aa r X i v : . [ m a t h - ph ] D ec Fourier Duality of Quantum Curves
Martin T. Luu, Albert Schwarz
Abstract
There are two different ways to deform a quantum curve along the flows of the KP hierarchy.We clarify the relation between the two KP orbits: In the framework of suitable connectionsattached to the quantum curve they are related by a local Fourier duality. As an application wegive a conceptual proof of duality results in 2D quantum gravity.
One way to define a quantum curve, see for example [20], is as a pair (
P, Q ) of ordinary differentialoperators in C [[ x ]][ ∂ x ] such that [ P, Q ] = ~ Here ~ might be viewed as a formal parameter or a fixed complex number depending on the situationof interest. For the remainder of this work we set ~ = 1. We say that the quantum curve has bi-degree( p, q ) if P is a differential operator of order p and Q is a differential operator of order q. We will workwith scalar differential operators, however, our methods can be applied also to matrix differentialoperators. The notion of quantum curve originates from the fact that given a complex algebraiccurve and two suitable functions f , f on it, it is known by work of Burchnall and Chaundy [4] andKrichever [11], that one can construct two commuting scalar differential operators P and Q . Hencean algebraic curve is related to the classical situation ~ = 0.Notice that there are two natural ways to deform a quantum curve along the flows of the Kadomtsev-Petviashvili (KP) hierarchy of partial differential equations. Namely, if P is a normalized differentialoperator in the sense that is has the form ∂ px + a p − ∂ p − x + · · · , then we can define a pseudodif-ferential operator L = P p . In terms of this operator we can define a family of quantum curves( P ( t , · · · , t p + q ) , Q ( t , · · · , t p + q )) of bi-degree ( p, q ) solving the KP-equations ∂∂t n L ( t , · · · , t p + q ) = [ L ( t , · · · , t p + q ) n + , L ( t , · · · , t p + q )]where the subscript + denotes the differential part of the pseudodifferential operator. The operator P ( t , · · · , t p + q ) is then obtained as L ( t , · · · , t p + q ) p and the operator Q ( t , · · · , t p + q ) can be calculatedby the equation ∂∂t n Q ( t , · · · , t p + q ) = [ L ( t , · · · , t p + q ) n + , Q ( t , · · · , t p + q )]Note that one can construct a larger family of quantum curves solving the same equations wherenow all the KP times t i for i ≥ Q is a normalized operator we can define another family of quantum curves( ˆ P (ˆ t , · · · , ˆ t p + q ) , ˆ Q (ˆ t , · · · , ˆ t p + q )) solving the KP-equations. These curves are of bi-degree ( q, p ) andare obtained by deforming the string equation [ Q, − P ] = 1 along a second set ˆ t , ˆ t , · · · of KP times.Our goal is to determine the relation between the two KP orbits by showing in Theorem 1 that theyare related by a type of Fourier duality.It can be seen that for both KP deformations there are a priori the same amount of relevant KPtime variables and this makes it at least conceivable that there is a good correspondence between thetwo theories. However, the two corresponding KP τ -functions appear to be very different and dependnon-trivially on different numbers of the time variables. The beautiful consequence, which is usefulalso in 2D quantum gravity, is that the duality between the two KP deformations of the quantumcurves allows to reduce the number of relevant time variables for one of the two families of quantumcurves and hence leads to a simplified description.To formulate the main result we notice that differential operators with power series coefficients canbe regarded as operators acting in the space of polynomials C [ z ]. Namely, we assume that ∂ x acts asmultiplication by z and multiplication by x acts as − dd z . It is clear that this construction gives us astructure of D -module on C [ z ] where D = C [ x, ∂ x ] is the one-variable Weyl algebra. In other words,one obtains a D-module on the plane A . Using this fact and assuming that the operator P is monicwe can assign a connection to a quantum curve ( P, Q ). To define this connection we can consider amatrix M of the operator Q in P -basis. Here we are saying that the elements e , ..., e p of C [ z ] forma P -basis if elements P k e i form a basis. If P is monic a P -basis exists: we can take, for example, e i = z i − . In other words, in this case the operator P specifies a structure of free module over thering of polynomials. The matrix M (the companion matrix of the quantum curve in the terminologyof [20]) can be considered as a matrix with polynomial coefficients. It specifies a connection ∇ = dd u − M ( u )We consider it as a connection on the punctured formal disc D × centered at the point ∞ , meaningthat we choose a presentation of the punctured disc as D × ∼ = Spec C ((1 /u )).If the operator Q is also monic, we can define another connection using the same construction; weprove that these two connections are related by local Fourier