GAMMA CONJECTURES FOR FANO MANIFOLDS |
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4. Gamma Conjecture II
In this section we restrict to a Fano manifold F with semisimple quantum cohomology and state a re nement of Gamma Conjecture I for such F . We represent the irregular monodromy of quantum connection by a marked re ection system given by a basis of asymptotically exponential at sections. Gamma Conjecture II says that the marked re ection system coincides
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(F ). This also reb nes Dubrovin's conjecture [18]. |
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4.1. Semiorthonormal basis and Gram matrix. Let V = (V ; [ ; ); v1; : : : ; vN ) be a triple of a vector space V , a bilinear form [ ; ) = [ ; )V , and a collection of vectors v1; : : : ; vN 2 V . The bilinear form [ ; ) is not necessarily symmetric nor skew-symmetric. We say that V is a semiorthonormal collection if the Gram matrix Gij = [vi; vj) is uni-uppertriangular: [vi; vi) = 1 for all i, and [vi; vj) = 0 for all i > j. This implies that G is non-degenerate, hence vectors vi are linearly independent so N 6 dim V . We say that V is a semiorthonormal basis (SOB) if N = dim V i.e. vi form a basis of vector space V , this implies that pairing [ ; ) is non-degenerate.
Set of SOBs admits an obvious action |
of (Z=2Z)N : if v1; : : : ; vN |
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v1; : : : ; vN , in this case we say that SOBs are the same up to sign. |
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ne the right mutation Ruv and the left mutation Luv by |
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Ruv := v [v; u)u; |
Luv := v [u; v)u: |
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The braid group BN on N strands acts on the set of SOBs of N vectors by: |
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i(v1; : : : ; vN ) = (v1; : : : ; vi 1; vi+1; Rvi+1 vi; vi+2; : : : ; vN ) |
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i 1(v1; : : : ; vN ) = (v1; : : : ; vi 1; Lvi vi+1; vi; vi+2; : : : ; vN ) |
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where 1; : : : ; N are generators of BN and satisfy the braid relation |
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4.2. Marked re ection |
system. A marked re ection system (MRS) of phase |
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tuple (V; [ ; ); fv1; : : : ; vN g; m; e |
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a complex vector space V ;
a bilinear form [ ; ) on V ;
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marking m: |
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such that the vectors vi are semiorthonormal in the sense that: |
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When the basis is ordered appropriately, (V; [ ; |
); v1; : : : ; vN ) gives an SOB. We say that |
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2 R is admissible for a multiset u = fu1; : : : ; uN g in C if e |
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di erence u |
u . An MRS is said to be admissible if the phase is admissible for u. By a |
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small perturbation of the marking, one can always make an MRS admissible. The circle S |
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acts on the space of all marked re ection systems by rotating simultaneously the phase and all markings: 7! + , ui 7!ei ui.
We write h : C ! R for the R-linear function h (z) = Im(e i z).
28 |
GALKIN, GOLYSHEV, AND IRITANI |
Remark 4.2.2 ([30]). The data of admissible marked re ection system may be used to produce a polarized local system on Cnu (cf. the second structure connection): by identifying the ber with V , endowing it with a bilinear form f ; g (either with the symmetric form fv1; v2g = [v1; v2) + [v2; v1) or with the skew-symmetric form fv1; v2g = [v1; v2) [v2; v1)), choosing \ei 1" for the base point, joining it with the points ui with the level rays h (z) = h (ui) as paths, trivializing the local system outside the union of the paths, and requiring that the turn around ui act in the monodromy representation by the re ection with respect to vi : rvi (v) = v fv; vigvi. In case ui is a multiple marking all the respective re ections commute, so the monodromy of the local system is well-de ned as a product of such re ections. The form f ; g could be degenerate, however construction above also provides local systems with ber the quotient-space V= Kerf ; g (use re ections with respect to images of vi in the quotientspaces).
When u1; : : : ; uN are pairwise distinct, this construction could be reversed with a little ambiguity: if there is such a local system and the paths are given, we take V to be a ber, de ne vi 2 V as a re ection vector of the monodromy at ui (which could be determined up to sign if f ; g is non-degenerate), and de ne [vi; vi) = 1 for all i, for i 6= j put [vi; vj) = 0 if h (ui) < h (uj) and [vi; vj) = fvi; vjg otherwise.
