aa r X i v : . [ m a t h . AG ] A ug FROM PRIMITIVE FORM TO MIRROR SYMMETRY
KYOJI SAITOA
BSTRACT . This is a report on the recent joint work [29] on LG-LG mirror symmetry forthe 14 exceptional unimodular singularities.
1. I
NTRODUCTION
In the early 80’s, the author introduced the theory of primitive forms [38, 40–42], whichstudies the period integrals of a primitive form over cycles which are vanishing to iso-lated critical points of a function. One important consequence of the theory is that a prim-itive form induces a flat structure [42] on the deformation parameter space of the function,including the flat metric , the flat coordinate system and the potential function (which is latercalled the prepotential). Later on, in the early 90’s, Dubrovin [12] studied the 2D topolog-ical field theory of genus zero curves and found the same structure, which is axiomatizedto, so-called, the
Frobenius manifolds structures.
Lots of Frobenius manifolds structuresare found. They include the examples constructed on the orbit spaces of Coxeter groups[39],[45],[13],[48] (which contain the first cases found before the primitive form theory),the one constructed by primitive forms [42],[10],[11],[23],[50], the quantum cohomologyrings, Barannikov-Kontsevich construction [4] using polyvector fields of a Calabi-Yaumanifold, and the FJRW-thoery (the A-model of quantum singularity theory) [14, 15].The Gromov-Witten theory [25] counts pseudoholomorphic curves in a given symplec-tic manifold. Its application to the symplectic structure on a K¨ahler manifold was exten-sively studied from a view point of the mirror symmetry. Here the mirror symmetry isone of the dualities in physics and had a strong impact on mathematics [21, 22]. Namely,it asserts that certain data counted from symplectic geometry (the A-model side) shouldbe equivalent to that from the complex structure of the mirror manifold (the B-model side).There are several different levels of formulation of the mirror symmetry such as the cate-gorical level [16, 24], the geometric level [49], or the equivalence of the genus zero theoryon the A-model side with the variation of Hodge structures on the B-model side [19, 30]. Present note is worked out with the help of the coauthors Changzheng Li, Si Li and Yefeng Shen, towhom the author expresses his deep gratitudes. rimitive forms are about universal deformations F of functions, giving flat structureson the deformation spaces. Hence, the theory is relevant in the complex geometric (B-model) aspects of N=(2,2) supersymmetric Landau-Ginzburg (LG) theory with the su-perpotential F . However, this pattern of the mirror symmetry was not mathematicallyrigorously worked out until recently. This is because of (1) lack of mathematical theoryof A-model LG-theory at that early time, and (2) the difficulty of calculating primitiveforms until recently, where explicit expressions of primitive forms were known only forweighted homogeneous polynomials of central charge less than or equal to 1. Both diffi-culties were resolved as follows.In around 2007, Fan, Jarvis and Ruan constructed a so-called quantum singularity theory by counting virtual cycles associated with a weighted homogeneous polynomial, whosepotential (generating series) gives again a Frobenius manifold structure on the so-called FJRW state space [14, 15]. This is considered as an A-model Landau-Ginzburg theory.They immediately realized that such Frobenius manifold for an ADE-polynomial W isactually isomorphic to the Frobenius manifold arising from the primitive form (i.e. B-side) of another ADE-polynomial W T . The superpotential polynomial W T on B-side isobtained by the transposition of exponents of monomials in the polynomial W on A-side[2, 3, 26], which is later called Berglund-H ¨ubsch-Krawitz mirror . As an application of thismirror theorem, they solved the so-called generalized Witten conjecture , which says thatthe generating functions arising from the Landau-Ginzburg model for ADE-singularitiesshould be governed by some ADE-integrable hierarchies. Also a similar observation forsimply elliptic singularities [37] was achieved [27, 32, 33]. We remark that the relationshipbetween FJRW theory and Gromov-Witten theory (both are in A-model side) is studiedunder the name of LG/CY-correspondence, for which one is referred to [6–8, 27, 32, 35].On the other hand, in 2013, jointly with Changzheng Li and Si Li, the author came toa new perturbative construction of primitive forms [28], where Birkhoff decompositiontheorem used in the original formulation [42] was replaced by the asymptotic expansionof oscillatory integrals. This enables us to calculate primitive forms explicitly as a powerseries in an algorithmic way (at least for weighted homogeneous polynomials). Withthe perturbative approach, we can calculate further the flat coordinate system and thepre-potential function up to any finite order. This will be sufficient to determine the flatcoordinate system and the pre-potential function with a help of WDVV-equations.These two new developments thoroughly changed the view on the LG-LG mirror sym-metry. Namely, up to a choice of primitive forms, one asks whether the pre-potential at-tached to FJRW theory for a weighted homogeneous polynomial could coincide with theprepotential associated to a primitive form for the mirror dual-polynomial. Such a mirror ymmetry is called the Landau-Ginzburg to Landau-Ginzburg (LG-LG) mirror symmetry .Its study has developped rapidly in the last years. In the present note, we briefly intro-duce the theories on both sides and the mirror map construction connecting them. Then,we confirm the LG-LG mirror symmetry for the 14 unimodular singularities, which arethe first case of weighted homogeneous polynomials whose central charge exceeds 1. Remark.
