aa r X i v : . [ m a t h . AG ] D ec SUSY N -supergroups and their real forms R. Fioresi ♭ , S. Kwok ⋆♭ Dipartimento di Matematica, Universit`a di BolognaPiazza di Porta S. Donato, 5. 40126 Bologna. Italy. e-mail: rita.fi[email protected] ⋆ Mathematics Research Unit, University of Luxembourg6, Rue Richard Coudenhove-Kalergi, L-1359, Luxembourg. e-mail: [email protected], [email protected]
Abstract
We study SUSY N -supergroups, N = 1 ,
2, their classification andexplicit realization, together with their real forms. In the end, we givethe supergroup of SUSY preserving automorphism of C | and we iden-tify it with a subsupergroup of the SUSY preserving automorphismsof P | . The papers [5] and [6] carry out a thorough study of the real compact su-pergroups S | and S | , called supercircles , in odd dimension 1 and 2, andtheir theory of representation, together with the Peter-Weyl theorem. Thesesupercircles are realized as real forms of ( C | ) × and ( C | ) × respectively, andin the case of S | , we have a precise relation between the real structures andreal forms of ( C | ) × and the SUSY preserving automorphism of the SUSY1-curve ( C | ) × . In this paper we want to study the SUSY N -curves, whichalso admit a supergroup structure leaving invariant their SUSY structure,namely the SUSY N -supergroups. SUSY N -curves have been the objectof study of several papers. After the foundational work by Manin [21], in[16, 22] Rabin et al. study families of super elliptic curves over non trivialodd bases, which are SUSY 1-curves, whose reduced part is an elliptic curve.These families of supercurves however do not admit a natural supergroupstructure leaving invariant their SUSY structure, hence they are not SUSY N -supergroups according to our terminology (see also Remark 2.6). On theother hand, studying the real forms and the supergroup structure of SUSY1urves gives the opportunity of exploiting the representation theory for phys-ical applications. We also point out that we are not merely studying SUSY- N curves that are also supergroups, but our requirement that the supergroupstructure preserves the SUSY N -structure is quite restrictive and reducesdrastically the possibilities for such supercurves, yet provides a useful localmodel for generic ones.Our paper is organized as follows.In Sections 2, 3 we give the definition of SUSY N -supergroups and weclassify them. We also give an interpretation of the supergroup SL(1 |
1) asthe SUSY 2-curve incidence supermanifold of the SUSY 1-curve ( C | ) × andits dual. In Section 4 we study of real forms of SUSY N -supergroups oftype 1 and classify them. Finally in Section 5 we compute the supergroup ofSUSY preserving automorphism of C | . Acknowledgments . We wish to thank prof. P. Deligne for helpful com-ments. We are indebted to our anonymous Referee, for valuable commentsthat helped us to considerably improve our paper. R. Fioresi, S. D. Kwokthank the University of Luxembourg and the University of Bologna for thewonderful hospitality while this work was done. S. D. Kwok was partly sup-ported by AFR grant no. 7718798 of the Luxembourgish National ResearchFoundation. N -supergroups We want to study SUSY N -curves which also have a supergroup structure,which preserves the SUSY N -structure. In the following we shall use thenotation and terminology as in [20] Ch. 2. X is a SUSY -curve if X is a1 | | D such that theFrobenius map D ⊗ D →
T X/ D given by Y ⊗ Y [ Y , Y ] ( mod D ) is anisomorphism. X is a SUSY -curve if it is a 1 | | D i such that [ D i , D i ] ⊂ D i and the Frobeniusmap D ⊗ D → T X/ [ D , D ] is an isomorphism. In [20] Ch. 2, Maninprovides local models for such distributions: D = ζ ∂ z + ∂ ζ on X a SUSY 1-curve D = ζ ∂ z + ∂ ζ D = ζ ∂ z + ∂ ζ on X a SUSY 2-curve2 efinition 2.1. Let X be a SUSY N -curve, with 0 | D i ,where i = 1 for N = 1 and i = 1 , N = 2. We say that X is a SUSY N -supergroup if X is a supergroup and the distribution(s) D i are left invariant.If X and Y are SUSY N -supergroups, we say that f : X −→ Y isa morphism of SUSY N -supergroups if f is a supergroup morphism and f ∗ ( D i ) = D j , that is f preserves the distributions or exchanges them.We can immediately compute the Lie superalgebra of SUSY N -supergroups. Proposition 2.2.
