aa r X i v : . [ m a t h - ph ] D ec Noname manuscript No. (will be inserted by the editor)
Hyperbolic Superspaces and Super-Riemann Surfaces
Zhi Hu · Runhong Zong
Received: date / Accepted: date
Abstract
In this paper, we will generalize some results in Manin’s paper m [1] to the supergeometric setting. Moreprecisely, viewing C | as the boundary of the hyperbolic superspace H | , we reexpress the super-Green functionson the supersphere ˆ C | and the supertorus T | by some data derived from the supergeodesics in H | . Keywords
Hyperbolic superspaces, Super-Riemann Surfaces, Super-Green functions, Supergeodesics
Contents
Let K be a number field, and O K be the ring of integers of K . The choice of a model X O K of a smooth algebraiccurve over K defines an arithmetic surface over Spec( O K ) . A closed vertical fiber of X O K over a prime p ∈ O K isgiven by X p , the reduction mod p of the model. The completion ¯ X O K of X O K is achieved by adding to Spec( O K ) the archimedean places represented by the set of all embeddings α : K ֒ → C . The Arakelov divisors on thecompletion ¯ X O K are defined by the divisors on X O K and by the formal real combinations of the closed verticalfibers at infinity. However, Arakelov geometry does not provide an explicit description of these fibers, and itprescribes instead a Hermitian metric on each Riemann surface X C for each archimedean prime α . The Hermitiangeometry on each X C is sufficient to develop an intersection theory on the completed model. For example, theintersection indices of divisors on the fibers at infinity are obtained via Green functions on X C l [2].The missing structure in Arakelov’s theory is the analog at the arithmetic infinity of the reductions modulopowers of p of the closed fibers of X O K . In m [1], Yu. Manin planned to enrich Arakelov’s metric structure byrealizing such missing structure. Inspired by Mumford’s p -adic uniformization theory of algebraic curve mu [3], hesuggested to construct a differential-geometric object –a certain hyperbolic 3-manifold– playing the role of a modelat infinity. Roughly speaking, by choice of a Schottky uniformization, the Riemann surface X C is the boundary at Zhi HuResearch Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, JapanDepartment of Mathematics, Mainz University, 55128 Mainz, GermanySchool of Mathematics, University of Science and Technology of China, Hefei, 230026, ChinaE-mail: [email protected]; [email protected] ZongDepartment of Mathematics, Nanjing University, , Nanjing, 21009, ChinaE-mail: [email protected] Zhi Hu, Runhong Zong the infinity of a hyperbolic 3-manifold described as the quotient of the hyperbolic 3-space H by the action of theSchottky group Γ . Furthermore, Manin corroborated his suggestion by interpreting Arakelov Green functions interms of geodesic configurations on this space.Along Manin’s pioneering work, there are several generalizations toward different aspects. The followings aresome interesting progresses on this topic. – A. Werner generalized Manin’s approach to higher dimensional cases w [4]. Namely, she interpreted certainArchimedean (non-Archimedean) Arakelov intersection numbers of linear cycles on P n − with Riemanniansymmetric space associated to SL ( n, C ) (Bruhat-Tits building associated to P GL ( n ) ). – Although when Manin’s work was published, one did not yet discover various novel dualities in string theory,from the physical point of view, Manin’s perspective has a heavy flavor of AdS/CFT-correspondence sinceGreen functions are related to the quantum correlation functions of boundary CFT s [5] and geodesic config-urations are related to the classical bulk gravity with the cosmology constant. In the paper mm [6], Yu. Maninand M. Marcolli considered certain hyperbolic 3-manifolds as analytic continuations of the known Lorentziansignature black holes (e.g. BTZ black hole, Krasnov black hole), and they demonstrated that the expressionsfor Green functions in terms of the geodesic configurations in the above hyperbolic 3-manifolds can be nicelyinterpreted in the spirit of AdS/CFT-correspondence. – In the paper cm [7], C. Consani and M. Marcolli consider the case of an arithmetic surface over Spec( O K ) with thefibers of genus g ≥ . They defined a cohomology of the cone of the local monodromy N at arithmetic infinitywhich is related to Delinger’s archimedean cohomology and regularized determinants de [8]. And by Connes’theory of spectral triples, they established a connection between such cohomology and the dual graph of thefiber at infinity in terms of the infinite tangle of bounded geodesics in the hyperbolic 3-manifold.In this paper, we will generalize Manin’s results to the supergeometric setting. The motivation is that in theoriginal AdS/CFT-correspondence coming from string theory, supersymmetries are the necessary ingredients bothappearing in the boundary and the bulk theories. Firstly, let us briefly collect some basic materials on geometry ofsupermanifolds. More details can be found in [9,10,11]. I. Supermanifolds.
There are two approaches to define supermanifolds: – Algebro-geometric definition: a supermanifold M p | q of dimension p | q is a pair ( M, O M ) consisting of a (Haus-dorff and second countable) topological space M together with a sheaf O M of commutative superalgebras withunity, such that – there exists an open cover { U α } of M , where for each α , O M ( U α ) ≃ C ∞ ( U α ) ⊗ Λ ( R q ) , – if N M is the sheaf of nilpotents of O M , then ( M, O M / N M ) is isomorphic to ( M, C ∞ ( M )) . – Differential geometric definition: let R p | q be a p | q -dimensional superspace over the real Grassmann algebra Λ ∞ R , and endow R p | q with the (non-Hausdorff) De Witt topology, then a supermanifold M p | q is obtained bygluing open sets of R p | q via superdiffeomorphisms. Denote by ǫ the projection map onto the zero-order partof the Grassmann algebra, and define an equivalence relation ∼ on the supermanifold: for x, y ∈ M p | q , x ∼ y iff ǫ ( x ) = ǫ ( y ) , then the space M = M p | q / ∼ is a p -dimensional C ∞ manifold, called the body of M p | q .The two categories of supermanifolds defined by the above two different manners respectively are essentiallyequivalent. However, the latter one is more geometric, so it is convenient to talk about the local coordinate { X A } A =1 , ··· ,p ; p +1 , ··· ,p + q = ( x , · · · , x p ; θ , · · · , θ q ) of M p | q with x , · · · , x p valued in the even part ( Λ ∞ R ) of the Grassmann algebra Λ ∞ R and θ , · · · , θ q valued in the odd part ( Λ ∞ R ) . Hence it is accepted commonly by thephysicists. Therefore, we will adopt the second approach throughout this paper. II. Super-Riemann geometry.
