A very short note on the (rational) graded Hori map
aa r X i v : . [ m a t h . A T ] M a r A VERY SHORT NOTE ON THE (RATIONAL) GRADEDHORI MAP
MATTIA COLOMA, DOMENICO FIORENZA, AND EUGENIO LANDI
Abstract.
The graded Hori map has been recently introduced by Han-Mathai in the context of T-duality as a Z -graded transform whose homoge-neous components are the Hori-Fourier transforms in twisted cohomologyassociated with integral multiples of a basic pair of T-dual closed 3-forms.We show how in the rational homotopy theory approximation of T-duality,such a map is naturally realised as a pull-iso-push transform, where theisomorphism part corresponds to the canonical equivalence between theleft and the right gerbes associated with a T-duality configuration. Contents
1. Foreword 12. Cocycles and extensions of DGCAs 23. Two equivalent rational gerbes 54. The graded Hori map from rational equivalences of gerbes 95. Hori transforms of meromorphic functions 126. Extending coefficients to the ring of Jacobi forms 13Acknowledgements 15References 151.
Foreword
The graded Hori map has been recently introduced in [HM20], by assem-bling together the Z -family of Hori maps associated with a certain Z -family ofT-duality configuration data naturally associated to a single T-duality configu-ration. This may at first sight appear as a rather ad hoc construction. The aimof this note is to show how, on the contrary, the graded Hori map as a wholenaturally emerges from the geometry associated with a T-duality configuration.One only needs to look at a step higher with respect to the T-dual bundles: thegraded Hori map is a manifestation of a canonical equivalence between the leftand the right gerbes associated with a T-duality configuration. More precisely,we show that, in the rational homotopy theory approximation of T-duality,such a map is naturally realised as a pull-iso-push transform, where the iso-morphism part corresponds to the left gerbe/right gerbe canonical equivalence.We will construct this pull-iso-push transform using only purely algebraicconstructions related to the category DGCA of differential graded commutativealgebras (DGCAs) over a characteristic zero field K , which can be assumed to be the field Q of rational numbers. In particular, we will heavily use the lan-guage of extensions of DGCAs associated with DGCA cocycles. The readerfamiliar with rational homotopy theory will immediately recognise every stepin the construction we are going to present as a translation of phenomena ap-pearing in the rational homotopy theory approximation of T-duality. We pointthe unfamiliar reader to [FSS18b] for an introduction very close to the spirit ofthis note. We also borrow from [FSS18a] the rational homotopy theory descrip-tion of the equivalence of the gerbes associated to a T-duality configuration.See [BS05] for the topological origin of this equivalence. Here we choose topresent the construction in purely algebraic terms, leaving to the reader thejob of connecting to rational homotopy theory.The note is organised as follows. First we review topological T -duality inrational homotopy theory, in particular, in Section 2 we recall a few basic con-structions on extensions of DGCAs and define the DGCA (co)classifying ratio-nal T -duality configurations, and in Section 3 we recall the definition of the twoisomorphic rational gerbes associated with a rational T -duality configuration,whose isomorphism will be the “iso” part in the “pull-iso-push” transform.After this review, in Section 4 we define the graded Hori map T L Ñ R asso-ciated with these data and extend it to Laurent series. In Section 5 we showhow, when the base field is the field C of complex numbers, this allows one todescribe the graded Hori map as an operator on rings of meromorphic func-tions with a single pole at the origin taking values in a DGCA A endowedwith a rational T-duality configuration. It turns out that in this translationthe graded Hori map becomes the antidiagonal matrix ˜ ´ q ddq ¸ where q is the complex coordinate on C . Finally, in Section 6 we show how onecan further extend coefficients to A -valued index Jacobi forms in the twovariables p z, τ q P C ˆ H , by means of their q -expansion, where q “ e πiz . Thisway we recover the original definition of the graded Hori map by Han-Mathai,as well as its main properties. In particular, one identifies the graded Horimap on Jacobi forms with the antidiagonal matrix ˜ ´ πi BB z ¸ , and therefore the composition of two graded Hori transforms as the operator ´ πi BB z on the ring of A -valued index Jacobi-forms [HM20, Theorem 2.2].2.
