Geometric quantization of almost toric manifolds

GEOMETRIC QUANTIZATION OF ALMOST TORIC MANIFOLDS EVA MIRANDA, FRANCISCO PRESAS, AND ROMERO SOLHA Abstract. Kostant gave a model for the geometric quantization via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed by counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one induced by an integrable system with focus-focus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme. 1. Introduction An important contribution of Kostant has been the definition of geometric quantization via the cohomology associated to the sheaf of sections of a chosen pre-quantum line bundle that are flat along a given polarization. This construction using real polarizations is an abstraction of Kähler quantization and has been used in connection to representation theory (see for instance [9]). Generalizations of this scheme considering non-degenerate singularities have also been obtained by Hamilton [10], Hamilton and Miranda [11], and Solha [25]. A toric manifold is a symplectic manifold endowed with an effective Hamiltonian action of a torus whose rank is half of the dimension of the manifold. A theorem of Delzant [4] establishes a one-to-one correspondence between closed toric manifolds in dimension 2mand a class of polytopes (now called Delzant polytopes) on Rm. The real geometric quantization of closed toric Date: April 9, 2019. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize and is partially supported by grants with reference MTM2015-69135-P (MINECO-FEDER) and 2017SGR932 (AGAUR). Romero Solha is supported by CAPES and partially supported by MTM2015-69135P (MINECO/FEDER). Francisco Presas is supported by the grant reference number MTM2016-79400-P (MINECO/FEDER). Eva Miranda and Francisco Presas are supported by an EXPLORA CIENCIA project with reference number MTM2015-72876-EXP and by the excellence project SEV-2015-0554. This article is written within the collaborative agreement of a Doctor Vinculado honorary position at ICMAT-CSIC held by Eva Miranda. 1 2 EVA MIRANDA, FRANCISCO PRESAS, AND ROMERO SOLHA manifolds can be read from the Delzant polytope, as proved by Hamilton [10] (generalizing previous results by Śniatycki [24] to the singular context): given a toric manifold, its real geometric quantization is completely determined by the count of integral points inside (boundary points are excluded) its associated Delzant polytope. Toric manifolds are central in the study of the geometry of symplectic manifolds and their symmetries, as are their generalizations, such as semitoric integrable systems [21] or almost toric manifolds [14], in which the rank of the torus is no longer half of the dimension of the manifold. Examples of almost toric manifolds are given by K3 surfaces, which are also of relevance in complex geometry. A semitoric integrable system (see for instance [22, 21]) is an integrable system admitting only non-degenerate singularities composed of elliptic and focus-focus components, but excluding any hyperbolic ones. As observed in [25] the quantization of almost toric manifolds can be reduced to the computation of the contribution of a neighborhood of a Bohr– Sommerfeld focus-focus singular fiber by the use of factorization tools1. This is because the geometric quantization of neighborhoods of Bohr–Sommerfeld fibers computes the geometric quantization of the whole manifold (by means of a standard Mayer–Vietoris sequence). In this article we apply Kostant’s model to focus-focus singularities and conclude its computation, showing that the first cohomology group associated to the real geometric quantization of a small neighborhood of a focusfocus fiber of a 4-dimensional semitoric integrable system is trivial, but not the second cohomology group, which is infinite dimensional when the singular fiber is Bohr–Sommerfeld (theorem 5.1). This determines completely the geometric quantization when the real polarization has focus-focus fibers (the cohomology group in degree zero is trivial and had already been computed in [25]) and, thus, brings to a close the problem of geometric quantization of integrable systems with non-degenerate singularities as initiated in [10] and [11] for 4-dimensional manifolds with no hyperbolic-hyperbolic fibers. As a motivation for these results, we present K3 surfaces as an example of almost toric manifolds and analyze the effect of nodal trades [14] in their real quantization. Other models of quantization for K3 surfaces have been recently obtained by Castejón [2] using the Berezin–Toepliz operators approach [1]. For this direction see also [20]. 