Chern-Simons theory on supermanifolds

We consider quantum field theories on supermanifolds using integral forms. The latter are used to define a geometric theory of integration and they are essential for a consistent action principle. The construction relies on Picture Changing Operators, analogous to the one introduced in String Theory. As an application, we construct a geometric action principle for N = 1 D = 3 super-Chern-Simons theory.


Introduction
suitable PCO with different properties, but belonging to the same cohomology class. As a playground, we choose 3D, N=1 super-Chern-Simons theory.
The conventional bosonic Chern-Simons theory is described by the geometrical action where A (1) is the 1-form gauge connection with values in the adjoint representation of the gauge group G, the trace is taken over the same representation and the integral integrates a 3-form Lagrangian over a three dimensional manifold M. As is well known, it provides a meaningful integral, independent of the parametrization of M and of its metric. The 3-form Lagrangian is closed by construction and its gauge variation is exact.
For the corresponding super Chern-Simons action on a supermanifold M (3|2) one needs a (3|2)-integral form that, however, cannot be built only by connections as A (1|0) . The latter are differential 1-superform with zero picture (as been explained in [1,2]), leading to a (3|0) superform Lagragian as (1.1) that cannot be integrated. Nonetheless, it can be converted to a (3|2)-integral form by multiplying it by a PCO belonging to Ω (0|2) for example where V a = dx a + θ α γ a αβ dθ β is the super-line element. γ a αβ , γ ab αβ are the Dirac gamma matrices and ι α is the usual contraction operators along the odd vector D α = ∂ α − (θγ a ) β ∂ a . The operator Y (0|2) new is closed, supersymmetric and not exact, namely it belongs to H (0|2) . Consequently the super Chern-Simons action reads where the integration is extended to the entire supermanifold SM. As can be checked, the result is gauge invariant, supersymmetric and leads to the well-known super Chern-Simons action in superspace. An obvious question is whether one can change the PCO Y new is achievable only if the (3|0) Lagrangian is closed. That request, for a (3|0) superform in the supermanifold M (3|2) , is non-trivial and indeed the action given in (1.3) has to be modified accordingly. It is easy to show that there is a missing term in the action and the closure implies the usual conventional constraints. Then, after that modification, we can change the PCO for getting new forms of the action with the same physical content, but displaying different properties.
In the present context, we provide a new geometrical perspective on QFT's superspace and on supermanifolds. We are able to prove that the Rheonomic action (see [9]) formulation of N = 1 D = 3 super Chern-Simons theory with rigid supersymmetry (the local supersymmetric case will be discussed separately) can be considered a "mother" theory which has built-in all possible superspace realizations for that theory. In particular we show that using a given PCO the action reduces to the usual action in terms of component fields and by another choice we get the superspace action written in terms of superfields. However, only for the choice (1.2) we are able to derive the conventional constraint by varying the action and without resorting to the rheonomic parametrization.
The paper is organised as follows: Sec. 2 deals with background material, the definition of integral forms and integration on supermanifolds. In Sec. 3, we introduce PCO's for spacetime quantum field theory. In Sec. 4, we discuss the action of super-Chern-Simons theory in 3d.
The relation between different types of PCO's and actions are given in Sec. 5.
