Week 16: 1 The following books and articles are useful for this discussion:

J. Wess and J Bagger “Supersymmetry and ”, chapters 1–7. • N. Seiberg,“The power of holomorphy: exact results in 4D SUSY field • theories” http://arXiv.org/abs/hep-th/9408013

1 Superspace

It is usual to consider field operators to be labeled by positions

φ(x) (1)

The commutation relations with the momentum operators are given exactly by [Pµ,φ(x)] = i∂µφ(x) (2) One can use these commutation relations repeatedly, to show that

φ(x) = exp( iP xµ)φ(0) exp(iP xµ) (3) − µ µ where expanding on the right hand side one clearly gets a Taylor series for φ(x). It is natural then to argue that space is obtained by doing group opera- tions starting from zero. Also, a natural basis of local operators associated to φ(0) are all the multi-derivatives of φ(0). The field φ(x) can be interpreted formally as a generating series for these multiderivative fields, via the usual Taylor expansion. The multiderivative fields are obtained from φ(0) by doing repeated com- mutators with P and can be argued to be a representation of the (trivial) Lie algebra of translations on operators. The fact that the P commute amongst themselves becomes the ‘obvious statement’ that partial derivatives com- mute. Also, the P act as derivations on the algebra of operators. This is

[P, 1(0) 2(0)] = [P, 1(0)] 2(0) + 1(0)[P, 2(0)] (4) O O O O O O 1 c 2009

1 which is the familiar rule for derivatives on fields. This is, we take

i∂ ( 1(x) 2(x)) 0 = i(∂ 1(x)) 2(x)+ i 1(x)∂ 2(x) =0 (5) µ O O | µO O O µO |x So, if we want to find the Taylor series of 1 2, we have to take the two generating series, and do the following operationO O

( 1 2)(z) = dxdy 1(x) 2(y)δ(x z)δ(y z) (6) O O Z O O − − dp dq = dxdy 1(x) 2(y) exp(ip(x z)+ iq(y z))(7) Z (2π)d (2π)d Z O O − − dp dq = ˜1(p) ˜2(q) exp( i(p + q)z) (8) Z (2π)d (2π)d O O − This just states that the Taylor series of a product is the product of the Taylor series (which is why we set x = z,y = z), and that one can obtain that by trivially multiplying fields. For example, changing equation 8 can lead to the notion of noncommutative space, where the labels x have a non-trivial algebra among themselves. The idea of superspace is similar. We notice that we have both P, Q, and that they give a generalization of a Lie algebra. Thus, we can also make commutation relations like equation 2, to obtain (for example)

[Qα,φ(0)] = ψα(0) (9) and then we have Q , ψ (0) = ǫ F (0) (10) { β α } αβ etc. The right hand side of equation 10 is obtained from noticing that the repeated commutation is antisymmetric due to the the generalized Jacobi identity

Q , [Q ,φ] = [ Q , Q ,φ] Q , [Q ,φ] = Q , [Q ,φ] (11) { α β } { α β} −{ β α } −{ β α } This way we see that repeated Q commutators vanish beyond this point, because they would need to be antisymmetric in labels (and there are no more than two spinor labels). We can similarly define the commutations with Q¯. However, we get an interesting relation when we try to do the following operation

Q¯ ˙ , ψ = [ Q¯ ˙ , Q ,φ] Q , [Q¯ ˙ ,φ] =2i∂ φ(0) Q , [Q¯ ˙ ,φ] (12) { α α} { α α} −{ α α } µ −{ α α } 2 If it weren’t for the derivative of φ appearing on the right hand side, we would obtain generally that the spinor indices from doing Q commutators would always be ’anticommuting’. However, on the right hand side, we get derivatives of φ again. So if we don’t count those as new fields, we get that the application of commutations with supersymmetry on any field give a finite number of associated fields. These objects can be assembled into one object (a field representation of supersymmetry). The fields ψ,φ,F would be the component fields of such representations. The idea of superfields and superspace is the same as that of a Taylor series in the variables x. We need one fermionic coordinate per Q, Q¯. Because the Q are fermionic, in order to get the statistics of the expansion parameters right, these Taylor expansion coordinates should be anticommuting. Thus, we would write

α ¯α˙ 1 a φ(x, θ, θ¯)= φ(x)+ θ ψ˜ (x)+ θ¯˙ ψ˜ + θ θ F˜(x)+ ... (13) α α 2 a The commutation with P is the usual Taylor series. Here we see that P is represented as a vector field on the parametrized by x. Similar, Qα should be some generalization of a vector field so that it acts on superfields as an obvious derivation. The most general such object would be a linear object in derivatives with respect to x and θ By comparing with above, we find that ∂ Q + ... (14) α ∼ ∂θα The commutation relation that needs to be solved is given by