Cocycles and extensions of DGCAs
We start with a (non-negatively graded) differential graded commutativealgebra p A, d q over the field K and with a -cocycle, i.e., a closed homogeneouselement of degree , t P A . We can extend our base DGCA A in such a wayto trivialise the -cocycle t by adding a formal generator e of degree and VERY SHORT NOTE ON THE (RATIONAL) GRADED HORI MAP 3 declaring our extension to be p A, d q A t t u : “ p A r e s , de “ t q , ι where the differential of A t t u coincides with the differential d on the subalgebra A .The choice of a -cocycle for the DGCA A is the same datum as a DGCAmap from the polynomial DGCA p K r x s , q to A , where p K r x s , q is the polyno-mial algebra over K on a single degree 2 generator x , endowed with the trivialdifferential. This in turn means regarding A as an object under p K r x s , q , apoint of view that will be useful later.More generally, the datum of a DGCA map from p K r x n s , q to A , where now x n is a degree n variable, is the same as that of an n -cocycle in A and, again,given such a cocycle t n P A it is possible to extend A to trivialise t n by p A, d q A t t n u : “ p A r e n ´ s , de n ´ : “ t n q . ι The construction of A t t n u out of the pair p A, t n q is universal: A t t n u togetherwith the embedding of the sub-DGCA A is the homotopy cofibre of t n , i.e., thehomotopy pushout of the diagram: K r x n s p A, d q ψ tn of DGCAs, where ψ t n is the unique DGCA morphism with ψ p x n q “ t n , in theprojective model structure on non-negatively graded DGCAs, see, e.g., [BG76].Indeed, in order to compute a model for this cofibre one has to replace thevertical map by a cofibration followed by a weak equivalence, and the easiestway of doing this is to consider K r x n s ã Ñ p K r x n , e n ´ s , de n ´ “ x n q – , and then compute the ordinary pushout of the diagram K r x n s p A, d qp K r x n , e n ´ s , de n ´ “ x n q ψ tn to obtain K r x n s p A, d qp K r x n , e n ´ s , de n ´ “ x n q p P, d P q , ψ tn with p P, d P q “ p A r e n ´ s , d P a “ da for a P A, d P e n ´ “ ψ t n p x n qq“ p A r e n ´ s , de n ´ “ t n q “ A t t n u . MATTIA COLOMA, DOMENICO FIORENZA, AND EUGENIO LANDI
Universality implies in particular that the construction p A, t n q ù A t t n u isnatural, a fact that can also be easily checked directly: if f : p A, t n q Ñ p B, s n q is a morphism of DGCAs endowed with n -cocycles, i.e., if f is a morphism ofDGCAs, f : A Ñ B , such that f p t n q “ s n , then we get a morphism of DGCAs ˆ f : A t t n u Ñ B t s n u by setting ˆ f p a q “ f p a q for any a P A and ˆ f p e n ´ A q “ e n ´ B .This is manifestly compatible with compositions of morphisms of DGCAs en-dowed with n -cocycles. Remark . If n is even, every degree k element a k in A t t n u can be uniquelywritten as a k “ α k ` e n ´ β k ´ n ` , for some degree k element α k and somedegree k ´ n ` element β k ´ n ` in A . The map π : A t t n u Ñ A r´ n ` s α k ` e n ´ β k ´ n ` ÞÑ β k ´ n ` is a map of chain complexes. Namely, we have d r´ n ` s p π p a k q “ d r´ n ` s p π p α k ` e n ´ β k ´ n ` qq“ d r´ n ` s β k ´ n ` “ p´ q p n ´ q dβ k ´ n ` and π p da k q “ π p d p α k ` e n ´ β k ´ n ` qq“ π p dα k ` t n β k ´ n ` ` p´ q n ´ dβ k ´ n ` q“ p´ q n ´ dβ k ´ n ` . Of course, π is not a map of DGCAs (the shifted complex A r´ n ` s does noteven have a natural DGCA structure). But it is a map of right DG- A -modules:if γ l is a degree l element in A , then π p a k γ l q “ π pp α k ` t n β