One can define the tangent sheaf T M p | q and the cotangent sheaf Ω M p | q over asupermanifold M p | q . The supermetric on M p | q is a graded-symmetric even non-degenerate O M -linear morphismof sheaves h• , •i : T M p | q × T M p | q → O M . Switching to of differential geometric point of view, one writesthe supermetric as g = dX A ( A g B ) B dX in terms of the local coordinates with B dX = ( − | B | dX B . Then thecorresponding super-Levi-Civita connection ∇ g is given by the super-Christoffel symbols Γ CAB = 12 ( − | D | C g D [ D g A,B + ( − | A || B | D g B,A − ( − | D | ( | A | + | B | ) A g B,D ] . yperbolic Superspaces and Super-Riemann Surfaces 3 A supercurve Υ : R | → M p | q with parameter ( u ; γ ) is called a super-geodesic with respect to ∇ g if and only if ∇ g du ( Υ ∗ ∂ u ) = 0 , which is locally equivalent to the system of differential equations d du Φ ∗ ( X A ) + X B,C ddu Φ ∗ ( X B ) ddu Φ ∗ ( X C ) Φ ∗ ( Γ ABC ) = 0 for any A . Super-Riemann curvature, super-Ricci curvature and super scalar curvature can be also easily general-ized to supermanifolds by formulas R DABC = − Γ DAB,C + ( − | B | ( | A | + | E | ) Γ DEB Γ EAC + ( − | B || C | Γ D AC, B − ( − | C | ( | A | + | B | + | E | ) Γ DEC Γ AB ,R AC = ( − | B | ( | A | +1) R BABC ,R = R AB g AB . Definition 1
1. A supermanifold M p | q is called a (positive/ negative/ zero) Einstein supermanifold if it admits asupermetric g = dX A ( A g B ) B dX such that the corresponding super-Ricci curvature satisfies the condition R AB = c A g B for a (positive/ negative/ zero) real number c .2. A supermanifold M p | q is called a (positive/ negative/ zero) Bosnic supermanifold if it admits a supermetric g = dX A ( A g B ) B dX such that the corresponding super scalar curvature satisfies the condition that ǫ ( R ) is a(positive/ negative/ zero) real constant.Next, in Sect. 2 we will consider the minimal supergeometric extension of some classical low dimensional hy-perbolic manifolds, in particular, the supermetrics and super-volume forms invariant under certain Lie supergroupsare constructed. In Sect. 3, we will define the super-Green functions for super-Riemann surfaces, then viewing C | as the boundary of the hyperbolic superspace H | , we reexpress the super-Green functions on the super-sphere ˆ C | and the supertorus T | by some data derived from the supergeodesics in H | . The main results ofthis paper are summarized as follows. Theorem 1 (=Proposition
8, Proposition
1. Viewing the super-Riemann sphere ˆ C | as the boundary of H | S {∞} , the super-Green function on ˆ C | canbe reexpressed as G P ( Z, Θ ) = log( 1cosh d Q ( P , P ) −
12 cosh d Q ( P , P ) (1 − d Q ( P , P ) ) Θ ¯ Θ )= log 1cosh d ( ˜ Q, ˜ P ) , where P = (0 , , P = ( Z, Θ ) are points in ˆ C | , Q = (0 , ,
1; 0 , is a point in H | , and ˜ P , ˜ Q are alsopoints in H | determined by P , P , Q . Zhi Hu, Runhong Zong
2. Viewing the supertorus T | as the boundary of H | / ˜ Γ , where ˜ Γ denotes the super-Poincáre extension ˜ Γ ofthe supertranslation group, the super-Green function on T | with supermoduli ( τ ; δ ) can be expressed as G ( Z, Θ ) = 12 d ( P , b Q ) B ( d ( P , b P ) d ( P , b Q ) ) + d ( P , c P − )+ ∞ X n =1 ( d ( P , c Q n ) + d ( P , d Q n − )) + 4 π d ( P , b Q ) Θ ¯ Θ = 12 d ( e P , b Q ) B ( d ( e P , b P ) d ( e P , b Q ) ) + d ( e P , c P − )+ ∞ X n =1 ( d ( e P , c Q n ) + d ( e P , d Q n − )) + 4 π d ( e P , b Q ) Θ ¯ Θ, where Q = ( q , Θ ) , P − = (1 − ρ ; Θ ) , Q n = (1 − q n ρ, Θ ) and Q n − = (1 − q n ρ − , Θ ) are all points lying onthe boundary of H | with q = e πi ( τ + Θδ ) . Some more related questions are worthy to be further studied. For example, – we should consider the boundary supermanifold with the body as a higher genus Riemann surface, and con-sider the bulk supermanifold with the body as an another type of hyperbolic 3-manifold in Thurston’s eightgeometries hh [12]; – we should add more odd degree of freedoms beyond the minimal supergeometric extension; – we should give an interpretation of these Manin-type formulas from the viewpoint of AdS/CFT-correspondencewith supersymmetries as done in mm [6]; – we should consider similar problems for supergeometries of non-Archimedean version, and consider the rela-tion with p -adic AdS/CFT-correspondence gk [13]; – we should consider the mathematical aspects of AdS/CFT-correspondence inspired by the Hyperbolic/Arakelovgeometry correspondence. In particular, we should provide an exact expression as the initial step; – we should ask if the Manin’s model at the arithmetic infinity is replaced by P. Scholze’s perfectoid version ofhyperbolic 3-manifolds pet [14], how does the story go? Acknowledgement
The author Z. Hu would like to thank Prof. Yanghui He and Prof. S. Mochizuki for theircontinuing supports and encouragements. The authors would like to thank Prof. Bailing Wang, Prof. Kang Zuoand Dr. Chunhui Liu, Dr. Ruiran Sun, Dr. Yu Yang for their helpful discussions and comments. The authors aresupported by the grant SFB/TR 45 of Deutsche Forschungsgemeinschaft.
Let us consider the superspace R p +1 | q endowed with a supermetric ds = − dx + dz + · · · + dx p + dθ dθ + · · · + dθ q − dθ q written in terms of matrix as g = (cid:18) η ,p J q (cid:19) for η ,p = (cid:18) − p (cid:19) , J q = − . . . − . The Lie supergroup preserving this supermetric is the or-thosymplectic group