4.3. Mutation of MRSs. We describe the change of an MRS under deformation of parameters (u; ). By perturbing the marking (or the phase) a little, we may start with an admissible MRS (see Remark 4.3.1 below). We draw a straight ray Li emanating from each marking ui in the direction of and obtain a picture as in Figure 5. We let the marking u and the phase vary. We allow that a multiple marking separates into distinct ones, but do not allow that two di erent markings ui 6= uj collide unless the corresponding vectors vi, vj are orthogonal [vi; vj) = [vj; vi) = 0. When a multiple marking separates into distinct ones, the vectors v1; : : : ; vN remain the same (under su ciently small deformation). When the parameter (u; ) crosses the codimension-one wall consisting of non-admissible con gurations (u; ), we transform the vectors v1; : : : ; vN by a left/right mutation. The situation is illustrated in Figure 6; see also Figures 1 and 2. In this Figure 6, a vector vl equipped with the marking ui is transformed to:
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The vectors that are not assigned the marking ui remain the same. Note that the vectors corresponding to the same marking uj are orthogonal to each other and hence the above product of mutations is well-de ned.
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Figure 5. Half-lines Li in the admissible direction ei = 1.
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Figure 6. Right and left mutation
Remark 4.3.1. Since we have [vi; vj) = [vj; vi) = 0 if h (ui) = h (uj) and i 6= j (see (4.2.1)), all admissible MRSs obtained from a non-admissible MRS by a small perturbation of the
marking or the phase are connected by mutation.
Remark 4.3.2. Let CN (C) denote the con guration space (2.7.1) of distinct N points on C Formally speaking the above procedure de nes a local system over CN (C) S1 such that theber at an admissible parameter (u; ei ) is the set of all MRSs on the vector space V with marking u and phase . The base space CN (C) S1 has a wall and chamber structure given by the wall W of non-admissible parameters:
W = f(u; ei ) 2 CN (C) S1 : h (ui) = h (uj) for some i 6= jg:
The locus of self-intersection of W gives a codimension-two stratum:
W 0 = f(u; ei ) 2 CN (C) S1 : h (ui) = h (uj) = h (uk) for some i 6= j 6= kg W:
Note that W is invariant under the S1-action on CN (C) S1 discussed above. To de ne a local system, it su ces to de ne a parallel transportation along a path connecting two points from (CN (C) S1) n W . For a given path, we can perturb the path so that it meets only the smooth part W n W 0 of the wall. Then we de ne the parallel transportation as the composition of right/left mutations corresponding to crossings of the wall. To check that the parallel transportation depends only on the homotopy class of the path, it su ces to recall that the mutations satisfy the braid relation (4.1.2). Note that the S1 symmetry for MRSs
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4.4. Exceptional collections. In this section we sketch only the necessary de nitions of (full) exceptional collections in triangulated categories and braid group action on them. For details we refer the reader to [7] or to the survey [34].
Recall that an object E in a triangulated category T over C is called exceptional if Hom(E; E) = C and Hom(E; E[j]) = 0 for all j 6= 0. An ordered tuple of exceptional objects E1; : : : ; EN is called an exceptional collection if Hom(Ei; Ej[k]) = 0 for all i; j; k with i > j and k 2 Z. If E1; : : : ; EN is an exceptional collection and k1; : : : ; kN is a set of integers then E1[k1]; : : : ; EN [kN ] is also an exceptional collection, in this case we say that two collections coincide up to shift. An exceptional object E gives a pair of functors called left mutation LE and right mutation RE which t into the distinguished triangles8:
LEF [ 1] ! Hom (E; F ) E ! F ! LEF;
REF ! F ! Hom (F; E) E ! REF [1]:
8What we write as LEF , REF here are denoted respectively by LEF [1], REF [ 1] in [7, 34].
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30 GALKIN, GOLYSHEV, AND IRITANI
The set of exceptional collections of N objects in T admits an action of the braid group BN in a way parallel to the BN -action (4.1.1) on SOBs, i.e. the generator i replaces fEi; Ei+1g
with fEi+1; REi+1 Eig and i 1 replaces fEi; Ei+1g with fLEi Ei+1; Eig.
The subcategory in T generated by Ei (minimal full triangulated subcategory that contains all Ei) stays invariant under mutations. An exceptional collection is called full if it generates the whole category T ; clearly braid group and shift action respects fullness. When
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Ei ! Ei[ki] described above descends to the action of BN and change of signs on the K- group described in x4.1.
