String Sigma Models on Curved Supermanifolds

We use the techniques of integral forms to analyse the easiest example of two dimensional sigma models on a supermanifold. We write the action as an integral of a top integral form over a D=2 supermanifold and we show how to interpolate between different superspace actions. Then, we consider curved supermanifolds and we show that the definitions used for flat supermanifold can also be used for curved supermanifolds. We prove it by first considering the case of a curved rigid supermanifold and then the case of a generic curved supermanifold described by a single superfield $E$.


Introduction
During the last years, some new structures in the geometry of supermanifolds have been uncovered. The conventional 1 exterior bundle of a supermanifold Λ * [M] is not the complete bundle needed to construct a geometric theory of integration, the Hodge dual operation and to study the cohomology. One has to take into account also the complexes of pseudo and integral forms.
We call the complete bundle the integral superspace. In the present notes, we consider some new developments and we study also the case of curved integral superspace.
One of the key ingredient of string theory (in the Ramond-Nevue-Schwarz formulation) is the worldsheet supersymmetry needed to remove the unphysical tachyon from the spectrum, to describe fermionic vertex operators and to construct a supersymmetric spectrum. All of these properties are deeply related to the worldsheet supersymmetry and they are clearly displayed by using the superspace approach in 2 dimensions. Pertubartive string theory is described by a non-linear sigma model for maps φ m (z,z), λ m (z,z),λ m (z,z) (1.1) from the worldsheet Riemann surface Σ (1) (with one complex dimension) to a 10 dimensional target space M (10) , (m = 0, . . . , 9) with an action given by (for a flat surface Σ = C) S[φ, λ,λ] = Σ d 2 zL(φ, λ,λ) . (1.2) Where φ, λ denote respectively the bosonic and the fermionic fields .
To generalize it to any surface, one has to couple the action to D = 2 gravity in the usual way, namely by promoting the derivatives to covariant derivatives and adding the couplings with the D = 2 curvature. That can be easily done by considering the action as a 2-form to be integrated on Σ using the intrinsic definition of differential forms, Hodge duals and the differential d. To avoid using the Hodge dual, one can pass to first order formalism by introducing some auxiliary fields. Then, we have S[φ, λ,λ] = Σ L (2) (φ, . . . , dφ, . . . , V ±± , ω) . (1.3) where the 2-form action L (2) (φ, . . . , dφ, . . . , V ±± , ω) depends upon the fields (φ m , λ m ,λ m ), their differentials (dφ m , dλ m , dλ m ), the 2d vielbeins V ±± and the SO(1, 1) spin connection ω. To make the supersymmetry manifest, one can rewrite the action (1.1) in the superspace formalism by condensing all fields into a superfield Φ m = φ m + λ m θ + +λ m θ − + f θ + θ − (1.4) (we introduce the two anticommuting coordinates θ ± and their corresponding derivatives D ± and the auxiliary field f ) as follows: The integration is extended to the superspace using the Berezin integration rules. In the same way as above, in order to generalise it to any super Riemann surface SΣ or, more generally, to any complex D = 1 supermanifold, we need to rewrite the action (1.5) as an integral of an integral form on SΣ. As will be explained in the forthcoming section and as is discussed in the literature [3,4,10], the action L(Φ) becomes a (2|0)-superform L (2|0) (Φ, dΦ, V ±± , ψ ± , ω). It is well wnown that a superform cannot be integrated on the supermanifold SΣ and it must be converted into an integral form by multiplying it by a PCO Y (0|2) (V ±± , ψ ± , ω). The latter is the Poincaré dual of the immersion of the bosonic submanifold into the supermanifold SΣ. The action is written as: The PCO Y (0|2) is a (0|2) integral form and is d-closed and not exact. If we shift it by an exact term Y + dΛ, the action is left invariant if dL (2|0) (Φ, dΦ, V ±± , ψ ± , ω) = 0. That can be obtained in presence of auxiliary fields and can be verified using the Bianchi identities for the torsion T ±± , the gravitinos field strengths ρ ± and the curvature R. The choice of the PCO allows to interpolate between different superspace frameworks with different manifest supersymmetries.