We remark that the primitive form theory depends only on the analytic equiv-alence class of the singularity of the function W T , although associated primitive formsmay not be unique but form a family. On the other hand, FJRW theory depends on thepolynomial W itself together with a symmetry group of W . Hence, to achieve the mirrorsymmetry to FJRW theory on W , both the analytic equivalence class of W T and the choiceof the primitive form for W T depend on the choice of the polynomial W . We do not yetunderstand this phenomenon conceptually (c.f. [29] Remark 4.9. (2)).2. P
RIMITIVE FORM THEORY .The origin of a Landau-Ginzburg B-model (with respect to trivial group symmetry) atgenus zero is the theory of primitive forms [28, 38, 40–42, 44]. The starting data of thetheory is a holomorphic function f : ( X , ) → ( C , 0 ) defined on a Stein domain X ⊂ C n with finite critical points. For our purpose on the LG-LG mirror symmetry, it is sufficientto consider a weighted homogeneous polynomial f = f ( x , · · · , x n ) with an isolatedcritical point at the origin ∈ X = C n , f ( λ q x , · · · , λ q n x n ) = λ f ( x , · · · , x n ) , ∀ λ ∈ C ∗ ,Here ( q , · · · , q n ) in Q n > are called the weights of the coordinates ( x , · · · , x n ) , and eachweight 0 < q i ≤ is unique [36]. In [41], the author introduced the formal completion ofthe Brieskorn lattice together with a semi-infinite z -adic filtration by a formal variable z :ˆ H ( ) f : = Ω nX , [[ z ]] / ( d f + zd ) Ω n − X , [[ z ]] ,and constructed a higher residue pairingK f : ˆ H ( ) f ⊗ ˆ H ( ) f → z n C [[ z ]] which satisfies a number of properties, and plays a key role in the theory of primitiveforms. A universal unfolding of f is given by F : ( X × S , × ) → ( C , 0 ) , F ( x , s ) = f ( x ) + µ (cid:229) α = s α φ α , here { φ , · · · , φ µ } ⊂ C [ x ] are weighted homogeneous polynomials representing anadditive basis of the Jacobian algebra Jac ( f ) , and s = { s α } α = ··· , µ parametrizes the de-formation space S ⊂ C µ . Using, so called, Kodaira-Spencer map: (cid:229) i a i ¶ s i (cid:229) i a i φ i , thetangent bundle of S is identified with the Jacobi ring of F , which gives a ring structure(Frobenius algebra structure) and a natural inner product J (the first residue pairing) onthe tangent bundle of S . There is a family version ˆ H ( ) F (resp. K F ) of ˆ H ( ) f (resp. K f ) withrespect to the universal unfolding F . We remark that in the recent work [28] by C. Li, S.Li and the author, an alternate complex differential geometric approach to the moduleˆ H ( ) F is developed. Therein we give a simple construction of the higher residue pairingby using integration of compactly supported polyvector fields.A primitive form is a section ζ ∈ Γ ( S , ˆ H ( ) F ) , represented by a relative holomorphicvolume form ζ = P ( x , s ) d n x ( d n x = dx · · · dx n ) on X × S , satisfying the properties of primitivity, orthogonality, holonomicity, and homogeneity , described by bilinear equations on ζ using the higher residue pairing K F together with Gauss-Manin connection on ˆ H ( ) F .Roughly speaking, the submodule of ˆ H ( ) F consisting of the covariant differentiations ofa primitive form by the tangent vectors of S forms a splitting factor to the adic filtration onˆ H ( ) F defined by (the powers of) z , i.e. ˆ H ( ) F ≃ T S ⊕ z · ˆ H ( ) F . In this way, properties of theprimitive form are transferred to the splitting factor i.e. to the tangent bundle of S , and,hence, the space S obtains a differential geometric structure, called the flat structure (= the Frobenius manifold structure ) associated with ζ . For instance, the orthogonality propertyof ζ gives a flat metric J (i.e. the first residue pairing) on S . Then the flat section of theLevi-Civita connection of that metric defines the flat coordinate system (see e.g. [44] formore details).For weighted homogeneous polynomials, { φ α d n x } α ⊂ ˆ H ( ) f is called a good basis ifthe vector subspace B = Span C { φ α d n x } α satisfies K f ( B , B ) ⊂ C z n , where we note thatthe space B is isomorphic to the Jacobi algebra