Study on Chohomology of Supermanifolds

BRAC University Journal, Vol. III, No. 1, 2006, pp. 59-63

STUDY ON CHOHOMOLOGY OF SUPERMANIFOLDS

Khondokar M. Ahmed Department of Mathematics University of Dhaka Dhaka -1000, Bangladesh

ABSTRACT

In this present paper de Rham cohomology of graded manifolds, cohomology of graded differential forms and cohomology of DeWitt supermanifolds are studied. The structure of a G-supermanifold in general is not acyclic. So, the cohomology is defined via the complex of graded differential forms. This situation is investigated together with the graded Dolbeault cohomology of complex G- supermanifolds. A theorem is established that the structure sheaf of any DeWitt G-supermanifold is acyclic.

Key words: Manifolds, chomology, supermanifolds.

I. INTRODUCTION d 1 d 0 → Ñ → A → Ω A → 2 d The aim of this paper is to unfold a basic Ω A → L (2.1) cohomology theory for supermanifolds. This is exact. This 'graded Poincare′ Lemma' is most cohomology does not embody only trivial easily proved by working in local coordinates and extensions of results valid for differentiable proceeding on the analogy of the usual Poincare′ manifolds. For instance the natural analogue of the Lemma. By defining the de Rham cohomology of de Rham theorem does not holds in general and • (X,A), denoted by H DR (X , A), as the also in the case of complex super-manifolds there is generally no analogue of the Dolbeault theorem. cohomology of the complex of graded vector As a result the sturecture sheaf of a supermanifolds spaces A • (X ) , from (2.1) and using the ordinary does not need to be cohomology trivial [2]. The de Rham theorem, we obtain the following result fact that the cohomology of the complex of global (cf. [8] Theorem 4.7.). graded differential forms on a G-supermanifold ( M,A) depends on the G-supermanifold structure ( Proposition 2.1. There is a canonical isomorphism M,A) so that super de Rham cohomology is a fine k k DR A)' DR for all invariant of the supermanifold structure [3]. H (X , H (X ) k ≥ 0. k Here H DR (X ) denotes the usual de Rham II. de Rham COHOMOLOGY OF GRADED cohomology of X. MANIFOLDS III. COHOMOLOGY OF GRADED Graded manifolds are not very interesting as far as DIFFERENTIAL FORMS their cohomology is concerned. In the real case, the structure sheaf of a graded manifold (X,A) is fine Let (M,A) be a G-supermanifold. The sheaves k and therefore A and all sheaves Ω A of graded Ω k ⊗ B of smooth B -valued differential differential forms are acyclic. This implies that the Ñ L L • forms on M provide a resolution of the constant cohomology of the complex Ω A (X) coincides sheaf B on M, in the sense that the differential with the de Rham cohomology of X. In the L complex analytic case, a similar argument allows complex of sheaves of graded-commutative BL - one to prove a Dolbeault-type theorem. • 0 ∞ algebras Ω ⊗ Ñ B (with Ω ⊗ Ñ B ´C L ) is a The complex of sheaves A • is exact and moreover, L L it is a resolution of the constant sheaf Ñ on X; i.e., resolution of the constant sheaf BL , i.e. the the sequence of sheaves of Ñ-modules sequence Khondokar M. Ahmed