2. Main definitions 2.1. Singular Lagrangian fibrations. The symplectic manifolds of interest to this article have a great deal of symmetry, and such symmetries are related to some particular classes of integrable systems: those admitting only non-degenerate singularities. Definition 2.1. An integrable system on a symplectic manifold (M, ω)of dimension 2mis a set of mfunctions, f1, . . . , fm∈C∞(M;R), satisfying df1∧ · · · ∧ dfm6= 0 over an open dense subset of Mand 1Which behave following a Künneth formula [17], as a simple sheaf cohomology. GEOMETRIC QUANTIZATION OF ALMOST TORIC MANIFOLDS 3 {fj, fk}ω= 0 for all j, k. The Poisson bracket is defined by {f, ·}ω=Xf(·), where Xfis the unique vector field defined by the equation ıXfω+ df= 0, called the Hamiltonian vector field of f. The next definition refers to the critical set of an integrable system, i.e. the set of points where df1∧ · · · ∧ dfmvanishes. Definition 2.2. A critical point of rank kr=m−ke−kh−2kfof an integrable system (f1, . . . , fm) : M→Rmis a non-degenerate singular point of Williamson type (ke, kh, kf)if the quadratic parts of f1, . . . , fmcan be written as: hj=xj(regular) 1≤j≤kr hj=x2 j+y2 j(elliptic) kr+ 1 ≤j≤kr+ke hj=xjyj(hyperbolic) kr+ke+ 1 ≤j≤kr+ke+kh hj=xjyj+xj+1yj+1 hj+1 =xjyj+1 −xj+1yj(focus-focus) j=kr+ke+kh+ 2l−1, 1≤l≤kf in some Darboux local coordinates (x1, y1, . . . , xm, ym). The fiber of an integrable system is the preimage of a point in the image of (f1, . . . , fm) : M→Rm, and such a map will be referred to as moment map. A singular fiber of an integrable system is said to be of Williamson type (ke, kh, kf)if all of its singular points are non-degenerate singular points of that same Williamson type. Another terminology is also used in this article: in dimension 2 an elliptic fiber and a hyperbolic fiber are singular fibers of Williamson type (1,0,0) and (0,1,0), in dimension 4 a focus-focus fiber is a singular fiber of Williamson type (0,0,1). When we refer to the foliation associated to an integrable system we refer to the foliation described by the orbits of the Hamiltonian vector fields. When we refer to the fibration associated to an integrable system we refer to the fibration defined by the moment map. The foliation associated to the last fibration and the first foliation do not necessarily coincide at the singular points. As it was proved by Eliasson [5, 6] and Miranda [15, 16, 19], non-degenerate singularities are characterized by the fact that the foliation associated to an integrable system is equivalent to the foliation described by its quadratic part. Let us define the notion of a singular Lagrangian fibration. Definition 2.3. A singular Lagrangian fibration is a symplectic manifold (M, ω)of dimension 2mtogether with a surjective map F:M→N, where Nis a topological space of dimension m, such that for every point in Nthere exist an open neighborhood V⊂Nand a homeomorphism χ:V→U⊂Rm satisfying that χ◦ FF−1(V)is an integrable system on (F−1(V), ωF−1(V)). When the integrable systems in a singular Lagrangian fibration do not have singularities, one refers to it as a regular Lagrangian fibration. The 4 EVA MIRANDA, FRANCISCO PRESAS, AND ROMERO SOLHA real geometric quantization of such manifolds was computed in [24], whereas (closed) locally toric manifolds were considered in [10] (see [25] for the noncompact case); these symplectic manifolds are singular Lagrangian fibrations whose singularities are of Williamson type (ke,0,0) only. Almost toric manifolds are singular Lagrangian fibrations admitting only singularities of Williamson type (ke,0, kf). In particular, regular Lagrangian fibrations and locally toric manifolds (which include toric manifolds), are examples of almost toric manifolds; as well as the semitoric integrable systems in dimension four, which are included in the almost toric manifolds whose bases are subsets of R2. The semi-local and global classification of these symplectic manifolds has been the object of study of [3, 14, 22, 21, 26]. 