OSp (1 , p | q, R ) = { X ∈ SMat( p + 1 | q, R ) : X st gX = g } , yperbolic Superspaces and Super-Riemann Surfaces 5 where expressing X = (cid:18) A (0) B (1) C (1) D (0) (cid:19) in terms of the even parts labelled by subscript (0) and the odd partslabelled by (1) , the supertranspose X st is given by X st = A t (0) C t (1) − B t (1) D t (0) ! . A p | q -dimensional hyperbolic totalsuperspace H p | q is defined as H p | q = { H := x ... x p θ ... θ q ∈ R p +1 | q : H t gH = − } . The supergroup
OSp (1 , p | q, R ) acts obviously on ( H p | q , ds ) . Proposition 1 H p | can be identified with OSp (1 , p | , R ) /OSp ( p | , R ) .Proof We consider a supervector H = x −→ x −→ θ ∈ H p | for −→ x = x ... x p and −→ θ = (cid:18) θ θ (cid:19) , then x = 1 + −→ x ·−→ x + θ θ , where −→ x · −→ x = P pi =1 x i , and we consider a supermatrix X = (cid:18) A (0) B (1) C (1) D (0) (cid:19) , where A (0) = (cid:18) x −→ x t −→ x Id p + −→ x −→ x t x (cid:19) ,B (1) = 12 θ θ x x θ x x θ ... ... x p x θ x p x θ ,C (1) = (cid:18) θ x x θ · · · x p x θ θ x x θ q · · · x p x θ (cid:19) ,D (0) = (cid:18) − θ θ − θ θ (cid:19) . One can check that X ∈ OSp (1 , p | , R ) , and it transforms the supervector H = ... into H . The isotropygroup of H is exactly the supergroup OSp ( p | , R ) by the natural embedding. The claim follows. ⊓⊔ From now on, we work only with two odd dimensions, and we call such setup the minimal supergeometricextension . Assume the zero-order part ǫ ( x ) of x is positive, then x has an inverse x − = 1 ǫ ( x ) [1 − x − ǫ ( x ) ǫ ( x ) + ( x − ǫ ( x ) ǫ ( x ) ) − · · · ] in the Grassmann algebra Λ ∞ R , hence we define the map α : x −→ x −→ θ x ′ −→ x ′ −→ θ ′ = x − x − −→ xx − −→ θ Zhi Hu, Runhong Zong such that P pi =0 ( x ′ i ) + θ ′ θ ′ = 1 . Then since P pi =0 ( ǫ ( x ′ i )) = 1 and ǫ ( x ′ ) > , x ′ p + 1 also has an inverse ( x ′ p + 1) − , hence we define the map β : x ′ −→ x ′ −→ θ ′ x ′′ −→ x ′′ −→ θ ′′ = ( x ′ p + 1) − x ′ ... ( x ′ p + 1) − x ′ p − x ′ p + 1) − −→ θ ′ . There are two supermetrics expressed in terms of the coordinates of supermanifolds as follows ds ′ = ( dx ′ ) + · · · + ( dx ′ p ) + dθ ′ dθ ′ ( x ′ ) ,ds ′′ = ( dx ′′ ) + · · · + ( dx ′′ p − ) + dθ ′′ dθ ′′ ( x ′′ ) such that α ∗ ( ds ′ ) = ds and β ∗ ( ds ′′ ) = ds ′ .In summary, enjoying the same supergeometry of the hyperbolic superspace H p | , we have three modelswith supermetrics listed as follows, whose bodies are the corresponding models of usual p -dimensional hyper-bolic space. The isometry group of the supermetrics on H p | is the subgroup of OSp (1 , p | , R ) , denoted by yp e r bo li c S up e r s p ace s a nd S up e r- R i e m a nn S u rf ace s O S p ( , p | , R ) . Model Definition MetricSuper Hyperboloid Model H p | = { H := x ... x p θ θ ∈ R p +1 | : H t gH = − , ǫ ( x ) > } ds = − dx + dx + · · · + dx p + dθ dθ Super Semisphere Model H p | = { H := x ... x p θ θ ∈ R p +1 | : H t (cid:18) Id p +1 J (cid:19) H = 1 , ǫ ( x ) > } ds = dx + dx + ··· + dx p + dθ dθ x Upper Half Superspace Model H p | = { ( x , · · · , x p − | θ , θ ) ∈ R p | : ǫ ( x ) > } ds = dx + dz + ··· + dx p − + dθ dθ x . Zhi Hu, Runhong Zong
We will focus on the cases of p = 2 , , which admit richer structures. Firstly, when p = 2 , H | (the upperhalf superplane model) can be made into a | -dimensional complex supermanifold (and redenoted by C H | )by introducing the complex coordinates ( Z ; Θ ) for Z = ix + x , Θ = iθ + θ with complex conjugates ¯ Z = − ix + x , ¯ Θ = iθ + θ satisfying the rules: • + ∗ = ¯ • + ¯ ∗ , •∗ = ¯ ∗ ¯ • . The superconformal changes of coordinatesare given by the supermatrix Γ = a b αb − βac e αe − βcα β βα ∈ OSp (1 | , C ) , with ae − bc = 1 + αβ , via the super-Möbius transformations Z Z ′ = aZ + b + ( αb − βa ) ΘcZ + e + ( αe − βc ) Θ = aZ + bcZ + e + Θ αZ + β ( cZ + e ) ,Θ Θ ′ = αZ + β + (1 − αβ ) ΘcZ + e + ( αe − βc ) Θ = αZ + βcZ + e + Θ cZ + e . The following proposition collects some important properties of C H | with respect to the super-Möbius transfor-mations. For the convenience of the readers, we give a brief proof of these properties, more details can be found in u,uy,sm,mmm [15,16,17,18]. Proposition 2
1. Assume Γ ∈ OSp (1 | , R ) , and Let Y = Im( Z ) + 12 Θ ¯ Θ = x − θ θ , then under the super-Möbius transformations, Y ′ = | F Γ ( Z, Θ ) | Y, for F Γ ( Z, Θ ) = 1 cZ + e + ( αe − βc ) Θ .
Hence ǫ ( Y ′ ) = | ǫ ( c ) ǫ ( z )+ ǫ ( e ) | ǫ ( Y ) > .2. The supermetric ds = dx + dx − θ dθ − θ dθ ) dx + 2( θ dθ + θ dθ ) dx + 4( t − θ θ ) dθ dθ x − x θ θ , (2.1) ss is invariant under any super-Möbius transformation Γ ∈ OSp (1 | , R ) , and gives rise to an OSp (1 | , R ) -invariant super-volume form d SVol = 12 ( 1 x + θ θ x ) dx dx dθ dθ . Moreover, this supermetric makes H | into a negative Einstein supermanifold.Proof The supermetric ( ss dX A ( A H ¯ B ) ¯ B dX = − dX A ( ∂ ∂X A ∂X ¯ B log Y ) ¯ B dX = − − ( | X A | + | X ¯ B | + | X A || X ¯ B | ) ( ∂ ∂X A ∂X ¯ B log Y ) dX A dX ¯ B = 1 Y ( dZd ¯ Z − iΘdZd ¯ Θ − i ¯ ΘdΘd ¯ Z − (2 Y + Θ ¯ Θ ) dΘd ¯ Θ ) , yperbolic Superspaces and Super-Riemann Surfaces 9 where { X A } = { Z, Θ } , { X ¯ A } = { ¯ Z, ¯ Θ } and ¯ B dX = ( − | ¯ B | dX ¯ B . It has super-Ricci curvature gm [19] R A ¯ B = − ∂ ∂X A ∂X ¯ B (log Sdet( A H ¯ B )) = − A H ¯ B , where Sdet( A H ¯ B ) = Sdet Y iΘ Y Y i ¯ Θ Y i ¯ Θ Y