Supergravity Gravity: on-shell Einstein gravity gauge theory of - PowerPoint PPT Presentation

Supergravity

Gravity: on-shell Einstein gravity ↔ gauge theory of local Lorentz/translation symmetry generators M ab , P a ↔ “gauge fields” ω ab µ , spin connection, e a µ vierbein where a, b = 0 , . . . , 3 are Lorentz gauge group indices µ, ν = 0 , . . . , 3 are spacetime indices e a µ and ω ab µ transform as collections of vectors gauge fields ↔ feld strengths R ab µν , (Riemann curvature), C a µν , (torsion) C a µν = 0, solve for ω ab µ in terms of e a µ counting: e a µ 16 components subtract 4 (equations of motion) subtract 4 (local translation invariance) subtract 6 (local Lorentz invariance) leaves 2 degrees of freedom: massless spin-2 particle

Gravity: on-shell Couplings to matter: ∇ µ = ∂ µ − e a µ P a − ω ab µ M ab feld strengths can be obtained from ∇ µ ∇ ν − ∇ ν ∇ µ Writing e = | det e m µ | invariant action with only two derivatives is linear in the field strength: � � S GR = M 2 ρλ = M 2 d 4 x e ǫ µνρλ ǫ abcd e a µ e b ν R cd d 4 x e R Pl Pl 2 2 where R is the curvature scalar

Supergravity: on-shell e a µ ↔ helicity 2 particle, N = 1 SUSY requires helicity 3 / 2 ψ α ν (gravitino) on-shell each has two degrees of freedom gravitino is gauge field ↔ Q α ↔ field strength D µνα C a µν = 0, solve for ω ab µ find on-shell supergravity action: � � S = M 2 d 4 x ǫ µνρσ ψ µ γ 5 γ ν D ρσ d 4 x e R + i Pl 2 4 call second term S gravitino metric: g µν = e a µ e b ν η ab in terms of a local inertial coordinate system ξ a at the point X µ ( X ) = ∂ξ a e a ∂x µ

Brans–Dicke Gravity first consider toy example, scale-invariant Brans–Dicke theory: � � e � 2 σ 2 R + e d 4 x 12 ∂ µ σ∂ µ σ S BD = treat scalar σ as a spurion field and set σ = M Pl break local conformal invariance to local Poincar´ e invariance ⇒ Einstein gravity

Superconformal Gravity in addition to the “gauge“ fields e a µ and ψ να we have A µ ↔ local U (1) R symmetry, and b µ ↔ local conformal boosts Counting degrees of freedom off-shell (subtracting gauge invariances): field d.o.f. e a µ : 16 − 4 − 6 − 1 = 5 ψ α ν : 16 − 4 − 4 = 8 A µ : 4 − 1 = 3 − 4 b µ : 4 = 0 e a µ subtract 4 (translation), 6 (Lorentz), 1 (dilations) ψ α ν subtract 4 (SUSY Q α , Q ˙ α ), 4 (conformal SUSY S β and S ˙ β ) A µ subtract 1(local R -symmetry) b µ subtract 4 (four conformal boost generators) no auxiliary fields for the superconformal graviton multiplet “gauge” fields, couple with gauge covariant derivatives

Supergravity: off-shell spurion chiral superfield to break the conformal symmetry: Σ = ( σ, χ, F Σ ) in global N = 1, Σ is a chiral superfield here it contains part of the off-shell graviton superfield Σ called conformal compensator assign conformal weight 1 to the lowest component of Σ ( x µ and θ have conformal weight − 1 and − 1 / 2) full superconformal gravity action is � � d 4 θ Σ † Σ + S gravitino d 4 x e 2 σ ∗ σR + e S scg = derivatives are covariant in “gauge“ fields ( e a µ , ψ να , A µ , b µ ) a superconformal Brans–Dicke theory

Supergravity: off-shell Treat σ , χ , and b µ as spurion fields σ = M Pl , χ = 0 , b µ = 0 local superconformal invariance → local super-Poincar´ e invariance resulting action is: � A µ A µ � � M 2 Σ − 2 M 2 2 R + F Σ F † d 4 x e S sg = + S gravitino Pl Pl 9 F Σ and A µ are auxiliary fields, counting: field d.o.f. e a µ : 16 − 4 − 6 = 6 ψ α − 4 ν : 16 = 12 A µ : 4 = 4 F Σ : 2 = 2 6 bosonic degrees of freedom from F Σ and A µ are just what is required to have N = 1 SUSY manifest off-shell

Superspace α ) ˙ eight-dimensional space z M = ( x µ , θ α , θ require super-general coordinate invariance z M → z ′ M = z M + ξ M where ξ M ( z M ) Superspace scalars transform φ ′ ( z ′ ) = φ ( z ) while fields with a superspace index ∂φ ψ M = ∂z M transform as M ( z ′ ) = ∂z N ψ ′ ∂z ′ M ψ N ( z )

Superspace construct a vielbein E A M relates the superspace world coordinate to a locally Lorentz covariant (tangent space) coordinate contains the off-shell multiplet ( e a µ , ψ να , A µ , F Σ ) we can choose a coordinate system where, for θ = 0, E a µ = e a µ , E α µ = 1 2 ψ α µ , E ˙ µ = 1 α 2 ψ ˙ α µ

Coupling to matter arbitrary global SUSY theory: � � � � d 4 θK (Φ † , e V Φ) + iτ d 2 θ 16 π W α W α L gl = W (Φ) − + h.c. define conformal weight 0 fields and mass parameters by Φ ′ = Σ Φ m ′ = Σ m dropping the primes, local superconformal-Poincar´ e invariant Lagrangian: � � � d 4 θf (Φ † , e V Φ) Σ † Σ d 2 θ Σ 3 d 2 θ iτ 16 π W α W α + h.c. L = Pl + Pl W (Φ) − M 2 M 3 Σ − 2 M 2 A µ A µ + L gravitino 6 f ( φ † , φ ) σ ∗ σR + F Σ F † − 1 Pl 9 action: � d 4 x e L S =