transform. (Note that the local Fouriertransform is defined only up to gauge equivalence, hence it would be more accurate to say that thelocal Fourier transform relates equivalence classes of connections.) Notice that this statement is truealso for quantum curves specified by matrix differential operators.To state the precise relation obtained by the Fourier transform it is best to work with the wholeKP orbits. We prove in Theorem 1 that the family of connections corresponding to the first family ofquantum curves ( P ( t , · · · , t p + q ) , Q ( t , · · · , t p + q )) is related by local Fourier transform with the familyof connections corresponding to the family of quantum curves ( ˆ P (ˆ t , · · · , ˆ t p + q ) , ˆ Q (ˆ t , · · · , ˆ t p + q )) wherethe parameters (ˆ t , · · · , ˆ t p + q ) are expressed in terms of the parameters ( t , · · · , t p + q ) in the followingway:Let s be an indeterminate. For p, q ≥
1, we say two functions f ∈ C ((1 /s /p )) and f ∈ C ((1 /s /q ))2re compositional inverses up to gauge equivalence if f − = f + element of s − C [[ s − /q ]] f − = f + element of s − C [[ s − /p ]]Then the following two functions are inverses up to gauge equivalence: f = 1 p · p + q X k =1 kt k s k − pp and f = − q · p + q X k =1 k ˆ t k s k − qq This is sufficient to describe each set of parameters in terms of the other one. The gauge ambiguityof the inverse functions comes from the fact that the Fourier transform only relates gauge equivalenceclasses of connections.In Section 4 we show that the duality relation between the quantum curves can be applied to giveconceptual proofs of the duality in 2D quantum gravity. This duality corresponds to a certain changein the matter content of the theory and is called T – duality or p – q duality. The latter notation stemsfrom the fact that for positive co-prime integers p and q the duality relates the partition function ofthe ( p, q ) minimal model coupled to gravity to the corresponding function of the ( q, p ) minimal modelcoupled to gravity.Previous approaches [8], [10] by physicists were of more computational nature. It seemed clearthat a more conceptual approach would be useful. The first step was taken in [15], where we showedhow to describe the duality as a local Fourier duality of connections on the formal punctured disc.However, the proofs were still of computational nature. As a consequence of our general duality resultsfor quantum curves obtained in the present work, the reason underlying the precise dynamics of theduality can now be much better understood as consequence of general properties of the local Fouriertransform.Consider for example the duality between the (5 ,
2) model and the (2 ,
5) model of 2D quantumgravity. The former depends a priori on the time variables t , · · · , t and the latter depends a priorion the time variables ˆ t , · · · , ˆ t . It is known that the τ -function of the (5 ,
2) model depends triviallyon the variable t and the τ -function of the (2 ,
5) model depends trivially on the variables ˆ t , ˆ t , ˆ t .The duality therefore leads to a simplified description of the (5 ,
2) model.As a consequence of the duality the two sets of times can be expressed in terms of each other. Forexample we will give a proof based on the local Fourier transform that the dynamics of the 2 – 5duality is given by t = ˆ t + 58 · ˆ t − · ˆ t · ˆ t + 567200 · ˆ t · ˆ t + 185 · ˆ t · ˆ t · ˆ t − · ˆ t · ˆ t − · ˆ t − · ˆ t · ˆ t + 8150 · ˆ t · ˆ t − · ˆ t · ˆ t − · ˆ t t = ˆ t + 3 · ˆ t · ˆ t − · ˆ t · ˆ t − · ˆ t · ˆ t + 38883125 · ˆ t + 7225 · ˆ t · ˆ t − · ˆ t · ˆ t t = ˆ t − · ˆ t − · ˆ t · ˆ t + 92 · ˆ t · ˆ t − · ˆ t = ˆ t − · ˆ t · ˆ t + 5425 · ˆ t t = ˆ t − · ˆ t t = ˆ t t = ˆ t · − n over local fields.Our results are based on [20], however, the familiarity with this paper is not necessary for thereading of present paper: all definitions and statements we are using are repeated. Suppose (
P, Q ) is a bi-degree ( p, q ) quantum curve. Using the action on C [ z ] described in the intro-duction, we can construct two D-modules associated to this curve, meaning two representations of theWeyl algebra D = C [ x, ∂ x ]. The first one D ( P, Q ) is given by ∂ x P and x Q and the second one D ( P, Q ) is given by ∂ x
7→ − Q and x P Hence, the first D-module is the analogue of the second D-module but for the string equation [ Q, − P ] =1 instead of [ P, Q ] = 1. In other words, one hasD ( Q, − P ) = D ( P, Q )The global Fourier transform F glob of a D-module is the D-module obtained by composing with themap ∂ x x and x
7→ − ∂ x It is clear from the definitions that the two D-modules are globally Fourier dual: F glob (cid:2) D ( P, Q ) (cid:3) ∼ = D ( P, Q )This global Fourier duality will be crucial in our approach to the duality of KP orbits of quantumcurves. 4 .1 KP-flows
In this section we describe, following [20], the KP-flows on the space of quantum curves. Via therelevant dressing operators attached to a solution to the string equation [