Y + Θ ¯ Θ Y iΘ Y − Y − Θ ¯ Θ Y = − Y . The super-volume form d SVol = p | Sdet( A H ¯ B ) | dZd ¯ ZdΘd ¯ Θ = dZd ¯ ZdΘd ¯ Θ Y is OSp (1 | , R ) -invariant.To show the OSp (1 | , R ) -invariance of the given supermetric, we firstly note that the super-Möbius transfor-mations are generated by the following transformations T : ( Z, Θ ) ( aZ + b, Θ ) ,T : ( Z, Θ ) ( − Z , ΘZ ) ,T : ( Z, Θ ) ( Z − αZΘ, Θ + αZ ) ,T : ( Z, Θ ) ( Z − βΘ, Θ + β ) . Therefore, we only need to check the invariance under T i , i = 1 , · · · , , which can be done easily. ⊓⊔ Next we consider the case of p = 3 . One takes H | = { ( x, y, t ; θ , θ ) : x, y, t ∈ ( Λ ∞ R ) , ǫ ( t ) > , θ , θ ∈ ( Λ ∞ R ) } as the subspace of C H | : { ( Z, T ; Θ , Θ : Z, T ∈ ( Λ ∞ C ) , Θ , Θ ∈ ( Λ ∞ C ) , ǫ (Im( T )) > } , and then one introduces ˜ Z = x + iy + jt ( or ˜ Z = Z + jT ) , ˜ Θ = jθ + θ ( or ˜ Θ = jΘ + Θ ) , ˜ Y = − j ˜ Z − ¯˜ Z − ˜ Z − ¯˜ Z j + 12 ˜ Θ ¯˜ Θ, where the imaginary unit j satisfies j = − , ij + ji = 0 . The OSp (1 | , C ) -transformations on C H | can beobtained by the super-Poincáre extension of those on H | . More precisely, replacing Z by ˜ Z and Θ by ˜ Θ in the previous super-Möbius transformations, we get the transformations Z ( aZ + b + ( αb − βa ) Θ )( cZ + e + ( αe − βc ) Θ ) | cZ + e + ( αe − βc ) Θ | + | cT + ( αe − βc ) Θ | + ( aT + ( αb − βa ) Θ )( cT + ( αe − βc ) Θ ) | cZ + e + ( αe − βc ) Θ | + | cT + ( αe − βc ) Θ | ,T (1 + αβ ) T + ( αb − βa ) Θ ( cZ + e + ( αe − βc ) Θ ) | cZ + e + ( αe − βc ) Θ | + | cT + ( αe − βc ) Θ | − ( αe − βc ) Θ ( aZ + b + ( αb − βa ) Θ ) | cZ + e + ( αe − βc ) Θ | + | cT + ( αe − βc ) Θ | ,Θ ( αd − βc ) T + (1 − αβ ) Θ ( cZ + e + ( αe − βc ) Θ ) | cZ + e + ( αe − βc ) Θ | + | cT + ( αe − βc ) Θ | − ( αe − βc ) Θ ( αZ + β + (1 − αβ ) θ ) | cZ + e + ( αe − βc ) Θ | + | cT + ( αe − βc ) Θ | Θ ( αZ + β + (1 − αβ ) Θ )( cZ + e + ( αe − βc ) Θ ) | cZ + e + ( αe − βc ) Θ | + | cT + ( αe − βc ) Θ | + ( αT + (1 − αβ ) Θ )( cT + ( αe − βc ) Θ ) | cZ + e + ( αe − βc ) θ | + | cT + ( αe − βc ) Θ | . The element Γ ∈ OSp (1 | , C ) preserving H | is called an R -element. The maximal super subgroup of OSp (1 | , C ) consisting of the R -elements is denoted by H . Proposition 3 H | can be equipped with an H -invariant supermetric such that it is a negative Einstein su-permanifold.2. H | can be equipped with a supermetric such that it is a positive Bosonic supermanifold.3. ( t + θ θ t ) dxdydtdθ dθ is an H -invariant super-volume form on H | .Proof By super-Poincáre extension described above, the H -invariant and negative Einstein supermetric on H | can be given by ds = dx + dy + dt − θ dθ + θ dθ ) dx + 2( θ dθ − θ dθ ) dt − t − θ θ ) dθ dθ t − tθ θ , which gives rise to an H -invariant super-volume form vuuuuuuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Sdet t + θ θ t − θ t − θ t t + θ θ t − t + θ θ t − θ t − θ t − θ t − θ t − tθ t θ t − t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdydtdθ dθ = 12 ( 1 t + 3 θ θ t ) dxdydtdθ dθ . We can also consider the following supermetric ds = dx + dy + dt + 2( θ dθ + θ dθ ) dx − θ dθ − θ dθ ) dt + 4( t − θ θ ) dθ dθ t − tθ θ . yperbolic Superspaces and Super-Riemann Surfaces 11 We calculate the corresponding super scalar curvature. In terms of X = x, X = y, X = t, X = θ , X = θ ,the nonzero super-Christoffel symbols are given by Γ = − t + θ θ t , Γ = θ t , Γ = θ t , Γ = Γ = − t + θ θ t ,Γ = Γ = 3 θ t , Γ = Γ = θ t , Γ = Γ = − t − θ θ t ,Γ = Γ = − θ t , Γ = Γ = θ t , Γ = Γ = 12 t + θ θ t ,Γ = − t + θ θ t , Γ = θ t , Γ = θ t ,Γ = Γ = − t − θ θ t , Γ = Γ = θ t , Γ = Γ = − θ t ,Γ = − t + θ θ t , Γ = θ t , Γ = − θ t ,Γ = Γ = − θ t , Γ = Γ = θ t , Γ = Γ = 12 t − θ θ t ,Γ = Γ = − θ t , Γ = Γ = − θ t , Γ = Γ = 12 t + θ θ t ,Γ = − Γ = 2 − θ θ t , Γ = − Γ = θ t , Γ = − Γ = θ t . Hence the non-vanishing components of the super-Ricci curvature are given by R = − t − θ θ t , R = R = θ θ t , R = R = 3 θ t ,R = R = 3 θ t , R = − t − θ θ t , R = − t − θ θ t ,R = R = θ t , R = R = − θ t , R = − R = − t + 25 θ θ t . Therefore, the super scalar curvature reads R = 2 − θ θ t , which means H | can be made into a positive Bosonic supermanifold. ⊓⊔ As the end of this section, we mention another important | -dimensional hyperbolic supermanifold, the su-pergroup OSp (1 | , R ) whose body is the non-compact Lie group SL (2 , R ) closely related to the BTZ black holein AdS gravity mm,hh [6,12]. The basis of the corresponding Lie superalgebra osp (1 | , R ) are given by three evengenerators L = − , L = , L = − , and two odd generators Q = − , Q = − − − . They satisfy the following (anti-)commutative relations: [ L i , L j ] = 2 ǫ ijl η kl L l , [ L i , Q α ] = ( σ i ) αβ Q β , { Q α , Q β } = 2( Cσ i ) αβ L i , where the indices i, j, k run over , , , and α, β run over , , and { σ i } denote the Pauli matrices, i.e. σ = (cid:18) − (cid:19) , σ = (cid:18) − (cid:19) and σ = (cid:18) (cid:19) , C = ( ǫ αβ ) . Let Str denote the super-Killing form on osp (1 | , R ) ,then Str( L i , L j ) = η ij , Str( L i , Q α ) = 0 , Str( Q α , Q β ) = − C αβ . We parameterize the elements of