Ak A1 k ∞ ω = dz ∧ ∧ dz ω ∈ Ω H (U) be ∞ d 1 d L A1LAk 0 → B → C L → Ω ⊗ Ñ B → L L an ∞ graded differential k-form on U (k>0); let 2 H Ω ⊗ Ñ BL → L (3.1) us set is exact. The cohomology associated with this ˆ k Ak −1 Kω(z) = (−1) dz ∧L∧ complex via the global section functor Γ(⋅, M ) , t • A1 B k −1 i.e. the cohomology of the complex Ω ⊗ Ñ BL , is dz z t ω (tz)dt. ∫ BA1LAk −1 • 0 denoted by H DR (M , B ) and is called the B - L L k We have an isomorphism [2] Ω G (U ) ' valued de Rham cohomology of M. Since BL is a k ∞ H ; it is therefore possible to introduce a finite dimensional real vector space, the universal Ω Ñ BL coefficient theorem [6] entails the isomorphism homotopy operator • • k k −1 H DR (M , BL ) ' H DR (M ) ⊗ Ñ BL . (3.2) K : Ω G (U ) → Ω G(U), defined by By virtue of the de Rham theorem, equation (3.2) K( a) Kˆ a. can be equivalently written as ω ⊗ = ω ⊗ • • One can indeed verify easily that H DR (M , B ) ' H (M , B ) . (3.3) L L dKλ + Kdλ = λ for any section λ ∈ Gk (U ) , By H • (M ,⋅) we designate interchangeable the so that, if dλ = 0 then λ = d(Kλ) . The case 1ech or sheaf cohomology functor, which coincide since the base space is paracompact. k = 0 has been left out. However, if f ∈ G (U ) , by writing f as f = f ⊗ a with f ∈ In order to gain information not on the topological ∑ i i i or smooth structure of M, but rather on its G- ∞ H (U ) and ai ∈ BL , the condition df = 0 implies supermanifold structure, we therefore need to define a new cohomology, obtained via a resolution directly that f is a constant in BL . of BL , different from the differential complex k Definition 3.2. Given a G-supermanifold (M,A), (3.1). We consider the sheaves Ω A of graded the cohomology of the complex differential forms. The following result is a d 1 d A(M) → Ω A generalization of the usual Poincare′ lemma (cf. (M ) → 2 [5]). Ω A (M ) → L (3.5) • Proposition 3.1. Given a G supermanifold (M,A), denoted by H SDR (M , A), is called the super de the differential complex of sheaves of graded B - Rham cohomology of (M,A). L The operation of taking the SDR cohomology of a algebras on M G-supermanifold is functorial. Indeed, given a G- d 1 d 0 → BL → A → Ω A → morphism ( f ,φ) : (M , A) → (N, B), it is easily 2 • Ω A → L (3.4) proved that the morphism Ω B (N) → is a resolution of B . • L f∗Ω A(M) induced byφ commutes with the Proof. Since the claim to be proved is a local exterior differential and therefore yields a m,n ¤ matter, we may assume that (M,A) = (BL ,G); morphism of graded BL -modules φ : moreover, it is enough to • • H SDR (N,B ) → H SDR (M ,A ) . It should be show that, if U is an open ball around the origin in • m,n noticed that the functor H SDR (⋅) does not fulfill BL then any closed graded differential k-form k the Eilenberg-Steenrod [10] axiomatics for λ ∈ Ω G(U) is exact; i.e., there exists a graded cohomology (if it did, it would coincide with the k−1 differential (k -1)-form η ∈ Ω G (U) such that BL -valued de Rham cohomology functor) since it λ = dη . Given coordinates (z1 , , z m+n ) in U, does not satisfy the excision axiom. Moreover, the L • let functor H SDR (⋅) does not give rise to

60 Study on Chohomology of Supermanifolds topological invariants. On the other hand, it is Proposition 3.3. provides a usefull tool for investigating the cohomological properties of the easily varified that the graded BL -modules k structure sheaf of a G- supermanifold. For instance, H SDR (M , A)are invariants associated with the it suffices to exhibit a G-supermanifold (M,A) such G-supermanifold structure of M. Indeed, if 1 1 that H SDR (M , A) ≠ H DR (M , BL ) to deduce ( f ,φ) : (M , A) → (N, B) is a G-isomorphism, it that, in general, the sheaf A can not be expected to ¤ • is easily proved that φ : H SDR (N, B) be acyclic. • → H SDR (M , A). is an isomorphism. IV. COHOMOLOGY OF DEWITT SUPERMANIFOLDS The most natural thing to do to gain insight into the geometric significance of the groups Considering in M the fine topology we study the k cohomology of a De Witt supermanifold (M,A); H SDR (M , A) - which, as a matter of fact, are this is advantageous because in this way M is para- graded BL -modules - is to compare them with the compact. Thus we continue to confuse the sheaf cohomology groups H k (M , B ) , which have a and 1ech cohomologies with coefficients in L sheaves on M. natural structure of graded BL -modules as well. k k We need the following Lemma, which is obtained The morphisms Ω A (M ) → Ω from a result given in [19] by strengthening certain induced by the morphism (M ) ⊗ Ñ BL hypotheses. ∞ δ : A → C L give rise to a morphism of Lemma 4.1. Let X and Y be topological spaces, differential complexes, which induces in with Y locally euclidean and F a sheaf of abelian cohomology a morphism of graded BL -modules groups on X; let we assume that all groups k k k · : H SDR (M , A) → H (X , F) are finitely generated. Then for all k n ≥ 0 there is an exact sequence of abelian H DR (M , BL ) ∀k ≥ 0. (3.6) 0 groups In degree zero, · is an isomorphism, in that one has manifestly j k 0 C 0 0 → ⊕ H (X , F) ⊗ Ù H (Y, Ù) → H SDR (M , A)' (BL ) ' H DR (M , BL ) , j+k=n where C is the number of connected components of H n (X ×Y, π −1 F) → M, which we assume to be finite. In degree higher → ⊕ Tor[H j (X , F), H k (X , Ù)] → 0 than zero, we have, as a straightforward application j+k =n+1 of the abstract de Rham theorem, the following result. whereTor [⋅,⋅] defines the torsion product [6], [7] and π : X ×Y → X is the canonical projection. Proposition 3.3. Let (M,A) be a G-supermanifold k p m,n and fix an integer q ≥ 1. If H (M ,Ω A ) = 0 Proposition 4.2. The G-supermanifold (BL , G) for 0 ≤ p ≤ q −1 and 1 ≤ k ≤ q , there are is cohomologically trivial: isomorphisms k m,n H (BL , G) = 0 ∀k > 0 (4.1) k k H SDR (M , A)' H (M , BL ) for 1 ≤ k ≤ q . Proof. In view of the definitions of the sheaves GH and G, one has an isomorphism