P, Q ] = 1, this is related tothe fact that KP-flows can be described geometrically in terms of the Sato Grassmannian. We startby recalling this:As a set, the big cell Gr of the Sato Grassmannian consists of the C -subspaces of H = C ((1 /z ))that are comparable to H + = C [ z ]. This simply means that the projection map to H + that takes aLaurent series to its polynomial part is an isomorphism. The algebra of pseudodifferential operators C [[ x ]](( ∂ − x )) acts on the space of Laurent series H = C ((1 /z )) in the following manner: For f ( z ) ∈ C ((1 /z )) let ( x m ∂ nx ) f ( z ) = (cid:0) − dd z (cid:1) m ( z n f ( z )). For differential operators this action was described inthe introduction. Note that differential operators can be characterized as pseudodifferential operatorstransforming C [ z ] into itself.Let us review shortly the elements of Sato theory and the results of [20]. The commutative Liealgebra γ + of polynomials P k ≥ t k z k acts on H by means of multiplication operators, hence it acts on Gr in a natural way. There exists a one-to-one correspondence between the elements of the group G of monic zeroth order pseudodifferential operators and points of Gr . Namely, every subspace V ∈ Gr has a unique representation in the form V = S H + where S ∈ G . It follows that γ + acts also on G : ∂S∂t n = ( S∂ nx S − ) − S (1)where the subscript “ − ” denotes the part of the pseudodifferential operator involving negative powersof ∂ x .Suppose now that P is a normalized pseudodifferential operator of order p , meaning it is of theform P = ∂ px + a p − ∂ p − x + · · · Every such operator can be represented in the form S − ∂ px S where S ∈ G and this representationis unique up to multiplication by an operator with constant coefficients. Using this statement wecan construct the action of Lie algebra γ + on the space of normalized pseudodifferential operatorsdifferentiating the relation P ( t ) = S − ( t ) ∂ px S ( t ) with respect to t n . As indicated in the introduction,the action on this space can be written in the form of differential equations ∂P∂t n = [ P np + , P ] (2)Notice that this formula determines also the action of Lie algebra γ + on the space of normalizeddifferential operators. All actions we described can be considered as different forms of the KP hierarchy.Integrating the actions of Lie algebra γ + we obtain the action of a commutative group Γ + , the elementsof this group can be written in the form g ( t ) = exp( P k ≥ t k z k ) . We now come back to our aim of describing the KP-flows on the space of quantum curves. Wewould like to solve the string equation [
P, Q ] = 1. We will recall the relevant results from [20], but incontrast to loc. cit. we will assume that P is a normalized differential operator instead of Q . One cansay that we apply the results of [20] to the string equation [ − Q, P ] = 1. This normalization is useful5hen working with the D-module D ( P, Q ) that we defined earlier.As we mentioned, we can construct the operator S ∈ G such that SP S − = ∂ px . Introducingthe notation V = S H + we obtain a subspace V ∈ Gr invariant with respect to multiplication by z p and with respect to the action of the operator − ˜ Q = pz p − dd z + b ( z ) where b ( z ) stands for themultiplication by a Laurent series denoted by the same letter. Here we use the fact that the actionof ∂ px can be interpreted as multiplication by z p and the fact that H + is invariant with respect to theaction of differential operators. The form of the operator − ˜ Q = S ( − Q ) S − follows from the relation[ − ˜ Q, ˜ P ] = 1 where ˜ P = SP S − = ∂ px . We can invert this consideration to obtain the followingstatement, see [20]: If the point V of the big cell of the Grassmannian satisfies z p V ⊆ V and (cid:18) pz p − dd z + b ( z ) (cid:19) V ⊆ V we can construct a differential operator P and a normalized differential operator Q obeying [ P, Q ] = 1.The leading term of the operator Q is determined by the leading term of the Laurent series b ( z ).To summarize, finding a solution to the string equation is equivalent to finding a suitable point ofthe Sato Grassmannian stabilized by two operators of the above form.It follows that the Lie algebra γ + acts on the space of pairs ( P, Q ) of differential operators obeying[
