aa r X i v : . [ m a t h - ph ] D ec FROBENIUS MANIFOLDS, SPECTRAL CURVES, ANDINTEGRABLE HIERARCHIES
JIAN ZHOU
Abstract.
We formulate some conjectures that relates semisim-ple Frobenius manifolds, their spectral curves and integrable hier-archies.
In this short note we will propose some conjectures to unify thereconstruction approach and the emergent approach in the study ofGromov-Witten type theories.As explained in an influential article by Anderson [2] and an inspiringbook by Laughlin [26], there are the reduction/reconstruction and theemergence approaches to science. In the former approach one tries toderive the collective behavior of large quantity of individual particlesfrom the fundamental laws obeyed by each individual particle. In thisapproach the difficulty is that one has to first find the fundamentallaws, and secondly the complexity of deriving the collective behaviorfrom these laws makes it very difficult and often out of reach. In thelatter approach there are fundamental laws at each level of complexity[2], and it is even possible that all the fundamental laws for individualparticles have their origins in their collective behavior [26, Preface,XV]. So fundamental laws might emerge from the collective behavior.In some recent papers [35, 36, 37, 38], we have embarked on astudy of Gromov-Witten type theories by an emergent approach. AGromov-Witten type theory is roughly speaking a topological field the-ory that describes topological matters coupled to 2D topological grav-ity. The topological matters give us a finite-dimensional space (calledthe small phase space) of topological observables (called the primaryobservables). When coupled to the topological gravity, each observablehas an infinite sequence of gravitational descendants. One then getsan infinite-dimensional space called the big phase space. A Gromov-Witten type theory defines a partition function on the big phase space.A reconstruction approach towards a Gromov-Witten type theory is toreconstruct the partition function on the big phase from the restrictionof its genus zero part on the small phase space.
In many examples, the genus zero free energy partition of a Gromov-Witten type theory satisfies the WDVV equations. A geometric struc-ture called a Frobenius manifold was introduced by Dubrovin [4] toencode such information. It forms the foundation of some reconstruc-tion approaches and plays the role of fundamental laws of individ-ual particles. Two formalisms have been proposed to reconstruct thewhole theory from a Frobenius manifold, under a technical condition ofsemisimpleness. One of them is Givental’s quantization formalism [18].It takes a product of finite many copies of the partition function of theGromov-Witten theory of a point, the number of which equals the num-ber of primaries, and obtains from it the partition function of the wholetheory by the action of some operators obtained from quantization of asymplectic structure on the formal loop space of the small phase space.The other formalism due to Dubrovin-Zhang [10] is to first extend thesystem of WDVV equation to an infinite system of equations calledthe principal hierarchy. This hierarchy is regarded as the dispersion-less limit (i.e. the genus zero part) of an integrable hierarchy satisfyingcertain axioms, called the Dubrovin-Zhang hierarchy. This is inspiredby Krichever’s observation [24] that certain solutions of dispersionlesslimit of integrable hierarchies lead to solutions of WDVV equations.Since the Dubrovin-Zhang hierarchy is some special deformation of theprincipal hierarchy, one can try to establish its existence and uniquenessby computing certain cohomology groups [29, 7]. One can also estab-lish the equivalence of these two formalisms [27]. Both the Giventalformalism and the Dubrovin-Zhang formalism are inspired by WittenConjecture/Kontsevich Theorem [33, 23]. Whereas in Dubrovin-Zhangformalism integrable hierarchy plays a key role, connections to inte-grable hierarchies can also be made in the Givental formalism in manyexamples [18, 20, 17, 15, 28].A different reconstruction approach, called Eynard-Orantin topolog-ical recursion [13], starts with a plane curve with some extra data anddefines n -point functions recursively. In appearance this looks quitedifferently from either the Givental formalism or the Dubrovin-Zhangformalism, since both of them starts with a semisimple Frobenius man-ifold. Nevertheless it has been shown that a local version of topologicalrecursion is equivalent to Givental’s quantizantion formalism [12]. Thisformalism turns out to be very successful in reformulating local mir-ror symmetry of toric Calabi-Yau 3-folds [6]. Both the proofs in thecase of toric Calabi-Yau 3-manifold [14] and 3-orbifolds [16] use com-parison with recursions in the Givental formalism. By these results,one may expect to establish some connections between the global EOtopological recursions and the Frobenius manifold theory by associating ROBENIUS MANIFOLDS, SPECTRAL CURVES, AND INTEGRABLE HIERARCHIES3 spectral curves to Froebnius manifolds. In fact, Dubrovin [4] defined asuperpotential function of a semisimple Frobenius manifold very longago , and people suspect that the partition function of the semisimpleFrobenius manifold obtained by Givental formalism or Dubrovin-Zhangformalism satisfies the Eynard-Orantin topological recursion using thespectral curve defined by this superpotential function. This has beenshown to be the case [11].One can consider the problem of going now in the reversed direc-tion, i.e., from topological recursions on a spectral curve to Frobeniusmanifold. A consequence of being able to do so is that one can clearlysee that a tau-function of some integrable hierarchy can be producedin this way. As pointed out by Eynard and Orantin [13], the partitionfunction defined by the topological recursion should be the tau-functionof some integrable hierarchy. This was further elaborated by Borot andEynard [5].We will refer to Givental formalism and Dubrovin-Zhang formalismas Type A reconstruction formalisms, to Eynard-Orantin formalism asa Type B reconstruction formalism. As we point out above, both typesof reconstruction formalisms lead to integrable hierarchies, this is whywe take integrable hierarchies as our foundation for an emergent ap-proach. We take the point of view of the Kyoto school, i.e., we treatgeneral integrable hierarchies as reductions of KP or n-component KPhierarchies [30, 21]. The KP case has been studied in [37, 38], the n-component KP [22] case will be reported in [39], and some reductionswill be reported in [40]. The main results of these work can be summa-rized as follows. Given a tau-function of one of these integrable hierar-chies, one can produce a spectral curve by studying the Kac-Schwarzoperators that specify it, together with some natural quantization anddeformation over the big phase space. One can derive from this theW-constraints satisfied by the tau-function. The quantization of thespectral curve leads to a reduction of the integrable hierarchy to a dis-persive universal Whitham hierarchy introduced by Tanisaki-Takebe[32] and Szablikowski-B laszak [31] associated with the spectral curvein genus zero. Recall the dispersionless universal Whitham hierarchywas introduced by Krichever [25] for a curve of arbitrary genera.This short note is driven by the hope that by putting all these resultstogether, one can find a unification of different approaches. In the caseat hand, the reconstruction approaches and the emergent approachare just two sides of the same coin. They nicely fit together as they The author thanks Professor Si-Qi Liu for bringing this to his attention a coupleof years ago.
JIAN ZHOU should. This can be best seen if we summarize the work mentionedabove about the relationship among three subjects in the title in thefollowing picture:Frobenius Manifolds ( I ) . . ( V ) ( ( Spectral Curves ( III ) s s ( II ) n n Integrable Hierarchies ( IV ) ( V I ) h h Here we content ourselves with semisimple Frobenius manifolds. Linksin the top row can be thought of as mirror symmetry, links from thetop row to the bottom row are the reconstruction approaches, and thelinks from the bottom row to the top row can be regarded as emergentapproaches. The link (I) is established in [11, 12]. A direct link (II)is missing, an indirect link is through (III) and (VI). For (III), see[13, 5] and this paper. We establish (IV) for KP and n-componentKP hierarchies in [37] and [39] respectively. For examples of (VI), see[25, 3]. For examples of (V), see [10, 19, 20, 17, 15, 28].To strengthen the above unification of reconstruction and emergentapproaches, let us now formulate some conjectures.
Conjecture 1.
One can generalize Krichever’s construction of uni-versal Whitham hierarchy on punctured Riemann surfaces of arbitrarygenera to the dispersive case.
Conjecture 2.
By Eynard-Orantin topological recursion one can con-struct a tau-function of the dispersive universal Whitham hierarchy as-sociated with the spectral curve.
By combining the above two conjectures with the construction ofspectral curve of semisimple Frobenius manifold [11], we make the fol-lowing:
Conjecture 3.
The Dubrovin-Zhang integrable hierarchy associatedwith a semisimple Frobenius manifold can be identified with the univer-sal Whitham hierarchy associated to the spectral curve of the Frobeniusmanifold constructed using Dubrovin’s potential function.
