“Problem Set Three”

(1) Here we will study the theory related to “differential forms on the loop space of a manifold.”

We let W^{2,2} be a supermanifold of the indicated dimension with one-dimensional odd distributions Λ_{+}, Λ_{−} generated by vector fields D_{+}, D_{−} such that D_{+}, D_{−}, D_{+}^{2}, D_{−}^{2} generate the tangent bundle of W^{2,2}. The flat model is R^{2,2} with even coordinates u, v, and odd coordinates θ_{−}, θ_{+}, with

\begin{aligned}D_{+} & = \frac{∂}{∂θ_{+}} − iθ_{+} \frac{∂}{∂v} \\ D_{−} & = \frac{∂}{∂θ_{−}} − iθ_{−} \frac{∂}{∂u}. \end{aligned}  (1)

R^{2,2} has an infinite dimensional supergroup of automorphisms, roughly G × G, where G is the symmetry group of the supermanifold R^{1,1} (with even and odd coordinates t, θ and distribution generated by ∂_{θ} + iθ∂_{t}). For instance, generators are

\begin{aligned}Q_{+} & = \frac{∂}{∂θ_{+}} + iθ_{+} \frac{∂}{∂v} \\ Q_{−} & = \frac{∂}{∂θ_{−}} + iθ_{−} \frac{∂}{∂u}. \end{aligned}  (2)

Let M be a fixed Riemannian manifold with metric g. Let Φ : W^{2,2} → M be a map, and let

L = ∫_{W^{2,2}} g_{IJ}D_{+}Φ^{I}D_{−}Φ^{J}.  (3)

(a) By now it should be routine to interpret the integrand as a section of the sheaf of volume forms and thereby show that the integral makes sense and that L is invariant under all of the symmetries of W^{2,2}.

Show that this has the following generalization: if B is a two-form on M, we can add another term

L′ = ∫_{W^{2,2}} B_{IJ}D_{+}Φ^{I}D_{−}Φ^{J}  (4)

which still preserves the symmetries of W^{2,2}. (Of course we can write a lot of other terms, some of which will be discussed later, if we pick a volume form on W^{2,2}.) Adding L′ will give a theory that involves not ordinary differential forms on the path or loop space of M, but differential forms with values in a certain line bundle. Show that (4) is unchanged (up to a total derivative) if one replaces B by B′ = B + dΛ.

(b) To understand more concretely what this theory describes, compute in the flat R^{2,2} and set

\begin{aligned}X & = Φ|_{θ_{±}=0} \\ ψ_{+} & = D_{+}Φ|_{θ_{±}=0} \\ ψ_{−} & = D_{−}Φ|_{θ_{±}=0} \\ F & = −iD_{+}D_{−}Φ|_{θ_{±}=0}. \end{aligned}  (5)

(So X is a map X : W_{red} → M, while ψ_{±} and F take values in the obvious line bundles on W_{red} tensored with X^{∗}(T M), and the definition of F uses the connection on X^{∗}(T M) as well as the explicit D_{±}.)

Calculate L = ∫ du dv dθ_{+} dθ_{−}S (with S as in (3) above – omit the generalization using the two-form) by writing it as L = ∫ du dv D_{+}D_{−}S|_{θ_{±}=0}.

Don’t be too careless in differentiating the metric – note the appearance of the Riemann tensor!

You should see that ψ_{+} are right-moving fermions, with values in X^{∗}(T M), and ψ_{−} are left-movers, again with values in X^{∗}(T M). So as in last week’s problem set, quantization of ψ_{+} and ψ_{−} will each give spin bundles on the loop or path space LM of M. Call these spin bundles S_{L}(LM) and S_{R}(LM). S_{L} and S_{R} are not isomorphic, unlike the finitedimensional case, because the positive energy condition gives opposite polarizations for left and right movers.

Nonetheless, the tensor product S_{L} ⊗ S_{R} would in finite dimensions be isomorphic to the de Rham complex Ω^{∗}(LM), so this theory gives a version of Hodge - de Rham theory of LM.

Show that this theory is Z_{2} × Z_{2} graded, where one Z_{2} acts by θ_{+} → −θ_{+} and the other by θ_{−} → −θ_{−}. The two Z_{2}’s would correspond for a finite-dimensional manifold to the Hodge ∗ and the operator that acts on p-forms by (−1)^{p}. The Z-valued grading of differential forms is lost in this (good!) version of Hodge theory of loop space.

(c) Let u = t−x, v = t+x, and consider only Φ’s that are invariant under translations of x. The theory thus reduces to a theory on R^{1,2} with coordinates t, θ_{+}, θ_{−}, and

L = ∫ dtdθ_{+}dθ_{−}g_{IJ}D_{+}Φ^{I}D_{−}Φ^{J}.  (6)

Q_{+} and Q_{−} in this theory correspond to d+∗d∗ and i(d−∗d∗) acting on differential forms.