2.2. Real geometric quantization. Let (M, ω)be a symplectic manifold of dimension 2mwhose de Rham class [ω]admits an integral lift. Such a symplectic manifold will be called pre-quantizable, and a complex line bundle over it with a connection ∇ωsatisfying curv(∇ω) = −iω is said to be a prequantum line bundle for (M, ω). Definition 2.4. A real polarization Pis an integrable (in the Sussmann’s sense) distribution of TM whose leaves are generically Lagrangian. The complexification of Pis denoted by Pand will be called the polarization. The most relevant real polarization for this work is hXf1, ..., XfmiC∞(M;R): the distribution of the Hamiltonian vector fields of an integrable system. The leaves of the associated (possibly singular) foliation are isotropic submanifolds and they are Lagrangian at points where the first integrals are functionally independent. Definition 2.5. Let Jdenote the sheaf of sections of a pre-quantum line bundle Lsuch that for each open set V⊂Mthe set J(V)is the module (over the ring of smooth leafwise constant complex-valued functions of V) of sections s∈Ldefined over Vsatisfying ∇ω Xs= 0 for all vector fields Xin Pdefined over V. Definition 2.6. The quantization of (M, ω, L, ∇ω, P)is given by Q(M) = M n≥0 Hn(M;J), where Hn(M;J)are the sheaf cohomology groups associated to J. The following definition plays a very important role in the computation of the cohomology groups appearing in geometric quantization: Definition 2.7. A leaf `of Pis Bohr–Sommerfeld if there exists a nonvanishing section s:`→Lsuch that ∇ω Xs= 0 for any complex vector field Xin the polarization P(restricted to `). Fibers that are a union of Bohr–Sommerfeld leaves are called Bohr–Sommerfeld fibers. GEOMETRIC QUANTIZATION OF ALMOST TORIC MANIFOLDS 5 3. A motivating example: K3 surfaces A K3 surface is an example of a total space of an almost toric manifold: it admits an almost toric fibration over the sphere with 24 focus-focus fibers. The base is a sphere with 24 marked points, and on the complement of these points one has a regular Lagrangian fibration with torus fibers. A way to construct such an almost toric manifold, as done in [14] (cf. [7]), is to consider two copies of a (symplectic and toric) blowup of the complex projective plane at 9 different points as toric manifolds, apply nodal trades to all of their elliptic-elliptic singular fibers, and take their symplectic sum along the symplectic tori corresponding to the preimage of the boundary of their respective bases (as almost toric fibrations). Starting with a pre-quantizable K3 surface, this construction (together with a gluing result described in section 7) allows one to obtain a K3 surface with up to 24 Bohr–Sommerfeld focus-focus fibers. Here is how the construction works. •Starting with a complex projective plane, understood as a toric manifold and described here by its Delzant polytope [4], one performs three blowups at different points, represented in their Delzant polytopes by cuts based on their three vertices (cf. [13]), followed by another six blowups at different points, represented in their Delzant polytopes by cuts based on their six new vertices formed after the first three blowups: see figure 1. - q q qpppppppppp ppppppppp pppppppp ppppppp pppppp ppppp p p p p ppp p p p p p p p p p p p p p p p p p p @ @ @ @ @ @ @ @ @ @ p p p p p p p p p p p p p p p ppppppppppppppp - q q q q q q p p p p ppppp pppppp ppppppp pppppp ppppp p p p p @ @ @ @ p p p p p p p p p p p @ @ @ @ pppppp p p p p p p p p p p p p p