3. Topological Field Theory From N = 4 Super Yang-Mills Theory
3.4. The Topological Lagrangian
Our goal now is to find a Lagrangian that possesses the topological symmetry for any value of t and reduces when M is flat to the Lagrangian of the underlying N = 4 super Yang-Mills theory. We could do this by hand, starting with (2.6) for flat M and asking what curvature dependent terms are needed to maintain the topological symmetry when M is not flat. We will follow a different approach.
A very useful first step is to compute the algebra generated on the fields by the the supersymmetries Qℓ andQr, or equivalently by Q=uQℓ+vQr. For the fieldsA, φ, ψ, ψ,e and σ, we compute
δT2A=−i(u2+v2)(−Dσ) δT2φ=−i(u2+v2)[σ, φ]
δT2ψ=−i(u2+v2)[σ, ψ]
δT2ψe=−i(u2+v2)[σ,ψ]e δT2σ= 0.
(3.34)
These results can be summarized by saying that if Φ is any field of K ≥ 0, then δT2Φ =
−i(u2 +v2)£σ(Φ), where £σ(Φ) is the first order change in Φ in a gauge transformation generated by σ (so £ stands for Lie derivative). For example, £σ(A) =−Dσ, and for the other fields here, £σ(Φ) = [σ,Φ]. The relationδT2Φ =−i(u2+v2)£σ(Φ) can be expanded in powers of u and v and is equivalent to the following:
δℓ2Φ =δ2rΦ =−i£σ(Φ)
{δℓ, δr}Φ = 0. (3.35)
If, however, one computesδT2Φ for a field ofK<0, one does not get−i(u2+v2)£σ(Φ) in an obvious way. In fact, the relation δT2Φ = −i(u2+v2)£σ(Φ) does hold for all Φ in the twisted N = 4 super Yang-Mills theory, but for some fields, it only holds upon using the equations of motion.
Construction of supersymmetric Lagrangians is generally much easier if one can in- troduce “auxiliary fields,” which are extra fields that can ultimately be eliminated by their equations of motion if one so chooses, such that the algebra, in the present case δT2Φ =−i(u2+v2)£σ(Φ), is satisfied without using equations of motion. In theories with a great deal of supersymmetry, such as N = 4 super Yang-Mills theory, it can be very difficult to find suitable auxiliary fields. The present example, as first noted in [53], is a case in which this problem arises.
However, the problem can be greatly alleviated by introducing an auxiliary fieldP, a zero-form with values in the Lie algebra, so as to close the algebra for η and ηe(but not χ). One takes the transformation laws of σ, η,eη, and P to be
δTσ =iuη+ivηe δTη =vP +u[σ, σ]
δTηe=−uP +v[σ, σ]
δTP =−iv[σ, η] +iu[σ,η].e
(3.36)
These equations will reduce to those in section 3.1 once we impose the equations of motion – which notably will set P =D∗φ.
Once we have a closed algebra on a set of fields – in this case all fields in the theory including P but excluding χ, since we have not closed the algebra on χ – it is generally a simple matter to find the possible invariant Lagrangians. In the present case, we would like to find a partial actionI0 for the the fields on which we have closed the algebra which obeys δℓI0 = δrI0 = 0, and hence, for any u, v, obeys δTI0 = 0 where δT = uδℓ +vδr. These properties will hold ifI0 =δℓδrV for some gauge-invariantV. Indeed, ifV is gauge- invariant, then £σ(V) = 0 and (3.35) reduces to δℓ2V = {δℓ, δr}V = δ2rV = 0. So δℓδrV will automatically be annihilated by δ.
In addition, we want to pick V so that the equation of motion for P is P =D∗φ, so as to be compatible with the formulas of section 3.1. So we take
V = 2 e2
Z
M
d4x√ g
−1
2Trηηe−iTrσD∗φ
, (3.37)
and compute that I0 =δℓδrV is I0 = 2
e2 Z
M
d4x√gTr 1
2P2−P D∗φ+ 1
2[σ, σ]2−Dµσ Dµσ−[φµ, σ][φµ, σ]
+ieηDµψeµ+iηDµψµ−ieη[ψµ, φµ] +iη[ψeµ, φµ]
−i
2[σ,η]eηe− i
2[σ, η]η+i[σ, ψµ]ψµ+i[σ,ψeµ]ψeµ
.
(3.38)
The Euler-Lagrange equation for P isP =D∗φ, as desired, and one can make the replace- ment
Tr P2−2P D∗φ
→ −Tr(D∗φ)2, (3.39)
if one wishes.
I0 manifestly possesses the topological symmetry for any value oft =v/u. Moreover, as it is of the form δℓδrV, where the metric of M does not enter in the definition of δℓ and δr but only in the choice of V, it also has the key property that leads to a topological field theory: its dependence on the metric ofM is of the form δℓδr(. . .).