k ´ n q γ l q “ π pp α k γ l q ` t n p β k ´ n γ l q “ β k ´ n γ l “ π p a k q γ l . As a side remark, by thinking of ι : A Ñ A t t n u as a pullback p ˚ and of π : A t t n u Ñ A r´ n ` s as the pushforward p ˚ , the above identity is the projec-tion formula: p ˚ p a k p ˚ p γ l qq “ p ˚ p a k q γ l . Finally, the map of right DG- A -modules π : A t t n u Ñ A r´ n ` s has an evidentsection e n ´ ¨ ´ : A r´ n ` s Ñ A t t n u given by the left multiplication by e n ´ .An example of the construction p A, t n q ù A t t n u we will be interested in isthe following. Consider the polynomial algebra K r x L , x R s – K r x L s b K r x R s VERY SHORT NOTE ON THE (RATIONAL) GRADED HORI MAP 5 on two degree 2 generators x L and x R , endowed with the trivial differential. Then the element x L x R is a 4-cocycle and so defines a DGCA map K r t s Ñ K r x L , x R s t ÞÑ x L x R The associated extension is the DGCA K r x L , x R s t x L x R u “ p K r x L , x R , y s , dx L “ dx R “ , dy “ x L x R q Notice that K r x L , x R s t x L x R u carries two distinguished 2-cocycles x L and x R and that σ : x L Ø x R is a DGCA automorphism of K r x L , x R s t x L x R u exchanging the two cocycles. We denote by p L , p R : K r x s Ñ K r x L , x R s t x L x R u the two maps corresponding to the cocycles x L , x R , respectively.3. Two equivalent rational gerbes
In order to get the DGCA construction corresponding to the rational ho-motopy description of the pull-iso-push transform between gerbes associatedwith a T-duality configuration, we consider a DGCA A together with a map K r x L , x R s t x L x R u f Ñ A . As we noticed above, the source of f has two dis-tinct -cocycles corresponding to maps p L , p R : K r x s Ñ K r x L , x R s t x L x R u sending the generator x in x L and in x R , respectively. Composing withthe map f we therefore get maps f L , f R : K r x s Ñ A , corresponding to twodistinct -cocycles in A , and we end up with following commutative diagramof DGCAs: K r x s A A K r x L , x R s t x L x R u K r x L , x R s t x L x R u p L p R f L f R σf f The previous diagram shows that the map f can be read in two differentways as a map in the under category K r x s{ DGCA of DGCAs endowed with adistinguished -cocycle. In particular, identifying a -cocycle with the mor-phism out of K r x s identifying it, we have that f is both a map between K r x L , x R s t x L x R u and A decorated with their left -cocycles p K r x L , x R s t x L x R u , p L q p A, f L q , f and with their right -cocycles p K r x L , x R s t x L x R u , p R q p A, f R q . f This will be crucial in order to define the equivalence between the algebraicstructures corresponding to the left and right gerbes of topological T-duality. Here and below, all tensor products are over K . MATTIA COLOMA, DOMENICO FIORENZA, AND EUGENIO LANDI
We begin with the following, which is a particular case of the “hofib/cyc ad-junction” of [FSS18a,FSS18b], and whose proof in this specific case we give forthe sake of completeness.
Proposition 3.1.
Let p A, t q be a DGCA with a distinguished -cocycle t .Then the association Hom K r x s{ DGCA ` p K r x L , x R s t x L x R u , p L q , p A, ψ t q ˘ Hom
DGCA ` K r x s , A t t u ˘ ϕ ˜ ϕ where ˜ ϕ is defined by ˜ ϕ : x ÞÑ ϕ p y q ´ e ϕ p x R q , is a natural bijection. Clearly, everything identically works if we exchange p L with p R and x R with x L .Proof. We begin by showing that ˜ ϕ p x q is a -cocycle. If ϕ : p K r x L , x R s t x L x R u , p L q Ñ p A, ψ t q is a map in the under category K r x s{ DGCA , then ϕ p x L q “ p ϕ ˝ p L qp x q “ ψ t p x q “ t . Therefore, d p ˜ ϕ p x qq “ d p ϕ p y q ´ e ϕ p x R qq ““ ϕ p x L q ϕ p x R q ´ t ϕ p x R q ` e ϕ p dx R q ““ . This shows that the map ϕ ÞÑ ˜ ϕ actually takes values in Hom
