aa r X i v : . [ m a t h . AG ] J un On SUSY curves
R. Fioresi, S. D. Kwok
Dipartimento di Matematica, Universit`a di BolognaPiazza di Porta S. Donato, 5. 40126 Bologna. Italy. e-mail: rita.fi[email protected], [email protected]
In this note we give a summary of some elementary results in the theory ofsuper Riemann surfaces (SUSY curves), which are mostly known, but arenot readily available in the literature. Our main source is Manin, who hasprovided with a terse introduction to this subject in [10]. More recentlyFreund and Rabin have given important results on the uniformization (see[12]) and Witten has written an account of the state of the art of this subject,from the physical point of view, in [15].The paper is organized as follows.In Sections 2 and 3, we are going to recall briefly the main definitions ofsupergeometry and study in detail the examples of super projective space andΠ-projective line, which are very important in the theory of SUSY curves.In Section 4 we discuss some general facts on SUSY curves, includingthe theta characteristic , while in Section 5 we prove some characterizationresults concerning genus zero and genus one SUSY curves.
We are going to briefly recall some basic definitions of analytic supergeometry.For more details see [10], [11], [14], [4], [7] and the classical references [2], [3].Let our ground field be C .A superspace S = ( | S | , O S ) is a topological space | S | endowed with asheaf of superalgebras O S such that the stalk at a point x ∈ | S | , denoted by O S,x , is a local superalgebra.A morphism φ : S −→ T of superspaces is given by φ = ( | φ | , φ ∗ ), where | φ | : | S | −→ | T | is a map of topological spaces and φ ∗ : O T −→ φ ∗ O S is such1hat φ ∗ x ( m | φ | ( x ) ) = m x where m | φ | ( x ) and m x are the maximal ideals in thestalks O T, | φ | ( x ) and O S,x respectively.The superspace C p | q is the topological space C p endowed with the follow-ing sheaf of superalgebras. For any open subset U ⊂ C p O C p | q ( U ) = Hol C p ( U ) ⊗ ∧ ( ξ , . . . , ξ q ) , where Hol C p denotes the complex analytic sheaf on C p and ∧ ( ξ , . . . , ξ q ) isthe exterior algebra in the variables ξ , . . . , ξ q .A supermanifold of dimension p | q is a superspace M = ( | M | , O M ) whichis locally isomorphic to C p | q , as superspaces. A morphism of supermanifoldsis simply a morphism of superspaces.We now look at an important example of supermanifold, namely the projective superspace .Let P m = C m +1 \ { } (cid:14) ∼ be the ordinary complex projective space ofdimension m with homogeneous coordinates z , . . . , z m ; [ z , . . . , z m ] denotesas usual an equivalence class in P m . Let { U i } i =1 ,...,m be the affine cover U i = { [ z , . . . , z m ] | z i = 0 } , U i ∼ = C m . On each U i we take the global ordinarycoordinates u i , . . . , ˆ u ii , . . . u im , u k := z k /z i (ˆ u ii means we are omitting thevariable u ii from the list). We now want to define the sheaf of superalgebras O U i on the topological space U i : O U i ( V ) = Hol U i ( V ) ⊗ ∧ ( ξ i , . . . ξ in ) , V open in U i where Hol U i is the sheaf of holomorphic functions on U i and ξ i , . . . ξ in are oddvariables.As one can readily check U i = ( U i , O U i ) is a supermanifold, isomorphicto C m | n . We now define the morphisms φ ij : U i ∩ U j
7→ U i ∩ U j , where thedomain is thought as an open submanifold of U i , while the codomain as anopen submanifold of U j . By the Chart’s Theorem the φ ij ’s are determinedby the ordinary morphisms, together with the choice of m even and n oddsections in O U i ( U i ∩ U j ). We write: φ ij : ( u i , . . . , ˆ u ii , . . . u im , ξ i , . . . , ξ in ) (cid:18) u i u ij , . . . , u ij , . . . , u im u ij , ξ i u ij , . . . , ξ in u ij (cid:19) (1)2here on the right hand side the 1 /u ij appears in the i th position and the j th position is omitted. As customary in the literature, the formula (1) is just asynthetic way to express the pullbacks: φ ∗ ij ( u jk ) = u ik u ij , ≤ k = j ≤ m, φ ∗ ij ( ξ jl ) = ξ il u ij , ≤ l ≤ n, One can easily check that the φ ij ’s satisfy the compatibility conditions: φ ij φ ki = φ jk , on U i ∩ U j ∩ U k hence they allow us to define uniquely a sheaf, denoted with O P m | n , hencea supermanifold structure on the topological space P m . The supermanifold( P m , O P m | n ) is called the projective space of dimension m | n .One can replicate the same construction and obtain more generally asupermanifold structure for the topological space: P ( V ) := V \ { } , where V in any complex super vector space.We now introduce the functor of points approach to supergeometry.The functor of points of a supermanifold X is the functor (denoted withthe same letter) X : (smflds) o −→ (sets), X ( T ) = Hom( T, X ), X ( f ) φ = f ◦ φ .The functor of points characterizes completely the supermanifold X : in fact,two supermanifolds are isomorphic if and only if their functor of points areisomorphic. This is one of the statements of Yoneda’s Lemma, for moredetails see [4] ch. 3.The functor of points approach allows us to retrieve some of the geometricintuition. For example, let us consider the functor P : (smflds) o −→ (sets)associating to each supermanifold T the locally free subsheaves of O m +1 | nT ofrank 1 |
0, where O m +1 | nT := C m +1 | n ⊗ O T . P is defined in an obvious wayon the morphism: any morphism of supermanifolds φ : T −→ S defines acorresponding morphism of the structural sheaves φ ∗ : O S −→ φ ∗ O T , so thatalso P ( φ ) is defined.The next proposition allows us to identify the functor P with the functorof points of the super projective space P m | n . Proposition 2.1.
