# HOMOLOGICAL GEOMETRY AND MIRROR SYMMETRY

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1 HOMOLOGICAL GEOMETRY AND MIRROR SYMMETRY Alexander B. GIVENTAL Dept. of Math., UC Berkeley Berkeley CA 94720, USA 0. A popular example. A homogeneous degree 5 polynomial equation in 5 variables determines a quintic 3-fold in CP 4. Hodge numbers of a non-singular quintic are known to be: h p,p = 1, p = 0, 1, 2, 3 (Kahler form and its powers), h 3,0 = h 0,3 = 1 (a quintic happens to bear a holomorphic volume form), h 2,1 = h 1,2 = 101 = (it is the dimension of the space of all quintics modulo projective transformations, and h 2,1 is responsible here for infinitesimal variations of the complex structure) and all the other h p,q = 0. Consider the family of quintics x 1...x 5 = λ 1/5 (x x 5 5) invariant to 5 4 multiplications of the variables by 5-th roots of unity. The quotient by these symmetries will generate singularities. Resolve the singularities. The result is known to be a family Y λ of 3-folds with the table of Hodge numbers mirror-symmetric to that of the quintics X: h p,q (Y ) = h 3 p,q (X). Manifolds with mirror-symmetric Hodge tables are called geometrical mirrors. Discovered accidentally in a computer experiment, such mirror 3-folds very soon took their place in various string models of the 10-dimensional Universe. As it is clear now, so called Arnold s strange duality of exceptional singularities [1] was probably the first manifestation of mirror phenomena for K3-surfaces. Current interest to mirror manifolds is due to the so called mirror conjecture and its first applications to enumerative algebraic geometry. The idea is that along with the equality h 1,1 (X) = h 2,1 (Y ) of moduli numbers of Kahler structures on X and of complex structures on Y, whole symplectic topology on X is equivalent to complex geometry on Y, and vice versa. This idea have led to a number of beautiful predictions (see for instance [6, 5]) in enumerative algebraic geometry, in particular for numbers of rational curves of each degree on the quintics. Supported by Alfred P. Sloan Foundation and by NSF Grant DMS

2 1. Singularity theory. Given a complex manifold Y n, a holomorphic volume form ω and a holomorphic function f : Y C, one can study exponential integrals I = Γ ef(y)/ ω, their asymptotics at 0 and their dependence on parameters. Example. Let f be a weighted-homogeneous polynomial of deg f = 1 on n complex variables (y 1,..., y n ) of some positive weights deg y i = α i > 0 with an isolated critical point y = 0 of multiplicity µ, a 1,..., a µ = 1 monomials representing a basis in the local algebra H = C[y]/( f/ y), f λ = f + λ 1 a λ µ a µ a miniversal deformation of the critical point. The formal stationary phase approximation gives I i (λ) = e f λ(y) a i (y)dy 1... dy n n/2 e f λ(y )/ a i(y ) Jλ (y ) for each of µ critical points y (λ) of f λ, where J λ = det( 2 f λ / y 2 ). These asymptotics satisfy I j / λ i k ck ij (λ)i k where c k ij are structural constants of the algebra H λ = C[y]/( f λ / y) of functions on the critical set: a i a j = c k ija k in H λ. The cycles of integration can be described as real n-dimensional Morse-theoretic cycles of the function Ref and thus correspond to the critical points and represent classes in the asymptotical homology group H n (C n, Ref = ). Then the residue paring (a, b) = y a(y )b(y ) J(y ) = 1 a(y)b(y)dy 1... dy n (2πi) n f/ y =const f λ / y 1... f λ / y n becomes an asymptotical intersection pairing between the cohomology for f and f and is known to extend without singularities to λ = 0. Theorem[16] These stationary phase asymptotics can be made exact by a suitable choice of the volume forms ω λ instead of dy 1... dy n and in special coordinates λ on the parameter space, instead of (λ 1,..., λ µ ). In particular this means that the differential equations i I = [ai ] I with [a i ], I( λ) H λ form a family = d [a i ]d λ i of connections flat for all. They are identified with the Gauss Manin connections in the cohomological bundle. The residue pairing therefore literally coincides with the intersection pairing and induces (see [17]) on the parameter space a flat complex metric. The coordinates λ are defined as flat coordinates of this metric. In the contemporary language this theorem means that the integrals define on the parameter space the structure of a Frobenius manifold [7] and thus satisfy axioms of Topological Conformal Field Theory (TCFT) (Landau Ginzburg models of TCFT). One can at least try to extend this theorem based on deep properties of variations of Hodge structures to arbitrary families (Y, f, ω) λ. Consider a degenerate case where Y is a compact Kahler manifold. For this, Y should bear a holomorphic volume form (h n,0 = 1). Then f is necessarily constant, and the exponential integrals turn into 2

