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1 Identities for Lorentz Group Representations

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1 Identities for Lorentz Group Representations SUSY Problem Set 2 1 Identities for Lorentz Group Representations We will use undotted and dotted spinor indices (α 2 f1; 2g,_α 2 f1; 2g) as well as vector indices (µ 2 f0; 1; 2; 3g). While the raising and lowering of vector indices is done with the mostly plus µν metric ηµν (or it's inverse η ), the spinor indices are raised and lowered with the epsilon symbols αβ α_ β_ " , "αβ, " , "α_ β_ . Objects with an odd number of spinor indices are Grassmann odd, i.e. they anticommute with other Grassmann odd objects. We pick the convention 12 21 αβ α " = −" = 1 and "12 = −"21 = −1 thus " "βγ = δγ ;::: (1) (i) Show the completeness relation γδ γ δ δ γ "αβ" = −δαδβ + δαδβ (2) The pulling up and down of indices is as follows α αβ β ¯ ¯β_ ¯α_ α_ β_ ¯ = " β ; α = "αβ and α_ = "α_ β_ ; = " β_ : (3) Remember under so(4) the following fields transform as α as (1=2; 0) and ¯α_ as (0; 1=2) : (4) Contracted spinor indices may be suppressed with the convention. The dot on top of the index is really a representation of a balloon filled with helium. The index holds on to the balloon and flies up. Without such a balloon the index falls down, naturally... α α β ¯ ¯α_ α_ β_ ¯ χψ = χ α(= χ "αβ ) andχ ¯ =χ ¯α_ (=χ ¯α_ " _ ) β (5) (0; 0) = Λ2(1=2; 0) and (0; 0) = Λ2(0; 1=2) : Note that it's important to stick to this convention! Show (ii) α α_ χα = −χψ andχ ¯ ¯α_ = −χ¯ ¯ (6) (iii) χψ = χ andχ ¯ ¯ = ¯χ¯ (7) (iv) 2 2 1 2 1 2 = = 2 2 1 = 2 and ¯ = ¯ ¯ = 2 ¯1 ¯2 = 2 ¯ ¯ (8) (v) 1 _ 1 _ α β = − 2"αβ and ¯α_ ¯β = ¯2"α_ β (9) 2 2 1 (vi) (χψ)∗ =χ ¯ ¯ (10) µ We can now introduce the σαα_ constants that help us switch between spinor and vector indices (Remember that (1=2; 0) ⊗ (0; 1=2) = (1=2; 1=2) ). ! ! ! ! −1 0 0 1 0 −i 1 0 σ0 = ; σ1 = ; σ2 = ; σ3 = : (11) 0 −1 1 0 i 0 0 −1 And their conjugate _ σ¯µαα_ = "α_ β"αβσµ : (12) ββ_ Show the completeness relations (vii) _ T r(σµσ¯ν) = −2ηµν and σµ σ¯ββ = −2δαδα_ : (13) αα_ µ β β_ We define µ Vαα_ = σαα_ Vµ : (14) Note, we are only changing the description, not the representation. The sigmas are just a bunch of constants after all (remember intertwiners?). Show that (viii) −1 V µ = σ¯µαα_ V (15) 2 αα_ We can use sigma matrices to compute tensor products of spinors. (ix) Show ( σµχ¯)∗ = χσµ ¯ and σµχ¯ = −χ¯σ¯µ : (16) What is the relevant tensor product? (x) Show ( σµσ¯νχ)∗ =χ ¯σ¯νσµ ¯ and σµσ¯νχ = χσνσ¯µ : (17) What is the relevant tensor product? (xi) Write the metric tensor gµν in spinor notation (xii) Write the 4-dimensional epsilon µνσρ tensor in spinor notation 1 ρσ (xiii) Defining (∗F )µν = 2 µνρσF , show that ± Fµν = Fµν ± i(∗F )µν ; (18) and express both parts in terms of E and B. (xiv) Show that 1 σµνβ := (σµ σ¯ναβ_ − σν σ¯µαβ_ ) (19) α 4 αα_ αα_ are generators of the Lorentz group acting on the (1=2; 0) representation. 2 (xv) Show that σµν = 0 : (20) Prove the Fierz identities (xvi) For 4 spinors 1; : : : ; 4 1 ( )( ¯ ¯ ) = ( σµ ¯ )( ¯ σ¯ ) (21) 1 2 3 4 2 1 4 3 µ 2 (xvii) 1 ( σµ ¯)( σν ¯) = − ηµν( )( ¯ ¯) : (22) 2 For 4 spinors 1; : : : ; 4 show (xviii) ( 1 2)( 3 4) + ( 1 3)( 2 4) + ( 1 4)( 2 3) = 0 : (23) (xix) µ µ µ ( 1 2)( 3σ ¯4) + ( 1 3)( 2σ ¯4) + ( 1σ ¯4)( 2 3) = 0 : (24) (xx) µ ν µ ν ν µ ( 1σ ¯2)( 3σ ¯4) − ( 1 3)( ¯2σ¯ σ ¯4) − ( 1σ ¯4)( ¯2σ¯ 3) = 0: (25) Show (xxi) µ ν ν µ α µν α (σ σ¯ + σ σ¯ )β = −2η δβ (26) (xxii) σµσ¯ν = −ηµν + 2σµν : (27) (xxiii) We know that (1=2; 0) ⊗ (1=2; 0) = (0; 0) ⊕ (1; 0). How can you decompose αχβ = ::: + ::: ? (28) (xxiv) Extra credit: Find any typos and report them! 2 Supermultiplets (i) Write down the content of massless multiplets with highest helicity 1=2 and 1. (ii) Write down the content of massive multiplets with highest spin 1=2 and 1. (iii) Count the number of bosonic versus fermionic degrees of freedom for each multiplet above and confirm that they match. (iv) Compare the degrees of freedom of the massless and the massive multiplets. How does this connect to the Higgs mechanism? 