OSp (1 | , R ) by g = exp ( αL ) exp ( λL ) exp ( βL ) exp ( θ R ) exp ( θ R ) (2.2) with α, β, λ ∈ ( Λ ∞ R ) , ǫ ( α ) , ǫ ( β ) ∈ [0 , π ) , −∞ < ǫ ( λ ) < + ∞ and θ , θ ∈ ( Λ ∞ R ) , where R , = ( Q ± Q ) .The Lie supergroup OSp (1 | , R ) can be endowed with the following pseudo-supermetric (where the phrase"pseudo" means that the ( Λ ∞ R ) -component gives rise to a pseudo-metric with signature ( − , , on the bodymanifold) invariant under the OSp (1 | , R ) left-action and the SL (2 , R ) × SL (2 , R ) bi-action ds = (1 + 2 θ θ )( dα + dλ + dβ + 2 cosh 2 λdαdβ )+ ( θ cosh 2 β sinh 2 λ + θ cosh 2 λ + 2 θ sinh 2 λ sinh 2 β ) dαdθ + θ dβdθ − ( θ sinh 2 β + 2 θ cosh 2 β ) dλdθ + θ (cosh 2 β sinh 2 λ − cosh 2 λ ) dαdθ − θ dβdθ − θ sinh 2 βdλdθ − (1 − θ θ ) dθ dθ , and the associated OSp (1 | , R ) × OSp (1 | , R ) bi-invariant super-volume form d SVol= vuuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det θ θ (1 + sinh β sinh λ ) − θ θ sinh 4 β sinh 2 λ (1 + θ θ ) cosh 2 λ − θ θ sinh 4 β sinh 2 λ θ θ (1 + sinh β ) 0(1 + θ θ ) cosh 2 λ θ θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · s(cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) − θ θ )2(1 + θ θ ) 0 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dαdλdβdθ dθ =2(1 + 3 θ θ ) sinh 2 λdαdλdβdθ dθ . Indeed, with the given parametrization ( g − dg = e i L i + E α Q α , where e = − cosh 2 β sinh 2 λdα − θ θ (sinh 2 λ cosh 2 β + cosh 2 λ ) dα + (1 + θ θ ) sinh 2 βdλ − θ θ dβ + θ dθ + θ dθ ,e = cosh 2 λdα + θ θ (sinh 2 λ cosh 2 β + cosh 2 λ ) dα − θ θ sinh 2 βdλ + (1 + θ θ ) dβ + θ dθ − θ dθ ,e = − (1 + θ θ ) sinh 2 β sinh 2 λdα + (1 + θ θ ) cosh 2 βdλ − θ dθ , E = [ θ − θ λ (cosh 2 β − sinh 2 β ) + θ + θ λ ] dα + θ − θ β − sinh 2 β ) dλ + θ + θ dβ + 1 − θ θ dθ + 12 dθ , E = − [ θ + θ λ (cosh 2 β + sinh 2 β ) + θ − θ λ ] dα + θ + θ β + sinh 2 β ) dλ − θ − θ dβ + 1 − θ θ dθ − dθ , yperbolic Superspaces and Super-Riemann Surfaces 13 then the super-Killing form provides a pseudo-supermetric ds = Str( g − dg, g − dg ) = − ( e ) + ( e ) + ( e ) + 2 E E − E E , which obviously has the desired invariance.The renormalized volume of a hyperbolic manifold is a quantity motivated by the AdS/CFT correspondenceand can be computed via certain regularization procedure ks [20]. Proposition 4
With respect to the above pseudo-supermetric,
OSp (1 | , R ) has the renormalized volume − π .Proof The volume of
OSp (1 | , R ) is calculated as Vol(
OSp (1 | , R )) = Z OSp (1 | , R ) d SVol= 6 Z π dα Z π dβ Z ∞−∞ sinh 2 λ dλ , where α = ǫ ( α ) , β = ǫ ( β ) , λ = ǫ ( λ ) . Let λ = ln t with < t ≤ , then Vol(
OSp (1 | , R )) = − π Z t ( 1 t − t t + t dt = − π lim z → z − Z (1 − t
16 )( t
16 ) z − = − π lim z → z − Γ (2) Γ ( z − ) Γ ( z + )= − π Γ (2)4 = − π , which gives the renormalized volume. ⊓⊔ A super-Riemann surface S | is a complex | -dimensional supermanifold with the following properties in termsof the local coordinate ( Z ; Θ ) [11,18,21,22] – (supercomplex structure) the transition functions are holomorphic: Z ′ = F ( Z, Θ ) , Θ ′ = Ψ ( Z, Θ ) , – (superconformal structure) the differential operator D = ∂∂Θ + Θ ∂∂Z transforms homogeneously: D ′ ∝ D .More explicitly, the general form of transition functions reads Z ′ = f ( Z ) + Θψ r ∂f∂Z ,Θ ′ = ψ ( Z ) + Θ r ∂f∂Z + ψ ∂ψ∂Z . In other words, the extra structure on super-Riemann surface S | is given by a | -dimensional subbundle D ofthe tangent bundle T X | such that the following sequence is exact → D → T S | → D → . There are three typical | -dimensional super-Riemann surfaces with simply-connected bodies: – the complex superplane C | ; – the super-Riemann sphere ˆ C | : covered by two open domains (in the De Witt topology) which are glued bythe transition functions ( Z ′ , Θ ′ ) = ( − Z , ΘZ ); – the upper half superplane C H | discussed in the previous section.The groups of superconformal automorphisms on ˆ C | and C H | are OSp (1 | , C ) / {± Id } and OSp (1 | , R ) / {± Id } , respectively. Proposition 5
Let S | be a super-Riemann surface with a compact Riemann surface S of genus g S ≥ asthe body of S | . Then the superconformal structure on S | produces irreducible representations ρ : π ( S ) → SL (2 , R ) of the fundamental group π ( S ) of S .Proof Manin et al. have showed that the superconformal structure on S | corresponds to a choice of the thetacharacteristic on S , namely, a line bundle L over S satisfying L ⊗ ≃ Ω S sm [17]. Then one can construct a bundle E by the following extension → L − → E → L → . The Higgs field φ ∈ H ( S, End( E ) ⊗ Ω S ) is defined by the composition φ : E → L ≃ L − ⊗ L ⊂ E ⊗ Ω S .It is obvious that the line subbundle of E preserved by the Higgs field φ is exactly L − , and deg( L − ) < E ) when g S ≥ . Hence ( E, φ ) is a stable Higgs bundle over S , which yields an irreducible representation ρ : π ( S ) → GL (2 , C ) by Simpson correspondence si [23]. Obviously, the image of this representation lies in thesubgroup SL (2 , R ) . ⊓⊔ Inspired by the definition of the classical Arakelov-Green function l,s,we [2,5,24], we propose the supergeometricversion as follows.
Definition 2
Let S | be a super-Riemann surface with local coordinates { X A } = ( Z ; Θ ) . A triple ( P, g, G P ) consisting of a fixed point P ∈ S | , a supermetric g = g A ¯ B dX A dX ¯ B , and a superfunction G P : S | → ( Λ ∞ R ) is called a super-Green triple on S | if it satisfies the following conditions: – ǫ ( G P ( Q )) ≥ for any Q ∈ S | , – writing G P ( X ) = G (0) P ( Z ) + ( G (1) P ( Z ) Θ + ¯ G (1) P ( Z ) ¯ Θ ) + G (2) P ( Z ) Θ ¯ Θ in the neighborhood centered at P =(0; 0) , each nonzero component G ( i ) P ( Z ) , i ∈ { , , } has a first order zero for Z = 0 , – − ( − | X ¯ B | dX A ( ∂ ∂X A ∂X ¯ B log G P ( X )) dX ¯ B coincides with g outside the singular locus of log G P ( X ) , – ( ǫ ( P ) , ǫ ( g ) , ǫ ( G P ( X ))) provides a classical Green triple on the body of S | .In particular, the superfunction G P = log G P is called a super-Green function on S | associated with the super-metric g .We first consider the super-Riemann sphere ˆ C | . Proposition 6 ˆ C | can be endowed with a supermetric ds = −