3 periods ω n,0 of the volume form. The periods depend on the complex structure on Y and satisfy some linear differential Picard Fuchs equations (describing variation of pure Hodge structures). The algebra H λ of the critical set Y should be replaced by its cohomology H n (Y ). It is a separate problem whether one can derive from these massless Landau Ginzburg data complete models of TCFT (they are called B-models, after E.Witten), but in many cases one can construct flat coordinates on moduli spaces of complex structures. The family of quintic-mirrors Y λ is one of them. 2. Symplectic topology. Let X n be a compact Kahler manifold. Given m cycles A 1,..., A m X, an integral homology class D H 2 (X) and a configuration (x 1,..., x m ) of m points on CP 1, one may ask: how many holomorphic maps ϕ : CP 1 X with ϕ [CP 1 ] = D and ϕ(x i ) A i are there? The answers (let us denote them F m,d [A]), being properly understood as intersection indices in certain moduli spaces of holomorphic maps, turn out to depend only on homology classes of A i and homotopy type of almost Kahler structure on X and provide symplectic invariants of X called Gromov-Witten invariants. They are not independent, and the universal identities for them can be interpreted as the associativity constraint of the quantum cohomology algebra H q (X) and compatibility of some linear PDEs. Pick an integral basis p 1,..., p k of symplectic classes in H 2 (X), denote (D 1,..., D k ) coordinates of D in the dual basis and put (a 1 a 2... a m ) = D qd F m,d [A 1,..., A m ] where A i are Poincare dual to cohomology classes a i. Theorem [14, 15]. Gromov - Witten invariants are well - defined at least if c 1 (X) 0, (a 1 a 2 ) coincides with the Poincare pairing (a 1, a 2 ) on H q (X) = H (X, C[[q]]), (a 1 a 2 a 3 ) are structural constants (a 1 a 2, a 3 ) of a skew - commutative associative multiplication on H q (X) which at q = 0 coincides with the usual cupproduct, and (a 1... a m ) = (a 1... a m, 1). Beside this, the differential equations q i q i I = pi I, i = 1,..., k, for a vector-function I(q) H (X) form a compatible system (i. e. a flat connection) for each. The -product is graded if one assigns usual degrees deg a j = codim A j to the cocycles and non-trivial degrees deg q i = 2d i to the parameters where d 1 p d k p k = c 1 (X). Actual definitions of Gromov Witten invariants involve non-integrable perturbations of the complex structure on CP 1 X, and rigorous computation of quantum cohomology is a non-trivial problem. The following examples, except for the first one, are rather reasonable conjectures than theorems. Examples. 1) H q (CP n 1 ) C[p, q]/(p n q), and the differential system is equivalent to the scalar equation n d n I/dt n = e t I where t = log q. The intersection pairing is given by the residue formula a(p)b(p)dp/(p n q) and similar formulas hold in all other examples below. 3

