Supersymmetric field theories Antoine Van Proeyen Antoine Van - PowerPoint PPT Presentation

Supersymmetric field theories Antoine Van Proeyen Antoine Van Proeyen KU Leuven Summer school on Differential Geometry and Supersymmetry, September 10-14, 2012, Hamburg Based on some chapters of the book ‘Supergravity’

Wess, Zumino Supersymmetry and supergravity � supersymmetry δ = ε ψ A x x ( ) ( ) Bosons and fermions ∂ δ ψ = γ µ ε x A x ( ) ( ) in one multiplet µ ∂ x � commutator gives general � commutator gives general coordinate transformations ∂ µ = γ Q Q P δ ε δ ε = ε γ ε µ { , } or [ ( ), ( )] µ ∂ µ x 1 2 2 1 ⇒ gauge theory contains gravity: Supergravity Freedman, van Nieuwenhuizen, Ferrara

1. Scalar field theory and its symmetries: A. Poincaré group Algebra SO(1, D-1) η νρ m [ �σ ] − η �ρ m [ νσ ] [ m [ �ν ] , m [ ρσ ] ] = Space with ( x � ) = ( t, � x ) − η νσ m [ �ρ ] + η �σ m [ νρ ] Metric d s 2 = − d t d t + d � x = d x � η �ν d x ν Act on fields: φ (x)= φ ´ (x ´ ) Act on fields: φ (x)= φ ´ (x ´ ) = − d t d t + d � x � d � x � d � d s x = d x η �ν d x φ ′ ( x ) = U (Λ) φ ( x ) = φ (Λ x ) Isometries (preserve metric) e − 1 x � = Λ �ν x ′ ν + a � 2 λ ρσ L [ ρσ ] U (Λ) ≡ Λ �ρ η �ν Λ νσ = η ρσ ≡ x ρ ∂ σ − x σ ∂ ρ L [ ρσ ] Expand More general if not scalar fields Λ �ν δ �ν + λ �ν + O ( λ 2 ) = J [ ρσ ] = L [ ρσ ] + m [ ρσ ] , � � � 1 2 λ ρσ m [ ρσ ] = e ν ψ ′ i ( x ) = U (Λ , a ) ij ψ j ( x ) � � i �ν δ � ρ η νσ − δ � �ν m [ ρσ ] ≡ σ η ρν = − m [ σρ ] e − 1 2 λ ρσ m [ ρσ ] j ψ j (Λ x + a ) =

B. Other symmetries and currents

Exercises on chapter 1 � Ex 1.5: Show that the action � � � η �ν ∂ � φ i ∂ ν φ i + m 2 φ i φ i � d D x L ( x ) = − 1 d D x S = 2 is invariant under the transformation Λ → φ ′ ( x ) ≡ φ (Λ x ) . → φ ′ i ( x ) ≡ φ i (Λ x ) . φ i ( x ) φ ( x ) − − Important: fields transform, not the integration variables [ L [ �ν ] , L [ ρσ ] ] � Ex.1.6: Compute the commutators and show that they agree with that for matrix generators. Show that to first order in λ ρσ 2 λ ρσ L [ ρσ ] φ i ( x � ) = φ i ( x � + λ �ν x ν ) φ i ( x � ) − 1

2. The Dirac field

Exercise on chapter 2 � Show using the fundamental relation of gamma matrices that [Σ �ν , γ ρ ] = 2 γ [ � η ν ] ρ = γ � η νρ − γ ν η �ρ � Prove the consistency of � Prove the consistency of 2 λ �ν � δ Ψ = − 1 Ψ = 1 2 λ �ν Σ �ν Ψ , δ � ΨΣ �ν � Prove then the invariance of the action � d D x � Ψ[ γ � ∂ � − m ]Ψ( x ) S [ � Ψ , Ψ] = −

3. Clifford algebras and spinors � Determines the properties of - the spinors in the theory - the supersymmetry algebra � We should know � We should know - how large are the smallest spinors in each dimension - what are the reality conditions - which bispinors are (anti)symmetric (can occur in superalgebra)

3.1 The Clifford algebra in general dimension 3.1.1 The generating � matrices Hermiticity γ � † � γ � γ � γ � (hermitian for spacelike ) (hermitian for spacelike ) Hermiticity representations related by conjugacy by unitary S γ ′ � � Sγ � S − �

3.1.2 The complete Clifford algebra γ � 1 ...� r = γ [ � 1 . . . γ � r ] , γ �ν = 1 2 γ � γ ν − 1 2 γ ν γ � e.g. 3.1.3 Levi-Civita symbol ε 012( D − 1) = − 1 ε 012( D − 1) = 1 , 3.1.4 Practical � -gamma matrix manipulation γ � γ � = D , γ �ν γ ν = ( D − 1) γ �

3.1.5 Basis of the algebra for even dimension D = 2 m { Γ A = , γ � , γ � 1 � 2 , γ � 1 � 2 � 3 , � � � , γ � 1 ��� � D } with � 1 < � 2 < ...< � r reverse order list { Γ A = { Γ = , γ � , γ � 2 � 1 , γ � 3 � 2 � 1 , . . . , γ � D ��� � 1 } . } . , γ , γ , γ , . . . , γ Tr(Γ A Γ B ) = 2 m δ A B expansion for any matrix in spinor space M � 1 m A Γ A , M = m A = 2 m Tr( M Γ A ) A

3.1.6 The highest rank Clifford algebra element

3.1.7 Odd spacetime dimension D=2m+1 � matrices dan be constructed in two ways from those in D=2m : γ � 1 ...� r The set with all is overcomplete

3.2 Supersymmetry and symmetry of bi-spinors (intro) � E.g. a supersymmetry on a scalar is a symmetry transformation depending on a spinor ε: � For the algebra we should obtain a GCT � For the algebra we should obtain a GCT ξ ξ µ ξ ξ µ µ µ � Then the GCT parameter should be antisymmetric in the spinor parameters Thus, to see what is possible, we have to know the symmetry properties of bi-spinors

3.2 Spinors in general dimensions 3.2.1 Spinors and spinor bilinears Majorana conjugate � with anticommuting spinors Since symmetries of spinor bilinears are important for supersymmetry, we use the Majorana conjugate to define λ .