There is a one-to-one correspondence between the two sets: P ( T ) ←→ P m | n ( T ) , T ∈ (smflds) which is functorial in T . In other words P is the functor of points of P m | n ( T ) . roof. We briefly sketch the proof, leaving to the reader the routine checks.Let us start with an element in P ( T ), that is a locally free sheaf F T ⊂ O m +1 | nT of rank 1 |
0. We want to associate to F T a T -point of P m | n that is a morphism T −→ P m | n . First cover T with V i so that F T | V i is free. Hence: F T ( V i ) = span { ( t , . . . , t m , θ , . . . , θ n ) } ⊂ O m +1 | nT ( V i )where we assume that the section t i ∈ O T ( V i ) is invertible without loss ofgenerality, since the rank of F T is 1 | F T ( V i ) = span { ( t /t i , . . . , , . . . , t m /t i , θ /t i , . . . , θ n /t i ) } Any other basis ( t ′ , . . . , t ′ m +1 , θ ′ , . . . , θ ′ n ) of F T ( V i ) is a multiple of ( t , . . . , t m , θ , . . . , θ n ) by an invertible section on V i , hence we have: t ′ j /t ′ i = t j /t i θ ′ k /t ′ i = θ k /t i Thus the functions t j /t i , θ k /t i , which a priori are only defined on open subsetswhere F T is a free module, are actually defined on the whole of the open setwhere t i is invertible, being independent of the choice of basis for F T ( V i ).We have then immediately a morphism of supermanifolds f i : V i −→ U i ⊂ P m | n : f ∗ i ( u i ) = t /t i , . . . , f ∗ i ( u im ) = t m /t i , f ∗ i ( ξ i ) = θ /t i , . . . , f ∗ i ( ξ in ) = θ n /t i (2)where V i = ( V i , O T | V i ) and U i = ( U i , O P m | n | U i ). It is immediate to check thatthe f i ’s agree on V i ∩ V j , so they glue to give a morphism f : T −→ P m | n .As for the vice versa, consider f : T −→ P m | n and define V i = | f | − ( U i ).The morphism f | V i by the Chart’s Theorem corresponds to the choice of m even and n odd sections in O T ( V i ): v , . . . , v m , η , . . . , η n . We can then defineimmediately the free sheaves F V i ⊂ O T | m +1 | nV i of rank 1 | V i as F V i ( V ) := span { ( v | V , . . . , , . . . v m | V , η | V , . . . , η n | V ) } (1 in the i th position). As one can readily check the F V i glue to give a locallyfree subsheaf of O m +1 | nT . 4e now want to define the Π -projective line which represents in somesense a generalization of the super projective space of dimension 1 | P = C \ { } / ∼ be the ordinary complex projective line with ho-mogeneous coordinates z , z . Define, as we did before, the following super-manifold structure on each U i belonging to the open cover { U , U } of P : O U i ( V ) = Hol U i ( V ) ⊗ ∧ ( ξ ), V open in U i , i = 1 ,
2, so that U i = ( U i , O U i ) isa supermanifold isomorphic to C | . At this point, instead of the change ofchart φ , we define the following transition map (there is only one such): ψ : U ∩ U −→ U ∩ U ( u, ξ ) (cid:0) u , − ξu (cid:1) As one can readily check, this defines a supermanifold structure on thetopological space P and we call this supermanifold the Π -projective line P | = ( P , O P ).In the next section we will characterize its functor of points. Π -pro jective line In this section we want to take advantage of the functor of points approach inorder to give a more geometric point of view on the Π-projective line and tounderstand in which sense it is a generalization of the super projective line,whose functor of points was described in the previous section. Let us startwith an overview of the ordinary geometric construction of the projectiveline.The topological space P consists of the 1-dimensional subspaces of C ,that is P = C \ { } / ∼ , where ( z , z ) ∼ ( z ′ , z ′ ) if and only if ( z , z ) = λ ( z ′ , z ′ ), λ ∈ C × . In other words, the equivalence class [ z , z ] ∈ P consistsof all the points in C which are in the orbit of ( z , z ) under the action of C × by left (or right) multiplication.Now we go to the functor of points of P | . A T -point of P | locallyis a 1 | O | T ( V ) ( V is a suitably chosen open in T ). So it islocally an equivalence class [ z , z , η , η ] where we identify two quadruples( z , z , η , η ) ∼ ( z ′ , z ′ , η ′ , η ′ ) if and only if z i = λz ′ i and η i = λη ′ i , i =0 , λ ∈ O T ( V ) × . In other words, exactly as we did before for the case of5 , we identify those elements in C | ( T ) that belong to the same orbit ofthe multiplicative group of the complex field G | m ( T ) ∼ = C × ( T ). It makesthen perfect sense to generalize this construction and look at the equivalenceclasses with respect to the action of the multiplicative supergroup G | m , whichis the supergroup with underlying topological space C × , with one global oddcoordinate and with group law (in the functor of points notation):( a, α ) · ( a ′ , α ′ ) = ( aa ′ + αα ′ , aα ′ + αa ′ ) . G | m is naturally embedded into GL(1 | G | m ( T ) −→ GL(1 | T )( a, α ) (cid:18) a αα a (cid:19) This is precisely the point of view we are taking in constructing the Π-projective line: we identify T -points in C | which lie in the same G | m orbit,but instead of looking simply at rank 1 | C | ( T ) we look ata more elaborate structure, which is matching very naturally the G | m actionon C | . This structure is embodied by the condition of φ -invariance for asuitable odd endomorphism φ of C | , that we shall presently see. For moredetails see Appendix A.Consider now the supermanifold C | , and the odd endomorphism φ on C | given in terms of the standard homogeneous basis { e , e |E , E } by: ! We note that φ = 1. All of our arguments here take place for an open cover of T in which a T pointcorresponds to a free sheaf and not just a locally free one. For simplicity of exposition weomit to mention the cover and the necessary gluing to make all of our argument stand.