Jac ( f ) as C -vector spaces. One key step toconstruct a primitive form is that the concept of the primitive forms is equivalent to the notionof good section [42] (cf. [46]). In order to show this, we need to extend a good basis in ˆ H ( ) f to a ”deformed good basis” in the deformed module ˆ H ( ) F , where, in the proof, we use aclassical analytic result known as Birkhoff decomposition theorem . In [28], we replaced therole of the Birkoff theorem by a multiplication of the ”holomorphic part of the oscillatoryintegral factor” e F − fz : B → B (( z ))[[ s ]] (here the first copy of B should be read off asubspace of the deformation ˆ H ( ) F of ˆ H ( ) f ), which is able to calculate in power series inthe local coordinate perturbatively. Inspired from this, we obtain the following, which isa combination of several propositions in section 3.2 of [29]. roposition 2.1. Given a good basis { [ φ α d n x ] } µα = ⊂ ˆ H ( ) f , there exists a unique pair ( ζ , J ) satisfying the following: ( ) ζ ∈ B [[ z ]][[ s ]] , ( ) J ∈ [ d n x ] + z − B [ z − ][[ s ]] ⊂ ˆ H f [[ s ]] , and ( ⋆ ) e ( F − f ) / z ζ = J . Moreover, we embed z − C [ z − ][[ s ]] ֒ → z − C [[ z − ]][[ s ]] and decompose J = [ d n x ] + − ∞ (cid:229) m = − z m J m , where J m = (cid:229) α J α m [ φ α d n x ] , J α m ∈ C [[ s ]] . Then ζ gives a formal primitive form, and {J α − } give a formal Frobenius manifold structure onS with flat coordinates {J α − } α . In particular, both ζ and J can be computed recursively by analgebraic algorithm via the above formula. Explicitly, let us denote by J ( · , · ) and ∗ the flat metric (the first residue pairing) and theproduct structure on the tangent bundle of S , respectively. For simplicity, let us denoteby t , · · · , t µ the flat coordinate system on S and by ¶ t , · · · , ¶ t µ their partial derivatives.Then, as a consequence of the flat structure, the following 3-tensor A ( ¶ t i , ¶ t j , ¶ t k ) : = J (cid:0) ¶ t i ∗ ¶ t j , ¶ t k (cid:1) = J (cid:0) ¶ t i , ¶ t j ∗ ¶ t k (cid:1) ∈ Γ ( S , O S ) ≤ i , j , k ≤ µ is symmetric in the three variables, and satisfies the following integrability conditions ¶ t l A ( ¶ t i , ¶ t j , ¶ t k ) = ¶ t i A ( ¶ t l , ¶ t j , ¶ t k ) for all 1 ≤ i , j , k , l ≤ µ .Therefore, there exists a function (formal power series in the flat coordinates) F SG0, f on S ,called the prepotential , such that ¶ t i ¶ t j ¶ t k F SG0, f = A ( ¶ t i , ¶ t j , ¶ t k ) = J (cid:0) ¶ t i ∗ ¶ t j , ¶ t k (cid:1) (where the quadratic terms are normalized to be 0).We are enabled to compute the prepotential F SG0, f of the associated formal Frobenius mani-fold structure in a perturbative way, for an arbitrary weighed homogeneous singularity.On the other hand, it is shown in [28] that the formal power series ζ is in fact the Taylorseries expansion of the associated (analytic) primitive form around the origin ∈ S . Thisexplains the geometric origin of the induced (formal) Frobenius manifold structure in theabove proposition together with the analyticity of its prepotental F SG0, f .Let us restrict our attention to the case of exceptional unimodular singularities now.Originally, there are 14 exceptional unimodular singularities by Arnold [1], which are oneparameter families of singularities with three variables. Each family contains a weightedhomogenous singularity characterized by the existence of only one negative degree butno zero-degree deformation parameter [43]. Hence in the present note, by exceptional nimodular singularities, we mean the weighted homogeneous polynomials in these oneparameter families, which are given in Table 1.T ABLE
1. Exceptional unimodular singularitiesPolynomial Polynomial Polynomial Polynomial E x + y W x + y U x + y + z Q x y + xy + z Z x y + y x S x y + y z + z xE x + xy + z E x + xy Z x + xy + yz W x + xy + yz Q x y + y + z Z x y + y Q x y + y z + z S x y + y z + z There is a partial classification [43] of weighted homogeneous polynomial with iso-lated singularity by using the central chargeˆ c f : = n (cid:229) i = ( − q i ) .The case ˆ c f ≤ c f <
1, or simple elliptic singular-ities if ˆ c f =