There are many different theories that produces semisimple Frobe-nius manifolds, so our conjecture proposes the integrable hierarchy sat-isfied by their partition functions. For example, Gromov-Witten theoryof Fano manifolds, Saito’s theory of primitive forms, etc. A particularlyinteresting application is to deformation theory of isolated singulari-ties. One first uses Saito’s theory to produce a semisimple Frobeniusmanifold, then uses Dubrovin’s superpotential to produce a spectral
ROBENIUS MANIFOLDS, SPECTRAL CURVES, AND INTEGRABLE HIERARCHIES5 curve, our conjecture then produces a tau-function of the integrablehierarchy associated to the spectral curve. Emergent geometry of theintegrable hierarchy then produces a quantum deformation theory ofthe integrable hierarchy.By combining Conjecture 1 with the proof of the BKMP RemodellingConjecture [14, 16], we make the following:
Conjecture 4.
Partition function of a toric Calabi-Yau 3-manifoldor 3-orbifold with Aganagic-Vafa outer D-branes is a tau-function ofuniversal Whitham hierarchy associated with its local mirror curve.
One can also compare Conjecture 3 and Conjecture 4 and make aconnection between them. It was conjectured in [1] that the partitionfunctions in Conjecture 4 are tau-functions of n-component KP hier-archy and this leads to a fermionic reformulation of the theory. Somespecial cases have been verified in [34]. For more partial results, seealso the more recent work [8, 9].
Conjecture 5.
There is a fermionic reformulation of dispersive uni-versal Witham integrable hierarchy associated with any spectral curve.
Conjecture 6.
There is a fermionic reformulation of the Dubrovin-Zhang integrable hierarchy for semisimple Frobenius manifolds.
Of course there is no reason that stops us from conjecturing that anyGromov-Witten type theory has a fermionic reformulation.
Acknoledgements . This research is partially supported by NSFCgrant 11171174. The author thanks Professors Si-Qi Liu and You-jin Zhang for explaining to him the Dubrovin-Zhang theory and itsrelationship to Givental formalism. The author also thanks them andProfessor Yongbin Ruan for explaining their joint work.
References [1] M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mari˜no, C. Vafa, Topologicalstrings and integrable hierarchies. Comm. Math. Phys. 261 (2006), no. 2,451–516.[2] P.W.Andrson,
More is different , Science 177 (1972), No. 4047, 393-396.[3] B. Dubrovin,
Hamiltonian formalism of Whitham-type hierarchies and topo-logical Landau-Ginsburg models . Comm. Math. Phys. 145 (1992), no. 1, 195–207.[4] B. Dubrovin,
Geometry of 2D topological field theories . Integrable systemsand quantum groups (Montecatini Terme, 1993), 120-348, Lecture Notes inMath., 1620, Springer, Berlin, 1996.
JIAN ZHOU [5] G. Borot, B. Eynard,
Geometry of spectral curves and all order dispersiveintegrable system . SIGMA Symmetry Integrability Geom. Methods Appl. 8(2012), Paper 100, 53 pp.[6] V. Bouchard, A. Klemm, M. Mari˜no, S. Pasquetti,
Remodeling the B-model .Comm. Math. Phys. 287 (2009), no. 1, 117–178.[7] G. Carlet, H. Posthuma, S. Shadrin,
Deformations of semisimple Poissonpencils of hydrodynamic type are unobstructed , arXiv:1501.04295.[8] F. Deng, J. Zhou,
Fermionic gluing principle of the topological vertex . J. HighEnergy Phys. 2012, no. 6, 166, front matter+26 pp.[9] F. Deng, J. Zhou,
On fermionic representation of the framed topological ver-tex , doi:10.1007/JHEP12(2015)019.[10] B. Dubrovin, Y. Zhang,
Normal forms of hierarchies of integrable PDEs,Frobenius manifolds and Gromov - Witten invariants , arXiv:math/0108160,[11] P. Dunin-Barkowski, P. Norbury, N. Orantin, A. Popolitov, S. Shadrin,
Dubrovin’s superpotential as a global spectral curve , arXiv:1509.06954.[12] P. Dunin-Barkowski, N. Orantin, S. Shadrin, L. Spitz,