6n analogy with the projective superspace, we now consider the functor P Π : (smflds) o −→ (sets), where P Π ( T ) := { rank 1 | φ -invariant subsheaves of O | T } Here the action of φ is extended to O | T = C | ⊗ C O T by acting on the firstfactor. Lemma 3.1.
Let F T ∈ P Π ( T ) . Then there exist an open cover { V i } of T ,where F T ( V i ) is free and a basis e , E of F T ( V i ) such that φ ( e ) = E , φ ( E ) = e .Proof. Since F T is locally free, there exist an open cover { V i } of T , where F T ( V i ) is free with basis, say, e ′ , E ′ . Let Ψ be the matrix of φ | V i in this basis.Since φ = 1, we have Ψ = 1, which implies that Ψ has the form:Ψ := (cid:18) α aa − − α (cid:19) with a ∈ O ∗ T ( V i ) , α ∈ O T ( V i ) . Let P ∈ GL(1 | O T ( V i )) be the matrix: P := (cid:18) a − − a − α (cid:19) P is invertible because a is, and one calculates that P Ψ P − = Φ, so P givesthe desired change of basis. Proposition 3.2.
There is a one-to-one correspondence between the two sets: P Π ( T ) −→ P | ( T ) , T ∈ (smflds) which is functorial in T . In other words P Π is the functor of points of P | .Proof. We briefly sketch the proof, leaving to the reader the routine checks.Let us consider a locally free sheaf F T ⊂ O | T of rank 1 | P Π ( T ), invariantunder φ . We want to associate to each such F T a T -point of P | ( T ), that is,a morphism f : T −→ P | . 7irst we cover T with V i , so that F T | V i is free. By Lemma 3.1 there exists abasis e , E of F T ( V i ) such that φ ( e ) = E , φ ( E ) = e .Representing e , E using the basis { e , E , e , E } of C | , we have: F T ( V i ) = span { e = ( s , σ , s , σ ) , E = ( σ , s , σ , s ) } for some sections s j , σ j in O T ( V i ). Since the rank of F T is 1 |
1, either s or s must be invertible. Let us call V the union of the V i for which s is invertibleand V the union of the V i for which s is invertible.Hence, we can make a change of basis of F T ( V i ) by right multiplying thecolumn vectors representing e and E in the given basis, by a suitable element g i ∈ GL(1 | O T ( V i )) obtaining: F T ( V ) = span { (1 , , s s − − σ σ s − , σ s − − s σ s − ) , (0 , , σ s − − s σ s − , s s − − σ σ s − ) } , F T ( V ) = span { ( s s − − σ σ s − , σ s − − s σ s − , , , ( σ s − − s σ s − , s s − − σ σ s − , , } ,g = (cid:18) s − − σ s − − σ s − s − (cid:19) , g = (cid:18) s − − σ s − − σ s − s − (cid:19) Suppose now { e ′ , E ′ } := { ( s ′ , σ ′ , s ′ , σ ′ ) , ( σ ′ , s ′ , σ ′ , s ′ ) } is another basisof F T ( V i ) such that φ ( e ′ ) = E ′ , φ ( E ′ ) = e ′ . The sections in O T ( V i ) we haveobtained, namely: v = s s − − σ σ s − , ν = σ s − − s σ s − v = s s − − σ σ s − , ν = σ s − − s σ s − are independent of the choice of such a basis. This can be easily seen withan argument very similar to the one in Prop. 2.1.Hence we have well-defined morphisms of supermanifolds f i : V i −→ U i ⊂ P | : f ∗ ( u ) = v , f ∗ ( ξ ) = ν f ∗ ( u ) = v , f ∗ ( ξ ) = ν V i = ( V i , O T | V i ) and U i = ( U i , O P | | U i ), while ( u i , ξ i ) are global coordi-nates on U i ∼ = C | . A small calculation shows that the f i ’s agree on V ∩ V ,in fact as one can readily check:(1 , , v , ν ) ∼ (cid:18) v , − ν v , , (cid:19) and similarly for ( v , ν , , P | we defined in Sec. 2. So the f i ’s glue to give a morphism f : T −→ P | .For the converse, consider f : T −→ P | and define V i = | f | − ( U i ). Wecan define immediately the sheaves F V i ⊂ O | V i on each of the V i as we did inthe proof of Prop. 2.1: F V = span { (1 , , t , τ ) , (0 , , τ , t ) }F V = span { ( t , τ , , , ( τ , t , , } where t i = f ∗ ( u i ) , τ i = f ∗ ( ξ i ). The F V i so defined are free of rank 1 | φ -invariant by construction. Finally, one checks that the relations t = t − , τ = − t − τ imply that the F V i glue on V ∩ V to give a locally free rank 1 | O | T . In this section we give the definition of super Riemann surface and we exam-ine some elementary, yet important properties.Much of the material we discuss in this section is contained, though not soexplicitly, in [11].
Definition 4.1.
A 1 | super Riemann surface is a pair ( X, D ), where X is a 1 | D is a locally direct (andconsequently locally free, by the super Nakayama’s lemma) rank 0 | T X such that:
D⊗D −→
T X/ D Y ⊗ Z [ Y, Z ] ( mod D )9s an isomorphism of sheaves. Here [ , ] denotes the super Lie bracket of vectorfields. The distinguished subsheaf D is called a SUSY-1 structure on X ,and 1 | SUSY-1curves . We shall refer to SUSY 1-structures simply as SUSY structures.We say that X has genus g if the underlying topological space | X | has genus g . Definition 4.2.