P, Q ] = 1 under the assumption that P is a normalized operator. The proof is based on the remarkthat for V as above, the space V ( t ) := g ( t ) V where g ( t ) = exp X k ≥ t k z k ∈ Γ + satisfies a similar condition as V but with − ˜ Q replaced by g − ( t )( − ˜ Q ) g ( t ): Namely, one has z p V ( t ) ⊆ V ( t ) and (cid:18) pz p − dd z + b ( z ) − X kp t k z k − p (cid:19) V ( t ) ⊆ V ( t ) (3)It follows that the KP-flows (2) are defined on the space of quantum curves.To simplify the formulas that we will obtain in Theorem 1, we normalize the KP times so that attimes t , t , · · · the point V ( t ) of the KP orbit of V is stabilized by z p and pz p − dd z − P kp t k z k − p . We now describe in detail how to attach a connection on the formal punctured disc to a quantumcurve. Recall first that the category Conn( D × ) of connections on the formal punctured disc D × canbe defined as follows: After choosing an isomorphism D × ∼ = Spec C (( t )) each object in this categorycan be described as a pair ( M, ∇ ) where M is a finite dimensional C (( t ))-vector space and ∇ is a C -linear map ∇ : M −→ M such that ∇ ( f · m ) = f · ∇ ( m ) + d f d t · m for all f ∈ C (( t )) and all m ∈ M . The morphisms in the category Conn( D × ) are C -linear maps thatalso commute with the maps ∇ . 6he category D-Mod of left D = C [ x, ∂ x ]-modules can be viewed as the category of D-modules over P ( C ) \{∞} . Hence the D-modules D and D that we associated to a quantum curve are D-moduleson the plane P ( C ) \{∞} .Let Hol(D-Mod) denote the full sub-category of holonomic D-modules on P ( C ) \{∞} . There is arestriction functor that captures the local information near ∞ : ψ ∞ : Hol(D-Mod) −→ Conn( D × )It is given on objects by N N ⊗ C [ x ] C (( t )) with t = 1 x where the C (( t ))-vector space structure comes from the second factor and the map ∇ : ψ ∞ N −→ ψ ∞ N is given by n ⊗ f ( ∂ x · n ) ⊗ ( − t f ) + n ⊗ dd t f Applying the functor ψ ∞ to D-modules related by the global Fourier transform we obtain connec-tions related by local Fourier transform. This statement could be considered as a definition of thelocal Fourier transform for connections that can be obtained from D-modules by means of the functor ψ ∞ . See the discussion preceding Theorem 1 for a precise statement of the result.Let us consider a solution to the string equation [ P, Q ] = 1 with P normalized. For SP S − = ∂ px let V = S H + be the associated point of the Grassmannian. Suppose e , ..., e p − is a P -basis of H + ,meaning that the collection of elements P k e i for varying k and i is a C -basis. Under this assumptionone has that v , ...., v p − with v i = Se i is a z p -basis of V . We obtain˜ P v i = M ji ( z p ) v j (4)where the entries of the matrix M are polynomials with respect to z p . The matrix M coincides withthe companion matrix of the pair ( P, Q ) defined as a matrix of P in Q -basis.We apply these constructions to the D-modules coming from a bi-degree ( p, q ) quantum curve ( P, Q )where P is monic. The D-module D ( P, Q ) is holonomic and we can apply the functor ψ ∞ to obtaina p -dimensional connection. One sees that B := { ⊗ , · · · , z p − ⊗ } is a C (( t ))-basis of the vector space D ( P, Q ) ⊗ C [ x ] C (( t )). The connection is nothing but the companionmatrix connection introduced in [20] and we will denote it by ∇ M ( P,Q ) .Since on D ( P, Q ) we defined the ∂ x action to be the action of − Q , it follows that apart from thefactor − /t , the matrix of the action of ∂ x with respect to the basis B is the matrix of the action of − Q with respect to the P -basis { , z, · · · , z p − } of C [ z ]. Hence this is simply − M ( P, Q ). Hence, by7riting the connection with respect to the basis B , one can writeD ( P, Q ) ⊗ C [ x ] C (( t )) ∼ = (cid:18) C (( t )) p , dd t − M ( P, Q )(1 /t ) · ( − /t ) (cid:19) Or, more naturally, when we write this connection in terms of the coordinate u such that the punctureddisc is given by D × ∼ = Spec C ((1 /u )) the connection is simply described bydd u − M ( P, Q )Hence, this is the companion matrix connection ∇ M ( P,Q ) . Furthermore, if p and q are co-prime it isknown, see [20], that one obtains an irreducible connection. The isomorphism classes of connections have been classified in work of Levelt and Turrittin, see forexample [5] for a detailed exposition. Therefore, one can ask what the description of the connectionof the quantum curve in terms of this classification is. In order to answer this, we describe theLevelt-Turrittin classification.Sometimes it is convenient to describe a connection with respect to a choice of basis of the vectorspace M . Then the map ∇ is simply of the formdd t + A ( t )with A ( t ) ∈ gl n C (( t )). This classification implies in particular that given an irreducible connection onecan simplify the connection matrix A ( t ) if one changes coefficients from C (( t )) to some suitable finiteextension C (( t /q )). For the extended coefficients there is g ∈ GL n ( C (( t /q ))) such that with respect tothe new basis obtained by multiplying by g the old one, the description of the connection becomesdd t + gA ( t ) g − + g · dd t g − with gA ( t ) g − + g · dd t g − = f . . . f n for some functions f i ∈ C (( t /q )). This type of simplification is conveniently described in terms of thepush-forward and pull-back of connections along suitable maps. Given a map ρ : C [[ t ]] −→ C [[ u ]]which takes t to some element in u C [[ u ]] one can define associated push-forward and pull-back oper-ations on the categories of connections on the formal punctured disc with local coordinate t and u respectively. Following the notation in [7] we denote by [ i ] for i ∈ Z ≥ the map ρ that takes t