Identification ofthe Givental formula with the spectral curve topological recursion procedure ,Comm. Math. Phys., Vol. 328 (2014), issue 2, pp 669–700.[13] B. Eynard, N. Orantin,
Invariants of algebraic curves and topological expan-sion . Commun. Number Theory Phys. 1 (2007), no. 2, 347–452.[14] B. Eynard, N. Orantin,
Computation of open Gromov-Witten invariants fortoric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP con-jecture , arXiv:1205.1103.[15] H. Fan, T. J. Jarvis, Y. Ruan,
The Witten equation and its virtual fundamen-tal cycle , arXiv:0712.4025, 108 pp.[16] B. Fang, C.-C. M. Liu, Z. Zong,
All genus open-closed mirror symmetry foraffine toric Calabi-Yau 3-orbifolds , arXiv:1310.4818.[17] E. Frenkel, A. Givental, T. Milanov,
Soliton equations, vertex operators, andsimple singularities . Funct. Anal. Other Math. 3 (2010), no. 1, 47–63.[18] A.B. Givental,
Gromov-Witten invariants and quantization of quadraticHamiltonians . Dedicated to the memory of I. G. Petrovskii on the occasionof his 100th anniversary. Mosc. Math. J. 1 (2001), no. 4, 551–568.[19] A. Givental, A n − singularities and n KdV hierarchies . Dedicated to VladimirI. Arnold on the occasion of his 65th birthday. Mosc. Math. J. 3 (2003), no.2, 475–505, 743.[20] A.B. Givental, T.E. Milanov, Simple singularities and integrable hierarchies .The breadth of symplectic and Poisson geometry, 173–201, Progr. Math., 232,Birkh¨user Boston, Boston, MA, 2005.[21] M. Jimbo, T. Miwa,
Solitons and infinite-dimensional Lie algebras . Publ. Res.Inst. Math. Sci. 19 (1983), no. 3, 943–1001.[22] V. G. Kac, J. W. van de Leur,
The n-component KP hierarchy and represen-tation theory . Integrability, topological solitons and beyond. J. Math. Phys.44 (2003), no. 8, 3245-3293.[23] M. Kontsevich,
Intersection theory on the moduli space of curves and thematrix Airy function . Comm. Math. Phys. (1992), no. 1, 1–23.[24] I.M. Krichever,
The dispersionless Lax equations and topological minimalmodels . Comm. Math. Phys. 143 (1992), no. 2, 415–429.
ROBENIUS MANIFOLDS, SPECTRAL CURVES, AND INTEGRABLE HIERARCHIES7 [25] I. M. Krichever,
The τ -function of the universal Whitham hierarchy, matrixmodels and topological field theories . Comm. Pure Appl. Math. 47 (1994), no.4, 437–475.[26] R. B. Laughlin, A different universe. Reinventing physics from the bottomdown. Basic Books, 2005.[27] S.-Q. Liu, Givental’s quantization method and integrable hierarchies , Lecturesat BICMR, April-May, 2012.[28] S.-Q. Liu, Y. Ruan, Y. Zhang,
BCFG Drinfeld-Sokolov hierarchies andFJRW-theory . Invent. Math. 201 (2015), no. 2, 711–772.[29] S.-Q. Liu, Y. Zhang,
Bihamiltonian cohomologies and integrable hierarchiesI: A special case . Comm. Math. Phys. 324 (2013), no. 3, 897–935.[30] M. Sato,
Soliton equations as dynamical systems on infinite dimensionalGrassmann manifolds ., Res. Inst. Math. Sci. Kokyuroku 439 (1981), 30–46.[31] B. Szablikowski, M. B laszak,
Dispersionful analog of the Whitham hierarchy .J. Math. Phys. 49 (2008), no. 8, 082701, 20 pp.[32] K. Takasaki, T. Takebe,
Universal Whitham hierarchy, dispersionless Hirotaequations and multicomponent KP hierarchy . Phys. D 235 (2007), no. 1-2,109–125.[33] E. Witten,
Two-dimensional gravity and intersection theory on moduli space ,Surveys in Differential Geometry, vol.1, (1991) 243–310.[34] J. Zhou,
Hodge integrals and integrable hierarchies . Lett. Math. Phys. 93(2010), no. 1, 55–71.[35] J. Zhou,
Quantum deformation theory of the Airy curve and mirror symmetryof a point , arXiv:1405.5296.[36] J. Zhou,
Emergent geometry and mirror symmetry of a point ,arXiv:1507.01679.[37] J. Zhou,
Emergent geometry of KP hierarchy , arXiv:1511.08257.[38] J. Zhou,
Emergent Geometry of KP Hierarchy. II , arXiv:1512.03196.[39] J. Zhou,
Emergent geometry of n -KP , in preparation.[40] J. Zhou, Emergent geometry from A to G , in preparation.
Department of Mathematical Sciences, Tsinghua University, Beijng,100084, China
E-mail address ::