Integral forms, integration on supermanifolds, the role of picture changing operators in QFT and applications to gauge theories was one of the last discussions with Raymond Stora during the last extended period spent by one of the authors at CERN, for that reason this note is dedicated to him. We recall that in 3d N=1, the supermanifold SM (3|2) (homeomorphic to R 3|2 ) is described locally by the coordinates (x a , θ α ), and in terms of these coordinates, we have the following two differential operators known as superderivative and supersymmetry generator, respectively. They have the proper- In 3d, with η ab = (−, +, +), we use real and symmetric Dirac matrices γ a αβ defined as The conjugation matrix is αβ and a bi-spinor is decomposed as follows R αβ = R αβ + R a γ a αβ where R = − 1 2 αβ R αβ and R a = tr(γ a R) are a scalar and a vector, respectively. In addition, it is easy to show that γ ab αβ ≡ 1 2 [γ a , γ b ] = abc γ cαβ . For computing the differential of Φ (0|0) , we can use the basis of (1|0)-forms defined as follows where V a = dx a + θγ a dθ and ψ α = dθ α which satisfy the Maurer-Cartan equations Given a (0|0)-form Φ (0|0) , we can compute its supersymmetry variation (viewed as a super translation) as a Lie derivative L with = α Q α + a ∂ a ( a = α γ a αβ β are the infinitesimal parameters of the translations and α are the supersymmetry parameters) and we have In the same way, acting on (p|q) forms, where p is the form degree and q is the picture number, we use the usual Cartan formula L = ι d + dι . It follows easily that δ V a = δ V α = 0 and The top form is represented by the expression which has the properties It is important to point out the transformation properties of ω (3|2) under a Lorentz transformation of SO(2, 1). Considering V a , which transforms in the vector representation of SO(2, 1), On the other hand, dθ α transform under the spinorial representation of SO(2, 1), say Λ β α = (γ ab ) β α Λ ab with Λ ab ∈ so(2, 1), and thus an expression like δ(dθ α ) is not covariant. Nonetheless, the combination αβ δ(dθ α )δ(dθ β ) = 2δ(dθ 1 )δ(dθ 2 ) is invariant using formal mathematical properties of distributions, for instance dθδ(dθ) and dθδ (dθ) = −δ(dθ). We recall that δ(ψ α ) ∧ δ(ψ β ) = −δ(ψ β ) ∧ δ(ψ α ). In addition, ω (3|2) has a bigger symmetry group: we can transform the variables (V α , dθ α ) under an element of the supergroup SL(3|2). The form ω (3|2) is a representative of the Berezinian bundle, the equivalent for supermanifolds of the canonical bundle on bosonic manifolds.

Integral Forms
Consider the generalized form multiplication as where 0 ≤ p, q ≤ n and 0 ≤ r, s ≤ m with (n|m) are the bosonic and fermonic dimensions of the supermanifold SM. Due to the anticommuting properties of the delta forms this product is by definition equal to zero if the forms to be multiplied contain delta forms localized in the same variables dθ.
Given the space of pseudo forms Ω (p|r) , a (p|r)-form ω formally reads where g(t) denotes the differentiation degree of the Dirac delta function corresponding to the 1-form dθ t . 1 If g(t) = 0 it means that the Dirac delta function has no derivative. The three indices l, h and r satisfy the relation where the last equation means that each α l in the above summation should be different from any β k , otherwise the degree of the differentiation of the Dirac delta function can be reduced and the corresponding 1-form dθ α k is removed from the basis. The components of ω are superfields.
In fig. 1, we display the complete complex of pseudo-forms. We notice that the first line and the last line are bounded from below and from above, respectively. This is due to the fact that in the first line, being absent any delta functions, the form number cannot be negative, and in the last line, having saturated the number of delta functions we cannot admit any power of dθ (because of the distributional law dθδ(dθ) = 0).
Before discussing the Chern-Simons action, we analyze the dimension of each space Ω (p|r) .
The dimension of Ω (p|0) is given by the power of the dx 1-forms and by the power of the dθ where we have decomposed the form degree p into l + h where the degree l is carried by dx and the degree h is carried by dθ. For that decomposition, we have n(n − 1) . . .
In the same way, if we consider the integral forms Ω (n−p|m) of the last line, we see that we can have powers of dx and derivatives on the Dirac delta functions as where g(t) is the order of the derivative on δ(t). The form degree is l − m k=1 g(α k ). For example, for n = 3, m = 2 the superspace is SM (3|2) and there are three complexes: , Ω (p|1) and Ω (p|2) . The first one is bounded from below being Ω (0|0) the lowest space 1 It is an easy exercise to rewrite ω in terms of the susy invariant superforms V a , ψ α .