1. The first examples of ˆ c f > ABLE E E E Z Z Z W W Q Q Q S S U ˆ c f For the 14 singularities f , the good basis is already known to be unique [20, 28, 47],and is simply given by a basis of Jacobian algebra Jac ( f ) . Hence, the primitive formis unique (up to a nonzero scalar). By Proposition 2.1, we can obtain the data on LGB-model at genus zero in a perturbative way, and in particular we can calculate thefour-point function F ( ) (that is, the degree 4 terms of the prepotential F SG0, f with re-spect to the flat coordinate system) of the Frobenius manifold structure associated tothe primitive form. For instance for U -singularity, f = x + y + z , we let { φ i } i = { z , x , y , z , xz , yz , xy , xz , yz , xyz , xyz } . By direct calculations, we obtain the four-point function in flat coordinates ( t , · · · , t ) with respect to the primitive form ζ = xdydz + O ( s ) , −F ( ) = t t t + t t t + t t t + t t t t + t t t + t t t t + t t t + t t t + t t t + t t t + t t t + t t + t t + t t t Here we make boxes for the last three monomials, which will be compared with the dataon the LG A -side, studied in the next section.3. M IRROR CONSTRUCTION AND THE F AN -J ARVIS -R UAN -W ITTEN THEORY
For the mirror symmetry purpose, we restrict our singularity into an invertible poly-nomial , where the number of variables is the same as the number of monomials in thepolynomial. We consider a pair ( W , G ) , where W is an invertible polynomial with n variables x , · · · , x n and has no monomials of the form x i x j for i = j . By rescaling thevariables, we can always write this polynomial by W = n (cid:229) i = n (cid:213) j = x a ij j .The matrix E W : = ( a i j ) n × n of exponents is called the exponent matrix of W . Let us useAut ( W ) to denote the group of diagonal symmetries of W ,Aut ( W ) : = { diag ( λ , · · · , λ n ) | W ( λ x , · · · , λ n x N ) = W ( x , · · · , x n ) , λ i ∈ C ∗ } .Then G is a subgroup in Aut ( W ) containing J W : = diag (cid:16) exp ( π √− q ) , · · · , exp ( π √− q n ) (cid:17) ,with q , · · · , q n are the weights of variables in W . Berglund and H ¨ubsch constructed amirror polynomial W T [2] by taking W T = n (cid:229) i = n (cid:213) j = x a ji j where E W T is the transpose matrix of E W . In general, the LG-LG mirror symmetry relatesthe pair ( W , G ) to a mirror pair ( W T , G T ) , where G T is constructed by [3, 26].In particular, if G = Aut ( W ) , then G T is the group with only an identity element. ALG-LG mirror symmetry conjecture can be formulated as the equivalence of Frobeniusmanifold structure associated with the primitive form theory of W T and that associatedwith the genus-0 Fan-Jarvis-Ruan-Witten theory (FJRW) theory of ( W , G = Aut ( W )) .The FJRW theory is introduced by Fan, Jarvis and Ruan in a series of papers [14, 15],based on a proposal of Witten [52]. The theory works for the pair ( W , G ) in general,where W is a weighted homogenous polynomial which has an isolated critical point at he origin and G is a subgroup in Aut ( W ) . The theory also requires that G contains J W .For technical reasons, W does not contain any monomial term xy . In the present note, wewill focus only on the case G = Aut ( W ) .For a pair ( W , Aut ( W )) , there is an FJRW state space H W which collects all Aut ( W ) -invariant part of middle dimensional Lefschetz thimble on the fixed locus of each groupelement γ in Aut ( W ) , H W : = M γ ∈ Aut ( W ) H mid ( Fix ( γ ) ; W ∞ γ ; C ) Aut ( W ) .Here W ∞ γ is the preimage of [ M , ∞ ) , for M ≫