12 ( − | X ¯ B | dX A ( ∂ A ∂ ¯ B log Z ¯ Z Z ¯ Z + Θ ¯ Θ ) dX ¯ B = 1(1 + Z ¯ Z + Θ ¯ Θ ) ((1 + Θ ¯ Θ ) dZd ¯ Z − Θ ¯ ZdZd ¯ Θ + Z ¯ ΘdΘd ¯ Z + (1 + Z ¯ Z + 2 Θ ¯ Θ ) dΘd ¯ Θ ) , (3.1) and a super-volume form d SVol= vuuuuuuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
Sdet Θ ¯ Θ Z ¯ Z + Θ ¯ Θ ) Θ ¯ Z Z ¯ Z + Θ ¯ Θ ) Θ ¯ Θ Z ¯ Z + Θ ¯ Θ ) − Z ¯ Θ Z ¯ Z + Θ ¯ Θ ) − Z ¯ Θ Z ¯ Z + Θ ¯ Θ ) − Z ¯ Z +2 Θ ¯ Θ Z ¯ Z + Θ ¯ Θ ) Θ ¯ Z Z ¯ Z + Θ ¯ Θ ) Z ¯ Z +2 Θ ¯ Θ Z ¯ Z + Θ ¯ Θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dZd ¯ ZdΘd ¯ Θ = dZd ¯ ZdΘd ¯ Θ Z ¯ Z + Θ ¯ Θ . yperbolic Superspaces and Super-Riemann Surfaces 15
Proof
We calculate the transformation of each summand with respect to the transition functions of ˆ C | Θ ¯ Θ (1 + Z ¯ Z + Θ ¯ Θ ) dZd ¯ Z Z ¯ Z + Θ ¯ Θ ) (1 + Θ ¯ ΘZ ¯ Z ) dZd ¯ Z, − Θ ¯ Z (1 + Z ¯ Z + Θ ¯ Θ ) dZd ¯ Θ Z ¯ Z + Θ ¯ Θ ) ( ΘZ dZd ¯ Θ − Θ ¯ ΘZ ¯ Z dZd ¯ Z ) ,Z ¯ Θ (1 + Z ¯ Z + Θ ¯ Θ ) dΘd ¯ Z Z ¯ Z + Θ ¯ Θ ) ( − ¯ Θ ¯ Z d ¯ ZdΘ − Θ ¯ ΘZ ¯ Z dZd ¯ Z ) , Z ¯ Z + 2 Θ ¯ Θ (1 + Z ¯ Z + Θ ¯ Θ ) dΘd ¯ Θ Z ¯ Z + Θ ¯ Θ ) ((1 + Z ¯ Z + 2 Θ ¯ Θ ) dΘd ¯ Θ − (1 + 1 Z ¯ Z ) Θ ¯ ZdZd ¯ Θ + (1 + 1 Z ¯ Z ) Z ¯ ΘdΘd ¯ Z + (1 + 1 Z ¯ Z ) Θ ¯ ΘdZd ¯ Z ) . Combining these results, the given metric is globally well-defined on the supersphere. ⊓⊔ d Corollary 1
Let P = (0; 0) ∈ ˆ C | , g is the supermetric given by ( , and G P ( X ) = q Z ¯ Z Z ¯ Z + Θ ¯ Θ , then ( P, g, G P ) forms a super-Green triple on ˆ C | . To obtain the supergeometric analog of Manin’s result connecting Green function and geodesic, we need tostudy the supergeodesics in the hyperbolic superspaces. The supergeodesic in C H | with respect to the superme-tric ( ss u,uy [15,16] d Zd u + iY ( dZdu ) + ¯ ΘY dZdu dΘdu = 0 ,d Θd u + iY dZdu dΘdu = 0 , and the complex conjugated ones. Proposition 7
Let P = ( Z , Θ ) , P = ( Z , Θ ) be two points in C H | , then one can join P and P bysupergeodesics piecewisely.Proof We have the following solutions for the supergeodesic equations: – Type-I: when dZdu ≡ , there is a solution Z = c, ¯ Z = ¯ c, Θ ( u ) = γu + ζ, ¯ Θ ( u ) = ¯ γu + ¯ ζ, for constants c ∈ ( Λ ∞ C ) , γ, ζ ∈ ( Λ ∞ C ) , – Type-II: when dZdu = 0 , there is a solution Z ( u ) = c [tanh ω ( u + u ) + i sech ω ( u + u )] + c , or ie ω ( u + u ) + c ,Θ ( u ) = ξZ ( u ) , ¯ Z ( u ) = Z ( u ) , ¯ Θ ( u ) = Θ ( u ) , for constants c , c , c , ω, u ∈ ( Λ ∞ R ) , and ξ ∈ ( Λ ∞ R ) .Firstly, we join P and P ′ = ( Z , ξZ ) for some ξ ∈ ( Λ ∞ R ) by virtue of a supergeodesic G I described by thetype-I solution. The same argument as that for the classical geodesics implies that there exists a supergeodesic G II described by the type-II solution such that it is parameterized by u with Z ( u ) = Z , Z ( u ) = Z and ǫ ( u ) ∈ [ ǫ ( u ) , ǫ ( u )] . Namely, G II joins the points P ′ and P ′ = ( Z , ξZ ) . Finally, we join the points P ′ and P via a supergeodesic G ′ I described by the type-I solution.The above proposition suggests the following definition. Definition 3
The bosonic superdistance d ( P , P ) between P and P is defined by the integral along the super-geodesic G II d ( P , P ) = Z u u r ( dsdu ) du = ω ( u − u ) , or defined by cosh d ( P , P ) = 1 + | Z − Z | Z )Im( Z ) . Now we view ˆ C | as the boundary of H | S {∞} . For two distinct point P , P ∈ ˆ C | , one has an upperhalf superplane C H | P P = { ( Z, Θ ) : Im( Z ) = t > } embedded in H | such that P , P lie on its boundary.According to Manin’s approach, one joins P and P piecewisely with supergeodesics in C H | P P as described inthe proof of Proposition
7. In particular, within these supergeodesics, the supergeodesic G II with a chosen constant ξ for odd coordinates is denoted by { P , P } ξ . Let Q be another point in H | , then one intoduces d Q ( P , P ) = d ( Q, P Q ) where P Q ∈ { P , P } ξ is uniquely determined by the following condition ǫ ( d ( Q, P Q )) = inf P ∈{ P ,P } ξ ǫ ( d ( Q, P )) , and the bosonic superdistances appearing here are calculated in the upper half superplane C H | QP Q with Q = Q | t =0 , P Q = P Q | t =0 . For two given pints P = (0 , ,
0; 0 , , P = ( x, y, θ , θ ) lying on the boundary of H | and Q = (0 , ,
1; 0 , ∈ H | , we have P Q = ( x x + y , y x + y , p ( x + y )(1 + x + y )2 + x + y ; xξ x + y , yξ x + y ) , hence cosh d Q ( P , P ) = 1 + ( | Z | | Z | ) + (1 − | Z | √ | Z | | Z | ) | Z | √ | Z | | Z | = s | Z | , where Z = x + iy . Consequently, we arrive at the following proposition. Proposition 8
The super-Green function on ˆ C | defined as in Corollary d G P ( Z, Θ ) = log( 1cosh d Q ( P , P ) −