. generated by constant functions, the last one is bounded from above with Ω (3|2) the highest space spanned by the top form and finally, the middle one is unbounded. In addition, the dimension of each space of the first and of the last one is finite, while for the middle one each Let us consider the space Ω (1|0) spanned by dx a , dθ α with dimensions (3|2) (which means 3 bosonic generators -instead of dx a , one can use the supersymmetric variables V a = dx a + θγ a dθ -and 2 fermionic generators ψ α ). The space Ω (2|2) , spanned by where ι α δ 2 (dθ) denote the derivative of δ 2 (dθ) with respect dθ α . It has dimensions (3|2) and therefore there should be an isomorphism between the two spaces. The construction of that isomorphism, which is the generalization of the conventional Hodge dual to supermanifolds, has been provided in [10].
Let us consider another example: the space Ω (2|0) is spanned by with dimension (6|6). The dual space is Ω (1|2) and it is spanned by which has again (6|6) dimensions. The last example is the one-dimensional space Ω (0|0) of 0-forms and its dual Ω (3|2) , a one-dimensional space generated by d 3 xδ 2 (dθ), the top form of the supermanifold SM (3|2) . Now, let consider the middle complex Ω (1|1) spanned (in the sense of formal series) by the following psuedo-forms where the number n is not fixed and it must be a non-negative integer. Due to the bosonic 1-forms dx a and due to the fact that the index α must be different from β for a non-vanishing integral form (we recall that dθ α δ (n) (dθ α ) = −nδ (n−1) (dθ α ), and δ (0) (dθ α ) = δ(dθ α )), the number of generators (monomial forms) at a given n is (8|8), but the total number of monomial generators in Ω (1|1) is infinite. The dual of Ω (1|1) is itself, but the isomorphism is realised by an infinite matrix whose entries are (8|8) × (8|8) supermatrices.
In the same way, for a general supermanifold M (n|m) any form belonging to the middle complex Ω (p|r) with 0 < r < m is decomposed into an infinite number of components as in (2.14).
In general, if ω is a poly-form in Ω • (M) this can be written as direct sum of (p|q) pseudo and its integral on the supermanifold is defined as follows: (in analogy with the Berezin integral for bosonic forms):

Picture Raising Operator
In the present section, we discuss a class of PCO's relevant to the study of differential forms in Ω (p|q) . In particular we define a new operator that increases the number of delta's (then, increases the picture number), the Picture Raising Operator. 2 It acts vertically mapping superforms into integral forms.
To start with, given a constant commuting vector v α , consider the following object which has the properties where η (−1|1) is a pseudo-form. Notice that Y v belongs to H (0|1) (which is the de-Rham cohomology class in Ω (0|1) ) and by choosing two independent vectors v (α) , we have where v β (α) is the β-component of the vector v (α) . The result is independent of v α . We can apply the PCO operator to a given integral form by taking the wedge product of forms. For example, given ω in Ω (p|0) we have If dω = 0 then d(ω ∧ Y (0|2) ) = 0 (by applying the Leibniz rule), and if ω = dη then it follows that also ω ∧ Y (0|2) = dU where U is an integral form of Ω (p−1|2) . In [1], it has been proved that Y (0|2) is an element of the de Rham cohomology and that they are also globally defined.
So, given an element of the cohomogy H . 2 We warn the reader the meaning of raising and lowering is opposite to that used in string theory literature. In that case the picture is carried by the delta of the superghost δ(γ) = e −φ and it is conventionally taken to be negative, and indentified with the φ charge.
At the end, we have where A a (x, 0) is the lowest component of the superfield A a appearing in the superconnection A (1|0) . This seems puzzling since we have "killed" the complete superfield dependence of A a (x, θ) leaving aside the first component A a (x, 0). This happens because Y (0|2) as defined in (3.3) has an obvious non-trivial kernel.