The only reason thatI0 is not a satisfactory action is that it is degenerate; it does not contain the kinetic energy for the gauge fields or any terms containingχ. It does not seem to be possible to complete the construction of the action with a construction as simple as that above. Though it is possible to add auxiliary fields so as to close the algebra on χ, it does not seem to be possible to do this in a way that incorporates both δℓ and δr and is useful for understanding the appropriate twisted N = 4 action. (The fact that a t-dependent topological invariant occurs on the right hand side of (3.32) appears to be an obstruction to this.) Instead, we will fix a particular value of t and consider only the differential δt = δℓ +tδr. We add an auxiliary field H, which is to be a two-form with values in the Lie algebra, and postulate
δtχ =H
δtH =−i(1 +t2)[σ, χ]. (3.40) This is compatible with
δ2t(Φ) =−i(1 +t2)£σ(Φ), (3.41) for Φ =χ, H; this is the specialization of δ2T =−i(u2+v2)£σ for (u, v) = (1, t). For (3.40) to agree with the transformation of χ found in section 3.1, the equations of motion will have to give H+ =V+(t), H− =tV−(t). We will construct the action to ensure this.
As before, an action annihilated by δt can be I1 = δtV1, for any gauge-invariant V1. We pick
V1 = 2 e2
Z
M
d4x√g 1 (1 +t2)Tr
χ+µν
1
2H+µν −V+(t)µν
+χ−µν 1
2H−µν −tV−(t)µν
. (3.42) Then
I1 = 1 e2
Z
M
d4x√g
2 (1 +t2)Tr
1
2Hµν+H+µν−Hµν+V+(t)µν
+ 2
(1 +t2)Tr 1
2Hµν−H−µν−tHµν−V−(t)µν
+ 2Tr χ+µν
iDψ+i[ψ, φ]e µν
+χ−µν
iDψe−i[ψ, φ]µν + Tr iχ+µν[σ, χ+µν] +iχ−µν[σ, χ−µν]
.
(3.43)
Upon integrating H± out of this, we can write the equivalent action I1 = 1
e2 Z
M
d4x√ g
− t−1
(t+t−1)TrV+(t)µνV+(t)µν− t
(t+t−1)TrV−(t)µνV−(t)µν
+ 2Tr χ+µν
iDψ+i[ψ, φ]e µν
+χ−µν
iDψe−i[ψ, φ]µν + Tr iχ+µν[σ, χ+µν] +iχ−µν[σ, χ−µν]
.
(3.44) Finally, the useful identity (3.32) enables us to write
I1 =− 1 e2
Z
M
d4x√g
Tr 1
2FµνFµν +DµφνDµφν +Rµνφµφν + 1
2[φµ, φν]2−(D∗φ)2
−2Tr χ+µν
iDψ+i[ψ, φ]e µν
+χ−µν
iDψe−i[ψ, φ]µν
−Tr iχ+µν[σ, χ+µν] +iχ−µν[σ, χ−µν] + t−t−1
e2(t+t−1) Z
M
TrF ∧F.
(3.45) Apart from the topological term, the part of I0+I1 that depends only on A and φ can also [53] be written
I(A,φ)=− 1 e2
Z
d4x√ gTr
1
2FµνFµν + (D∗φ)2
, (3.46)
as one can show with some integration by parts similar to what is used in proving the vanishing theorems. The analogous terms involving σ can be found in (3.38) and are
Iσ = 2 e2
Z
d4x√ gTr
1
2[σ, σ]2−Dµσ Dµσ−[φµ, σ][φµ, σ]
. (3.47)
The key point – which we already exploited in proving the vanishing theorems – is thatI1 is independent oft except for the last term. But that last term, being a topological invariant, is automatically annihilated by δt, and indeed by both δℓ and δr, all by itself.
So without spoiling the topological symmetry, we can add another term to the action, I2 =−
t−t−1
e2(t+t−1) −i θ 8π2
Z
M
TrF ∧F. (3.48)
We have simply chosen the coefficient to cancel the t-dependence in I1, leaving us with the standard θ parameter multiplying this term in the action.
Finally, then, the overall action isI =I0+I1+I2. The construction makes it manifest that I is annihilated by δt for the specific value of t used in constructingI1. But since in fact I is independent of t (up to a topological invariant), it is annihilated by δt for all t.
Moreover,I is the action of a topological field theory. The change inI in a change in the metric of M is of the form δt(. . .), since I0 andI1 are of this form even before varying the metric, and I2 does not depend on the metric ofM at all.
We have accomplished our goal of constructing an action that has the fullCP1 family of topological symmetries – making it clear, among other things, that the partition function of this theory on a closed four-manifold M without operator insertions is independent oft.
One can readily verify that (after eliminating P via (3.39), whereupon the (D∗φ)2 terms cancel), the bosonic part of I reduces for M = R4 to (2.10). The topological symmetry implies that the fermionic terms also agree. So this theory is a generalization of N = 4 super Yang-Mills theory to a curved four-manifold.