Let ( X, D ), ( X ′ , D ′ ) be SUSY-1 curves, and F : X → X ′ abiholomorphic map of supermanifolds. F is a isomorphism of SUSY curves ,or simply a SUSY isomorphism , if ( dF ) p ( D p ) = D ′| F | ( p ) for all reduced points p ∈ | X | . Here ( dF ) p denotes the differential of F at p , D p ⊂ T p X (resp. D ′ q )the stalk of the subsheaf D (resp. D ′ ) at p (resp. q ). Example 4.3.
Let us consider the supermanifold C | , with global coordi-nates z , ζ together with the odd vector field: V = ∂ ζ + ζ ∂ z If D = span { V } , D is a SUSY structure on C | since V , V span T C | . Aswe will see, this is the unique (up to SUSY isomorphism) SUSY structure on C | .We now want to relate the SUSY structures on a supermanifold and thecanonical bundle of the reduced underlying manifold. It is important toremember that for a supermanifold X of dimension 1 | O X, = O X, red ; thatis, the even part of its structural sheaf coincides with its reduced part. Thisis of course not true for a generic supermanifold.We start by showing that any SUSY structure can be locally put into a canonical form . Lemma 4.4.
Let ( X, D ) be a SUSY-1 curve, p a topological point in X red .Then there exists an open set U containing p and a coordinate system W =( w, η ) for U such that D| U = span { ∂ η + η∂ w } .Proof. Since D is locally free, there exists a neighborhood U of p on which D = span { D } , where D is some odd vector field; by shrinking U , we mayassume it is also a coordinate domain with coordinates ( z, ζ ). Since X has10nly one odd coordinate, we have D = f ( z ) ∂ ζ + g ( z ) ζ ∂ z for some holomorphiceven functions f , g . So: D = [ D, D ] / g ( z ) ∂ z + gf ′ ζ ∂ ζ . Since D , D form a free local basis for the O X -module T X , we have a D + a D = ∂ z , a D + a D = ∂ ζ If we substitute the expression for D and D we obtain: g ( a ζ + a ) = 1 , a f = 1 − a gf ′ ζ from which we conclude that both f and g must be units.We now show that we can find a new coordinate system (possibly shrinking U ) so that D can be put in the desired form. We will assume such a coordinatesystem exists, then determine a formula for it and this formula will give usthe existence. Let w = w ( z ), η = h ( z ) ζ be the new coordinate system, where w and h are holomorphic functions. By the chain rule, we have: ∂ z = w ′ ( z ) ∂ w + h ′ ( z ) ζ ∂ η , ∂ ζ = h ( z ) ∂ η We now set D = ∂ η + η∂ w and substituting we have: D = ∂ η + η∂ w = f h∂ η + gh − ηw ′ ( z ) ∂ w which holds if and only if the system of equations: f h = 1 , gw ′ = h has a solution for w, h . By shrinking our original coordinate domain, we mayassume it is simply connected. Then since f and g are units, the system hasa solution, by standard facts from complex analysis. We leave to the readerthe easy check that ( w, η ) is indeed a coordinate system. Definition 4.5.
Let ( X, D ) be a SUSY-1 curve, U an open set. Any co-ordinate system ( w, η ) on U having the property of Lemma 4.4 is said tobe compatible with the SUSY structure D . Any open set U that admits a D -compatible coordinate system is said to be compatible with D .11 efinition 4.6. Let X red be an ordinary Riemann surface, K X red its canon-ical bundle. A theta characteristic is a pair ( L , α ), where L is a holomor-phic line bundle on X red , and α a holomorphic isomorphism of line bundles α : L ⊗ L −→ K X red . An isomorphism of theta characteristics ( L, α ), ( L ′ , α ′ )is an isomorphism φ : L → L ′ of line bundles such that α ′ ◦ φ ⊗ = α .Some authors also call a theta characteristic of X red a square root of thecanonical bundle K X red . Definition 4.7. A super Riemann pair , or SUSY pair for short, is a pair( X r , ( L , α )) where X r is an ordinary Riemann surface, and ( L , α ) is a thetacharacteristic on X r . An isomorphism of SUSY pairs F : ( X r , ( L , α )) → ( X ′ r , ( L ′ , α ′ )) is a pair ( f, φ ) where f : X r → X ′ r is a biholomorphism ofordinary Riemann surfaces, and φ : L ′ → f ∗ ( L ) is an isomorphism of thetacharacteristics on X ′ r .For the sake of brevity, we will occasionally omit writing the isomorphism α in describing a super Riemann pair. The following theorem shows that thedata of super Riemann surface and of super Riemann pair are completelyequivalent. Theorem 4.8.