The action (1.6) is invariant under superdiffeomorphisms by construction since it is an intergral of a top integral form. Therefore, it can be written for any solution of the Bianchi identity for any supermanifold compatible with them. As will be show in the following, we can write the most general solution of the Bianchi identities in terms of an unconstrained superfield E.
The paper has the following structure: in sec. 2, we summarize the geometry of integral forms. In sec. 3, we discuss the PCO's and their properties. In sec. 4, we discuss the action (1.2) in components and the Bianchi identities for the field strengths for the superfield Φ. In subsec. 4.1, we derive the action (1.6) and we show the relation between the component action and the superfield action. In sec. 5, we consider the preliminary case of rigid curved supermanifold based on the supercoset space Osp(1|2)/SO(1, 1). We show the relation between the volume form and the curvature. In sec. 6, we study the general case of 2d supergravity N=1. In particular, it is shown that the PCO in the curved space are closed because of the torsion constraints.

Integral forms and integration
The integral forms are the crucial ingredients to define a geometric integration theory for supermanifolds inheriting all good properties of differential forms integration theory in conventional (purely bosonic) geometry. In the following section we briefly describe the notations and the most relevant definitions (see [9], [2] and also [4,3,5]).
We consider a supermanifold with n bosonic and m fermionic dimensions, denoted here and in the following by M (n|m) , locally isomorphic to the superspace R (n|m) . The local coordinates in an open set are denoted by (x a , θ α ). When necessary, in sections 4, 5 and 6, we introduce supermanifolds locally isomorphic to the complex superspace C (n|m) . In this case the coordinates will be denoted by (z a ,z a , θ α ,θ α ); the formalism, mutatis mutandis, is the same.
The two integer numbers p and q correspond respectively to the form number and the picture number, and they range from −∞ to +∞ for p and 0 ≤ q ≤ m. The index b on the delta δ (b) (dθ α ) denotes the degree of the derivative of the delta function with respect to its argument. The total picture of ω (p|q) corresponds to the total number of delta functions and its derivatives. We call ω (p|q) a superform if q = 0 and an integral form if q = m; otherwise it is called pseudoform. The total form degree is given by p = r + s − i=q i=1 b i since the derivatives act effectively as negative forms and the delta functions carry zero form degree. We recall the following properties: The index α is not summed. The indices a 1 . . . a r and β 1 . . . β q are anti-symmetrized, the indices α 1 . . . α s are symmetrized because of the rules of the graded wedge product: As usual the module of (p|q) pseudoforms is denoted by Ω (p|q) ; if q = 0 or q = m it is finitely generated.
It is possible to define the integral over the superspace R (n|m) of an integral top form ω (n|m) that can be written locally as: where sdet(E) is the superdeterminant of the supervielbein (E a , E a ).
The integral form ω (n|m) can be also viewed as a superfunction ω(x, θ, dx, dθ) on the odd dual 2 T * (R (n|m) ) acting superlinearly on the parity reversed tangent bundle ΠT (R (n|m) ), and its integral is defined as follows: (2.14) where the order of the integration variables is kept fixed. The symbol [dxdθ d(dx)d(dθ)] denotes the Berezin integration "measure" and it is invariant under any coordinate transformation on It is a section of the Berezinian bundle of T * (R (n|m) ) (a super line bundle that generalizes the determinant bundle of a purely bosonic manifold). The sections of the determinant bundle transform with the determinant of the jacobian and the sections of the Berezinian with the superdeterminant of the super-Jacobian. The berezinian bundle of T * M (n|m) is always trivial but the berezinian bundle of M (n|m) in general is non trivial. The integrations over the fermionic variables θ and dx are Berezin integrals, and those over the bosonic variables x and dθ are Lebesgue integrals (we assume that ω(x, θ, dx, dθ) has compact support in the variables x and it is a product of Dirac's delta distributions in the dθ variables). A similar approach for a superform would not be possible because the polynomial dependence on the dθ leads to a divergent integral.
