includes certain non-perturbative effects e.g. 10.10.5 Yukawas

Although eventually desirable, fully compact models are involved Most relevant information already in local geometry around branes SU(5) GUT 7-brane on 4-cycle, matter from 2d intersections with other branes, Yukawas from intersection points Phenomenology of D-brane models

String scale: Recall brane world relation - Susy models, can choose until specific susy breking mechanism - Non-susy models, can lower to TeV by using large volume lose to 4d Planck scale a) b) W Supersymmetry: Optional

Unification: Present in models where all SM from single brane stack No natural gauge unification in models with several SM brane stacks 1 VΠa 2 = Proton decay: ga gs Very often prevented by global symmetries e.g. U(1) in U(3)

Extra dimensions: Models have extra dimensions, and could be large! Some more advanced questions

Yukawa couplings Q 1 L SU(2) Q 2 L Mediated by open string worldsheet Q 3 L instantons AHjk+iφjk H Yjk e− ! U 3 U 2 U(1) SU(3) U1 SU(3) SU(3) Exponential dependence may explain fermion mass hierarchy Computation of results yields complicated function of moduli Need to face moduli stabilization, see later

Geometrical intuitions possibly useful in search for textures [Heckman, Vafa, ’08 in F-theory context] Moduli stabilization

String compactifications, as previously described, have sets of continuous parameters:

- string coupling gs - parameters of the CY geometry: Kahler (sizes of 2- and 4-cycles) Complex structure (3-cycles) - D-brane positions, etc These parameters are not external, they are vevs of certain scalar fields in the 4d theory, called moduli These fields have no potential, and the vevs are undetermined Having no potential, these scalars are massless and hence a phenomenological disaster for the models Fifth forces, cosmological moduli problem, ... Need mechanisms to fix moduli vevs and give them masses Flux compactifications

Can enrich the ansatz for compactification: Include background not only for metric, but also for some of the p- form fields of the 10d theory For instance, in type IIB there are 2-forms B2, C2 They are generalized gauge potentials, e.g. BMN BMN + ∂ Λ → [M N] Gauge invariant backgrounds involve their field strengths H3, F3 Turn on field strength backgrounds H3, F3 along 3-cycles of CY [Dasgupta, Rajesh, Sethi ‘99; Giddings, Kachru, Polchinski ‘01] H3 X6 F3

M4

Fluxes produce a potential stabilizing many moduli! Flux potential

- Flux energy density depends on the size of the 3-cycles 4d vacuum energy depends on the flux energy density Potential for the (complex structure) moduli zi 1 1 - At large L, scale of moduli masses is 2 3 , below KK scale ! Ms L ! L Effect of fluxes described in 4d effective action, via superpotential i W = (F3 + H3 ) Ω3(zi) [Gukov, Vafa, Witten] CY gs ∧ ! i - Minimum of the potential: requires that G = F + H satisfies 3 3 g 3 (imaginary) self-duality s Fixes gs and 3-cycle sizes

3 3 Ex: T x T H3 x F3 H3 x F3

6d G3 = i G3 G = i G ∗ ∗6d 3 " 3 Supersymmetry breaking

[Graña, Polchinski] - Flux potential can break supersymmetry spontaneously ISD G3 allows a (2,1) component Gijk and a (0,3) form Gijk (2,1) component preserves supersymmetry, while (0,3) breaks it In terms of 4d effective theory, correspond to vevs for auxiliary fields in moduli multiplets - Focus on overall Kahler modulus T (overall CY size L) At large L, Kahler potential K(T, T¯) = 3 log(T + T¯) − F = D W G Ω T T ! 3 ∧ 3 !CY - No-scale supersymmetry breaking, vanishing vacuum energy at minimum (in classical, large L approx) K 2 K 2 V = e ( DW DW 3 W ) = e Fz = 0 for any F · − | | | | T Gravity mediation

- Supersymmetry breaking is transmitted to visible sector [Graña;...] e.g. gauge kinetic function on D7-branes given by T 4 2 α 4 ⇒ Gaugino masses d x d θ T WαW d x FT λλ → - Gravity mediation ! ! All soft terms of MSSM are generated F M 2 11 T s 3 Need to pick Ms=10 GeV so that Msoft = = 10 GeV MP MP ! - μ-term can be generated by supersymmetric component of flux Around similar scale - Flavor problem: Flavour physics decoupled from susy breaking. Yukawa couplings independent of moduli triggering susy breaking Underlies squark mass universality, etc A quite successful and tractable model of susy breaking in string theory The landscape

- Generalizations of previous mechanism can lead to stabilization of all moduli in AdS, dS and Minkowski vacua - Many possible minima, depending on discrete choices of flux

[Bousso, Polchinski; Susskind] [Schellekens]

- Estimate: for a fixed CY with b3 independent 3-cycles and N possible values of flux on each, number of vacua N (b3) For typical values, recover the toy estimate of 10500∼ of NYT - Can identify vacuum realized in Nature? Needle in (huge) haystack Viewpoints on the landscape

- Statistical analysis [Douglas;...] - Anthropic principle [Weinberg] - Cosmological evolution [Linde;...] - ... A personal thought: Landscape is less worrisome if interested in specific aspect e.g. particle physics models: Only geometry around D-branes relevant, global structure irrelevant (analogy: SM also has infinite number of parameters fixing higher dimensional operators, but they are irrelevant!) Possible to identify right string structure leading to correct relevant observables (analogy: physicists identified SM as right theory among infinite landscape of quantum field theories! or gravitational description of Solar system among infinite landscape of solutions of General Relativity) Cosmology

Being a gravitational theory, expect string theory to help understand origin and cosmological evolution of the Universe

Full understanding of time-dependent (rather than static) backgrounds in string theory is lacking Have to rely on approximations using effective field theory Hard to address questions on Big Bang singularities

Still there are UV sensitive cosmological phenomena which can be studied in string theory A prototypical case is inflation: Slow roll in a theory requires good knowledge of inflaton potential, sensitive to UV contributions D-brane inflation

A very intuitive realization of inflation D3-brane at a point in CY moves slowly, attracted by anti D3-brane Inflaton is worldvolume scalar, whose vev fixes D-brane location

At short distance, a tachyon develops, triggering annihilation (hybrid inflation)

Most successful models involve flux backgrounds and warping [Kallosh, Kachru, Linde, McAllister, Maldacena, Trivedi] Still, some amount of fine tuning required Conclusion:What is string theory?

physics? mathematics? theory of everything? theory of nothing? Conclusion:What is string theory?

Theory of something... Theory of many things... Theory of many interesting things

Particle physics Cosmology and inflation Quantum gravity and black holes Strongly coupled gauge theories ...