to u i .If N is a d -dimensional connection over C (( u )) then [ i ] ∗ N is a d · i -dimensional connection over C (( t )).8f M is a d -dimensional connection over C (( t )) then [ i ] ∗ M is a d -dimensional connection over C (( u )).Now suppose f ∈ C (( t /q )) and q is the minimal positive such integer. The connection denotedby E f in [9] denotes the push-forward along the map C (( t /q )) −→ C (( t )) that takes t /q to t of theone-dimensional connection dd t /q + qt ( q − /q f ( t /q )over C (( t /q )). The classification result is then that every irreducible connection is isomorphic to some E f and f is uniquely determined up to adding an arbitrary element in1 q · Z + t /q C [[ t /q ]]We will also use Fang’s results [7] later on and there a different notation is used, so we introduce itnow:Given a function α ∈ C (( t )) one denotes by [ α ] the connection( C (( t )) , dd t + αt )The Levelt-Turrittin classification in this language implies that every irreducible connection is iso-morphic to one of the form [ q ] ∗ ([ t∂ t ( α )] ⊗ C (( t )) R )where R is a suitable regular connection R = ( C (( t )) , dd t + rt ) for some r ∈ C and the tensor product of two connections ( V i , ∇ i ) has underlying vector space V ⊗ C (( t )) V and theconnection is given via v ⊗ v
7→ ∇ ( v ) ⊗ v + v ⊗ ∇ ( v )for all v ∈ V and v ∈ V .In order to conform with usual notational conventions for Kac-Schwarz operators, it will often beuseful to express connections on the formal punctured disc with coordinate u in terms of the reciprocalcoordinate z = 1 /u . Hence, for h ( z ) ∈ C ((1 /z )) we use the notation( C (( u )) , dd z + h ( z )) := ( C (( u )) , dd u − u h (1 /u ))Note that in the following we will write u for 1 /z without further comment. Our goal is to analyze the behavior of KP-flows under duality (
P, Q ) → ( − Q, P ) . We know alreadythat on the connection corresponding to the companion matrix M this duality acts as local Fouriertransform. From the other side we know the action of KP-flows on b (see (3)).We now describe the relation between b ( z ) and M in the case when P is normalized assuming that9 and q are co-prime. In this case the companion matrix connection ∇ M ( P,Q ) is irreducible. It followsfrom the Levelt-Turrittin classification and the irreducibility of ∇ M ( P,Q ) that[ p ] ∗ ∇ M ( P,Q ) ∼ = ( C (( u )) p , dd z + ξ ( z ) . . . ξ p ( z ) )for suitable ξ i ∈ C (( z )) that satisfy ξ i ( z ) ∈ z C [[ z ]]It is shown in [19] that up to a p ’th root of unity the coefficient of 1 /z is given by (1 − p ) /
2. Moreover,it is known that the gauge transformation can be taken to be of the form R ( z ) ∈ GL p ( C [[1 /z ]])Let R i,j ( z ) denote the ( i + 1 , j + 1) entry of R ( z ) and let t i ( z ) = exp (cid:18)Z b ( z ) pz p − d z (cid:19) p − X j =0 R i,j ( z ) v j z j One obtains that 1 pz p − dd z t i ( z ) = Λ i ( z ) t i ( z ) where Λ i ( z ) = ξ i ( z ) pz p − Let the constants c i be such that t i ( z ) = c i exp( Z Λ i ( z ) pz p − d z )Since v i = z i + lower order termsit follows that c exp( R (Λ ( z ) − b ( z )) pz p − d z )... c n exp( R (Λ n ( z ) − b ( z )) pz p − d z ) = R · µ , z − + · · · ...1 + µ n, z − + · · · for suitable constants µ i,j . Note that since R is invertible not all c i ’s can vanish. It also follows fromthe above equation that for each i such that c i = 0 one needsΛ i ( z ) − b ( z ) ∈ z p +1 C [[1 /z ]]Since Λ i ( z ) ∈ z p C [[ z ]] and since b ( z ) ∈ z p C [[ z ]] it follows that b ( z ) = Λ i ( z )10his holds for all i such that c i = 0. Now define a one-dimensional connection over C (( u )) by ∇ KS := ( C (( u )) , dd z + ˜ b ( z )) where 1 pz p − ˜ b ( z ) = b ( z )It follows from the previous calculations thatHom C (( u )) ( ∇ KS , [ p ] ∗ ∇ M ( P,Q ) ) = (0)Furthermore, one can define a connection on this space of homomorphisms and via the projectionformula for pull-back and push-forward, see for example [18] (Section 1), one has[ p ] ∗ Hom C (( u )) ( ∇ KS , [ p ] ∗ ∇ M ( P,Q ) ) ∼ = Hom C (( u p )) ([ p ] ∗ ∇ KS , ∇ M ( P,Q ) )Since ∇ M ( P,Q ) is irreducible it follows that ∇ M ( P,Q ) ∼ = [ p ] ∗ ∇ KS as desired.It can be seen from the description of the KP flows in Section 2.1 and the calculations in [19] thatthe KP times t , · · · , t p + q can be normalized so that ∇ M ( P ( t , ··· ,t p + q ) ,Q ( t , ··· ,t p + q )) ∼ = [ p ] ∗ ( C (( u )) , dd z − pz p − ( 1 − p p z p + 1 p p + q X i =1 it i z i − p )) ! In this normalization, the connection only depends on p and q and we will denote it by ∇ M p,q ( t , ··· ,t p + q ) . As explained previously, if P is normalized one can deform a bi-degree ( p, q ) quantum curve [ P, Q ] = 1along the first p + q flows of the KP hierarchy. If Q is normalized we have another deformation comingfrom the quantum curve [ − Q, P ] = 1 . We would like to consider the case when after the deformationboth P and Q remain normalized. As can be seen from the arguments in Section 2.1, this meansin particular that no time evolution is taking