12 cosh d Q ( P , P ) (1 − d Q ( P , P ) ) Θ ¯ Θ ) with P = ( Z ; Θ ) . The above approach is not sensitive to the odd coordinates. However, we can do some modifications for takingthe odd part into account. To achieve that, one introduces the superdistance function d : C H | × C H | → ( Λ ∞ R ) ,which was firstly defined by physicists Uehara and Yasui uy [16], as follows cosh d ( P , P ) = 1 + 12 R ( P , P ) − r ( P , P ) yperbolic Superspaces and Super-Riemann Surfaces 17 where P = ( Z ; Θ ) , P = ( Z ; Θ ) ∈ C H | , and R ( P , P ) = | Z − Z − Θ Θ | Y Y ,r ( P , P ) = 2 Θ ¯ Θ + i ( Θ − i ¯ Θ )( Θ + i ¯ Θ )4 Y + 2 Θ ¯ Θ + i ( Θ − i ¯ Θ )( Θ + i ¯ Θ )4 Y + ( Θ + i ¯ Θ )( Θ + i ¯ Θ )Re( Z − Z − Θ Θ )4 Y Y for Y i = Im( Z i ) + Θ i ¯ Θ i , i = 1 , . It is easy to see that this superdistance function enjoys the properties d ( P , P ) = d ( P , P ) = d ( Γ · P , Γ · P ) for any Γ ∈ OSp (1 | , R ) sm [17]. In our setting, the inputs are two given points P = (0; 0) , P = ( Z ; Θ ) lying onthe boundary of H | . Then in the upper half superplane C H | P P , we join P and P by the supergeodesic ˜ G II governed by the following solution Z ( u ) = ( | Z | − iΘ ¯ Θe ω ( u + u ) ω ( u + u ) )[tanh ω ( u + u ) + i sech ω ( u + u )] + | Z | ,Θ ( u ) = Θ ω ( u + u ) + i sech ω ( u + u )] , ¯ Z ( u ) = Z ( u ) , ¯ Θ ( u ) = Θ ( u ) , with ǫ ( u ) ∈ [ −∞ , + ∞ ] . The point ˜ P ∈ ˜ G II determined by ǫ ( ˜ P ) = ǫ ( P Q ) is given by ˜ P = ( | Z | | Z | + p | Z | (2 + | Z | ) Θ ¯ Θ + i ( | Z | p | Z | | Z | + | Z | | Z | ) Θ ¯ Θ ); Θ − | Z | | Z | + i p | Z | | Z | )) , as a point in C H | P P , hence the superdistance between ˜ Q = (0 , , | Z | + √ | Z | | Z | (2+ | Z | −| Z | √ | Z | ) Θ ¯ Θ ; 0 , and ˜ P isgiven by cosh d ( ˜ Q, ˜ P )=1 + ( | Z | | Z | + √ | Z | (2+ | Z | ) Θ ¯ Θ ) + (1 − | Z | √ | Z | | Z | + ( | Z | + √ | Z | | Z | (2+ | Z | −| Z | √ | Z | ) − | Z | | Z | ) ) Θ ¯ Θ ) | Z | √ | Z | | Z | + | Z | )(2+ | Z | ) Θ ¯ Θ − Θ ¯ Θ | Z | | Z | √ | Z | | Z | + | Z | (2+ | Z | ) Θ ¯ Θ = s | Z | + Θ ¯ Θ | Z | p | Z | = s | Z | + Θ ¯ Θ | Z | . Aa a result, we obtain the following proposition. Proposition 9
The super-Green function defined as in Corollary d G P ( Z, Θ ) = log 1cosh d ( ˜ Q, ˜ P ) for the points ˜ Q, ˜ P given as above. Next, we investigate the same problem for the supertorus. A supertorus T | = C | /Γ is defined as thequotient of the complex superplane C | , with the coordinates ( Z, Θ ) , by the supertranslation group Γ generatedby the transformations T : Z Z + 1 , Θ Θ,S : Z Z + τ + Θδ, Θ Θ + δ, where the pair ( τ, δ ) ∈ ( Λ ∞ C ) × ( Λ ∞ C ) with ǫ (Im( τ )) > is called the supermoduli of T | f [25]. The Jacobitheta function on the ordinary torus C / ( Z + τ Z ) is given by ϑ ( Z ; τ ) = − i X n ∈ Z ( − n q ( n + ) e (2 n +1) πiZ = iρ q ∞ Y n =1 (1 − q n )(1 − ρq n )(1 − ρ − q n − ) for ρ = e πiZ , q = e πiτ , which satisfies ϑ ( Z + 1; τ ) = − ϑ ( Z ; τ ) = ϑ ( − Z ; τ ) ,ϑ ( Z + τ ; τ ) = − ( ρq ) − ϑ ( Z ; τ ) . As an analog, we have the super-theta function, which was firstly introduced by Rabin and Freund f,ra [25,26], T ( Z, Θ ; τ, δ ) = ϑ ( Z ; τ + Θδ ) , then one easily checks the following proposition. Proposition 10
The super-theta function satisfies the properties: T ( Z, Θ ; τ, δ ) = ϑ ( Z ; τ ) + Θδ ˙ ϑ ( Z ; τ ) , T ( Z + 1 , Θ ; τ, δ ) = −T ( Z, Θ ; τ, δ ) = T ( − Z, Θ ; τ, δ ) , T ( Z + τ + Θδ, Θ + δ ; τ, δ ) = − (1 − πiΘδ ) q − e − πiZ T ( Z, Θ ; τ, δ ) , T ′ (0 , Θ ; τ, δ ) = − πq ∞ Y n =1 (1 − q n ) (1 + πi Θδ − πi ∞ X m =1 mq m − q m Θδ ) , where the symbols dot and prime denote the partial derivatives with respect to τ and Z , respectively. Following the classical Arakelov-Green function on the usual torus l [2], we introduce the super-Green function G ( Z, Θ ) on T | as follows G ( Z, Θ ) = log | T ( Z, Θ ; τ, δ ) T ′ (0 , Θ ; τ, δ ) | − π ( (Im( Z )) + 2 Θ ¯ Θ τ + Θδ ) − π ∞ X n =1 log | − q n − πiΘδnq n | + 112 Im( τ + Θδ ) − log 2 π π ) . By the same manner of supergeometric extension, one can generalize the classical Faltings invariant on the torus m,we,mb [1,24,27] to a superfunction F ( Θ ) that depends only on the odd coordinate of the supertorus F ( Θ ) = − π log | (Im( τ + Θδ )) ( T ′ (0 , Θ ; τ, δ )) | = − π log | Im( τ + Θδ ) | − π ∞ X n =1 log | − q n | + Im( τ + Θδ ) − Im( ∞ X m =1 mq m − q m Θδ ) − log 2 ππ . yperbolic Superspaces and Super-Riemann Surfaces 19 Proposition 11 G ( Z, Θ ) is invariant under the supertranslations.2. lim ( Z,Θ ) → (0 , G ( Z,Θ )log | Z | = 1 .3. G ( Z, Θ ) can be rewritten as G ( Z, Θ ) = log | q B
2( Im( Z )Im( τ ) )2 (1 − ρ ) ∞ Y n =1 (1 − ρq n )(1 − ρ − q n ) | + 2 π ( ∞ X m =1 Im( (2 − ρq m − ρ − q m ) mq m (1 − ρq m )(1 − ρ − q m ) Θδ ) + ( 12 ( Im( Z )Im( τ ) ) −
112 )Im( Θδ ) − ( 1Im( τ ) + (Im( Z )) τ )) δ ¯ δ ) Θ ¯ Θ ) , where B ( y ) = y − y + is the second Bernoulli polynomial, and the first term on the right-hand side of theequal sign is recognized as the Néron function l [2].4. The second-order partial derivatives of G ( Z, Θ ) outside the singular locus of G ( Z, Θ ) read − π ∂ G ( Z, Θ ) ∂Z∂ ¯ Z = 14Im( τ ) − Im( Θδ )4(Im( τ )) + δ ¯ δΘ ¯ Θ τ )) , − π ∂ G ( Z, Θ ) ∂Z∂ ¯ Θ = Im( Z )4(Im( τ )) ¯ δ + i Im( Z )4(Im( τ )) δ ¯ δΘ, − π ∂ G ( Z, Θ ) ∂Θ∂ ¯ Z = − Im( Z )4(Im( τ )) δ + i Im( Z )4(Im( τ )) δ ¯ δ ¯ Θ, − π ∂ G ( Z, Θ ) ∂Θ∂ ¯ Θ = − τ ) − (Im( Z )) τ )) δ ¯ δ. Proof (1) It obviously follows from the invariance of the classical Arakelov-Green function.(2) We only need to note that T ( Z, Θ ; τ, δ ) T ′ (0 , Θ ; τ, δ ) ( Z,Θ ) ∼ (0 , ∼ Z. Hence G ( Z, Θ ) ( Z,Θ ) ∼ (0 , ∼ log | Z | .(3) Letting q = e πi ( τ + Θδ ) , we have the identity ∞ Y n =1 (1 − ρ q n )(1 − ρ − q n ) = − πiρ (1 − ρ − ) q Q ∞ n =1 (1 − q n ) (1 − ρ q n )(1 − ρ − q n ) − πiρ (1 − ρ − ) q Q ∞ n =1 (1 − q n ) = 2 πi Q ∞ n =1 (1 − q n ) T ( Z, Θ ; τ, δ ) ρ (1 − ρ − ) T ′ (0 , Θ ; τ, δ ) , which yields G ( Z, Θ ) = log | q B