m,n −1 ∞ m n From equation (3.3), still working under the G' (σ ) (C Ñ ⊗ ∧ Ñ ⊗BL ). hypotheses of Proposition 3.3., we obtain Therefore, applying Lemma 4.1. with the following isomorphisms identifications: k k m m,n H SDR (M , A)' H (M , BL ) X = Ñ , Y = n L , for 0 ≤ k ≤ q.. ∞ m n F = C Ñ ⊗ ∧ Ñ ⊗BL ,

61 Khondokar M. Ahmed

k m,n We obtain ( since H (n L , Ù) = 0 for k > 0 and Proof. Any p ∈ M B has a system of 0 m,n neighbourhoods m such that for all H (n L , Ù) = Ù ) −1 W ∈ m the supermanifold (Φ (W ), A −1 ) is k m,n |Φ (W ) H (BL , G) ' m,n k m ∞ m n isomorphic to (BL , G); therefore, A −1 is H (Ñ , C Ñ ⊗ ∧ Ñ ⊗BL ). |Φ (W ) ∞ m acyclic. We are then in the hypotheses of the Now, since the sheaf of rings C Ñ is fine, the ∞ m n ∞ m inverse image in cohomology [2] and hence sheaf C Ñ ⊗ ∧ Ñ ⊗BL of C Ñ -modules is soft and therefore is acyclic, which yields the k k H (M , A)' H (M ,Φ A ), k ≥ 0. sought result. B ∗ is fine by Lemma 4.3., and hence acyclic, so that Coarse partitions of unity. DeWitt supermani- we achieve the thesis. folds do not admit partitions of unity in a strict sense, that is to say, there cannot exist partitions of We notice that the same procedure that brought to unity subordinated to any locally finite cover, since Theorem 4.5. can be applied to the structure the structure sheaf of a DeWitt supermanifold is sheaves of an H ∞ or GH ∞ DeWitt not soft and therefore is not even fine. However, supermanifolds, which are therefore acyclic as any DeWitt supermanifold has a particular kind of well. partition of unity, that we call a coarse partition of unity. Corollary 4.6. Any locally free A module F is acyclic. Proof. Let us at first assume that F Lemma 4.3. Let (M,A) be a DeWitt supermanifold, trivializes on a coarse cover. Then, since Φ ∗ F is a with body M B and projection φ : M → M B . Φ ∗ A -module, the same proof of the previous For any locally finite coarse cover U = {U j } of M Proposition applies. Now we must prove that F there exists a family {g }of global sections of A actually trivializes on a coarse cover. Without any j loss of generality we may assume that such that m,n (M,A) = (B , G) and that F trivializes on subsets (a) Supp g ⊂ U ; L j j m,n of B which are diffeomorphic to open balls. (b) g = 1. L ∑ j j Let U be one of these subsets; then F(U)'G p / q (U ) . In view of the definition of the Corollary 4.4. Let C be a locally finite coarse ( sheaf G, if V is any other set of this kind such that cover of M. Then H ( U,F) = 0, k > 0, for any Φ −1Φ(U ) = Φ −1Φ(V ) = W , then F(U)' sheaf F of A-modules. p|q If we consider in M the coarse topology (let us F (V ) , so that one has F |W = G |W . denote the resulting space by M DW , the sheaf A is apparently fine; however, this does not allow us to For instance, the sheaf of derivations DerA and k conclude that the sheaf cohomology of A is trivial, sheaves Ω A of graded differential forms on A since M DW is not paracompact. In any case, one (M, ) are acyclic. can conclude that the 1ech cohomology ( • SDR cohomology of DeWitt supermanifolds. H (M DW , A) (or the cohomology The previous results have an immediate ( • H (M , F), where F is any A-module) is trivial, consequence in connection with the super de Rham DW cohomology of DeWitt supermanifolds. since the direct limit over the covers involved in the definition of the 1ech cohomology can be taken Proposition 4.7. [9] The super de Rham on coarse covers. cohomology of a DeWitt supermanifold (M,A) is

isomorphic with the B -valued de Rham Theorem 4.5. The structure sheaf A of a DeWitt G- L supermanifold (M,A) is acyclic. cohomology of the body manifold M B :