µ Q , Q¯ ˙ =2iσ ∂ (15) { α β} αβ˙ µ By dimensional analysis, Q is of dimension one half, so we should get some- thing of the right dimension. It should also be translationally invariant ([P, Q] = 0 after all). So it can be linear in ∂x, but it needs to fix the dimensionalilty. We should also have the hermiticity condition Q† = Q¯ in a natural way. Thus, we find that for Lorentz invariance to make sense

∂ ˙ Q = kσµ θ¯β(i∂ ) (16) α αβ˙ µ ∂θα −

3 The complex conjugate gives us

∂ ∗ µ ∗ β Q¯ ˙ = k (σ ) θ ( i∂ ) (17) α ∂θα˙ − αβ˙ − µ which requires complex conjugating the σ matrices. Also, it is natural to have the Q have the conjugate structure to θ¯α˙ , so we need to raise and lower some indices. A complete solution occurs with kk∗ = 1, and choosing a rephasing, we can make k = 1. Finally, we have that

β˙ α µ α˙ β˙ Q¯ = ∂¯ iθ σ ǫ ∂ (18) θα˙ − αα˙ µ A straightforward computation (remembering how signs are supposed to act) gives the required relations between the Q and P . The extended set of coordinates x, θ, θ¯ is called superspace. We can obtain the component superfields of a multiplet by taking Q derivatives and setting fermionic coordinates equal to zero. In this way, taking products of superfields gives new superfields. Also, the Q can be exponentiated to generate some group translations on superspace. The parameters are Grassmanian. Indeed, Superspace can be thought of as the group super-manifold associ- ated the the supersymmetry Lie algebra. The action of Q is by left multipli- cation. The Q are a left covariant vector ’fields’ on the . Since the Q do not commute with each other, the corresponding supermanifold can not be considered to be flat. It turns out that the manifold has torsion. However, because it is a group manifold there are left invariant vector fields that commute with Q: these are the vector fields that generate group operations by right multiplication. As usual, left and right multiplications commute with each other. We call these objects Dα, D¯ α˙ . These right invari- ant vector fields have the same commutation relations than the Q. However, we have that Q, D = Q, D¯ =0 (19) { } { } They are the same as the Q, but we reverse the sign of k everywhere. The D are covariant derivatives with respect to supersymmetry. They allow us to differentiate superfields and obtain new superfields. This is similar to the situation familiar in gravity on a curved manifold, where we need to covariantize derivatives in order to have derivative operations that take to tensors. Having these objects is very convenient, because they make all the algebra essentially trivial.

4 The torsion is manifested in the calculation

D , D¯ ˙ =2i∂ ˙ (20) { α β} αβ This is of the form c [Da,Db]= TabDc + Rab (21)

The T is called the torsion tensor, and Rab is the familiar curvature tensor. We see that superspace has torsion but not curvature. After all, we started with a scalar, which is usually trivial when we con- sider . However, if the (lowest) component φ(0) was a spinor, we would not know immediate how to get the full multiplet structure. Especially if we consider dimensional analysis, as derivatives of φ could appear in the Q transformation of ψ. Also, the presence of D, D¯ permits us to define smaller representations of supersymmetry rather than full superfields. Because D, D¯ commute with Q, any field equation defined with these objects and in terms of superfields is compatible with supersymmetry (the Q variation of a field automatically satisfies the equation of motion). Definition: A chiral superfield is a superfield such that

Dφ¯ (x, θ, θ¯)=0 (22)

If there was no ∂x in the definition of D, this would mean that φ would be independent of θ¯, and an arbitrary function of x, θ. Clearly θ satisfies this equation. Consider the variable

+ µ α µ ¯α˙ y = x + iθ σα,α˙ θ (23) which is homogeneous of the right dimension. It is easy to show that Dy+ = 0. If we change variables from x to y+, θ, θ¯, we find that

D¯ ∂¯ (24) ∼ θ so that the first intuition is correct in this shifted variables. Thus, a chiral superfield is a general function of y+, θ.