Let X be a SUSY N -supergroup. Then: Lie( X ) = h Z, C i , [ Z, Z ] = 0 , for N = 1Lie( X ) = h Z , Z , C i , [ Z , Z ] = C, for N = 2 where Z , Z i are suitably chosen odd elements and we assume to be zero allthe brackets we do not write.Proof. For N = 1 see [5]. Let N = 2, D i the left invariant distributionson the SUSY 2-supergroup X . Let Z i ∈ Lie( X ) be a left invariant (odd)generator of D i . We have Lie( X ) = h Z , Z , [ Z , Z ] i . Notice that in general[ D i , D i ] = 0, however since Z i ∈ Lie( X ) and the bracket must preserve theparity, we have [ Z i , Z i ] = 0. Let C := [ Z , Z ]. The Jacobi identity givesimmediately that C is central.Since SUSY N -supergroups are in particular supergroups, we shall usefreely the formalism of Super Harish Chandra Pairs (SHCP), see [3] Ch. 7for more details. Proposition 2.3.
Let X , X ′ be SUSY N -supergroups. e X , e X ′ their under-lying reduced groups. Then X ∼ = X ′ if and only if e X ∼ = e X ′ .Proof. Suppose f : X → X ′ is an isomorphism. Then e X and e X ′ are isomor-phic.Conversely, suppose | f | : e X → e X ′ is an isomorphism. | f | : e X → e X ′ lifts to the universal covers, giving an isomorphism e f : C → C which fixesthe origin. By a standard result from one-variable complex analysis, e f ismultiplication by a nonzero scalar λ . If N = 1, define the super Lie algebramorphism ϕ : g → g by C λC ′ , Z
7→ √ λZ ′ . Clearly ϕ | g = d | f | so F := ( | f | , ϕ ) is an isomorphism of SHCP, hence X ∼ = X ′ as supergroups.3y construction dF ( D e ) = D ′ e at the identity e . However, as D , D ′ are left-invariant and F is a supergroup isomorphism, we have dF ( D p ) = D ′ p at everypoint p of X , hence F is an isomorphism of SUSY N -supergroups.If N = 2, define the super Lie algebra morphism ϕ : g → g by C λC ′ , Z
7→ √ λZ ′ , Z
7→ √ λZ ′ . Again, ϕ | g = d | f | so F := ( | f | , ϕ ) is anisomorphism of SHCP. This shows that X ∼ = X ′ as supergroups, however,reasoning as above, we obtain our result. Corollary 2.4.
Let X be a SUSY N -supergroup. Then, X = C | N or X ∼ = C | N /G , where G is either a rank free abelian subgroup of C | N , or a latticein C | N . Furthermore, X inherits its SUSY N -structure from its universalcover C | N .Proof. If X is simply connected, then X = C | N . Assume X is not simplyconnected. The universal cover of X is readily seen to be C | N . The kernel ofthe covering morphism π : C | N → X is a 0 | G of C | N and,by a classical argument, is either a rank 1 free abelian subgroup of C | N , ora lattice in C | N . So X ∼ = C | N /G . Definition 2.5.
We say that X is a SUSY N -supergroup of type 1 (resp. type 2 ) if X = C | N /G , where G is a rank 1 free abelian subgroup (resp. alattice) in C | N .We end this section with a remark relating our treatment with [16, 22]. Remark 2.6.