p p p p p p p p p pppppp ppppppppppp q q q q q q q q q q q q p p ppppp pppppp ppppp pppppp ppppp p p @ @ H H H @ @   A A A H H H   AA A Figure 1. CP2,CP2#3CP2, and CP2#9CP2. •Now, following [14], one can perform nodal trades to all the vertices of the resulting Delzant polytope. In figure 2 each nodal trade is being represented by a vector based at a vertex, and the monodromy around each of the resulting focus-focus fibers can be read from those vectors. •The resulting manifold, CP2#9CP2, is endowed with an almost toric fibration, and the preimage of the boundary of its base is a symplectic torus. Thus, one can consider two copies of this symplectic manifold and perform a symplectic sum along these tori (cf. [7]), obtaining a K3 surface, (CP2#9CP2)#T2(CP2#9CP2); together with an almost toric fibration whose base is the sphere formed by gluing two copies 6 EVA MIRANDA, FRANCISCO PRESAS, AND ROMERO SOLHA - q q q q q q q q q q q q p p ppppp p p p p p p ppppp p p p p p p ppppp p p -  -  6 ? 6 ? @ @R @ @I @ @R @ @I @ @ H H H @ @   A A A H H H   AA A p p p p p p p ppppp p p p p p p p q q q q q q q q q q q q c c c c c c c c c c c c Figure 2. Nodal trades on CP2#9CP2. of the previously constructed disk (with its twelve marked points) along their boundary (see figure 3). K3 Figure 3. K3 surface as a singular fiber bundle over the sphere. Before the nodal trades, the toric manifold CP2#9CP2of figure 1 admits a pre-quantum line bundle such that all the integer lattice points belonging to its Delzant polytope are images of Bohr–Sommerfeld fibers [9, 25]. Nodal trades produce a one parameter family of symplectomorphic manifolds (via nodal slides [14]), and the almost toric manifold constructed in figure 2 is related by a symplectomorphism isotopic to the identity (the same is true for different nodal trades resulting in up to 12 focus-focus fibers outside the integer lattice). Therefore, it inherits a pre-quantum line bundle whose Bohr–Sommerfeld fibers are still given by the integer lattice points in the base (which now includes up to 12 focus-focus fibers, depending on the size of the nodal trades). When gluing two copies of those almost toric manifolds by a symplectic sum, one can glue the pre-quantum line bundles to obtain a pre-quantum line bundle on a K3 surface having any number (between 0 and 24) of focusfocus Bohr–Sommerfeld fibers. This last assertion is justified by applying lemma 7.2 and corollary 7.1 (which can be found, together with their proofs, in section 7). GEOMETRIC QUANTIZATION OF ALMOST TORIC MANIFOLDS 7 4. Poincaré lemma and Künneth formula This section collects results from the literature needed for the proof of the main theorems of this article, and they are included here for the convenience of the reader. Theorem 4.1 (Solha [25]).The cohomology groups Hn(M;J)vanish for all n≥1in any sufficiently small contractible open neighborhood of a focus-focus singularity. Remark 4.1. The only property of Lbeing used here is the existence of flat connections along P; thus, the results here work if metaplectic correction is considered. The classical Künneth formula also holds for the geometric quantization scheme [17]. Let (M1,P1)and (M2,P2)be a pair of pre-quantizable symplectic manifolds endowed with Lagrangian foliations. The natural Cartesian product for the foliations is Lagrangian with respect to the product symplectic structure. The induced sheaf of flat sections associated to the product foliation will be denoted J12. Note that we use the pre-quantum line bundle defined as pull-backs of the ones defined over M1and M2. Theorem 4.2 (Miranda and Presas [17]).There is an isomorphism Hn(M1×M2,J12)∼ =M p+q=n Hp(M1,J1)⊗Hq(M2,J2), whenever M1admits a good cover, the geometric quantization associated to (M2,J2)has finite dimension and M2is a submanifold of a compact manifold. As an illustration, and to anticipate some needed results, let us mention what is the geometric quantization for M=T∗I×(Is×S1)with ω= dx1∧dy1+ dx2∧dy2, endowed with a trivial pre-quantum line bundle with