However, we can modify the PCO given in (3.3) with a more general construction. If we consider a set of anticommuting superfields Σ α (x, θ) such that Σ α (x, 0) = 0. They can be where (1 + DΣ) is a m × m invertible matrix and it should be obvious from the above formula how the indices are contracted. Expanding the Dirac delta function and recalling that the bosonic dimension of the space is 3, we get the formula where the superfields H, K α a , L is closed as can be easily verified by using dV a = ψγ a ψ and dψ α = 0. It is not exact, it is invariant under rigid supersymmetry and it differs from Y (0|2) by exact terms. This PCO can be expanded in different pieces by decomposing V a and by taking the derivatives ι α from δ 2 (ψ) to V 's: where the coefficients a i are fixed by simple Dirac matrix algebra. We notice that all pieces have zero form degree and picture number +2. Another property of Y (0|2) new is its duality with ω (3|0) = ψγ a ψV a . The latter is an element of the Chevalley-Eilenberg cohomology (see [9] for a complete discussion and references) and therefore it is closed (by using the Fierz identities γ a ψ(ψγ a ψ) = 0) and is not exact. The duality with Y where abc V a ∧ V b ∧ V c δ 2 (ψ) is the volume form belonging to Ω (3|2) .
If the gauge group is non-abelian, the field strength F (2|0) has to be modified in where the wedge product of two superform (at picture zero) gives a superform again at picture zero. However, to define a field strength at picture number 2, we immediately see that the product of A (1|2) ∧ A (1|2) = 0 independently of the non-abelianity of the gauge group, but because δ 3 (dθ) = 0 .

Super Chern-Simons Action
Let's begin by reviewing the standard superspace construction of Chern-Simons. We start from a 1-super form A (1|0) = A a V a + A α ψ α , (where the superfields A a (x, θ) and A α (x, θ) take value in the adjoint representation of the gauge group) and we define the field strength In order to reduce the redundancy of degrees of freedom because of the two components A a and A α of the (1|0) connection, one imposes (by hand) the conventional constraint from which it follows that F aα = γ a,αβ W β with W α = ∇ β ∇ α A β and ∇ α W α = 0. The gaugino field strength W α is gauge invariant under the non-abelian transformations δA α = ∇ α Λ.
The field strengths satisfy the following Bianchi's identities and by expanding the superfields A a , A α and W α at the first components we have where a a (x) is the gauge field, λ α (x) is the gaugino and f αβ = γ ab αβ f ab is the gauge field strength In terms of those fields, the super-Chern-Simons lagrangian becomes which in component reads That coincides with the bosonic Chern-Simons action with free non-propagating fermions.
In order to obtain an action principle by integration on supermanifolds we consider the natural candidates for a super-Chern-Simons lagrangian where A (1|0) is the superconnection and d is the differential on the superspace, and then we multiply it by a PCO, for example by Y (0|2) = θ 2 δ 2 (dθ). That leads to (3|2) integral form that can be integrated on the supermanifold, that is However, this action fails to give the correct answer yielding only the bosonic part of the action of S SCS . The reason is that the supersymmetry transformations of the PCO is and by integrating by parts, we find that the action is not supersymmetric invariant. On the other hand, as we observed in the previous section, we can use the new operator which is manifestly supersymmetric. Computing the expression in the integral, we see that The equations of motion correctly imply F αβ = 0 (which is the conventional constraint) and W α = 0 which are the super-Chern-Simons equations of motion. The second condition follows from F αβ = 0 and by the Bianchi identities which implies that F aα = γ aαβ W β .
Notice that this formulation allows us to get the conventional constraint as an equation of motion. In particular we find that the equation of motion, together with the Bianchi identity imply the vanishing of the full field-strenght.
Then, it is easy to show that The second equation is obvious since it is expressed in terms of supersymmetric invariant quantities. The first equation follows from the MC equations and gamma matrix algebra.
Chern-Simons theory on this group supermanifold share interesting similarities with a particular version of open super string field theory [12]. The reason for this is that the supergroup Osp(1|2) is infact the superconformal Killing group of an N = 1 SCFT on the disk. There is however an important difference wrt to [12]. Our choice of the picture changing operator Y applied to the field strength (dA (1|0) + A (1|0) ∧ A (1|0) ) leads to equation (4.13) and it directly implies the vanishing of the full field strength. In particular the kernel of the picture-changing operator is harmless in our case. It would be interesting to search for an analogous object in the RNS string.