As usual, this definition can be extended to supermanifolds M (n|m) by using bosonic partitions of unity. See again Witten [9] for a more detailed discussion on the symbol [dxdθd(dx)d(dθ)] and many other important aspects of the integration theory of integral forms.
According to the previous discussion, if a superform ω (n|0) with form degree n (equal to the bosonic dimension of the reduced bosonic submanifold M (n) ֒→ M (n|m) ) and picture number zero is multiplied by a (0|m) integral form γ (0|m) , we can define the integral on the supermanifold of the product: This type of integrals can be given a geometrical interpretation in terms of the reduced bosonic submanifold M (n) of the supermanifold and the corresponding Poincaré dual (see [4]).
2 In order to make contact with the standard physics literature we adopt the conventions that d is an odd operator and dx (an odd form) is dual to the even vector ∂ ∂x . The same holds for the even form dθ dual to the odd vector ∂ ∂θ . As clearly explained for example in the appendix of the paper [7] if one introduces also the natural concept of even differential (in order to make contact with the standard definition of cotangent bundle of a manifold) our cotangent bundle (that we consider as the bundle of one-forms) should, more appropriately, be denoted by ΠT * .

PCO's and their properties.
In this section we recall a few definitions and useful computations about the PCO's in our notations. For more details see [8] and [10].
We start with the Picture Lowering Operators that map cohomology classes in picture q to cohomology classes in picture r < q.
Given an integral form, we can obtain a superform by acting on it with operators decreasing the picture number. Consider the following integral operator: where D is an odd vector field with {D, D} = 0 3 and ι D is the contraction along the vector D.
The contraction ι D is an even operator.
For example, if we decompose D on a basis D = D α ∂ θ α , where the D α are even coefficients and {∂ θ α } is a basis of the odd vector fields, and take ω = ω β dθ β ∈ Ω (1|0) , we have In addition, due to {D, D} = 0, we have also that ι 2 D = 0. The differential operator δ(ι α ) ≡ δ (ι D ) -with D = ∂ θ α -acts on the space of integral forms as follows (we neglect the possible introduction of derivatives of delta forms, but that generalization can be easily done): where the sign ± is due to the anticommutativity of the delta forms and it depends on the index α. We have used also the fact that exp itι α represents a finite translation of dθ α . The result contains m − 1 delta forms, and therefore it has picture m − 1. It follows that δ(ι α ) is an odd operator.
We can define also the Heaviside step operator Θ (ι D ) : The operators δ (ι D ) and Θ (ι D ) have the usual formal distributional properties: In order to map cohomology classes into cohomology classes decreasing the picture number, we introduce the operator (see [8]): (3.20) In the simplest case D = ∂ θ α we have: The operator Z α is the composition of two operators acting on different quantities: ∂ θ α acts only on functions, and δ(ι α ) acts only on delta forms.
In order to further reduce the picture we simply iterate operators of type Z. An alternative description of Z in terms of the Voronov integral transform can be found in [10].
The Z operator is in general not invertible but it is possible to find a non unique operator Y such that Z • Y is an isomorphism in the cohomology. These operators are the called Picture Raising Operators. The operators of type Y are non trivial elements of the de Rham cohomology.
We apply a PCO of type Y on a given form by taking the graded wedge product; given ω in More general forms for Z and Y can be constructed, for example starting with the vector Q = ∂ θ + θ∂ x .
For example, if ϕ = g(x)θdxδ(dθ) is a generic top integral form in Ω (1|1) R (1|1) , the explicit computation using the formula Z = [d, Θ(ι Q )] is: The last expression is clearly closed. Note that in the above computations we have introduced formally the inverse of the (commuting) superform dθ. Using a terminology borrowed from superstring theory we can say that, even though in a computation we need an object that lives in the Large Hilbert Space, the result is still in the Small Hilbert Space.
Note that the negative powers of the superform dθ are well defined only in the complexes of superforms (i.e. in picture 0). In this case the inverse of the dθ and its powers are closed and exact and behave with respect to the graded wedge product as negative degree superforms of picture 0. In picture = 0 negative powers are not defined because of the distributional relation dθδ (dθ) = 0.