Coupling to matter M Pl → ∞ (global SUSY) limit, choose Pl e − K (Φ † ,e V Φ) / 3 M 2 f (Φ † , e V Φ) = − 3 M 2 Pl rescaling the vierbein by a Weyl (local scale) transformation µ → e − K/ 12 M 2 e a Pl e a µ one finds bosonic piece of the action: � � M 2 d 4 x e 2 R + K i j ( φ † , φ )( ∇ µ φ i ) † ∇ µ φ j S B = Pl � 16 π ( F µν F µν + iF µν � iτ −V ( φ † , φ ) + F µν ) + h.c. where K i and K i j (the K¨ ahler metric) are given by ∂ 2 K K i ( φ † , φ ) = ∂K ∂φ i , K i j ( φ † , φ ) = ∂φ j † ∂φ i

Coupling to matter scalar potential: �� � � � � � K − 1 � j j + W ∗ K j W i + W K i − 3 | W | 2 e K/M 2 V ( φ † , φ ) W ∗ = Pl M 2 M 2 M 2 i � � 2 Pl Pl Pl + g 2 K i T a φ i 2 last term is just the D -term potential in supergravity the energy density can be negative usually tune tree-level vacuum energy to zero by adding the appropriate constant to W

Coupling to matter auxiliary components of chiral superfields (no fermion bilinear VEVs): � � Pl � K − 1 � j j + W ∗ K j F i = − e K/ 2 M 2 W ∗ ( ∗ ) M 2 i Pl from fermionic piece of Lagrangian, ∇ µ � φ i contains a gravitino term µ Q α � 1 1 M Pl ψ α M Pl ψ α µ F i + O ( σ µ ∂ µ φ i ) φ i = so the K¨ ahler function contains a term: j M Pl ¯ θ � θ 2 ψ µ F i σ µ ¯ iK i 1 φ θ j in analogy to the ordinary Higgs mechanism, that the gravitino eats the goldstino if there is a nonvanishing F component in flat spacetime, goldstino adds right number of degrees of freedom to make a massive spin 3/2 particle

Gravitino Mass in flat spacetime F ∗ j K i j F i m 2 3 / 2 = 3 M 2 Pl use (*) and V = 0 ⇒ Pl | W | 2 3 / 2 = e K/M 2 m 2 M 4 Pl taking a canonical K¨ ahler function K = Z Φ i † Φ i and M Pl → ∞ reproduces usual global SUSY results

Maximal Supergravity massless supermultiplet with helicities ≤ 2 SUSY charges change the helicity by 1 2 ⇒ N ≤ 8 arbitrary dimension cannot have more than 32 = 8 × 4 real SUSY charges maximal dimension: spinor in 11 dimensions has 32 components supergravity theory must have e a µ and ψ α µ massless gauge fields D dimensions: “little“ group SO ( D − 2) graviton: symmetric tensor of SO ( D − 2) has ( D − 1)( D − 2) / 2 − 1 dof 44 dof for D = 11 gravitino is a vector-spinor and a vector has D − 2 dof spinor of SO ( D ) has d S components, where d S = 2 ( D − 2) / 2 (for D even) , d S = 2 ( D − 1) / 2 (for D odd)

11 dimensions Majorana spinor has d S = 32 real components, 16 dof on-shell tracelessness condition Γ µ ψ α µ = 0 leaves ( D − 3) d S / 2 degrees of freedom for the vector-spinor gravitino has 128 real on-shell dof gravitino - gaviton = 84 more fermionic dof than bosonic difference made up by three index antisymmetric tensor A µνρ antisymmetric tensor with p indices (i.e. rank p ) has 1 p ! ( D − 2) . . . ( D − p − 1) dof on-shell, also called a p -form field (11 − 2)(11 − 3)(11 − 4) = 3 · 4 · 7 = 84 6

11 dimensions: BPS solitons The SUSY algebra of 11-D supergravity has two central charges two Lorentz indices, five Lorentz indices ↔ BPS solitons central charge acts as a topological charge, spatial integral at fixed t preserve index structure, solitons extend in two and five spatial directions called p -branes for p spatial directions e.g. monopole is a 0-brane, couples to a 1-form gauge field A µ p -brane couples to a ( p + 1)-form gauge field 2-brane couples to 3-form gauge field A µνρ a p -form gauge field has a ( D − p − 2)-form dual gauge field field strength ↔ A µνρ is a 4-form: F µνρλ contract with ǫ tensor gives dual 7-form ↔ 6-form dual gauge field couples to the 5-brane

10 dimensions compactify 1 dimension on a circle decompose D = 11 fields into massless D = 10 fields (constant on circle) e a e a µ (44) → µ (35) , B µ (8) , σ (1) → A µνρ (84) A µνρ (56) , A µν (28) (56) , λ + α (8) , λ − α (8) ψ α ψ + α (56) , ψ − α → µ (128) µ µ 32 supercharges of D = 11 → two D = 10 spinors spinors have opposite chirality gravitino splits into states of opposite chirality, labeled by + and − this is Type IIA supergravity two other supergravity theories in D = 10 Type I: single spinor of supercharges Type IIB: supercharges are two spinors with the same chirality

Low-Energy Effective Theories Type IIA ↔ Type IIA string theory Type IIB ↔ Type IIB string theory Type I with E 8 × E 8 or SO (32) ↔ heterotic string theory D = 11 supergravity ↔ M-theory