In [16, 22], Rabin et al. consider families X → B of SUSY1-elliptic curves over a base superspace B = ( pt, Λ), where Λ is a non trivialGrassmann algebra. In this setting, families of SUSY 1-curves do not admitany natural structure of supergroup over B .This is consistent with our treatment, because we work in the absolute setting, i.e. taking B = { pt } , so it is possible to endow a SUSY 1-ellipticcurve with the supergroup structure inherited from its universal cover C | (Prop. 2.4). N -supergroups of type 1 We want to classify the SUSY N -supergroups of type 1 and relate themto Manin’s approach to SUSY curves. We shall use interchangeably the for-malisms of functor of points and also of super Harish-Chandra pairs (SHCP).4 roposition 3.1. Up to isomorphism, for N = 1 , fixed, we have only twoSUSY N -supergroups of type 1.1. For N = 1 they are ( C | ) × and C | with group law respectively: ( w, η ) · ( w, η ′ ) = ( ww ′ + ηη ′ , wη ′ + ηw ′ )( z, ζ ) · ( z ′ , ζ ′ ) = ( z + z ′ + ζ ζ ′ , ζ + ζ ′ ) (1)
2. For N = 2 they are ( C | ) × and C | with group laws: ( v, ξ, η ) · ( v ′ , ξ ′ , η ′ ) = ( vv ′ + ηξ ′ , vξ ′ + ξv ′ + ξv − ηξ ′ ,ηv ′ + vη ′ + ηξ ′ v ′− η ′ )( z, ζ , χ ) · ( z ′ , ζ ′ , χ ′ ) = ( z + z ′ + ζ χ ′ , ζ + ζ ′ , χ + χ ′ ) (2) Proof.
For N = 1 the statements are contained in [5], provided that oneverifies left invariance, which is a straightforward check. Let N = 2, D i the left invariant distributions on the SUSY 2-supergroup X . Let D i ∈ Lie( X ) be a left invariant (odd) generator of D i . By 2.2 we have Lie( X ) = h D , D , [ D , D ] i . The given group laws correspond to the Lie superalgebrawe have obtained respectively for G = C × and G = C . For examplelet us compute the Lie superalgebra structure for ( C | ) × (the case of C | is similar). The tangent space is C | and let e , E , E be the canonicalbasis. The corresponding left invariant vector fields in the global coordinates( v, ξ, η ) are: D = ( dℓ ( u,µ,ν ) ) (1 , , E = v∂ η D = ( dℓ ( u,µ,ν ) ) (1 , , E = − η∂ v + ( v + ξv − η ) ∂ ξ E = ( dℓ ( u,µ,ν ) ) (1 , , e = v∂ v + ξ∂ ξ + η∂ η As one can readily check [ D , D ] = − E (so we set C = − E ) and [ D i , D i ] =0. Remark 3.2.
We can interpret the multiplicative and additive SUSY 2-supergroups using matrix supergroups. ( C | ) × is SL(1 | C | ) × ( T ) = SL(1 | T ) = (cid:26)(cid:18) u ξη v (cid:19) | v − ( u − ξv − η ) = 1 (cid:27) C | is the subgroup of SL(2 |
1) given in the functor of points notation by: C | ( T ) = z ζ χ For example, let us check the second of these statements, the first one beingthe same. We can verify the claim reasoning in terms of the functor of points.So we have to compute: z ζ χ · z ′ ζ ′ χ ′ = z ′ + z + ζ χ ′ ζ + ζ ′ χ + χ ′ which is precisely the multiplication as in (2). This approach can be helpfulin calculations. We leave to the reader the straightforward checks regardingthe group law of the first statement.We now want to interpret some of the discussion in [20] Sec. 6 in theframework of SUSY supergroups. Let X be a SUSY 1-curve and b X itsdual. The T -points of b X are the 0 | X ( T ). Let ∆ be thesuperdiagonal subscheme of X × b X . It is locally defined by the incidencerelation: z − z ′ − ζ ′ ζ = 0 (3)( z, ζ ′ ) and ( z ′ , ζ ′ ) local coordinates of X and b X (see Def. 6.2 in [20]). ∆is a SUSY 2-curve, with distributions D , D and we have the commutativediagram: ∆ $ $ ❏❏❏❏❏❏❏❏❏ z z tttttttttt X = ∆ / D / / ∆ / D = b X (4)where ∆ / D i means the superspace whose reduced space is | ∆ | , and whosestructure sheaf is the subsheaf of O ∆ consisting of sections which are invariantunder D i . Specifying a SUSY-1 structure on X gives an isomorphism X ∼ = b X .6n ( C | ) × we have the global SUSY 2-structure: D = ∂ ζ + ζ ∂ z D = ∂ ζ + ζ ∂ z (5)Clearly D = D = 0 and [ D , D ] = 2 ∂ z . In Remark 3.2, we have viewed( C | ) × as the supergroup SL(1 | |
1) with ∆ for X = ( C | ) × and GL(1 |
1) with X × b X . Notice that in our special case, namelyfor X = b X = ( C | ) × , we have that X, b X ⊂ SL(1 |
1) as the subsupergroups: X ( T ) = (cid:26)(cid:18) x ξξ x (cid:19) (cid:27) , b X ( T ) = (cid:26)(cid:18) y η − η y (cid:19) (cid:27) These inclusions correspond to the Lie superalgebra inclusions: h C, U = E + F i , h C, V = E − F i ⊂ h C, E, F i = sl(1 | C , E , F are the usual generators for sl(1 | C, E ] = [
C, F ] = [
E, E ] = [
F, F ] = 0 , [ E, F ] = C (6) We want to study the real forms of SUSY N -supergroups of type 1. In [5] and[6] we proved that, up to isomorphism, there is one real form of the SUSY1-supergroup ( C | ) × and the corresponding involution is the composition ofcomplex conjugation and the SUSY preserving automorphisms P ± . We wishto prove a similar result for the SUSY 2-supergroups. Definition 4.1.
Let X be a SUSY 2-curve with distributions D i . We saythat an automorphism φ : X −→ X is SUSY preserving if φ ∗ ( D i ) = D j ,that is, if φ preserves individually each distribution or exchanges them. If X is a SUSY 2-supergroup we additionally require φ to be a supergroupautomorphism.Notice that in a SUSY 2 curve the roles of D and D are interchangeable;this forces us to give such a definition of SUSY preserving automorphism.We start our discussion by observing that, up to isomorphism, there isonly one real form of the Lie superalgebra sl(1 |
1) with compact even part.7n fact, assume g R = span R { iC, U, V } is such real form, with central evenelement iC (see (6)). If U = aE + bF , V = cE + dF , there is no loss ofgenerality in assuming a = 1 because E a − E , F aF , C C is a Liesuperalgebra automorphism of sl(1 | U, U ] = 0 (whenboth [ U, U ] = [
V, V ] = 0 we leave to the reader the easy check of what thereal form is). Then we have that b = i , up to a constant, that we absorbe in C . An easy calculation shows that V = iE − F , hence we have proven thefollowing proposition. Proposition 4.2.
Up to isomorphism, there is a unique real form of sl(1 | with compact even part, namely su(1 |
1) = span R { iC, U = E + iF, V = iE − F } ⊂ sl(1 | . We can then state the theorem giving all real forms of SUSY 2-supergroupswith compact support.
Theorem 4.3.
Up to isomorphism, there exists a unique real form of theSUSY -supergroup SL(1 | and it is obtained with an involution σ = c ◦ φ where φ is a SUSY preserving automorphism and c is a complex conjugation.Explicitly: σ (cid:18) a βγ d (cid:19) = d − − ia − γ − ia − β a − ! Proof.