DGCA p K r x s , A t t u q .Now we define a map in the opposite direction. For a DGCA morphism ψ : K r x s Ñ A t t u , let t be the -cocycle t “ ψ p x q in A t t u . The -cocycle t can be uniquely written as t “ a ´ e b with a , b P A . The association y ÞÑ a , x R ÞÑ b , x L ÞÑ t defines a map ˜ ψ : p K r x L , x R s t x L x R u , p L q Ñ p A, t q in K r x s{ DGCA . It is im-mediate to check that ˜˜ ϕ “ ϕ and ˜˜ ψ “ ψ , so the two maps are inverse eachother. (cid:3) Now, let us go back to our DGCA A endowed with a DGCA morphism K r x L , x R s t x L x R u f Ñ A . To avoid confusion, let us denote by e L and e R the additional degree 1 generators in the extensions A L : “ A f p x L q and A R : “ A f p x R q of A , respectively. By the above proposition, and looking at f both asa morphism from p K r x L , x R s t x L x R u , p L q to p A, f L q and as a morphism from p K r x L , x R s t x L x R u , p R q to p A, f R q , we end up with distinguished -cocycles f p y q ´ e L f p x R q P A L , f p y q ´ e R f p x L q P A R VERY SHORT NOTE ON THE (RATIONAL) GRADED HORI MAP 7 and again, we can define extensions of A L and A R by trivialising the above -cocycles. We define the left rational gerbe G L and the left rational gerbe G R of the rational T -configuration f as the DGCAs G L : “ A L t f p y q´ e L f p x R qu G R : “ A R t f p y q´ e R f p x L qu . Again, to avoid confusion, we denote by ξ L and ξ R the additional degree generators of G L and G R as extensions of A L and of A R , respectively. Both G L and G R are extensions of A (since both A L and A R were extensions), and thistower of extensions of A can be depicted in the diagram G L G R A L A R A i L i L ι L ι R We can add to this diagram the DGCA A LR : “ A L b A A R , i.e., the DGCA p A r e L , e R s , de L “ f p x L q , de R “ f p x R qq , obtaining the diagram G L A LR G R A L A R A ι R i L ι L i L ι L ι R where the central square commutes. As a matter of notation, in the abovediagram we are writing ι L (resp. ι R ) wherever the extension is made by meansof the -form e L (resp. e R ) and i L (resp. i R ) whenever the extension is madeby means of the -form ξ L (resp. ξ R ).We can extend G L and G R by computing the obvious (homotopy) pushoutsto get the further extensions G L t f p x R qu G R t f p x L qu G L A LR G R A L A R A ι R i R i L ι L ι R i L ι L i R ι L ι R Explicitly, G L t f p x R qu “ ¨˚˚˝ A r e L , e R , ξ L s , $’’&’’% de L “ f p x L q de R “ f p x R q dξ L “ f p y q ´ e L f p x R q ˛‹‹‚ MATTIA COLOMA, DOMENICO FIORENZA, AND EUGENIO LANDI and G R t f p x L qu “ ¨˚˚˝ A r e L , e R , ξ R s , $’’&’’% de L “ f p x L q de R “ f p x R q dξ R “ f p y q ´ e R f p x L q ˛‹‹‚ . We can now make explicit the iso part of our pull-iso-push transform.
Proposition 3.2.