3.2.2 Spinor indices NW-SE convention

3.2.4 Reality Complex conjugation can be replaced by charge conjugation, an operation that acts as complex conjugation on scalars, and has a simple action on fermion bilinears. For example, it preserves the order of spinor factors.

3.3 Majorana spinors � A priori a spinor ψ has 2 Int[ D /2] (complex) components � Using e.g. ‘left’ projection P L = (1+ γ * )/2 if D is even (otherwise trivial) ‘Weyl spinors’ P L ψ = ψ � In some dimensions (and signature) there are reality conditions ψ =ψ C = B −1 ψ * ψ =ψ C = B −1 ψ * consistent with Lorentz algebra: ‘Majorana spinors’ � consistency requires t 1 = -1.

Other types of spinors � If t 1 =1: Majorana condition not consistent � Define other reality condition (for an even number of spinors): � ‘Symplectic Majorana spinors’ � In some dimensions Weyl and Majorana can be combined, e.g. reality condition for Weyl spinors: ‘Majorana-Weyl spinors’

Dim Spinor min.# comp Possibilities for susy depend on the properties of irreducible spinors 2 MW 1 in each dimension 3 M 2 4 M 4 � Dependent on signature. 5 S 5 S 8 8 Here: Minkowski Here: Minkowski 6 SW 8 � M: Majorana MW: Majorana-Weyl 7 S 16 S: Symplectic 8 M 16 SW: Symplectic-Weyl 9 M 16 10 MW 16 11 M 32

3.4 Majorana OR Weyl fields in D=4 � Any field theory of a Majorana spinor field Ψ can be rewritten in terms of a Weyl field P L Ψ and its complex conjugate. � Conversely, any theory involving the chiral field � Conversely, any theory involving the chiral field χ =P L χ and its conjugate χ C =P R χ C can be rephrased as a Majorana equation if one defines the Majorana field Ψ =P L χ +P R χ C . � Supersymmetry theories in D=4 are formulated in both descriptions in the physics literature.

Exercise on chapter 3 � Ex. 3.40: Rewrite � S [Ψ] = − 1 d D x � Ψ[ γ � ∂ � − m ]Ψ( x ) 2 as � � � � � ( P L + P R )Ψ � � S [ ψ ] = − 1 d 4 x Ψ γ � ∂ � − m − − 2 � � � � � � d 4 x Ψ γ � ∂ � P L Ψ − 1 Ψ P L Ψ − 1 � 2 m � 2 m � = − Ψ P R Ψ . and prove that the Euler-Lagrange equations are / / ∂P L Ψ = mP R Ψ , ∂P R Ψ = mP L Ψ . Derive � P L,R Ψ = m 2 P L,R Ψ from the equations above

4. The Maxwell and Yang-Mills Gauge Fields 4.1 The Abelian gauge field A � (x)

4.3 Non-abelian gauge symmetry � Simplest: act by matrices and � Gauge fields for any generator � � ∂ � + gt A A A D � Ψ = Ψ cov. derivative: � � ( x ) = 1 g∂ � θ A + θ C ( x ) A B needs transform: δA A � ( x ) f BCA δA � ( x ) = g∂ � θ + θ ( x ) A � ( x ) f BC [ D � , D ν ]Ψ = gF A �ν t A Ψ � Curvatures F A �ν = ∂ � A A ν − ∂ ν A A � + gf BCA A B � A C ν � Typical action � � S [ A A Ψ α , Ψ α ] = d D x − 1 4 F A�ν F A � , � �ν Ψ α ( γ � D � − m )Ψ α � − �

Exercise on chapter 4 � Ex. 4.17: Use the Jacobi identity to show that the C t C and D satisfy [t A ,t B ]= f AB matrices (t A ) D E =f AE therefore give a representation � Ex 4.21: Show that � Ex 4.21: Show that D � F A νρ + D ν F A ρ� + D ρ F A �ν = 0 A is written in the form is satisfied identically if F � ν F A �ν = ∂ � A A ν − ∂ ν A A � + gf BCA A B � A C ν

6. N=1 Global supersymmetry in D=4 � Classical algebra

6.2. SUSY field theories of the chiral multiplet � Transformation under SUSY � Algebra � Simplest action � Potential term

6.2.2 The SUSY algebra • A transformation is a parameter times a generator • Calculating a commutator • Calculating a commutator bosonic

Calculating the algebra � Very simple on Z � On fermions: more difficult; needs Fierz rearrangement � With auxiliary field: algebra satisfied for all field configurations Without auxiliary field: satisfied modulo field equations. Without auxiliary field: satisfied modulo field equations. � auxiliary fields lead to - transformations independent of e.g. the superpotential - algebra universal : ‘closed off-shell’ - useful in determining more general actions - in local SUSY: simplify couplings of ghosts

6.3. SUSY gauge theories 6.3.1 SUSY Yang-Mills vector multiplet gauge)

6.3.2 Chiral multiplets in SUSY gauge theories