Note that in reducing from R^{2,2} to R^{1,2} we have lost the infinite-dimensional symmetry algebra of R^{2,2}. Show however, that one new symmetry appeared in the reduction – an SO(2) that rotates θ_{+}, θ_{−}. Show that this corresponds in the quantum theory to the Z-grading of differential forms. (A Z_{2} -subgroup of SO(2), which acts by sign change of both θ_{+} and θ_{−}, is present in the two-dimensional version, as seen in (b).)

(d) Now we want to compare this theory to a theory on R^{2,1} which is obtained from R^{2,2} by setting θ_{−} = 0.

We set Φ′ = Φ|_{θ_{−}=0}, and Λ_{−} = D_{−}Φ|_{θ_{−}=0}. Write ∫ du dv dθ_{+} dθ_{−} S = ∫ du dv dθ_{+} D_{−}S|_{θ_{−}=0} to convert the Lagrangian on R^{2,2} to an equivalent Lagrangian on R^{2,1}.

Compare with the theory of “spinors on LM with values in a vector bundle” that was in problem 5 of problem set 2, and show that – as befits “differential forms on LM” – the theory discussed here is a special case of “spinors on LM with values in a vector bundle.”

(e) So far we have seen that the theories of Dirac and Hodge-de Rham operators on a manifold M as customarily studied are the one-dimensional versions of two-dimensional theories. (The physics that goes with this is of course string theory.) For Dirac, that is as far as we can go – the chain that leads in this way to the Dirac operator on a manifold begins in two dimensions. But I want to point out that the Hodge-de Rham theory can come by reduction of a three-dimensional theory (generally less important in physics, so far at least, as it is less well-behaved quantum mechanically).

Here we start with a supermanifold W^{3,2} of the indicated dimension, with a twodimensional odd distribution A which is generic in the sense that A and {A, A} generate the tangent bundle of W^{3,2}. A sharp difference from the two-dimensional case is that we need a volume form ϵ on A. Letting Φ : W^{3,2} → M be a map as before to a fixed Riemannian manifold M, and D_{1}, D_{2} a basis of A such that ϵ(D_{1}, D_{2}) = 1, the Lagrangian is

L = ∫ g_{IJ}D_{1}Φ^{I} D_{2}Φ^{J}.  (7)

Show that this is well-defined, and invariant under all symmetries that preserve the distribution and volume form.

The flat model is R^{3,2} with odd coordinates θ^{A}, A = 1, 2 and even coordinates y^{AB} (with y^{AB} = y^{BA} so there are three y’s) and

D_{A} = \frac{∂}{∂θ^{A}} − iθ^{B} \frac{∂}{∂y^{AB}} .  (8)

Given a real structure and volume form on the space of the θ^{A}’s the group that can act linearly on the θ^{A}’s is SL(2, R). This is the double cover of SO(2, 1), which acts on the y’s, preserving a metric of signature + − −. The model lacks the infinite-dimensional symmetry group of the two-dimensional case, but has the following finite-dimensional group of symmetries. Let

Q_{A} = \frac{∂}{∂θ^{A}} + iθ^{B} \frac{∂}{∂y^{AB}} .  (9)

Show that the Q_{A} preserve the distribution, and that together with the {Q_{A}, Q_{B}} and the SL(2, R), they generate three-dimensional super-Poincaré. Together with some orientation and time-orientation reversing symmetries, this is the full symmetry group of R^{3,2} with the volume form.

Show that, if we impose invariance under translation of one of the y’s, the model reduces to the two-dimensional model introduced above.

If one imposes invariance under translations of two of the y’s – in fact, pick coordinates in which the metric is dt^{2} − dx_{1}^{2} − dx_{2}^{2} and impose invariance under translation of x_{1}, x_{2} – then we get the theory on R^{1,2} that is related upon quantization to ordinary differential forms on M with the d and d^{∗} operators. This theory therefore has an SO(2) symmetry of rotation of the x_{1} − x_{2} plane – it is the SO(2) noted above that rotates θ_{+} and θ_{−} in the original presentation; the generator acts on p-forms by multiplication by p (or p− \frac{1}{2} dim M).

(2) On differential forms on M there acts the SO(2) just noted, commuting with the Laplacian. It comes by dimensional reduction from a 1 + 2 = 3 dimensional theory. Differential forms on a generic M cannot come from reduction of a four-dimensional theory, as the SO(2) - which is better called Spin(2) - would then have to extend to a Spin(3).