2( Im( Z )Im( τ + Θδ ) )2 (1 − ρ ) ∞ Y n =1 (1 − ρ q n )(1 − ρ − q n ) | − π Θ ¯ Θ Im( τ + Θδ )= log | (1 − ρ ) ∞ Y n =1 (1 − ρ q n )(1 − ρ − q n ) |− π ( (Im( Z )) + 2 Θ ¯ Θ τ + Θδ ) − Im( Z )2 + 112 Im( τ + Θδ )) . Hence (3) follows.(4) These second-order partial derivatives can be directly calculated by means of the formula in (3). ⊓⊔ From the above proposition, we see that in order for illustrating G ( Z, Θ ) as a super-Green function we alsoneed the following proposition. Proposition 12
The supertorus T | can be equipped with a supermetric ds = 1Im( τ ) [(1 − Im( Θδ )Im( τ ) + δ ¯ δΘ ¯ Θ τ )) ) dZd ¯ Z − ( Im( Z )Im( τ ) ¯ δ + i Im( Z )(Im( τ )) δ ¯ δΘ ) dZd ¯ Θ + ( Im( Z )Im( τ ) δ − i Im( Z )(Im( τ )) δ ¯ δ ¯ Θ ) dΘd ¯ Z + (1 + (Im( Z )) (Im( τ )) δ ¯ δ ) dΘd ¯ Θ ] . Proof
By the concentrated expression of the supermetric ds = 1Im( τ + Θδ ) [ dZd ¯ Z − Im( Z )Im( τ + Θδ ) ¯ δdZd ¯ Θ + Im( Z )Im( τ + Θδ ) δdΘd ¯ Z + (1 + (Im( Z )) (Im( τ + Θδ )) δ ¯ δ ) dΘd ¯ Θ ] , one checks the invariance under the transformation S . Indeed, we have dZd ¯ Z dZd ¯ Z + ¯ δdZd ¯ Θ − δdΘd ¯ Z + δ ¯ δdΘd ¯ Θ, Im( Z )Im( τ + Θδ ) ¯ δdZd ¯ Θ Im( Z )Im( τ + Θδ ) ¯ δdZd ¯ Θ + Im( Z )Im( τ + Θδ ) δ ¯ δdΘd ¯ Θ + ¯ δdZd ¯ Θ + δ ¯ δdΘd ¯ Θ, Im( Z )Im( τ + Θδ ) δdΘd ¯ Z Im( Z )Im( τ + Θδ ) δdΘd ¯ Z − Im( Z )Im( τ + Θδ ) δ ¯ δdΘd ¯ Θ + δdΘd ¯ Z − δ ¯ δdΘd ¯ Θ, (Im( Z )) (Im( τ + Θδ )) δ ¯ δdΘd ¯ Θ (Im( Z )) (Im( τ + Θδ )) δ ¯ δdΘd ¯ Θ + 2Im( Z )Im( τ + Θδ ) δ ¯ δdΘd ¯ Θ + δ ¯ δdΘd ¯ Θ. Thus the conclusion follows. ⊓⊔ Assume the odd moduli δ is valued in ( Λ ∞ R ) , then the super-Poincáre extension ˜ Γ of the supertranslationgroup generated by the transformations ˜ S : x x + 1 , y y, t t, θ θ , θ θ , ˜ T : x + iy x + iy + τ + θ δ, t t + θ δ, θ θ , θ θ + δ acts on H | . Hence one regards T | as the boundary of H | / ˜ Γ , and the latter one is treated as a solid supertorus.To avoid extra requirements on the odd moduli, we adopt the following approach of extension instead of the super-Poincáre extension. One defines the supertorus by the equivalence relation ( ρ ; Θ ) ∼ ( q n ρ ; Θ + nδ ) for n ∈ Z whichcan be extended to H | as ( ρ, t ; Θ ) ∼ ( q n ρ, | q | n t ; Θ + nδ ) . Hence one views T | as the boundary of H | / ∼ . Fortwo equivalent points P = (0 ,
1; 0) and Q = (0 , | q | ; δ ) lying in H | , one calculates the superdistance between P and Q in some upper half superplane C H | as follows cosh d ( P , Q ) = 1 + (1 − | q | ) − δ ¯ δ | q | + δ ¯ δ ) , namely, we have d ( P , Q ) = d ( P , Q ) + d ( P , Q ) δ ¯ δ = log | q | + 1 + | q | | q | (1 − | q | ) δ ¯ δ. For a point P = ( ρ, Θ ) on the boundary of H | , there is a supergeodesic in C H | P P determined by the followingequations ρ ( u ) = ( | ρ | − iΘ ¯ Θe ω ( u + u ) ω ( u + u ) )[tanh ω ( u + u ) + i sech ω ( u + u )] ,Θ ( u ) = Θ ω ( u + u ) + i sech ω ( u + u )] , ¯ ρ ( u ) = Z ( u ) , ¯ Θ ( u ) = Θ ( u ) , yperbolic Superspaces and Super-Riemann Surfaces 21 which joins P and b P = ( Θ ¯ Θ , | ρ | ; i Θ ) . Then the superdistance between P and b P is given by d ( P , b P ) = d ( P , b P ) + d ( P , b P ) Θ ¯ Θ = log | ρ | + 1 + | ρ | | ρ | (1 − | ρ | ) Θ ¯ Θ. Similarly, for the point e P = (0 , − | ρ | +14 | ρ | ( | ρ | − Θ ¯ Θ ; 0) ∈ H | , we have d ( e P , b P ) = log | ρ | . As a consequence, we arrive at Proposition 13
The super-Green function on the supertorus T | can be expressed as G ( Z, Θ ) = 12 d ( P , b Q ) B ( d ( P , b P ) d ( P , b Q ) ) + d ( P , c P − )+ ∞ X n =1 ( d ( P , c Q n ) + d ( P , d Q n − )) + 4 π d ( P , b Q ) Θ ¯ Θ = 12 d ( e P , b Q ) B ( d ( e P , b P ) d ( e P , b Q ) ) + d ( e P , c P − )+ ∞ X n =1 ( d ( e P , c Q n ) + d ( e P , d Q n − )) + 4 π d ( e P , b Q ) Θ ¯ Θ, where Q = ( q , Θ ) , P − = (1 − ρ ; Θ ) , Q n = (1 − q n ρ, Θ ) and Q n − = (1 − q n ρ − , Θ ) are all points lying on theboundary of H | . References
1. Yu. Manin, Three-dimensional hyperbolic geometry as ∞ -adic Arakelov geometry, Invent. Math. 104, 223-243 (1991).2. S. Lang, Introduction to Arakelov theory, Springer, 1988.3. D. Mumford, An analytic construction of degenerating curves over complete local rings, Composito. Math. 24, 129-172 (1974).4. A. Werner, Arakelov intersection indices of linear cycles and the geometry of buildings and symmetric spaces, Duke Math. Jour. 111,319-355 (2002).5. D. Smit, String theory and algebraic geometry of moduli spaces, Commun. Math. Phys. 114, 645-685 (1988).6. Yu. Manin and M. Marcolli, Holography principle and arithmetic of algebraic curves arXiv: hep-th/0201036.7. C. Consani, M. Marcolli, Noncommutative geometry, dynamics, and ∞ -adic Arakelov geometry, Sel. Math. New Ser. 10, 167-251 (2004).8. C. Deninger, On the Γ -factors attached to motives, Invent. Math. 104, 245-261 (1991).9. B. De Witt, Supermanifolds (2nd Edition), Cambridge Uni. Press, 1992.10. A. Rogers, Supermanifolds: theory and applications, World Scientific, 2007.11. E. Witten, Notes on supermanifolds and integration, arXiv:1209.2199.12. S. Hu and Z. Hu, On SL (2 , R ) and AdS gravity, Intern. Jour. Modern Phys. A 27, 1250138 (2012).13. S. Gubser, J. Knaute, S. Parikh, A. Samberg and P. Witaszczyk, pp