The DGCAs G L t f p x R qu and G R t f p x L qu are isomorphic viaan isomorphism G L t f p x R qu G R t f p x L qu ν that is the identity on A LR and acts as ξ L ÞÑ ξ R ` e L e R , on the degree two generator. The inverse isomorphism is, clearly, ν ´ : ξ R ÞÑ ξ L ´ e L e R .Proof. The map ν is a map of graded commutative algebras, and it is of coursea bijection since an explicit inverse is given by the map of graded commutativealgebras ν ´ which is the identity on A LR and sending ξ R to ξ L ´ e L e R .To see that ν is a map of DGCAs we need to show that it is a map of chaincomplexes. This can be checked on the generators of the polynomial algebra G L t f p x R qu , so we only need to compute dν p ξ L q . We have dν p ξ L q “ d p ξ R ` e L e R q ““ f p y q ´ e R f p x L q ` f p x L q e R ´ e L f p x R q ““ f p y q ´ e L f p x R q ““ ν p f p y q ´ e L f p x R qq ““ ν p dξ L q , where we used that f p y q ´ e L f p x R q P A LR and ν is the identity on A LR . (cid:3) The isomorphisms ν and ν ´ complete our previous diagram to the commu-tative diagram G L t f p x R qu G R t f p x L qu G L A LR G R A L A R A νν ´ ι R i L i R ι L ι R i L ι L i R ι L ι R VERY SHORT NOTE ON THE (RATIONAL) GRADED HORI MAP 9 The graded Hori map from rational equivalences of gerbes
All the maps and the DGCAs appearing in the upper part of our diagram G L t f p x R qu G R t f p x L qu G L G Rνι R ι L can be extended to the rings of (bounded above) formal Laurent series in thedegree generators. For instance, as a graded commutative algebra the DGCA G L is the polynomial algebra A L r ξ L s over A L and so embeds as a subalgebrainto the ring of Laurent series x G L : “ A L rr ξ ´ L ssr ξ L s “ : A L rr ξ ´ L , ξ L s . The ring ˆ G L has moreover a natural DGCA structure, by setting dξ ´ L “ ´ ξ ´ L p f p y q ´ e L f p x R qq , making G L ã Ñ ˆ G L an inclusion of DGCAs. One similarly extends the otherDGCAs G R , G L t f p x R qu and G R t f p x L qu appearing in the above diagram.The maps ι R , ι L obviously extend to the rings of Laurent series. We denoteby ˆ ι L and ˆ ι R these extensions. We notice that ν extends too, we only needto be careful in defining the extension ˆ ν . As e L e R is nilpotent, this is doneby using the formal power series inverse for ´ η , i.e., by declaring that theaction of ˆ ν on ξ ´ L is given by ˆ ν p ξ ´ L q “ p ξ R ` e L e R q ´ “ ÿ i ě p´ q i p e L e R q i ξ ´ i ´ R “ ξ ´ R ´ e L e R ξ ´ R , where we used that p e L e R q “ . One easily checks that ˆ ν is indeed a DGCAmorphism: it is compatible with the relation ξ ´ L ξ L “ as ˆ ν p ξ ´ L q ˆ ν p ξ L q “ p ξ ´ R ´ e L e R ξ ´ R qp ξ R ` e L e R q “ and with the differential as ˆ ν p dξ ´ L q “ ˆ ν p´ ξ ´ L p f p y q ´ e L f p x R qqq“ ´p ξ ´ R ´ e L e R ξ ´ R q p f p y q ´ e L f p x R qq“ ´p ξ ´ R ´ e L e R ξ ´ R qp f p y q ´ e L f p x R qq“ ´ ξ ´ R f p y q ` e L e R ξ ´ R f p y q ` ξ ´ R e L f p x R q and d ˆ ν p ξ ´ L q “ d p ξ ´ R ´ e L e R ξ ´ R q“ ´ ξ ´ R p f p y q ´ e R f p x L qq ´ p de L e R q ξ ´ R ` e L e R ξ ´ R p f p y q ´ e R f p x L qq“ ´ ξ ´ R p f p y q ´ e R f p x L qq ´ p f p x L q e R ´ e L f p x R qq ξ ´ R ` e L e R ξ ´ R f p y q“ ´ ξ ´ R f p y q ` e L f p x R q ξ ´ R ` e L e R ξ ´ R f p y q As f p x L q is an even cocycle, by Remark 2.1 we have a projection π : G R t f p x L qu Ñ G R r´ s mapping α k ` e L β k ´ to β k ´ , which is a morphism of right DG- G R -modules.Also the projection π : : G R t f p x L qu Ñ G R r´ s naturally extends to formalLaurent series modules to a map ˆ π : { G R t f p x L qu Ñ x G R r´ s and so it is possible to build a pull-iso-push transform T L Ñ R as the composition { G L t f p x R qu { G R t f p x L qu x G L x G R r´ s . ˆ ν ˆ π ˆ ι R T L Ñ R The transform T L Ñ R : x G L Ñ x G R r´ s associated to the initial rational T -duality configuration K r x L , x R s t x L x R u f Ñ A is seen to coincide with the graded Hori map introduced by Han and Mathaiin [HM20]. Namely, the