A PCO of type Y invariant under the rigid supersymmetry transformations (generated by the vector Q) δ ǫ x = ǫθ and δ ǫ θ = ǫ is, for example, given by: We have:
Computing the differential of each superfield we have the following relations: The last field F is the auxiliary field and therefore it vanishes when the theory is on-shell. Before writing the rheonomic Lagrangian for the multiplet, we first write the equations of motion. If we set F = 0, then we see from the last equation that They implies that the superfield W is holomorphic W = W (z) and the superfieldW is antiholomorphic. Then, we can write eqs. (4.30) with these constraints: The consistency of the last two equations (d 2 = 0), implies that ∂ ++ ∂ −− Φ = 0, Then, we get that the rheonomic equations (4.30) are compatible with the set of the equations of motion which are the free equations of D = 2 multiplet. The Klein-Gordon equation in D = 2 implies that the solution Φ = Φ h (z) + Φh(z) is splitted into holomorphic and anti-holomorphic parts and therefore we get the on-shell matching of the degrees of freedom. In particular we can write on-shell holomorphic and anti-holomorphic superfields factorizing into left-and right-movers.
Let us now write the action. We introduce two additional superfields ξ andξ. Then, we have [1] The equations of motion are given by They imply the on-shell differentials (4.32), the equations of motion (4.33), and the relations expressing the additional auxiliary fields ξ andξ in terms of Φ. It is easy to check that they are consistent: acting with d on the third and on the fourth equations, and using the fifth equation, one gets a trivial consistency check. In the same way for all the others.
The action is a (2|0) superform, it can be verified that it is closed by using only the algebraic equations of motion for ξ andξ, which are solved in (4.37) and using the curvature parametrization dΦ, dW, dW and dF given in (4.30). Note that those equations are off-shell parametrizations of the curvatures and therefore they do not need the equations of motion of the lagrangian (4.35).

Sigma Model on Supermanifolds
To check whether this action leads to the correct component action we use the PCO Y (0|2) = θ + θ − δ(ψ + )δ(ψ − ). Then we have 4 where ξ 0 andξ 0 are the first components of the superfields ξ andξ. Eliminating ξ 0 andξ 0 one finds the usual equations of motion for the D = 2 free sigma model.
Choosing a different PCO of the form 5 which has again the correct picture number and is cohomologous to the previous one, leads to the superspace action (listing only the relevant terms) where ι ± are the derivatives with respect to ψ ± . The contractions give ι + dΦ = D + Φ and ι − dΦ =D − Φ. Then we get the superspace action  1). The MC equations can be easily computed by using the notation V ±± , ψ ± for the MC forms and ∇ for the SO(1, 1) covariant derivate. The MC forms V ±± have charge ±2, while ψ ± have charge ±1. Then, we have Computing the Bianchi identities, we have All the expressions have been constructed to respect the charge assignements. The superform R (2|0) is neutral and invariant.
The volume form is computed by observing that where Sdet(E) is the Berezinian of the supervielbein E of the super-coset manifold OSp(1|2)/SO(1, 1). It is susy invariant and it is closed. This can be checked by observing that Figure 1: The pseudoform complexes.
The first equality follows from the neutrality of the volume integral form Vol (2|2 and therefore, we can use the covariant derivative instead of the differential d. The covariant differential ∇ acts as derivative, and this leads to the last two lines. The third line cancels because of the Dirac delta functions multiplied by ψ ± , the fourth line vanishes since V ±± ∧ V ±± = 0. The relevant set of pseudo-forms are contained in the rectangular diagram in fig. (1). The vertical arrows denote a PCO which increases the picture number. There are additional sets outsides the present rectangular set, but are unessential for the present discussion since they do not contain non trivial cohomology classes (see [6]).
Let us discuss the relevant cohomology spaces.
It is easy to check the closure of all generators. In addition, all generators are neutral, for instance in V ++ ∧ δ ′ (ψ + ) the charge +2 is compensated by the negative charge −1 of δ(ψ + ) and the negative charge of the derivative of the delta form. It can be shown that This equation is rather suggestive. If we consider the cohomology class in H (0|2) as the total PCO Y (0|2) and if we consider the cohomology class in H (2|0) as the Kähler form K (2|0) of our complex supermanifold, we find which is the super-Liouville form.