+ + α + 2 + φ(x, θ, θ¯)= φ(y , θ)= φ(y )+ √2θ ψα(y )+ θ F (y ) (25)

5 remember that the θ are anticommuting variables. Here we are introducing standard factors of √2. Since ψ is a complex field, φ and F are necessarily complex. It is obvious that products of chiral superfields are chiral superfields. This is because D¯ is a derivation, and the condition of being chiral is stating that a superfield is covariantly constant with respect to some superderivatives. Also, in this case, we have that

Q = ∂θ (26) and that µ Q¯ ∂¯ +2iθσ ∂ µ (27) ∼ θ y Thus, we find that + √ + β + δQα φ(y , θ) = 2ψα(y )+2ǫαβθ F (y ) (28) = [Q, φ(y+, θ)] (29)

+ we find this way that δQF (y ) = 0. Also, from Q¯, we find that

+ µ + δ ¯ φ(y , θ)= 2iθσ ∂ φ(y , θ) (30) Qα˙ − µ Again, we notice that F transforms into a total derivative under Q, Q¯. Thus, we have that d4xF (y+)= d4xF (x) (31) Z Z for any chiral superfield transforms into a total derivative under application of Q. The anitchiral ssuperfields are defined by complex conjugation. They satisfy Dφ¯ = 0. Similarly, for a a general superfield, the top component (the coefficient of θ¯2θ2) transforms into a total derivative. This becomes evident because Q an Q¯ either remove a θ, θ¯, or they add a θ, θ¯ together with a derivative. A quick computation shows that the standard kinetic term for φ, ψ can be obtained from the superfield expansion

+ − µ µ ∗ φ(y , θ)φ¯(y , θ¯) 2 ¯2 ∂ φ(x)∂ φ¯(x)+ iψσ¯ ∂ ψ F (x)F (x) (32) |θ θ ∼ µ µ − We expand for y± in Taylor series around x. We use the fact that θ3 = θ¯3 =0 everywhere (this is in full detail in the book by Wess and Bagger). The right hand side has been integrated by parts.

6 Notice that D¯ 3 = 0, so that D¯ 2φ¯ is a chiral superfield also. Thus, the kinetic term action is also given by

+ 2 − φ(y , θ)D¯ φ¯(y , θ¯) 2 (33) |θ

For fermionic variables we have that dθ ∂θ. The top components we have used can be written neatly as a superspaceR ∼ integral as well:

S = d4x d2θd2θ¯φφ¯ + d4xd2θW (φ)+ d2barθW¯ (φ¯) (34) Z Z Z Z here, W is any function of the fields φ (we are grouping them all in one for simplifying notation). Because φ is complex, this function is necessarily holomorphic (depends only on φ, but not on φ¯. Reality of the action requires that W¯ (φ¯) = [W (φ)]∗. The (usually poly- nomial) function W is called the superpotential. The most general action with up to two derivatives can be written easily

S = d4x d2θd2θK¯ (φ,φ¯ )+ d4xd2θW (φ)+ d2barθW¯ (φ¯) (35) Z Z Z Z The function K is called a K¨ahler potential. If we modify K by K K + f(φ)+ f¯(φ¯) the action does not change at all. This is because the θ2→θ¯2 component of a chiral superfield is always a total derivative. The metric in the φ, φ¯ coordinates becomes

gi¯j = ∂φi ∂φ¯j K (36) with all other components vanishing. So the kinetic term in the action looks like 4 i µ j d xg ¯∂ φ ∂¯ φ¯ + ... (37) Z ij µ Such a structure has a natural place in the theory of complex geometry (geometry that has a metric compatible with a complex structure). Such a situation has natural holomorphic coordinates (the φ). If we do holomorphic changes of variables the structure is preserved. The chiral superfields furnish the basic building blocks for matter fields (spin zero and one half). The F fields do not have a kinetic term. They are auxiliary (they act as lagrange multipliers). Because of that they can always be integrated out.

7 The equations of motion can be written nicely in superspace, and they follow from writing the kinetic action in the form

S = d4y+d2θD¯ 2(K)+ d4y+d2θW (φ)+ d4y+d2θW¯ (φ¯) (38) kin Z Z Z

Now φ is an arbitrary function of θ,y+, so it can be varied freely (independent of φ¯. For the simplest kinetic term K = φiφ¯i, we have that

2 ¯i D¯ φ + ∂φi W =0 (39)

1.1 Vector superfields The conceptually most economical way to introduce vector fields so that it is ¯ easy to make sense of them is to consider generalizing derivatives Dα, Dβ˙ , ∂µ to covariant versions of themselves with a connection. Call these α, ¯α˙ , µ. It is naturalD toD haveD supercorvariantly chiral matter. This is, that the set of equations ¯α˙ φ = 0 can be consistently solved. This would imply that ¯ 2 D D φ = Fα˙ β˙ φ = 0, this is, the curvature in the antichiral directions should vanish. Similar with chiral derivatives. Furthermore, we can require that the torsion be the same as before, with no extra curvature. This is, we can reduce the number of fields if we ask that