Let ( X, D ) be a SUSY-1 curve. Then ( X red , O X, ) is a SUSYpair, where O X, is regarded as an O X, -line bundle. Furthermore, if F :=( f, f ) : ( X, D ) → ( X ′ , D ′ ) is a SUSY-isomorphism, then ( f, f | O X, ) :( X red , O X, ) → ( X ′ red , O X ′ , ) is an isomorphism of SUSY pairs.Conversely, suppose ( X r , ( L , α )) is a SUSY pair. Then there exists astructure of SUSY-1 curve ( X L , D L ) on X r , such that the SUSY pair as-sociated to ( X L , D ) equals ( X r , ( L , α )) . Any isomorphism of SUSY pairs ( X r , ( L , α )) → ( X ′ r , ( L ′ , α ′ )) induces a SUSY-isomorphism X L → X L ′ .Proof. First we show that if X is a 1 | O X, -line-bundle O X, is a square root of the canonicalbundle K X red .By Lemma 4.4, X has an open cover by compatible coordinate charts. If( z, ζ ) and ( w, η ) are two such coordinate charts, then D z = ∂ ζ + ζ ∂ z , D w = ∂ η + η∂ w with D z = h ( z ) D w , h ( z ) = 0. If w = f ( z ) and η = g ( z ) ζ a small calculationimplies that f ′ ( z ) = g , that is, O ⊗ X, and K X red have the same transition12unctions for this covering, hence there is an isomorphism O ⊗ X, → K X red .If F := ( f, f ) : ( X, D ) → ( X ′ , D ′ ) is a SUSY-isomorphism of SUSY-1curves with underlying Riemann surface X red , then one checks that f | O X ′ , : O X ′ , → f ∗ ( O X, ) is an isomorphism of line bundles. Covering X with anatlas of compatible coordinate charts, transferring this atlas to a compatibleatlas on X ′ by F , and comparing the transition functions for f ∗ ( O X, ) and O X ′ , in this atlas as above, we obtain the desired isomorphism of thetacharacteristics.Conversely, if we have a theta characteristic α : L ⊗ → K X red , we de-fine a sheaf of supercommutative rings O X L on | X | , the topological spaceunderlying X red , by setting: O X L = O X red ⊕ L with multiplication ( f, s ) · ( g, t ) = ( f g, f t + gs ). One checks that O X L sodefined is a sheaf of local supercommutative rings, using the standard factthat a supercommutative ring A is local if and only if its even part A islocal. By taking a local basis χ of L in a trivialization, and sending ( f, gχ )to f + gη , we see that O X L so defined is locally isomorphic to O X red ⊗ Λ[ η ]and hence ( X red , O X L ) is a supermanifold.The SUSY structure is defined as follows. Let z be a coordinate for X red on an open set U . By shrinking U we may assume O L ( U ) is free. Thenthere is some basis ζ of O L ( U ) such that α ( ζ ⊗ ζ ) = dz ; then z, ζ so definedare coordinates for X L on U . We set the SUSY structure on U to be thatspanned by D Z := ∂ ζ + ζ ∂ z .We will show the local SUSY structure thus defined is independent of ourchoices and hence is global on X . Suppose w is another coordinate on U ,and η a basis of O L ( U ) such that α ( η ⊗ η ) = dw . Then w = f ( z ) , η = g ( z ) ζ ,with f ′ a unit in U . Since dw = f ′ ( z ) dz , we have: α ( η ⊗ η ) = g α ( ζ ⊗ ζ )= f ′ ( z ) dz from which it follows that g = f ′ ; in particular, g is also a unit. Then bythe chain rule, ∂ ζ + ζ ∂ z = g ( ∂ η + η∂ w ), hence D Z and D W span the sameSUSY structure on U .Now suppose ( X ′ r , L ′ ) is another SUSY pair, isomorphic to ( X, L ) by ( f, φ ).Then φ will induce an isomorphism of analytic supermanifolds ψ : X L −→ L ′ , since f : X L , red → X L ′ , red is an isomorphism, and f ∗ ( O X L , ) ∼ = O X L′ , viathe isomorphism φ of theta characteristics. Now we check we have a SUSYisomorphism. This may be done locally: given a point p ∈ X red one choosescoordinates ( z, ζ ) and ( z ′ , ζ ′ ) around p that are compatible with the SUSY-1structures on X L and X L ′ , so that D Z := ∂ ζ + ζ ∂ z (resp. D Z ′ := ∂ ζ ′ + ζ ′ ∂ z ′ )locally generate the SUSY structures. In these coordinates the reader maycheck readily that dψ ( D Z | p ) = D Z ′ | p .Theorem 4.8 has the following important immediate consequence. Corollary 4.9. A -dimensional complex manifold X red carries a SUSYstructure if and only if X admits a theta characteristic. Remark 4.10.
One can prove, via a direct argument using cocycles, thatany compact Riemann surface S admits a theta characteristic, using thefact that the Chern class is c ( K S ) = 2 − g (i.e. it is divisible by 2 hence K S admits a square root). Hence SUSY-1 curves exist in abundance: anycompact Riemann surface admits at least one structure of SUSY-1 curve.Theorem 4.8 has also the following important consequences: Proposition 4.11.
Up to SUSY-isomorphism, there is a unique SUSY-1structure on C | , namely, that defined by the odd vector field: V = ∂ ζ + ζ ∂ z where ( z, ζ ) are the standard linear coordinates on C | .Proof. The reduced manifold of C | is C . It is well known that all holomor-phic line bundles on C are trivial. This implies there exists only one thetacharacteristic for C up to isomorphism, namely ( O C , ⊗ ). The essentialpoint in verifying this uniqueness is that any automorphism of trivial linebundles on C is completely determined by an invertible entire function on C , and such a function always has an invertible entire square root. Hence byTheorem 4.8, there is only one SUSY-1 structure on C up to isomorphism.For the last statement of the theorem, see Example 4.3.The next example shows that Lemma 4.4 is a purely local result.14 xample 4.12. Consider the vector field: Z = ∂ ζ + e z ζ ∂ z Z is an odd vector field on C | defining a SUSY structure on C | . The pre-vious proposition implies that the SUSY structure defined by Z is isomorphicto that defined by V = ∂ ζ + ζ ∂ z . However, it does not imply that there existsa global coordinate system ( w, η ) for C | in which Z takes the form ∂ η + η∂ w .In fact suppose such a global coordinate system w = f ( z ) , η = g ( z ) ζ existed.Then: (cid:18) f ′ g ′ ζ g (cid:19)(cid:18) e z ζ (cid:19) = (cid:18) η (cid:19) from which we conclude that g = 1 , f ′ = e − z . Hence f = − e − z + c , but since f is not one-to-one, this contradicts the assumption that ( w, η ) is a coordinatesystem on all of C | . This shows that Lemma 4.4 cannot be globalized evenin the simple case of C | , even though C | has a unique SUSY structure upto SUSY isomorphism. In this section we want to provide some classification results on SUSY curvesof genus zero and one. The next proposition provides a complete classificationof compact super Riemann surfaces of genus zero and shows the existence ofa genus zero 1 | Proposition 5.1. P | admits a unique SUSY structure, up to SUSY-isomorphism. More generally, if X is a supermanifold of dimension | of genus zero, then X admits a SUSY structure if and only if X isisomorphic to P | .2. P admits no SUSY structure. roof. To prove (1), recall the well-known classification of line bundles on P : P ic ( P ) is a free abelian group of rank one, generated by the isomorphismclass of the hyperplane bundle O (1), and K P ∼ = O ( − P , namely ( O ( − , ψ ) where ψ is any fixed isomorphism of line bundles O ( − ⊗ → O ( − O ( − ⊗ ∼ = O ( −
2) to an automorphism of O ( − End ( L ) = L ∗ ⊗ L = O for any line bundle L , and that H ( P , O ) = C . In particu-lar, any global automorphism of O ( − ⊗ is given by multiplication by aninvertible scalar, which has an invertible square root in C ; this is the desiredautomorphism of O ( − X = P | admitteda SUSY-1 structure, we would have O X, ∼ = O ( − O X, ∼ = O ( − X of genus one, that is anelliptic curve, is obtained by quotienting C by a lattice L ∼ = Z . It is easilyseen that any such lattice L is equivalent, under scalar multiplication, to alattice of the form L := span { , τ } , where τ lies in the upper half plane.Two lattices L = span { , τ } and L ′ = span { , τ ′ } , are equivalent, i.e., yieldisomorphic elliptic curves, if and only if τ and τ ′ lie in the same orbit of thegroup Γ = PSL ( Z ), where the action is via linear fractional transformations: τ aτ + bcτ + d A fundamental domain for this action is: D = { τ ∈ C | Im ( τ ) > , | Re ( τ ) | ≤ / , | τ | ≥ } We now want to generalize this picture to the super setting. Our mainreference will be [12].We start by observing the ordinary action of Z ∼ = L = h A , B i on C isgiven explicitly by: A : z z + 1 , B : z z + τ
16n [12] Freund and Rabin take a similar point of view in constructing a superRiemann surface: they define even super elliptic curves as quotients of C | by Z = h A, B i , acting by: A :( z, ζ ) ( z + 1 , ± ζ ) B :( z, ζ ) ( z + τ, ± ζ ) , In this section, we will justify their choice of these particular actions byshowing they are the only reasonable generalizations of the classical actionsof Z on C .In Sec. 4 we proved that on C | there exists, up to isomorphism, onlyone SUSY structure, corresponding to the vector field V = ∂ ζ + ζ ∂ z . We nowwant to characterize all possible SUSY automorphisms preserving this SUSYstructure.We start with some lemmas. Lemma 5.2.
Let X be a | complex supermanifold and ω, ω ′ be holomorphic | differential forms on X such that ker ( ω ) , ker ( ω ′ ) are | distributions.Then ker ( ω ) = ker ( ω ′ ) if and only if ω ′ = tω for some invertible evenfunction t ( z ) .Proof. The ⇐ implication is clear. To prove the ⇒ implication, we canreduce to a local calculation. Suppose now that ker ( ω ) = ker ( ω ′ ). Givenany point p in X , fix an open neighborhood U ∋ p where T X | U is free. As D is locally a direct summand, it is locally free of rank 0 |
1, by the superNakayama’s lemma (see [14]) and D| U has a local complement E ⊂
T X | U (shrinking U , if needed).Let us use the notation O U = O X | U and O ( U ) = O U ( U ). As E is also a directsummand of T X | U , it is also a free O U -module (again possibly shrinking U )hence must be of rank 1 |
0. Hence we have a local splitting
T X | U = D ⊕ E offree O U -modules. Let Z be a basis for D| U , W a basis for E ; then W, Z forma basis for
T X | U . ω | U : O T X ( U ) → O U ( U ) induces an even linear functional ω p : T p X → C onthe tangent space at p , and the splitting T X | U = D ⊕E induces a correspond-ing splitting T p X = D p ⊕ E p of super vector spaces, with dim ( D p ) = 0 | dim ( E p ) = 1 |
0, such that ker ( ω p ) = D p and span( W p ) = E p . By linearalgebra, ω p | E p is an isomorphism; in particular, ω p ( W p ) is a basis for C = O p / M p . The super Nakayama’s lemma then implies that ω ( W ) generates17 U as O U -module (again shrinking U if necessary), which is true if and onlyif ω ( W ) is a unit; the same is true for ω ( W ′ ).We now show that the ratio ω ′ ( W ) /ω ( W ) ∈ O ∗ U, is independent of the localcomplement E and the choice of W , so that it defines an invertible evenfunction t on all of X . Suppose E ′ is another local complement to D on U ,and W ′ a local basis for E ′ . We have as before that Z, W ′ form a basis of T X | U . Then ω ( W ′ ) , ω ′ ( W ′ ) are invertible in O U by the above argument, and W ′ = bW + βZ with b ∈ O ∗ U, , β ∈ O U, . ω ( W ′ ) ω ′ ( W ′ ) = bω ( W ) + βω ( Z ) bω ′ ( W ) + βω ′ ( Z )= ω ( W ) ω ′ ( W )Note here that we have used the hypothesis ker ( ω ) = ker ( ω ′ ) to conclude ω ( Z ) = ω ′ ( Z ) = 0.Finally, we verify that ω ′ = tω ; this can again be done locally since t isnow known to be globally defined. The argument is left to the reader.The odd vector field V = ∂ ζ + ζ ∂ z defining our SUSY structure D isdual to the differential form ω := dz − ζ dζ . As one can readily check D =span { V } = ker ( ω ). Lemma 5.3.