action of ˆ ν on a generic degree k element ω k “ ÿ n P Z p α n ` k ` e L β n ` k ´ ` e R γ n ` k ´ ` e L e R δ n ` k ´ q ξ ´ n L in { G L t f p x R qu is given by ˆ ν p ω k q “ ÿ n P Z p α n ` k ` e L β n ` k ´ ` e R γ n ` k ´ ` e L e R δ n ` k ´ q ˆ ν p ξ ´ n L q ““ ÿ n P Z p α n ` k ` e L β n ` k ´ ` e R γ n ` k ´ ` e L e R δ n ` k ´ qp ξ ´ n R ´ ne L e R ξ ´ n ´ R q ““ ÿ n P Z p α n ` k ` e L β n ` k ´ ` e R γ n ` k ´ ` e L e R δ n ` k ´ q ξ ´ n R ´ ne L e R α n ` k ξ ´ n ´ R ““ ÿ n P Z p α n ` k ` e L β n ` k ´ ` e R γ n ` k ´ ` e L e R p δ n ` k ´ ´ p n ´ q a n ` k ´ qq ξ ´ n R , hence the action of ˆ ν on the coefficients of a generic degree k Laurent seriesin { G L t f p x R qu is given by ¨˚˚˚˝ α n ` k β n ` k ´ γ n ` k ´ δ n ` k ´ ˛‹‹‹‚ ˆ ν ÞÑ ¨˚˚˚˝ α n ` k β n ` k ´ γ n ` k ´ δ n ` k ´ ´ p n ´ q α n ` k ´ ˛‹‹‹‚ The inclusion x G L ˆ ι R Ñ { G L t f p x R qu and the projection ˆ π : { G R t f p x L qu Ñ x G R r´ s canbe displayed in a similar way ˜ α n ` k b n ` k ´ ¸ ˆ ι R ÞÑ ¨˚˚˚˝ α n ` k β n ` k ´ ˛‹‹‹‚ ; ¨˚˚˚˝ α n ` k β n ` k ´ γ n ` k ´ δ n ` k ´ ˛‹‹‹‚ ˆ π ÞÑ ˜ β n ` k ´ δ n ` k ´ ¸ VERY SHORT NOTE ON THE (RATIONAL) GRADED HORI MAP 11
The left-to-right transform T L Ñ R therefore act on the coefficients of a genericdegree k element ř n P Z p α n ` k ` e L β n ` k ´ q ξ ´ n L P x G L as ˜ α n ` k β n ` k ´ ¸ ˆ ι R ÞÑ ¨˚˚˚˝ α n ` k β n ` k ´ ˛‹‹‹‚ ˆ ν ÞÑ ¨˚˚˚˝ α n ` k β n ` k ´ ´p n ´ q α n ` k ´ ˛‹‹‹‚ ˆ π ÞÑ ˜ β n ` k ´ ´p n ´ q α n ` k ´ ¸ i.e., it acts on the degree k element ř n P Z p α n ` k ` e L b n ` k ´ q ξ ´ n L P x G L as ÿ n P Z p α n ` k ` e L β n ` k ´ q ξ ´ n L ÞÑ ÿ n P Z p β n ` k ´ ´ p n ´ q e R α n ` k ´ q ξ ´ n R “ ÿ n P Z β n ` k ´ ξ ´ n R ` e R ÿ n P Z ´ nα n ` k ξ ´ n ´ R The above expressions can be conveniently packaged by introducing, for everysequence t η n ` k u n P Z of elements of A with deg p η n ` k q “ n ` k , the Laurentseries in a degree variable ξη p k q p ξ q “ ÿ n P Z η n ` k ξ ´ n . We have manifest isomorphisms of graded vector spaces { G L t f p x R qu ¨˚˚˚˚˚˚˚˚˚˚˝ A rr ξ ´ , ξ s‘ A rr ξ ´ , ξ sr´ s‘ A rr ξ ´ , ξ sr´ s‘ A rr ξ ´ , ξ sr´ s ˛‹‹‹‹‹‹‹‹‹‹‚ { G R t f p x L qu„ „ and x G L ¨˚˝ A rr ξ ´ , ξ s‘ A rr ξ ´ , ξ sr´ s ˛‹‚ x G R . „ „ In terms of these isomorphisms, the maps ˆ ν , ˆ ι R and ˆ π are represented by thefollowing matrices: ˆ ν ÞÑ ¨˚˚˚˝ ddz ˛‹‹‹‚ ; ˆ ι R ÞÑ ¨˚˚˚˝ ˛‹‹‹‚ ; ˆ π ÞÑ ˜ ¸ , so that the graded left-to-right Hori transform T L Ñ R is represented in matrixform as: T ÞÑ ˜ ddξ ¸ . One similarly defines the right-to-left Hori transform T R Ñ L . As ˜ ddξ ¸ ˜ ddξ ¸ “ ˜ ddξ ddξ ¸ one sees that T R Ñ L ˝ T L Ñ R “ ddξ L : x G L Ñ x G L r´ s and T L Ñ R ˝ T R Ñ L “ ddξ R : x G R Ñ x G R r´ s . Hori transforms of meromorphic functions
Before extending the ring of coefficients to the ring of Jacobi forms we startwith a one variable intermediate step. We will need an extra degree variablein order to keep the following computations within the context of graded maps.So we assume that our base DGCA A is of the form A : “ A r u ´ , u ss where u is a degree variable and A is a DGCA endowed with a rationalT-duality configuration K r x L , x R s t x L x R u f Ñ A . Notice the a T-dualityconfiguration on A induces a T-duality configuration K r x L , x R s t x L x R u f Ñ A ã Ñ A on A simply by composing f with the inclusion A ã Ñ A . All the extensionand gerbes below are computed with respect to this T-duality configurationon A . For instance, the extended left gerbe x G