2i ˙ = , ¯ ˙ (40) Dαβ {Dα Dβ}

Usually, solving [Dµ,Dν] = 0 can be done by saying that the gauge con- nection is a gauge transformation of the null connection. Similarly, we can use a gaege transformation to set

¯Dα˙ = D¯ α˙ (41)

This way, we have that

= exp( V )D exp(V ) (42) Dα − α the chiral derivatives are also a gauge transform of the identity. However, it is a different gauge transform, with a superfield V that is a compensator for the choice that makes ¯ = D¯. from here, one can compute Dµ straightforwardly in terms of V . D

8 Notice that the gauge choice is preserved if we make a chiral gauge trans- formation exp( U)D¯ ˙ exp(U)= D¯ ˙ (43) − α α if U is a general chiral superfield. This would modify 42, so that

= exp( U) exp( V )D exp(V ) exp(U) (44) Dα − − α But we can also use the expression

= exp( U) exp( V ) exp( U¯)D exp(U¯) exp(V ) exp(U) (45) Dα − − − α where U¯ is a general antichiral superfield. Thus, the general gauge transformations compatible with this structure are reduced chiral gauge transforms

exp(V ) exp(U¯) exp(V ) exp(U) (46) → where U is chiral. A natural constraint for hemiticity would have V real, so that U ∗ = U¯. Given , it is easy to build a chiral superfield out of V . Take D D¯ 2( )= W [D¯ , ] (47) Dα α ≃ α Dµ This object has no derivatives acting on it and represents a general curvature of the field compatible with having set , ¯ ˙ . Being a curvature, Dµ ∼ {Dα Dα} Wα is gauge covariant, and transforms as

W exp( U)W exp(U) (48) α → − α Let us simplify the case to a U(1) action. Then V V + U + U¯, with U, U¯ chiral and antichiral. We can check that UU¯ can cancel→ the terms with components of order 1, θ, θ,¯ θ2, θ¯2 (we call this the Wess-Zumino gauge). The simplest component not eliminated has a θθ¯

α α˙ θ θ¯ Aαα˙ (49) which is a vector field. Also, for general chiral transformations it will trans- form as expected for the gauge connection. Thus, we have a superfield with (leftover) components linear in θθ¯, θ2θ¯, 2 2 2 θ¯ θ and θ¯ θ . We call these Aµ, λ,¯ λ, D. One can easily compute Wα in the

9 Wess-Zumino gauge, finding that schematically W λ + θ(F + D)+ θ2(∂λ¯). For this one just matches dimensions and Lorentz indices.∼ The standard kinetic terms for A,λ arise from 1 1 d2θ W W α + d2θ¯ W W α (50) Z 4g2 α Z 4¯ α In general g should be chiral to make sense of this expression. Just like the F were auxiliary, the D term is also auxiliary. For the case of U(1), we have also that the following is invariant

d4ξθV ξD (51) Z ∼ A term like this in the action is called the Fayet-Illiopoulos term (FI-term)

1.2 Coupling to matter: Ask that matter be made of chiral superfields and that is transforms so that the are all covariant. This is they should take group tensors to group tensorsD φ exp( t U a)φ (52) → − a so that φ exp( U T a) φ (53) DA → − a DA Obviously, to write the Kahler potential, we have that chiral superfields and anti-chiral superfields transform with U and U¯. Fortunately V can compen- sate for this, so that we use

d4θφ¯ exp(V T a)φ (54) Z a

This gives the general kinetic term (we can write it more generally for an arbitrary Kahler potential, replacing φ by exp(V )φ everywhere in some standard expression).

2 General guidelines to build theories:

1. Choose gauge group (eg. SU(3) SU(2) U(1)). × ×

10 2. Introduce the corresponding vector superfields (these contain the gauge connections, and by quantization, the ordinary photons, gluons, etc. The supersymmetry implies a Weyl partner λ for each, called the photi- nos, gauginos, etc. There are also auxiliary terms.

3. Add matter in linear representations of the gauge group. These will have chiral fermins and their complex conjugates.

4. Write the most general effective action compatible with symmetries.

5. The data consists of a Kahler potential (with exp V ) sprinkled in vari- ous places).

6. The kinetic term for gauge fields, where g2 can be a holomorphic func- tion of the fields (chiral).

7. A general superpotential made of chiral superfields only, compatible with all the gauge symmetries.

8. All of these are written economically in superspace.

9. Only after all of this is done, one can integrate out the auxiliary fields (if desired)

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