The PCO Y (0|2)
new is related to the product of two non-covariant operators, each shifting the picture by one unit.
with v · w = 0 and by a little a bit of algebra, one gets Acting on the complete set of differential form Ω (p|q) , with the PCO's, for ω (p|q) ∈ Ω (p|q) with q > 0, we have Y (0|2) ∧ ω (p|q) = 0 due to the anticommuting properties of δ(dθ).

Changing the PCO and the relation between different superspace formulations
During the last thirty years, we have seen two independent superspace formalisms taking place, aiming to describe supersymmetric theories from a geometrical point of view. They are known as as superspace technology, whose basic ingredients are collected in series of books (see for example [13,14]) and the rheonomic (also known as group manifold) formalisms (see the main reference book [9]). They are based on a different approach and they have their own advantages and drawbacks. Without entering the details of those formalisms, we would like to illustrate some of their main features on the present example of super-Chern-Simons theories. A basic difference is that in the superspace few superfields contain the basic fields of the theory as components, while in the rheonomic approach any basic field of the theory is promoted to a superfield.
Let us start from the rheonomic action. This is given as follows where M 3 is a three-dimensional surface immersed into the supermanifold SM (3|2) and is a three-form Lagrangian constructed with superform A, their derivatives without the Hodge dual operator (that is without any reference to a metric on the supermanifold SM (3|2) ). Notice that the fields A are indeed superforms whose components are superfiels. namely the Lagrangian is a function of (x a , θ α , V a , ψ α ).
The rules to build the action (5.1) are listed and discussed in the book [9] in detail.
An important ingredient is the fact that for the action to be supersymmetric invariant, the Lagrangian must be invariant up to a d-exact term and, in addition, if the algebra of supersymmetry closes off-shell (either because there is no need of auxiliary fields or because it exists a formulation with auxiliary fields), the Lagrangian must be closed: dL (3) (A) = 0, upon using the rheonomic parametrization. One of the rules of the geometrical construction for supersymmetric theories given in [9] is that by setting to zero the coordinates θ α and its differential ψ α = dθ α , the action In order to express the action (5.1) in a more geometrical way by including the dependence upon the embedding into the integrand, we refer to [15] and we introduce the Poincaré dual form Y (0|2) = θ 2 δ 2 (dθ). As already discussed in the previous section, Y (0|2) is closed and its supersymmetry variation is d-exact. The action can be written on the full supermanifold as Therefore, by choosing the PCO Y (0|2) = θ 2 δ 2 (dθ), its factor θ 2 projects the Lagrangian identified only with the bosonic term A ∧ dA, but that turns out to be not closed. Therefore, that has to be modified as follows: as discussed above the physical fields of Chern-Simons theory are the gauge field a µ and the gaugino λ α which are the zero-order components of the supergauge field A(x, θ) and of the spinorial superfield W α (x, θ), the complete closed action reads which is a (3|2) form. 4 Imposing the closure of L (3|0) we get the rheonomic parametrizations of the curvatures, or differently said, the conventional constraints. Once this is achieved, we are free to choose any PCO in the same cohomology class. If we choose the PCO Y (0|2) = θ 2 δ 2 (dθ) we get directly the component action (4.7) and the third term in the action is fundamental to get the mass term for the non-dynamical fermions. On the other hand, by choosing Y (0|2) new , (1.2) we see that the last term is unessential becasue, due to the powers of V a , this term cancels out and we get the superspace action (4.6). This is the most general action and the closure of L (3|0) implies that any gauge invariant and supersymmetric action can be built by choosing Y (0|2) inside of the same cohomology class. Therefore, starting from the rheonomic action, one can choose a different "gauge" -or better said a different embedding of the submanifold M 3 inside the supermanifold SM (3|2) -leading to different forms of the action with the same physical content. It should be stressed, however, that the choice of Y (0|2) new , (1.2), is a preferred "gauge" choice, which allows us to derive the conventional constraint by varying the action without using the rheonomic parametrization.