place along the flows of the p + q ’th and p + q − t , · · · , t p + q of the first string equation andthe KP times ˆ t , · · · , ˆ t p + q of the second string equation: t p + q = pp + q = − pq · ˆ t p + q t p + q − = 0 = ˆ t p + q − Our aim is to relate the companion matrix connections ∇ M p,q ( t , ··· ,t p + q ) and ∇ M q,p (ˆ t , ··· , ˆ t p + q ) associatedwith the two deformations.Our main tool is the local Fourier transform: Bloch and Esnault [3] and Lopez [14] developed an11nalogue over C of the ℓ -adic local Fourier transform constructed by Laumon in [13]. See also thework of Arinkin [1] for an alternate approach.The stationary phase principle for the local Fourier transform is crucial for our proof of the maintheorem. This is a central part of the theory of local Fourier functors. In fact, from its very intro-duction in the ℓ -adic context, the guidance in defining the local Fourier transform is the search for astationary phase principle for the global Fourier transform. We now describe the relevant results.Fix a point ∞ ∈ P ( C ) and let ( M, ∇ ) be a connection on the formal punctured disc centered at ∞ .A Katz extension M of this connection is a certain connection on the punctured plane P ( C ) \{ , ∞} which is regular singular at 0. See for example [2], Theorem 2.8, for a precise definition of thisextension. For the connections of interest to our considerations, namely the irreducible objects ofConn( D × ) this can be described in the following manner: Consider the connection E f with f = X i ≫−∞ a i t i/p The Katz extension E f of this connection is then simply given bydd t + X − ≥ i ≫−∞ a i t i/p It is clear from the description of Katz extension in [2] that a Katz extension of the previouslyintroduced companion matrix connection ∇ M ( P,Q ) is given by D ( P, Q ).Let Conn( D × ) > denote the full subcategory of Conn( D × ) consisting of connections with slopesstrictly bigger than 1. The condition slope( E f ) > i < − p with a i = 0.The local Fourier transform is then a functor F ( ∞ , ∞ ) : Conn( D × ) > −→ Conn( D × ) > defined in the following way: Consider the Katz extension of the connection as a D-module, takeglobal Fourier, apply the functor ψ ∞ . In particular, see ([3], Proposition 3.12 (v)), ψ ∞ (cid:0) F glob ( E f ) (cid:1) ∼ = F ( ∞ , ∞ ) ( E f )This is an incarnation of the stationary phase principle for the Fourier transform and it is a keytool in proving the following theorem. Note that this result relates two connections up to gaugeequivalence. Furthermore, it is useful to recall the following definition from the introduction: For p, q ≥ f ∈ C ((1 /s /p )) and f ∈ C ((1 /s /q )) are compositional inverses up to gauge equivalence if f − = f + element of s − C [[ s − /q ]] and f − = f + element of s − C [[ s − /p ]]. For the statement of thefollowing result recall that we defined the suitably normalized family of connections ∇ M p,q ( t , ··· ,t p + q ) associated to differential operators P and Q at the end of Section 2.4. Theorem 1.
Let p and q be positive co-prime integers and consider a quantum curve ( P, Q ) of bi-degree ( p, q ) with P and Q normalized. Let t , t , · · · and ˆ t , ˆ t , · · · be two sets of KP times suchthat t p + q = pp + q = − pq · ˆ t p + q nd t p + q − = 0 = ˆ t p + q − Then F ( ∞ , ∞ ) ∇ M p,q ( t , ··· ,t p + q ) ∼ = ∇ M q,p (ˆ t , ˆ t , ··· , ˆ t p + q ) (5) where the relation between the times t , · · · , t p + q and ˆ t , · · · , ˆ t p + q is given by the fact that the followingtwo functions are inverses up to gauge equivalence: f = 1 p · p + q X k =1 kt k s k − pp and f = − q · p + q X k =1 k ˆ t k s k − qq where s is an indeterminate and f and f are viewed as elements of C ((1 /s /p )) and C ((1 /s /q )) .Proof. The relation between t p + q and ˆ t p + q has already been shown. We now prove the remainingparts of the theorem. From the previously mentioned global Fourier duality F glob (cid:2) D ( P ( t , t , · · · , t p + q ) , Q ( t , t , · · · , t p + q )) (cid:3) ∼ = D ( P ( t , t , · · · , t p + q ) , Q ( t , t , · · · , t p + q ))it follows from the stationary phase principle that ψ ∞ D ( P ( t , t , · · · , t p + q ) , Q ( t , t , · · · , t p + q )) ∼ = F ( ∞ , ∞ ) ∇ M p,q ( t , ··· ,t p + q ) Furthermore, by the discussion of Section 2.4, one also has ψ ∞ D ( P ( t , t , · · · , t p + q ) , Q ( t , t , · · · , t p + q )) ∼ = ∇ M q,p (ˆ t , ··· , ˆ t p + q ) for some choices of ˆ t , · · · , ˆ t p + q . We now relate these KP times concretely to the other set of KP times t , · · · , t p + q . To do so, we formulate the explicit description of the local Fourier transform as obtainedby Fang [7], Graham-Squire [9], and Sabbah [18].Let us first describe Fang’s version of the result. For Z = t − /p and Z ′ = t ′− /q and α ∈ C (( Z )) itfollows from [7] (Theorem 1.3) that for every regular one-dimensional connection R one obtains F ( ∞ , ∞ ) (cid:0) [ p ] ∗ (cid:0) [ Z∂ Z ( α )] ⊗ C (( Z )) R (cid:1)(cid:1) ∼ = [ q ] ∗ (cid:18) [ Z ′ ∂ Z ′ ( β ) + p + q ⊗ C (( Z ′ )) R (cid:19) where α and β are related in the following manner: ∂ t α + t ′ = 0 α + tt ′ = β Note that since p + q > p the connection to which we apply the local Fourier transform has slopestrictly bigger than one and so the results of [7] do indeed apply. It follows from the above equations13hat ∂ t ′ β = ∂ t α · ∂ t ′ t + t + t ′ ∂ t ′ t = ( ∂ t α + t ′ ) ∂ t ′ t + t = t and ∂ t ′ β = ( − ∂ t α ) − ( t ′ )where we consider the compositional inverse of − ∂ t α viewed as an element in C ((1 /t /p )). Note thatfor example by [9] (Lemma 5.1) such an inverse does indeed exists as a formal Laurent series. We nowuse this to obtain the time dynamics.For this purpose, only the irregular part of the relevant connections matters. This part is definedin the following manner: For an irreducible connection E f with f = P a i t i/p the sum P i< a i t i/p only depends on the isomorphism class of E f . Therefore one can define in a well-defined manner theassociated connection ( E f ) irreg := E ˜ f where ˜ f = X i< a i t i/p Note that for z = 1 /Z one has [ Z∂ Z ( α )] = ( C (( u )) , dd z − ∂ Z ( α ) z )Hence, if a function H satisfies pz p − H ( z ) = − ∂ Z ( α ) z then ∂ t α = H It follows that the explicit form of the local Fourier transform implies (cid:18) F ( ∞ , ∞ ) (cid:18) [ p ] ∗ ( C (( u )) , dd z + pz p − H ) (cid:19)(cid:19) irreg ∼ = (cid:18) [ q ] ∗ ( C (( u )) , dd z + qz q − ( − H ) − ) (cid:19) irreg where ( − H ) − denotes the compositional inverse of − H viewed as a function of 1 /z , hence as afunction of 1 /s /p where s := z p The above described explicit formula for the local Fourier transform can also be verified by comparisonto the formulas given by Graham-Squire in [9] which were obtained by a different method than Fang’s.Let f be such that E f ∼ = [ p ] ∗ (cid:0) [ Z∂ Z ( α )] ⊗ C (( Z )) R (cid:1) One has, compare to the discussion in [9] (Section 5.1), that z = t , ˆ z = t ′ and − t∂ t α = 1 t ∂ /t α = f and − t ′ ∂ t ′ β = 1 t ′ ∂ /t ′ β = g + regular terms14ence one obtains f = z ˆ z which is the first of the equations obtained by Graham-Squire. Furthermore, one has z = ( fz ) − (ˆ z ) = − g ˆ z + regular termsand this yields g = − f + regular termsand this is the second of the two equations given in [9].We now apply these generalities concerning the local Fourier transform to our concrete situation:Since p and q are co-prime, at least one of the numbers p and q is odd. Assume first that p is odd.Recall that ∇ M p,q ( t , ··· ,t p + q ) ∼ = [ p ] ∗ ( C (( u )) , dd z − pz p − ( 1 − p p z p + 1 p p + q X i =1 it i z i − p )) ! Hence, since 1 − p p ∈ p Z , it follows that ψ ∞ D ( P ( t , t , · · · , t p + q ) , Q ( t , t , · · · , t p + q )) ∼ = ∇ M q,p (ˆ t , ··· , ˆ t p + q ) ∼ = F ( ∞ , ∞ ) [ p ] ∗ ( C (( u )) , dd z + pz p − ( − p p + q X i =1 it i z i − p )) ! Therefore, if we define H ( s ) := − p p + q X i =1 it i s ( i − p ) /p , then the coefficent of 1 /s in − H ( s ) is zero. It then follows for example from [9] (Lemma 5.3) that thecoefficient of 1 /s in ( − H ( s )) − is zero as well. Therefore there are isomorphisms[ q ] ∗ (cid:16) ( C (( u )) , dd z + qz q − ( p P p + qi =1 it i s ( i − p ) /p ) − (cid:17)(cid:16) F ( ∞ , ∞ ) [ p ] ∗ (cid:16) ( C (( u )) , dd z + pz p − ( − p P p + qi =1 it i s ( i − p ) /p )) (cid:17)(cid:17) irreg ∼ = O O ∼ = (cid:15) (cid:15) [ q ] ∗ (cid:16) ( C (( u )) , dd z + qz q − ( − q P p + qi =1 i ˆ t i s ( i − q ) /q )) (cid:17) p p + q X k =1 kt k s k − pp and − q p + q X k =1 k ˆ t k s k − qq Assume now that q is odd.We first recall a general result about the local Fourier transform: Let a, b be indeterminates anddenote by ι the pull-back map of connections on the formal punctured disc along the map C (( a )) −→ C (( b )) with a
7→ − b Suppose given a connection ( M, ∇ ). The pull back is then given by the C (( a ))-vector space C (( a )) ⊗ C (( b )) M with the connection map that satisfies1 ⊗ m
7→ − ⊗ ∇ ( m )Therefore one obtains ι (cid:18) [ p ] ∗ ( C (( u )) , dd z + h ( z )) (cid:19) ∼ = [ p ] ∗ ( C (( u )) , dd z − h ( ζz ))where ζ p = − ι is related to the local Fourier transform: By [3] (Proposition 3.12 (iv)) one has F ( ∞ , ∞ ) ◦ F ( ∞ , ∞ ) = ι It follows that there are isomorphisms F ( ∞ , ∞ ) ∇ M q,p (ˆ t , ˆ t , ··· , ˆ t p + q ) ∼ = (cid:15) (cid:15) ι ∇ M p,q ( t ,t , ··· ,t p + q ) ∼ = (cid:15) (cid:15) [ p ] ∗ (cid:16) C (( u )) , dd z + pz p − ( p P p + qi =1 it i ( ζz ) i − p )) (cid:17) where ζ p = −
1. Here for the second isomorphism we have used that q is odd. It now follows by asimilar reasoning as before that the following two functions are compositional inverses up to gaugeequivalence: 1 p p + q X k =1 kt k ( − s ) k − pp and 1 q p + q X k =1 k ˆ t k s k − qq This again implies the desired result. 16
Application to p – q duality of 2D quantum gravity