L will be x G L “ starting DGCA hkkikkj A r u ´ , u ss looomooon additionalvariabletrivialise f p x L q hkkikkj r e L s rr ξ ´ L , ξ L s loooomoooon trivialise f p y q ´ e L f p x R q and ex-tend to Laurent se-ries . Assume now the base field K to be the field C of complex numbers and let M be the C -algebra of meromorphic functions on C that are holomorphic onthe punctured plane C zt u , i.e. meromorphic functions that admit at mosta polar singularity in the origin. By looking at the algebra M as a DGCAconcentrated in degree zero, we can then consider the DGCA M p A q : “ M b A , that we will call the DGCA of meromorphic functions with valuesin A and with at most polar singularities in the origin. A degree k elementin M p A q has a Laurent series expansion around the origin of the form f p q q “ ÿ n f n ; k q n where the f n ; k are degree k elements in A , with f n ; k “ for n ! . For any i P Z we have an isomorphism µ i of graded vector spaces M p A q µ i ÝÝÑ A rr ξ ´ , ξ sr i s f p q q ÞÑ ξ i f p uξ ´ q VERY SHORT NOTE ON THE (RATIONAL) GRADED HORI MAP 13
Notice that there exists a commutative diagram M p A q M p A q A rr ξ ´ , ξ s A rr ξ ´ , ξ sr´ s ´ q ddq µ µ ´ ddξ As remarked at the end of the previous section, the natural isomorphism ofgraded vector spaces of x G L and x G R with A rr ξ ´ , ξ s ‘ A rr ξ ´ , ξ sr´ s identifiesthe graded Hori map T L Ñ R with the antidiagonal matrix ´ ddξ ¯ , i.e., we havea commutative diagram A rr ξ ´ , ξ s ‘ A rr ξ ´ , ξ sr´ s A rr ξ ´ , ξ sr´ s ‘ A rr ξ ´ , ξ sr´ s x G L x G R r´ s ˆ ddξ ˙ ≀ ≀T L Ñ R Therefore, we see that the graded Hori map T L Ñ R participates into a commu-tative diagram of graded vector spaces M p A q ‘ M p A qr´ s M p A qr´ s ‘ M p A q A rr ξ ´ , ξ s ‘ A rr ξ ´ , ξ sr´ s A rr ξ ´ , ξ sr´ s ‘ A rr ξ ´ , ξ sr´ s x G L x G R r´ s ˆ ´ q ddq ˙ µ ‘ µ r´ s ≀ ≀ µ r´ s‘ µ ´ ˆ ddξ ˙ ≀ ≀T L Ñ R The same happens for the graded Hori map T R Ñ L , so that the composition T R Ñ L ˝ T L Ñ R is identified with the endomorphism ˜ ´ q ddq ´ q ddq ¸ of M p A q ‘ M p A qr´ s , and similarly for T L Ñ R ˝ T R Ñ L .6. Extending coefficients to the ring of Jacobi forms
In this concluding section we extend the ring of coefficients for our extendedgerbes to the graded ring of Jacobi forms of index . We address the reader tothe classic [EZ85] for a complete and detailed account of the general theory ofJacobi forms of arbitrary index, and here we content us in briefly recalling thedefinition of a (meromorphic) Jacobi form of index 0. Definition 6.1. A (meromorphic) Jacobi form of weight s and index is afunction C ˆ H C J which is meromorphic in the variable z and holomorphic in the variable τ , suchthat J ˝ is modular in τ i.e. J p zcτ ` d , aτ ` bcτ ` d q “ p cτ ` d q s J p z, τ q for any ` a bc d ˘ in SL p , Z q ; ˝ is elliptic in z i.e. J p z ` λτ ` µ, τ q “ J p z, τ q for any p λ, µ q in Z ; ˝ has a polar behaviour for z Ñ ` i .We notice two important features of Jacobi forms. First, by applying theoperator B{B z to both sides of the modularity and of the ellipticity equations,one sees that if J p z, τ q is a Jacobi form of weight s and index then BB z J p z, τ q is a Jacobi form of weight s ` and index .Secondly, from the ellipticity condition for the pair p , q P Z one sees thatevery Jacobi form is periodic in z of period , hence it has a series expansionin the variable q “ e πiz of the form J p z, τ q “ ÿ n “ n α n p τ q q n , for some n P Z , where the fact that this Laurent series is