Likewise, the Dirac equation on a general M cannot come by reduction from a theory above two dimensions. If it came from a three-dimensional theory, we would need to find a Spin(2) action on the spin bundle, commuting with the Laplacian and acting non-trivially on the Dirac operator. So d = 2 is the “starting point” in this sense for the Dirac equation.

But suppose M is a Kahler manifold. Then the Spin(2) action on differential forms does extend to a Spin(3) = SU(2) action (sometimes called SL(2), but SU(2) is the group that acts on the Hilbert space of differential forms preserving the structure). d and d^{∗} can be decomposed into four operators ∂, ¯∂, ∂^{∗}, ¯∂^{∗}. These are (complex) linear combinations of four self-adjoint operators that transform as the four dimensional real spinor representation of Spin(3). This suggests that differential forms on a Kahler manifold M can come from a theory in 1 + 3 = 4 dimensions.

Likewise if M is hyper-Kahler then the Spin(3) extends further to a Spin(5) with d and d^{∗} generating eight self-adjoint operators that transform as the eight dimensional real spinor of Spin(5). This suggests that differential forms on M can come from a theory in 1 + 5 = 6 dimensions.

Recall that, according to Nahm’s theorem, six is the maximum dimension for supersymmetric sigma models. The supersymmetric theory in six dimensions of maps to a hyper-Kahler manifold (plus odd variables) does exist – but like all the limiting cases allowed by Nahm’s theorem (the others being gauge theory in ten dimensions and gravity in eleven dimensions) there is no known construction via supermanifolds.

For the four-dimensional case, however, with M a Kahler manifold, a superspace construction does exist. For the time being, we will only describe it locally. (It takes a few more steps to describe a more global version.)

We consider a (4,4) supermanifold W with the structure described in Bernstein’s lectures – two complex conjugate integrable distributions A_{+} and A_{−} of dimension (0, 2) such that {A_{+}, A_{−}} generates T M/(A_{+} ⊕ A_{−}), A_{+} and A_{−} are endowed with volume forms. A chiral superfield is a field ϕ that is a function on W/A_{−}. If M is a complex manifold, a chiral map Φ : W → M is one such that, for f a local holomorphic function on M, Φ^{∗}(f) is a chiral function on W. In that case, Φ^{∗}(f̄) is antichiral, that is, it is a function on W/A_{+}.

Let K be a function on W and consider the Lagrangian

L = ∫_{W} d^{4}yd^{4}θ Φ^{∗}(K).  (10)

d^{4}y d^{4}θ is the section of the Berezinian coming from the volume forms on A_{±}.

(a) Show that L is invariant under K → K + f + f̄, where f is a local holomorphic function on W.

(b) Recall that any K determines the two-form ω = −i¯∂∂K which for a suitable class of K’s is the Kahler class of a Kahler metric g on M. Of course, ω is invariant under the transformation considered in (a).

Show that L can be written just in terms of ω (or equivalently g) and that the “reduced” part of L is the harmonic map action, specialized to maps to a Kahler manifold:

L = ∫_{W_{red}} d^{4}yg_{ij̄}(dX^{i}, dXj̄)  (11)

where X^{i} are local holomorphic coordinates on M.

(c) Specialize now to the flat model with W = R^{4,4}. A_{+} is generated by

D_{A} = \frac{∂}{∂θ^{A}} − iσ_{AȦ}^{m}θ̄Ȧ \frac{∂}{∂y^{m}}  (12)

where θ^{A} are odd coordinates transforming as “positive” spinors of Spin(1, 3) = SL(2, C), θ̄Ȧ are their complex conjugates, and y^{m} are even coordinates. σ is the isomorphism S_{+} ⊗ S_{−} = V. A_{−} is generated by

DȦ = \frac{∂}{∂θȦ} − iσ_{AȦ}^{m}θ̄^{A} \frac{∂}{∂y^{m}} .  (13)

Impose invariance under a space-like translation of W, say a translation of y^{3} (the metric being (dy^{0})^{2} − ∑_{i=1}^{3}(dy^{i})^{2} to get a model on R^{3,4}. Show that this is equivalent to the R^{3,2} σ model described in problem 1(e) (but specialized to the case that M is Kahler). Here is an attempt to explain the strategy: under Spin(1, 2), the two spinor representations of Spin(1, 3) become isomorphic. So after the reduction, A_{+} and A_{−} are naturally isomorphic. One can in an SO(1, 2)-invariant way take the “real” combination χ^{A} = θ^{A} + θ̄^{A} of the θ’s and θ̄’s. The R^{3,2} we want has even coordinates y^{i}, i = 0, 1, 2, and odd coordinates χ^{A}. By “integrating over the fibers” of a map from R^{3,4} to R^{3,2}, one reduces (11) (specialized to y^{3}-independent maps) to the R^{3,2} Lagrangian of problem 1(e).