An automorphism F : C | → C | is a SUSY automorphismif and only if F ∗ ( ω ) = t ( z ) ω for some invertible even function t ( z ) .Proof. Unraveling the definitions, one sees that F preserves the SUSY struc-ture if and only if ker ( F ∗ ( ω )) p = ker ( ω ) p for each p ∈ X . We claim that the latter is true if and only if ker ( F ∗ ( ω )) = ker ( ω ). One implication is clear. Conversely, suppose that ker ( F ∗ ( ω )) p = ker ( ω ) p for each p ∈ X . By a standard argument using the super Nakayama’sLemma, ker ( F ∗ ( ω )) = ker ( ω ) in a neighborhood of p for any point p , hence ker ( ω ) = ker ( F ∗ ( ω )). The result then follows by Lemma 5.2.We are now ready for the result characterizing all of the SUSY automor-phisms of C | . 18 roposition 5.4. Let ( z, ζ ) be the standard linear coordinates on C | , andlet C | have the natural SUSY-1 structure defined by the vector field V = ∂ ζ + ζ ∂ z . The SUSY automorphisms of C | are precisely the endomorphisms F of C | such that: F ( z, ζ ) = ( az + b, ±√ aζ ) where a ∈ C ∗ , b ∈ C , and √ a denotes either of the two square roots of a .Proof. Let F be such an automorphism, and z, ζ the standard coordinateson C | . Then by the Chart Theorem, F ( z, ζ ) = ( f ( z ) , g ( z ) ζ ) for someentire functions f, g of z . Similarly, F − ( z, ζ ) = ( h ( z ) , k ( z ) ζ )) for someentire functions h, k . Since F and F − are inverses, f is a biholomorphicautomorphism of C | , hence linear by standard facts from complex analysis: f ( z ) = az + b for some a, b ∈ C , a = 0; the same is true for h .So by the Lemma 5.3, F preserves the SUSY-1 structure on C | if andonly if F ∗ ( ω ) = t ( z ) ω . We calculate: F ∗ ( dz − ζ dζ ) = df − F ∗ ( ζ ) d ( gζ )= f ′ dz − g ζ dζ . Equating this with t ( z ) ω , we see t = f ′ = g . Thus g = a , so in particular g is constant. Hence: F ( z, ζ ) = ( az + b, cζ )where c = a , a ∈ C \{ } , b ∈ C . Conversely, one checks that anymorphism C | → C | of the above form is an automorphism, and that itpreserves the SUSY structure.From our previous proposition, we conclude immediately that the onlyactions of Z on C | that restrict to the usual action on the reduced space C are of the form: A :( z, ζ ) ( z + 1 , ± ζ ) B :( z, ζ ) ( z + τ, ± ζ ) , A and B must be by automorphisms of the form:( z, ζ ) ( az + b, ±√ aζ )and in this case, a must be taken to be 1. This justifies the choice made in[12]. Remark 5.5.
Using Theorem 4.8, we see that the SUSY structures on X red correspond one-to-one to isomorphism classes of theta characteristics on X red .It is well-known from the theory of elliptic curves over C that an elliptic curve X red has four distinct theta characteristics, up to isomorphism. Regarding X red as an algebraic group, these theta characteristics correspond to theelements of the subgroup of order 2 in X red .As noted in [9], one can define the parity of a theta characteristic L as dim H ( X red , L ) ( mod O X red is distinguished from the other three by its parity: it has odd parity, theothers have even parity. The odd case is therefore fundamentally differentfrom the perspective of supergeometry, and is best studied in the context of families of super Riemann surfaces; families of odd super elliptic curves areconsidered in, for instance, [12], [9], [15].In [12], Rabin and Freund also describe a projective embedding of the SUSYcurve defined by C | / h A, B i using the classical Weierstrass function ℘ andthe function ℘ defined as ℘ = ℘ − e (as usual e = ℘ ( ω i ) with ω = 1 / ω = τ / ω = (1 + τ ) / U , U , U is the open cover of P | ( C )described in Sec. 2, on U the embedding is defined as: C | / h A, B i −→ U ⊂ P | ( C )( z, ζ ) [ ℘ ( z ) , ℘ ′ ( z ) , , ℘ ( z ) ζ , ℘ ′ ( z ) ζ , ℘ ( z ) ℘ ( z ) ζ ]In [12], they describe also the equations of the ideal in U corresponding tothe SUSY curve in this embedding: y = 4 x − a x − a , x − e ) η = yη yη = 2( x − e )( x − e ) η η = xη x, y, η , η , η ) are the global coordinates on U ∼ = C | . One can read-ily compute the homogeneous ideal in the ring C [ x , x , x , ξ , ξ , ξ ] associ-ated with the given projective embedding. It is generated by the equations: x x = 4 x − a x x − a x , x x − e x ) ξ = x x ξ x x ξ = 2( x − e x )( x − e x ) ξ ξ x = x ξ A Π -Pro jective geometry revisited We devote this appendix to reinterpret the Π-projective line, discussed inSec. 3, through the superalgebra D .Let D denote the super skew field , D = C [ θ ], θ odd and θ = −
1. As acomplex super vector space of dimension 1 | D = { a + bθ | a, b ∈ C } , thus ithas a canonical structure of analytic supermanifold, and its functor of pointsis: T D ( T ) := ( D ⊗ O ( T )) = D ⊗ O ( T ) ⊕ D ⊗ O ( T ) Let D × be the analytic supermanifold obtained by restricting the structuresheaf of the supermanifold D to the open subset D \ { } . D × is an analytic supergroup and its functor of points is: T D × ( T ) := ( D ⊗ O ( T )) ∗ where ( D ⊗ O ( T )) ∗ denotes the invertible elements in ( D ⊗ O ( T )) ;As a supergroup D × is isomorphic to G | m , which is the supergroup withunderlying topological space C × , described in Sec. 3. The isomorphismbetween G | m and D × simply reads as:( a, α ) a + θα Notice that G | m (hence D × ) is naturally embedded into GL(1 | D × ( T ) ∼ = G | m ( T ) −→ GL(1 | T ) a + θα ∼ = ( a, α ) (cid:18) a αα a (cid:19) G | m ∼ = D × are commutativesupergroups, the commutative algebra D = C [ θ ] is not a commutative super-algebra, because if it were, then θ = 0 and not θ = − D . Lemma A.1.