We first check that the given σ is of the prescribed type, namely that φ (cid:18) a βγ d (cid:19) = (cid:18) d − − ia − γ − ia − β a − (cid:19) is a SUSY preserving automorphism. The fact that φ is a supergroup auto-morphism is a simple check, that can be verified using the functor of pointformalism, namely one sees that: φ (cid:18)(cid:18) a βγ d (cid:19) · (cid:18) a ′ β ′ γ ′ d ′ (cid:19)(cid:19) = (cid:18) d − − ia − γ − ia − β a − (cid:19) (cid:18) d ′− − ia ′− γ − ia ′− β a ′− (cid:19) Since φ is a supergroup morphism, dφ preserves left invariant vector fields(see [3] Ch. 7), hence it preserves the SUSY structure.As for uniqueness, by Prop. 4.2 we know there exists a unique realform su(1 |
1) of sl(1 |
1) = Lie(SL(1 | |
1) =Lie(SU(1 | The SUSY preserving automorphisms of C | C | we have the globally defined SUSY structure given by the vectorfield: D = ∂ ζ + ζ ∂ z where ( z, ζ ) are the global coordinates. This structure is unique up to iso-morphism (see [13] Sec. 4). We want to determine the supergroup of auto-morphism of C | preserving such SUSY structure. We will denote it withAut SUSY ( C | ). In the work [13] we have provided the C -points of such super-group; they are obtained by looking at the transformations leaving invariantthe 1-form: s = dz − ζ dζ and are given by the endomorphisms F ( z, ζ ) = ( az + b, √ aζ )We can identify the C -points of the supergroup of SUSY preserving auto-morphism with the matrix groupAut SUSY ( C | )( C ) = c d c −
00 0 1 | a, b ∈ C ⊂ Aut
SUSY ( P | )( C ) (7)This is a subgroup of the C -points of the SUSY-preserving automorphisms ofthe SUSY 1-curve P | , namely those fixing the point at infinity (see Sec. 5[13] and Sec. 5). In such identification a = c and b = dc . Notice that, thoughAut SUSY ( C | )( C ) is a matrix group, it is not obvious that also Aut SUSY ( C | )should be, since we are looking at the SUSY preserving automorphism of C | as supermanifold morphisms. Neverthless we will show that this is the case.An automorphism F : C | −→ C | induces an automorphism F ∗ : O ( C | ) −→ O ( C | ) of the superalgebra of global sections. F is SUSYpreserving if and only if F ∗ ◦ D = kD ◦ F ∗ (8)where D is now interpreted as a derivation of O ( C | ) and k is a suit-able constant. We first consider the infinitesimal picture and compute9ie(Aut SUSY ( C | )). By (7), Lie(Aut SUSY ( C | )) is 2 dimensional, and asone can readily check, it is spanned by the two even vector fields: U = 2 z∂ z + ζ ∂ ζ , U = ∂ z We hence only need to compute Lie(Aut
SUSY ( C | )) . Proposition 5.1.
Lie(Aut
SUSY ( C | )) is the Lie subsuperalgebra of the vectorfields on C | spanned by U = 2 z∂ z + ζ ∂ ζ , U = ∂ z , V = ζ ∂ z − ∂ ζ . with brackets: [ V, V ] = 2 U , [ U , U ] = − U , [ U , V ] = − V, [ U , V ] = 0 Proof.
Consider I + θχ , for χ ∈ Lie(Aut
SUSY ( C | )) . The condition (8) givesimmediately that the odd derivation χ ∗ of O ( C | ) induced by χ must satisfy[ χ ∗ , D ] = 0. A small calculation gives then the result.In the Super Harish-Chandra pair (SHCP) formalism, we can immediatelywrite the supergroup of SUSY preserving automorphism:Aut SUSY ( C | ) = (Aut SUSY ( C | )( C ) , Lie(Aut
SUSY ( C | ))The next proposition identifies such supergroup with a natural subsuper-group of Aut SUSY ( P | ) = SpO(2 |
1) (Ref. [14]) using the more geometricfunctor of points approach.
Proposition 5.2.