connection ∇= d−i(x1dy1+x2dy2), and Pgenerated by ∂ ∂y1and ∂ ∂y2, where x1is the coordinate function along the fibers of T∗I,y1is the coordinate function along the open interval I⊂R,x2is the coordinate function along the open interval Is⊂R, and y2is the periodic coordinate function along S1. Proposition 4.1. The geometric quantization of M=T∗I×(Is×S1), with the extra structures described above, is given by H1(T∗I×(Is×S1); J12)∼ =(H0(T∗I;J1))n∼ =(C∞(R;C))n, where nis the number of integers inside Is. Remark 4.2. For some (possibly infinite dimensional) vector space Hand positive integer k,Hkstands for the direct sum of kcopies of H. A proof of proposition 4.1 can be found in [17, 24, 25]. 8 EVA MIRANDA, FRANCISCO PRESAS, AND ROMERO SOLHA 5. Contribution from focus-focus singularities Let V⊂Mbe an open neighborhood of a non-degenerate focus-focus fiber `f(compact or not) over which a Hamiltonian S1-action is defined [27]. Note that a focus-focus fiber might have more than one singular point (also called a node, or nodal point [14]). Lemma 5.1 (Solha [25]).In the neighborhood of `fover which a Hamiltonian S1-action is defined, there exists a neighborhood Vcontaining only `fas a Bohr–Sommerfeld fiber such that H0(V;JV) = {0}. Therefore, without loss of generality, one can assume that Vcontains no Bohr–Sommerfeld fiber, or only one if `fis itself Bohr–Sommerfeld. Such a neighborhood Vof a focus-focus fiber `fis called a saturated neighborhood (since it is saturated by the orbits of the S1-action). Theorem 5.1. The geometric quantization of a saturated neighborhood of a focus-focus fiber with nnodes is: •0if the singular fiber is not Bohr–Sommerfeld. •isomorphic to (C∞(R;C))nf, if the singular fiber is Bohr–Sommerfed, where nf=n(for compact fibers) and nf=n−1otherwise. Proof. Let p1, . . . , pn∈`fbe nsingular points on the focus-focus fiber. Take W1, . . . , Wn⊂Vcontractible open neighborhoods of the singular points such that Wj∩Wk=∅for j6=k, and V0⊂Van open (not connected) neighborhood satisfying p1, . . . , pn/∈V0, as well as, V=V0∪W1∪ · · · ∪ Wn, and V0∩Wj=W− jtW+ jfor each Wj. The neighborhood Vis the total space of a singular Lagrangian fibration over an open disk D2∼ =R×Is(with Is⊂Ran open interval representing the circle action direction), as well as Vn=W1∪· · ·∪Wn(which is diffeomorphic to a disjoint union of open 4-balls centered in the nodal points), while W− j, W+ j, and V0are regular trivial Lagrangian fibrations. Indeed, V0∼ =(I0×S1)t · · · t (Ij×S1)t · · · t (I2π×S1)×D2, with I0= (0, b− 1),Ij= (a+ j, b− j+1),I2π= (a+ n,2π), and a+ j> b− j−1, W− j∼ =(I− j×S1)×D2 with I− j= (a− j, b− j)and a− j∈(b+ j−1, b− j), and W+ j∼ =(I+ j×S1)×D2 with I+ j= (a+ j, b+ j)and b+ j∈(a+ j, a− j+1)(see figure 4). For a compact fiber `f, one connects I0and I2πvia 0∼2π. Let us represent the trivial regular Lagrangian fibrations as products of two cotangent bundles V0∼ =(T∗(I0t · · · t Ijt · · · t I2π)) ×(Is×S1), GEOMETRIC QUANTIZATION OF ALMOST TORIC MANIFOLDS 9 I0 z }| { 0 b |{z} a− 1 r b− 1 b I− 1 Ij z }| { |{z} I+ j a+ j b b+ j r a− j+1 r | {z } I− j+1 b− j+1 b I2π z }| { |{z} I+ n a+ n b b+ n r 2π b Figure 4. Intervals along a focus-focus fiber. W− j∼ =T∗I− j×(Is×S1), and W+ j∼ =T∗I+ j×(Is×S1). We do so in order to use lemma 5.1, theorem 4.1, and proposition 4.1, which give: H0(V;JV) = {0}, H0(Vn;JVn) = H1(Vn;JVn) = H2(Vn;JVn) = {0}, H0(V0;JV0) = H0(W− j;JW− j ) = H0(W+ j;JW+ j ) = {0}, H2(V0;JV0) = H2(W− j;JW− j ) = H2(W+ j;JW− j ) = {0}, H1(V0;JV0)∼ =   {0}, if `fis not Bohr–Sommerfeld (C∞(R;C))n+1 , if `fis non-compact (C∞(R;C))n, if `fis compact , and H1(V0∩Vn;JV0∩Vn)∼ =(H1(W− j;JW− j )⊕H1(W+ j;JW+ j ))n ∼ =(C∞(R;C))2n, if `fis Bohr–Sommerfeld {0}, otherwise . Considering the open covering {V0, Vn}of V, one has the following Mayer– Vietoris sequence (see [17] for a proof of its existence): 0→H0(V0∪Vn;JV0∪Vn) →H0(V0;JV0)⊕H0(Vn;JVn)→H0(V0∩Vn;JV0∩Vn) →H1(V0∪Vn;JV0∪Vn) →H1(V0;JV0)⊕H1(Vn;JVn)→H1(V0∩Vn;JV0∩Vn) →H2(V0∪Vn;JV0∪Vn) →H2(V0;JV0)⊕H2(Vn;JVn)→H2(V0∩Vn;JV0∩Vn) →H3(V0∪Vn;JV0∪Vn)→ · · ·