bounded below is aconsequence of the polar behaviour of J for z Ñ ` i .As the weight s ranges over the integers, Jacobi form of index form agraded ring J “ à s P Z J p s q , (with degree given by the weight). The fact that BB z maps weight s index Jacobi forms to weight s ` index Jacobi forms then can be expressed bysaying that BB z is a degree 1 derivation of the graded ring J . Moreover, fromthe identity ´ q BB q “ ´ πi BB z we see that the ring of q -expansions of index Jacobi forms (a subring of thering of bounded below Laurent series in the variable q with coefficients in thering of holomorphic function on H ) is closed under the action of the operator ´ q BB q .We can now verbatim repeat the construction of Section 5. For a DGCA B over C , let us write B p τ q for the DGCA B p τ q : “ B b H ol p H q where the ring H ol p H q of holomorphic function in the variable τ P H is seen asa DGCA concentrated in degree zero. Also, let us write J p A q “ à s P Z J p s q p A q for the bigraded ring J p A q : “ J b A of index Jacobi forms with values ina DGCA A . Then the commutative diagram at the end of Section 5 induces VERY SHORT NOTE ON THE (RATIONAL) GRADED HORI MAP 15 the commutative diagram J p s q p A q ‘ J p s q p A qr´ s J p s q p A qr´ s ‘ J p s ` q p A q M p A p τ qq ‘ M p A p τ qqr´ s M p A p τ qqr´ s ‘ M p A p τ qq A p τ qrr ξ ´ , ξ s ‘ A p τ qrr ξ ´ , ξ sr´ s A rr ξ ´ , ξ sr´ s ‘ A rr ξ ´ , ξ sr´ s x G L p τ q x G R p τ qr´ s ˆ ´ πi BB z ˙ q -expansion q -expansion ˆ ´ q BB q ˙ µ ‘ µ r´ s ≀ ≀ µ r´ s‘ µ ´ ˆ ddξ ˙ ≀ ≀T L Ñ R That is, the graded Hori transform T L Ñ R induces the morphism ˜ ´ πi BB z ¸ : J p s q p A q ‘ J p s q p A qr´ s Ñ J p s q p A qr´ s ‘ J p s ` q p A q at the level of index A -valued Jacobi forms, for any weights s , s in Z .The same holds for the graded Hori transform T R Ñ L , so that the composition T L Ñ R ˝ T R Ñ L acts as ˜ ´ πi BB z ´ πi BB z ¸ : J p s q p A q ‘ J p s q p A qr´ s Ñ J p s ` q p A q ‘ J p s ` q p A qr´ s and similarly for T L Ñ R ˝ T R Ñ L . This reproduces [HM20, Theorem 2.2]. Acknowledgements. d.f thanks NYU-AD for support on occasion of theworkshop
M-theory and Mathematics during which the idea of this note orig-inated, and Hisham Sati and Urs Schreiber for discussions and comments onan early version of this note.
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Reviews in Math-ematical Physics , 17(01):77–112, Feb 2005.[EZ85] Martin Eichler and Don Zagier.
The theory of Jacobi forms , volume 55. Springer,1985.[FSS18a] Domenico Fiorenza, Hisham Sati, and Urs Schreiber. T-duality from super Lie n -algebra cocycles for super p -branes. Advances in Theoretical and MathematicalPhysics , 22(5):1209–1270, 2018.[FSS18b] Domenico Fiorenza, Hisham Sati, and Urs Schreiber. T-duality in rational ho-motopy theory via L -algebras. Geometry, Topology and Mathematical PhysicsJournal , 1, 2018.[HM20] Fei Han and Varghese Mathai. T-duality, Jacobi Forms and Witten Gerbe Modules,2020. https://arxiv.org/abs/2001.00322 . Università degli Studi di Roma "Tor Vergata"; Dipartimento di Matematica,Via della Ricerca Scientifica, 1 - 00133 - Roma, Italy;
E-mail address : [email protected] Sapienza Università di Roma; Dipartimento di Matematica “Guido Casteln-uovo”, P.le Aldo Moro, 5 - 00185 - Roma, Italy;
E-mail address : [email protected] Università di Roma Tre; Dipartimento di Matematica e Fisica Largo SanLeonardo Murialdo, 1 - 00146 - Roma, Italy;
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