4.5. Asymptotic basis and MRS of a Fano manifoldq . Let F be a Fano manifold such that ? is convergent and semisimple for some 2 H (F ). Choose a phase 2 R that is admissible for the spectrum of (E? ). By Proposition 2.5.1, we have a basis y1( ; z); : : : ; yN ( ; z) of asymptotically exponential at sections for the quantum connection rj de ned over the sector 2 < arg z < + 2 + for some > 0. They are canonically de ned up to sign and ordering. The basis is marked by the spectrum u = fu1; : : : ; uN g of (E? ). We can naturally identify yi( ; z) on the universal cover of the z-plane C . By r-parallel transportation of yi( ; z) to = 0, arg z = 0 andq jzj 1, we can identify the basis y1; : : : ; yN of at sections with a basis A1; : : : ; AN of H (F ) via the fundamental solution S(z)z z in Proposition 2.3.1, i.e.
dim F
= (Ai)(z) := (2 ) 2 S(z)z z Ai
where was given in (3.4.1). The basis fA1; : : : ; AN g is de ned up to sign and ordering and is semiorthonormal with respect to the pairing [ ; ) in (3.4.4){(3.4.5) by Proposition 2.6.4. We call it the asymptotic basis of F at with respect to the phase .
The marked re ection system (MRS)q of F is de ned to be the tuple: MRS(F; ; ) := (H (F ); [ ; ); fA1; : : : ; AN g; m: Ai 7!ui; ei ):
The marking m assigns to Ai the corresponding eigenvalue ui of (E? ). It is de ned up to sign. Note that MRS(F; ; ) and MRS(F; ; + 2 ) are not necessarily the same: they di er by the monodromy around z = 0. By the discussion in x2.6, when ( ; ) varies, the corresponding MRSs change by mutation as described in the previous section x4.3.
Remark 4.5.1. When ?0 is semisimple, we have the two MRSs MRS(F; 0; ) de ned at the canonical position = 0 for a su ciently small > 0. When = 0 is admissible for the spectrum of (c1(F )?0), the two MRSs coincide.
4.6. Dubrovin's (original) conjecture and Gamma conjecture II. We recall Dubrovin's Conjecture (see also Remark 4.6.5). Let F be a Fano manifold. Dubrovin [18, Conjecture 4.2.2] conjectured:
(1)the quantum cohomology of F is semisimple if and only if Dcohb (F ) admits a full exceptional collection E1; : : : ; EN ;
and if the quantum cohomology of F is semisimple, there exists a full exceptional collection E1; : : : ; EN of Dcohb (F ) such that:
(2) the Stokes matrix S = (Sij) is given by Sij = (Ei; Ej), i; j = 1; : : : ; N;
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(3)the central connection matrixq has the form Cq = C0C00 whenq the columns of C00 are the components of Ch(Ej) 2 H (F ) and C0 :q H (F ) ! H (F ) is some operator satisfying C0(c1(F )a) = c1(F )C0(a) for any a 2 H (F ).
The central connection matrix C above is given by |
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where the column vectors A1; : : : ; AN 2 H q (F ) are an asymptotic basis in x4.5. Gamma Conjecture II makes precise the operator C0 in Part (3) of Dubrovin's conjecture.
Conjecture 4.6.1 (Gamma Conjecture II). Let F be a Fanoq manifold such that the quantum
product ? is convergent and semisimple for some 2 H (F ) and that Dcohb (F ) has a full exceptional collection. There exists a full exceptional collection fE1; : : : ; EN g such that the
asymptotic basis A1; : : : ; AN of F at with respect to an admissible phase (see x4.5) is given by:
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4.6.3. The validity of Gamma Conjecture II does not depend on the choice of and |
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Remark 4.6.4. If F satis es Gamma Conjecture II, the quantum di erential equations of F specify a distinguished mutation-equivalence class of (the images in the K-group of) full exceptional collections of F given by an asymptotic basis. It is not known in general whether any two given full exceptional collections are connected by mutations and shifts.
Remark 4.6.5. While preparing this paper, we are informed that Dubrovin [19] himself formulated the same conjecture as Gamma Conjecture II.
Proposition 4.6.6. Gamma Conjecture II implies Parts (2) and (3) of Dubrovin's conjecture.
Proof. Recall from Proposition 2.6.4 that the Stokes matrix is given by the pairing [yi; yj) of asymptotically exponential at sections. Under Gamma Conjecture II, we have [yi; yj) = [Ai; Aj) = (Ei; Ej), where we used the fact that the pairing [ ; ) is identi ed with the Euler pairing (Lemma 3.4.6).
A relationship between Gamma Conjectures I and II is explained as follows.
Proposition 4.6.7. Suppose that a Fano manifold F satis es Property O and that the quantum product ?0 is semisimple. Suppose also that F satis es Gamma Conjecture II. If the exceptional object corresponding to the eigenvalue T and an admissible phase with j j < =2
is the structure sheaf O (or its shift), then F satis es Gamma Conjecture I. |
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Conjecture I holds (see Conjecture 3.4.3, |
Theorem 3.6.8). |
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