The PCO operator Y (0|2) easily factorizes as . This factorization is very useful for the equations below.
Finally, we want to show that acting with the PCO Z we can map the volume form Vol (2|2) into the Kähler form K (2|0) . For that we define the PCO's Acting with the first one on Vol (2|2) (and using the fact that dVol (2|2) = 0, we have It can be noticed that the final expression is chargeless, it is d-closed and it is expressed in terms of supersymmetric invariant quantities. Furthermore, in the first step of the computation we have used objects in the Large Hilbert Space, but the final result is again in the Small Hilbert Space (there are no inverse of ψ's) 6 .
Let us act with the second PCO, Z . Again, we use the fact that the result of (5.50) is d-closed. We have Again, in the intermediate steps we have expressions living into the Large Hilbert Space, but the final result is in the Small Hilbert Space and it is polynomial in the MC forms. This clearly shows how to act with the PCO's on the cohomology classes mapping from cohomology to cohomology. The same result is obtained by exchanging the two PCO's. The final result is mapping the volume form into the curvature of the manifold.

D=2 Supergravity
As is well know, there are no dynamical graviton and gravitino in 2 dimensions, nonetheless the geometric formulation of supergravity is interesting and it is relevant in the present work.
The definitions are where ω is the SO(1, 1) spin connection. These curvatures satisfy the following Bianchi identities The Bianchi identities can be solved by the following parametrization There are no dynamical constraint on E(x, θ) since there are no equations of motion. Nonetheless if we impose that D + E = D − E = 0 (6.56) we immediately get ∂ ++ E = ∂ −− E = 0, and therefore E = const. If we set E = Λ we get the well-known anti-de-Sitter solution T ±± = 0 , (6.57) describing the coset space Osp(1|2)/SO(1, 1).
Going back to a generic E, we consider the volume form which is closed since it is a top integral form Now we act with the PCO Z + = [d, Θ(ι + )] and we get Notice that the Dirac delta's do not carry any charges and the PCO Z + has negative charge. Therefore, the result is consistent. In addition, the r.h.s. is closed, as can be easily verified by using the Bianchi identites: since ρ − ∧ ρ − = 0 and ψ − δ(ψ − ) = 0.
Let us now act with the second PCO Z − . There are two ways to perform the computation: either using the complete expression given in the first line of (6.59) or using the Bianchi identities. With the second proposal we observe: Notice that acting both with Z − and Z + the total charge is zero as for R. The result is closed, dR = 0 , and it confirms the formula obtained for the curved rigid supermanifold (5.52). The result (6.61) is valid for any superfield E.
The PCO Y are defined as in the flat case We can easily check their closure: ∇Y + = ∇V ++ δ ′ (ψ + ) + V ++ δ ′′ (ψ + )∇ψ + (6.63) Therefore, it is closed if T ++ = 0. In the same way we get for Y − . Finally, we observe that This can also be obtained by observing that since V ++ ∧ ρ + = 0. In the same way, Z − Y − = 1/2. These equations are valid for any E. Using eq. (6.64), one can define an integral over the supermanifold: that might be interpreted as the Euler characteristic for supermanifolds.

Conclusions
To complete the program, one has to use the PCO's (6.62), for a generic background E to rewrite the action (1.6) in that background. Choosing a different PCO gives an equivalent string sigma model with different manifest supersymmetry.
Finally, we would like to point out the relation between the PCO used in the action, and the conventional PCO used for correlation computations in string theory. The latter can be written as follows for the left-and right-moving sector, where c ±± are the Einstein's ghosts and γ ± are the superghosts. They should be compared with (6.62). We further notice that BRST transformations of the D=2 supervielbeins V ±± , ψ ± are given by QV ±± = dc ±± + . . . , Qψ ± = dγ ± + . . . (7.68) where the ellipsis denotes non-linear terms, and therefore there should be a relation between the two types of PCO. We leave this to further investigations.