In this section we apply the previous results to give a conceptual proof, based on the local Fouriertransform, of the duality results of Fukuma-Kawai-Nakayama [8] concerning 2D quantum gravity.For positive co-prime integers p and q there exists the so-called ( p, q ) model of 2D quantum gravity(more precisely one should talk about ( p, q ) minimal model coupled to 2D gravity). This theory hasa partition function Z p,q that can be expressed in terms of a τ -function of the KP hierarchy. Thecrucial fact is that the partition function Z p,q is the square of a function τ p,q that is the τ -function ofthe KP hierarchy satisfying certain Virasoro constraints.The ( p, q ) models for varying p and q are not unrelated: There exists a certain duality between the( p, q ) and the ( q, p ) theory. This so-called p – q duality can be expressed as a relation between thetwo τ -functions τ p,q and τ q,p . One of the nice consequences of the duality is that it allows to describea theory by a simpler one: The ( p, q ) model depends non-trivially only on the KP times t i with1 ≤ i ≤ p + q and i Z p Therefore, the p – q duality can simplify the study of the ( p, q ) models. For example, the (3 ,
2) modeldepends a priori on four time variables but via the 3 – 2 duality it can be expressed in terms of threeparameters.In [15] it was shown that the p – q duality can be expressed in terms of the local Fourier duality ofcertain connections. We now use the results of the previous section to give a more conceptual proofof this fact and furthermore we will give a Fourier theoretic proof of the results of Fukuma-Kawai-Nakayama concerning the dynamics of the duality.It is known that ( p, q ) theory corresponds to a family of solutions to the string equation[ P ( t , · · · , t p + q ) , Q ( t , · · · , t p + q )] = 1 where P ( t , · · · , t p + q ) = ∂ px + a p − ∂ p − x + · · · + a is a differential operator of order p and Q is a differential operator of order q . The method that allowsus to describe the corresponding points of the Sato Grassmannian was given in [12]. For more detailssee [20], [10] or [8]. In this description, the τ -function τ p,q of the theory is known to satisfy ∂ ∂t ln τ p,q = a p − p It is shown in [8] that one simply has ∂ ∂t ln τ q,p = ∂ ∂t ln τ p,q + C ( t , · · · , t p + q )where the correction term is given by C ( t , · · · , t p + q ) = 1 − q · p + q − q ( p + q ) · (cid:18) t p + q − t p + q (cid:19) + p + q − q ( p + q ) · t p + q − t p + q Note in particular the case where p and q are such that one of the following holds:17i) q = 1 and q ≡ p (ii) q ≡ p In these cases the function ∂ ∂t ln τ p,q does not depend on one of the variables t p + q − or t p + q − andone can specialize the value of this time variable in such a manner that the correction term vanishes: C ( t , · · · , t p + q ) ≡ τ -functions it is crucial to obtain a relation between thetwo sets of KP times. This time dynamics of the duality was obtained by Fukuma-Kawai-Nakayamain [8]. We now give a conceptual proof of the relation between the ( p, q ) times and the ( q, p ) timesbased on properties of the local Fourier transform. Theorem 2 (Fukuma-Kawai-Nakayama [8]) . Let p and q be positive co-prime integers. Define a p + q − k = kt k ( p + q ) t p + q and ˆ a p + q − k = k ˆ t k ( p + q )ˆ t p + q Then ˆ t p + q = − qp · t p + q and there are values a n , ˆ a n extending the above definition to all n ≥ such that following two functionsare compositional inverses: g = z ∞ X n =1 a n z − n ! /q and g = z ∞ X n =1 ˆ a n z − n ! /p Proof.
We will be able to deduce this result from Theorem 1. It is clear that by Theorem 1 the times t , · · · , t p + q and ˆ t , · · · , ˆ t p + q can be related via the local Fourier transform (see (5)). Let f and f be as in the statement of Theorem 1. Namely f = 1 p · p + q X k =1 kt k u k − pp and f = − q · p + q X k =1 k ˆ t k u k − qq Setting ˆ a n = 0 for all n ≥ p + q one sees g ( u ) = ( f ( u q )) /p Then one sees that g ( u ) − = ( f − ( u p )) /q = f ( u p ) + ∞ X n = p b n u − n ! /q b n . Hence g ( u ) − = u p + q − X n =1 a n z − n + ∞ X n = p b n u − n − q ! /q Therefore one can define g by g := g − and one obtains the desired result.Via standard techniques involving formulas for inverse functions one can make the time relationeven more explicit. This has been carried out by Fukuma-Kawai-Nakayama in [8] and the generalformula is given by a n = − qp · X k ≥ k ( n − p − q ) /pk − ! X m , ··· ,m k ≥ , P m i = n ˆ a m · · · ˆ a m k Consider for example the 2 − q = 2 one can let the correction term vanish. The time variables t , · · · , t of the (5 , t , · · · , ˆ t of the (2 ,
5) model. We set t = 57and hence ˆ t = − Acknowledgements:
It is a pleasure to thank V. Vologodsky, A. Graham-Squire, M. Bergvelt for useful exchanges andthe referee for helpful remarks and corrections.
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Department of Mathematics, University of Illinois at Urbana-Champaign, IL 61801, USA
E-mail address: [email protected]
A. Schwarz,
Department of Mathematics, University of California, Davis, CA 95616, USA
E-mail address: [email protected]@math.ucdavis.edu