0, under the real part of W restricted onthe fixed locus Fix ( γ ) .Fan, Jarvis and Ruan [14, 15] studied the space of solutions of Witten equations for W ¶ u i ¶ z + ¶ i W ( u , · · · , u n ) = i = · · · , n where z is a local coordinate of the curve in consideration (but not the formal variable inprimitive form theory) and u i (1 ≤ i ≤ n ) is a section of a line bundle L i with suitable de-grees over the curve (for algebraic construction, see [5, 34]), and constructed a cohomolog-ical field theory (in the sense of Kontsevich-Manin [25]) { Λ Wg , k : ( H W ) ⊗ k → H ∗ ( M g , k , C ) } on moduli space of stable curves M g , k . As a consequence, this gives the FJRW invariants(3.1) h α ψ ℓ , . . . , α k ψ ℓ k k i Wg , k = Z M g , k Λ Wg , k ( α , . . . , α k ) k (cid:213) j = ψ ℓ j j , α j ∈ H W .Here ψ j is the j -th psi-class on M g , k . The genus-0 invariants without ψ -class involvedgive a formal Frobenius manifold structure on H W . The prepotential of this formal Frobe-nius manifold is F FJRW0, W = (cid:229) k ≥ k ! h t , . . . , t i W k , t = µ (cid:229) j = t α j α j .It is a formal power series of t α j , j = · · · , µ . More generally, the FJRW total ancestorpotential A FJRW W is defined to be A FJRW W = exp (cid:229) g ≥ ¯ h g − (cid:229) k ≥ k ! h t ( ψ ) + ψ , . . . , t ( ψ k ) + ψ k i Wg , k ! .Here t ( z ) = (cid:229) m ≥ (cid:229) µ j = t m , α j α j z m .4. M IRROR SYMMETRY FOR EXCEPTIONAL UNIMODULAR SINGULARITIES
In [29], the following isomorphism between two types of Frobenius manifolds is proven. heorem 4.1. Let W T be one of the 14 exceptional unimodular singularities in Table 1. Thereexists a mirror map Ψ : Jac ( W T ) ∼ = H W , which induces an equality (4.1) F SG0, W T = F FJRW0, W .The mirror map Ψ : Jac ( W T ) → H W is constructed by Krawitz [26] and proven thatit is a ring isomorphism under a technical condition that W (in the FJRW side) is notallowed to be a chain type polynomial with one weight 1 /
2. For exceptional unimodularsingularities, this condition excludes two examples, W T = x y + y z + z ( Q ) , x y + y z + z ( S ) . However, in [29], the technical condition is removed by using the Getzler’srelation in M [17].The proof of Theorem 4.1 mainly uses the axioms of cohomological field theories, inparticular, the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Combined withthe special properties on the weights of the exceptional unimodular singularities, it wasproved in [29] that both F SG0, W T and F FJRW0, W are determined by the underlying ring isomor-phism and a few initial invariants h· · ·i . The invariants on the primitive form theoryside can be calculated by the perturbative formula. On the other hand, again by someWDVV equations, the invariants on the FJRW side can be reduced to invariants whichcan be calculated by the so-called orbifold-Grothendieck-Riemann-Roch formula. Underthe mirror map which identifies the deformation parameter space in primitive form sideto the FJRW state space together with the ring structure and the inner product, the in-variants on both sides are identified up to a scale −
1. Then, by rescaling mirror mapappropriately, we obtain the desired equality (4.1).This equality of the pre-potentials in genus 0 is lifted to the equality of higher genuspotentials as follows. For the generic point s ∈ S in the universal unfolding F of W T ,the F ( x , s ) is a Morse function in x so that its Jacobian ring is a direct sum of the onedimensional algebra C . That is, after the Kodaira-Spencer map identification, the Frobe-nius algebra structure on the tangent space of S at s is semi-simple . Such a generic pointis called semisimple . Givental defined a total ancestor potential (or a higher genus formula)[18] using only the genus zero data near the generic point together with the knowledgeof the Witten-Kontsevich tau-function. The later is just also called the total ancestor poten-tial of the Gromov-Witten theory with the target being a point. Teleman [51] proved thatthis higher genus formula in a cohomological field theory is uniquely determined by theunderlying Frobenius manifold at the semisimple point. The origin in the universal un-folding space S is not semisimple, however, Givental’s formula can be uniquely extendedto A SG W T at the origin by Milanov [31] (see also Coates-Iritani [9]). The uniqueness of the xtension will upgrade Theorem 4.1 to an identity of higher genus potential function: A SG W T = A FJRW W .This completes a proof of LG-LG mirror symmetry. Acknowledgement : The author is partially supported by JSPS Grant-in-Aid for ScientificResearch (A) No. 25247004. R
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