62 Study on Chohomology of Supermanifolds

• • ∞ ∞ H SDR (M ) ' H DR (M B ) ⊗ Ñ BL . (4.2) ' G by the very definition of the sheaf G , the Proof. We have already seen that the sheaves of m,n result is proved for the supermanifold (BL , k graded differential forms Ω A are acyclic, G ∞ ) . From ( cf. [1] k p H (M ,Ω A)=0 for all k > 0 and p ≥ 0 . Proposition 4.11.) the result for a generic G ∞ Accordingly, Proposition 1.2. [9] implies DeWitt supermanifold now follows.

• • H SDR (M ) ' H (M , BB ) . (4.3) REFERENCES On the other hand, M is a fibration over with M B [1] K. M. Ahmed: A note on sheaves and a contractible fibre, so that cohomology on manifold, Ganit: J. • • Bangladesh Math. Soc., 22, pp 41-50. (2002) H DR (M ) ' H DR (M B ) and equation (4.3) is equivlent to equation (4.2) [2]. [2] C. Bartocci and U. Bruzzo: Cohomology of supermanifolds, J. Math. phys. 28, pp 2363- Dolbeault theorem. Let (M,B) be an 2368. (1987) (m, n)-dimensional complex G-supermanifold. p We have that Ω B is the sheaf of holomorphic [3] C. Bartocci and U. Bruzzo: Cohomology of the p,q structure sheaf of real and complex graded p-forms on(M,B), while Ω I is the sheaf supermanifolds, J. Math. phys 29, pp 1789- of graded differential forms of type (p, q). Here I is 1795. (1988) the complexification of the sheaf A, i.e. I =A ⊗ ÑÂ. Proposition 4.8. Let (M,B) be a complex DeWitt G- [4] G.E. Bredon: sheaf theory, New York: supermanifold. There are isomorphisms of graded McGraw-Hill. (1967)

CL -modules [5] U. Bruzzo: Supermanifolds, super-manifold p,q q p H (M , B)' H (M ,Ω B). cohomology, and super vector bundles, in : ∂ Differential geometric methods in theoretical physics, K. Bleuler and M. Werner eds., NATO Cohomology of G ∞ DeWitt supermanifolds. ASI Ser. C Kluwer, Dordrecht, 250, pp 417- Theorem 4.5., which states the acyclicity of the 440. (1988) structure sheaf of a DeWitt G-supermanifold can be shown to hold true also in the case of the sheaf [6] R. Godement: The′orie des faisceaux, Hermann, Paris. (1964) A ∞ of G ∞ functions on a DeWitt supermanifold. [7] P. J. Hilton and U. Stammbach: A course in Theorem 4.9. The structure sheaf of a G ∞ DeWitt homological algebra, New York: Springer- supermanifold is acyclic. Verlag. (1971)

[8] B. Kostant: Graded manifolds, graded Lie Proof. Working as in Lemma 4.3., one can theory, and prequantization, in: Differential ∞ construct a coarse G partition of unity on M, so geometric methods in mathematical physics, that the sheaf Φ A ∞ is fine and therefore is K. Bleuler and A. Reetz eds., Lecture Notes ∗ Math. Springer-Verlag, Berlin, 570, pp 177- acyclic. Let us now consider for a while 306. (1977) ∞ m,n ∞ the G DeWitt supermanifold (BL , G ) . Lemma 4.1. implies [9] J. M. Rabin: Supermanifold cohomology and the Wess-Zumino term of the covariant superstring action, Commun. Math. Phys. 108, k m,n m,n −1 m,n ∞ H (BL ,(σ ) (σ )∗ G ) ' pp 375-389. (1987) k m m,n ∞ H ( Ñ (σ )∗ G ) =0 [10] E. H. Spanier: Algebraic topology, New York: m,n −1 m,n ∞ McGraw-Hill. (1966) for all k > 0. Since (σ ) (σ )∗ G