3 3 Superspace and Superfields for 4d N = 1 µ Superspace for 4d N = 1 has coordinates (x ; θα; θ¯α_ ). The SUSY generators Qα and Q¯α_ can be expressed as differential operators on superspace ¯α_ @ µ ¯α_ Qα = + @α − iθ @αα_ = + α − iσαα_ θ @µ @θ (29) @ Q¯ = − @¯ + iθα@ = − + iθασµ @ α_ α_ αα_ @θ¯α_ αα_ µ (i) Check that fQα;Qβg = 0 (30) and fQα; Q¯α_ g = 2Pαα_ : (31) The covariant derivatives on superspace are defined as µ α_ Dα = + @α + iσ θ¯ @µ αα_ (32) ¯ ¯ α µ Dα_ = − @α_ − iθ σαα_ @µ α µ µ µ (ii) Show that D¯α_ annihilates both θ and y = x + iθσ θ¯. A superfield C is called chiral if D¯α_ C = 0 : (33) (iii) Show that any chiral field may be written as p µ α α_ 2 µ i 2 µ 1 2 2 µ C(x ; θ ; θ¯ ) = φ(x)+ 2θ (x)+θ F (x)+iθσ¯ θ@¯ µφ(x)−p θ @µ (x)σ θ¯+ θ θ¯ @ @µφ(x) 2 4 (34) (iv) Show that if C is chiral, then QαC and Q¯α_ C are chiral. (35) (v) Count the off-shell degrees of freedom of the chiral superfield. Do fermionic and bosonic degrees of freedom cancel? How about on-shell? (vi) What is an anti-chiral superfield? Show that C¯ is antichiral. There is a second kind of common superfield. The vector superfield. A superfield V is called a vector superfield if V¯ = V (36) (vii) Write the most general vector superfield. (viii) Count the off shell degrees of freedom of the vector superfield. Do fermionic and bosonic degrees of freedom cancel? How about on-shell? 4 Let's turn to integration on superspace. Remember that we're dealing with Berezin integration (i.e. differentiation really). (ix) Show that Z Z d4θK(Ci; C¯¯i;X) = d2θd2θK¯ (Ci; C¯¯i;X) ; (37) where Ci are chiral and X is a collection of arbitary superfields, is invariant under Susy transformations. You can show that Q acting on a superfield has a θ2θ¯2 term which is a total derivative @(::: ). (x) Show that Z d4θf(C) = @(::: ) : (38) 4 K¨ahlerPotentials, optional, you can hand it in K¨ahlerpotentials are objects from differential geometry. If you open e.g. Nakahara you will find a chapter on K¨ahlergeometry. Let's consider a complex, Riemannian, symplectic manifold M with metric g, complex structure I and symplectic form !. The manifold is said to be K¨ahlerif the following relation holds g(I(X);Y ) = !(X; Y ) for X; Y 2 Γ(TM) : (39) K¨ahlermanifolds have nice properties. Locally (i.e. on an open patch U) The complex structure allows us to introduce holomorphic coordinates φi and φ¯¯i. We will from now on work on this coordinate chart in complex coordinates. Partial derivatives are @ @ @i = and @¯ = (40) @φi i @φ¯¯i It can be shown that the metric can be expressed as ¯ ¯ gi¯i(φ, φ) = @i@¯iK(φ, φ) : (41) This is also called the K¨ahlermetric. The symplectic form (K¨ahlerform) can also be expressed in terms of derivatives of the K¨ahlerpotential. Show that K is not unique in the sense that gi¯i is invariant under K(φ, φ¯) 7! K(φ, φ¯) + (f(φ) + complex conjugate) : (42) The Christoffel symbols are i ¯li ¯¯i ¯li Γjk = g @kgj¯l and Γ¯jk¯ = g @k¯gl¯j ; (43) and the Riemann tensor is m m¯ Ri¯jk¯l = @k@¯lgi¯j − Γikgmm¯ Γ¯j¯l : (44) 5 Let's turn back to Supersymmetry! Interpreting the first components φi and φ¯i of our chiral superfields Ci and their aintichiral conjugates C¯¯i as coordinates on a manifold (also called the target space), show that we can express the supersymmetric Lagrangian Z 4 i ¯i i i 1 i ¯i ¯j k¯ 1 ¯i i i k L = d θK(C ; C¯ ) =g ¯F F¯ − F g ¯Γ ¯ ¯ − F¯ g ¯Γ ii 2 ıi ¯jk¯ 2 ii jk (45) i µ ¯i ¯i µ i 1 i k ¯j ¯l − g ¯@ φ @ φ¯ − ig ¯ ¯ σ¯ D + (@ @¯g ¯) ¯ ¯ ; ii µ ii µ 4 k l ij where i i i k j Dµ = @µ + Γjk@µφ : (46) Vary F¯ and find the equations of motion 1 F i = Γi j k ; (47) 2 jk assuming that gi¯i is invertible. By substituting the e.o.m. back into the Lagrangian, show that i µ ¯i ¯i µ i 1 i k ¯j ¯l L = −g ¯@ φ @ φ¯ − ig ¯ ¯ σ¯ D + R ¯ ¯ ¯ ¯ : (48) ii µ ii µ 4 ijkl What is the interpretation of the 's on the target space side? 6.
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