Aut
SUSY ( C | ) is the stabilizer subsupergroup in SpO(2 | of the point at infinity: Aut
SUSY ( C | )( T ) = c d γ c − c − γ | c, d ∈ O ( T ) , γ ∈ O ( T ) T ∈ (smflds) R . roof. The first statement is an immediate consequence of Proposition 5.1.As for the second one, consider the subgroup G of SpO(2 |
1) = Aut
SUSY ( P | )that fixes the point at infinity. Its functor of points is given by: G ( T ) = c d γ c − c − γ | c, d ∈ O ( T ) , γ ∈ O ( T ) G is representable and its SHCP coincides with Aut SUSY ( C | ), because Lie( G )= Lie(Aut SUSY ( C | ), | G | = Aut SUSY ( C | )( C ) and we have the compatibilityconditions. References [1] F. A. Berezin,
Introduction to superanalysis . D. Reidel Publishing Com-pany, Holland, 1987.[2] F. A. Berezin., Leites,
Supermanifolds , Dokl. Akad. Nauk SSSR, Vol.224, no. 3, 505-508, 1975.[3] 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.[4] C. Carmeli, R. Fioresi,
Superdistributions, analytic and algebraic superHarish-Chandra pairs . Pacific J. Math. 263 (2013), no. 1, 29-51.[5] C. Carmeli, R. Fioresi, S. D. Kwok,
SUSY structures, representationsand Peter-Weyl theorem for S | , J. Geom. Phys. 95, 144-158, 2015.[6] C. Carmeli, R. Fioresi, S. D. Kwok, The Peter-Weyl theorem for SU (1 | Notes on supersymmetry (following J. Bernstein) ,in: “Quantum fields and strings. A course for mathematicians”, Vol. 1,AMS, 1999.[8] R. Donagi, E. Witten,
Supermoduli Space Is Not Projected ,arXiv:1304.7798. 119] R. Fioresi,
Compact forms of Complex Lie Supergroups
J. Pure Appl.Algebra 218 (2014), no. 2, 228-236.[10] R. Fioresi, F. Gavarini,
Algebraic supergroups with Lie superalgebras ofclassical type . J. Lie Theory 23 (2013), no. 1, 143-158.[11] R. Fioresi, F. Gavarini,
Chevalley supergroups . Mem. Amer. Math. Soc.215 (2012), no. 1014.[12] R. Fioresi, F. Gavarini,
On the construction of Chevalley supergroups .Supersymmetry in mathematics and physics, 101-123, Lecture Notes inMath., 2027, Springer, Heidelberg, 2011.[13] R. Fioresi, S. D. Kwok,
On SUSY curves , in ”Advances in Lie Superal-gebras” M. Gorelik, Papi, P. (Eds.), Springer INdAM Series, (2014).[14] R. Fioresi, S. D. Kwok, The Projective Linear Supergroup and the SUSY-preserving automorphisms of P | , arXiv:1504.04492, 2015.[15] R. Fioresi, M. A. Lledo, The Minkowski and conformal superspaces ,World Sci. Publishing, 2014.[16] P. G. O. Freund, J. M. Rabin,
Supertori are elliptic curves , Comm.Math. Phys. Vol. 114(1), 131-145 (1988).[17] S. D. Kwok,
Some results in supersymmetric algebraic geometry , Ph. D.thesis, UCLA, 2011.[18] D. A. Leites,
Introduction to the theory of supermanifolds , Russian Math.Surveys : 1 (1980), 1-64.[19] A. M. Levin, Supersymmetric elliptic curves , Funct. Analysis and Appl.21 no. 3, 243-244, 1987.[20] Y. I. Manin,
Topics in non commutative geometry ; Princeton UniversityPress, 1991.[21] Y. I. Manin,
Gauge field theory and complex geometry ; translated by N.Koblitz and J.R. King. Springer-Verlag, Berlin-New York, 1988.[22] J. M. Rabin,
Super elliptic curves , J. Geom. Phys. Vol. 15, 252-80 (1995).1223] V. S. Varadarajan,
Lie groups, Lie algebras, and their representations .Graduate Text in Mathematics. Springer-Verlag, New York, 1984.[24] V. S. Varadarajan,