A right action of D on a complex super vector space V isequivalent to the choice of an odd endomorphism φ of V such that φ = 1 .Proof. Let V be a right D -module. A right action of D = C [ θ ] is an anti-homomorphism f : D → End( V ), which corresponds to a left action of theopposite algebra D o = C [ θ o ], ( θ o ) = 1 (End( V ) denotes all of the endomor-phisms of V , not just the parity preserving ones). Such actions are specifiedonce we know the odd endomorphisms ψ and φ corresponding respectivelyto θ and θ o . Hence explicitly right multiplication by θ gives rise to an oddendomorphism φ such that φ = 1, by: φ ( v ) := ( − | v | v · θ Conversely, given a super vector space V and an odd endomorphism φ ofsquare 1, we can define a right D -module structure on V by: v · ( a + bθ ) := v · a + ( − | v | φ ( v ) · b Given any complex supermanifold X , there is a sheaf D of superalgebras,defined by D ( U ) := O X ( U ) ⊗ C D , for any open set U ⊆ | X | . Then a sheaf ofright (resp. left) D -modules on X is a sheaf of right (resp. left) modules forthe sheaf D ; a morphism F → F ′ of sheaves of D -modules is simply a sheafmorphism that intertwines the D -actions on F , F ′ . A sheaf of D -modules F is locally free of D -rank n if F is locally isomorphic to D n . Lemma A.2.
Let X be a complex supermanifold and let U be an open set.If V is a free D ( U ) -module of D -rank , Aut D ( V ) ∼ = D × ( U ) . roof. Since V is free of D -rank 1, we may reduce to the case V = D ( U ) asright D -modules, where this identification is obvious: f f (1) ∈ D × ( U ) for f ∈ Aut D ( V ).We are now ready to reinterpret the functor of points of the Π-projectiveline. Proposition A.3.
Let the notation be as above. P Π ( T ) = { locally free, D -rank 1 right D -subsheaves F T ⊆ O | T } In other words, the functor of points of the Π -projective line associates toeach supermanifold T the set of locally free right D -subsheaves of rank | of O | T .Proof. (Sketch). The right action on F T of D corresponds to the right actionof D on C | ⊗ O T occurring on the first term through the left multiplica-tion by the odd endomorphism φ , (see Lemma A.1). Hence the φ -invariantsubsheaves are in one to one correspondence with the right D -subsheaves of O | T . Notice furthermore that by Lemma A.2, the change of basis of thefree module F T ( V i ) we used in 3.2, corresponds to right multiplication by anelement of G | m ∼ = D × , that is the natural left action of D × on the locallyfree, D -rank 1 sheaf F T ( V i ) by automorphisms. Remark A.4.
The generalization from C × action on C n (see the construc-tion of ordinary projective space Sec. 2) to G | m ∼ = D × action on C n | n givesus naturally the odd endomorphism φ , which is used to construct the Π-projective space and ultimately it is the base on which Π-projective geome-try is built. The introduction of the skew-field D , D × ∼ = G | m is not merely acomputational device, but suggest a more fundamental way to think aboutΠ-projective geometry. We are unable to provide a complete treatment here,but we shall do so in a forthcoming paper. References [1] M. F. Atiyah.
Spin structures on Riemann surfaces . Ann. Sci. ´EcoleNorm. Sup. Vol.4, no. 4, 4762, 1971.232] F. A. Berezin.
Introduction to Superanalysis . D. Reidel Publishing Com-pany, Holland, 1987.[3] F. A. Berezin., Leites,
Supermanifolds , Dokl. Akad. Nauk SSSR, Vol.224, no. 3, 505–508, 1975.[4] C. Carmeli, L. Caston, R. Fioresi,
Mathematical Foundation of Super-symmetry , with an appendix with I. Dimitrov, EMS Ser. Lect. Math.,European Math. Soc., Zurich, 2011.[5] L. Crane, J. Rabin,
Super Riemann surfaces: uniformization and Teich-muller theory , Comm. Math. Phys. Vol. 113, no. 4, 601-623, 1988.[6] P. Deligne, personal communication to Y.I. Manin, 1987.[7] P. Deligne, J. Morgan,
Notes on supersymmetry (following J. Bernstein) ,in: “Quantum Fields and Strings. A Course for Mathematicians”, Vol. 1,AMS, 1999.[8] D. A. Leites,
Introduction to the theory of supermanifolds , Russian Math.Surveys : 1 (1980), 1-64.[9] A. M. Levin, Supersymmetric elliptic curves , Funct. Analysis and Appl.21 no. 3, 243-244, 1987.[10] Y. I. Manin,
Topics in Noncommutative Geometry ; Princeton UniversityPress, 1991.[11] Y. I. Manin,
Gauge Field Theory and Complex Geometry ; translated byN. Koblitz and J.R. King. Springer-Verlag, Berlin-New York, 1988.[12] P. G. O. Freund, J. M. Rabin,
Supertori are elliptic curves , Comm.Math. Phys. 114(1), 131-145 (1988).[13] V. S. Varadarajan.
Lie Groups, Lie Algebras, and Their Representations .Graduate Text in Mathematics. Springer-Verlag, New York, 1984.[14] V. S. Varadarajan,