4 2) For the space F n of complete flags in C n, denote A n the n n matrix with u 1,..., u n on the diagonal, q 1,..., q n 1 right above, 1 s right under the diagonal, and zeroes otherwise. Put Σ i = tra i n. Then (see [11]) Hq (F n ) C[u, q]/(σ). In fact Σ i coincide with conservation laws of a Toda lattice on n particles with potentials q i = e t i t i+1 (see [11]). The question why the algebra Hq (F n ) is isomorphic to the algebra of functions on the singular invariant variety of the Toda lattice, is open. 3)Let X = C N //T k be a toric manifold obtained by the Marsden Weinstein reduction from the standard Hermitian space by a subtorus in the maximal torus T N. Denote (m ij ) the integral k N matrix of the natural projection Lie T N Lie T k. Then the quantum cohomology algebra of X is given by the generators (u 1,..., u N, p 1,..., p k, q 1,..., q k ) and relations u j = i m ijp i, q i = j um ij (V.Batyrev, see also [9] where a discrete version of quantum cohomology of toric manifolds had been computed as a by-product of a symplectic fixed point theorem). 4)Let X 3 be a non-singular quintic in CP 4. Its hyperplane section p generates in H (X) a subalgebra H even = C[p]/(p 4 ) with the intersection form (p i, 1) = 0 for i 3 and (p 3, 1) = 5. Its quantum deformation is almost the same except for p p = K(q)p 2 or, equivalently, (p p, p) = K(q) where K(q) = 5 + d=0 n dd 3 q d /(1 q d ) (see [2] ). Here n d is the number of degree-d rational curves in X: on a generic (almost)-kahler 3-fold with c 1 = 0 rational curves are discrete and all contribute to the quantum cupproduct (which now respects the usual grading i. e. deg q = 0). The corresponding differential system is equivalent to (I /K(q)) = 0 with = qd/dq. It is degenerate in the sense that it is independent on and easy to solve, except for the numbers n d with d > 3 are unknown! 1 In fact we have described only few of all Gromov - Witten invariants (see [18]), which form a complete set of correlation functions of a sigma-model, or A-model of TCFT and when computable, provide algebraic geometry with very non-trivial new enumerative information [13]. 3. The mirror conjecture. Mirror Conjecture predicts equivalence of A and B models of TCFT on an algebraic Calabi Yau manifold to B and A models on its geometrical mirror. In our down-to-earth language: for geometrical mirrors X and Y the differential system of Hq even (X) should coincide with the Picard Fuchs equation for Y taken in flat coordinates (and vice versa). Authors [6] of this formulation exploited it in order to predict numbers n d for quintics. They start with one of the periods I 1 = ω 3,0 λ = (5k)!λ k /(k!) 5, reconstruct the Picard Fuchs equation: D 4 I = 5λ(5D + 1)(5D + 2)(5D + 3)(5D + 4)I, where D = λd/dλ, bring it in a neighborhood of the singular point λ = 0 to the form (J /k(q)) = 0, conjecture that k(q) = K(q) and find from this n 1 = 2875, n 2 = , n 3 = (eventually in coherence with available data) and predict n 4 1 See however [12] 4

5 to be Bringing equation to the simple form involves: 1) finding the solution I 2 = log(λ)i 1 + Ĩ with Ĩ holomorphic and vanishing at λ = 0, 2) introducing the new local coordinate q = λexp(ĩ/i 1) and 3) computing the equation satisfied by J i = I i /I 1, i = 1, 2, 3, 4, as functions of q. Our previous discussion suggests the following generalization of the mirror problem: Is there a natural map (functor?) from (a class of) TCFT-models (symplectic sigma-models, or Frobenius manifolds, or quantum cohomology algebras) to generalized Landau - Ginzburg data? Simpler, how to solve the differential equations di = A I by means of exponential holomorphic integrals? We will partially answer this question for the class of toric manifolds. From such a point of view the Picard Fuchs equation for Y should have an intrinsic interpretation in terms of the problem of computing Gromov-Witten invariants for X. We will point out some. The (open) question why the above computational procedure (for an equation that had already been intrinsicly attributed to X) yields quantum cohomology of X, is probably related to the problem in what sense the above functor is an involution on its invariant subset that the class of algebraic Calabi-Yau manifolds seems to constitute. And all these problems lead to the same question: What is the intrinsic meaning of solutions of the quantum cohomology differential system? 4. A project: Equivariant Floer cohomology. Let LX be the space of contractible loops in a compact Kahler manifold X. It inherits the Kahler structures from X and additionally carries the action of S 1 by isometries (translations in the source). The Hamiltonian of this action is the action functional: to a contractible loop it assigns the symplectic area of a disk, contracting the loop, and can be multiplevalued. Doing Morse - Novikov theory for the action functionals H : LX R on the universal covering of LX, one comes to definition of Floer homology FH of X (isomorphic to H (X, C[q ±1 ])) [8]: gradient trajectories of H in LX are holomorphic cylinders in X. If one introduces multiplication in FH using the map LX LX LX of composition of loops (= holomorphic pants in X), it leads to the construction of quantum multiplication in Hq (X) (see [11]). Project: Construct S 1 -equivariant Floer cohomology FH S 1( LX). Let ω 1,..., ω k be Kahler forms LX corresponding to the integral basis of Kahler forms on X, let H 1,..., H k be corresponding Hamiltonians (of the same S 1 -action!) on the covering LX LX with the group of covering transformations Z k = H 2 (X, Z). The generators q 1,..., q k of the group Z k commute with the S 1 -action, preserve the forms ω i lifted to LX from LX, but transform H i : q i (H j ) = H j δ ij. 5

6 Denote C[ ] the coefficient algebra H (BS 1 ) = H (CP ) of the equivariant theory and introduce Duistermaat - Heckman equivariantly closed forms p i = ω i + H i see [3]). Proposition: p i q j q j p i = q i δ ij as operators on an equivariant Floer complex. Corollary: FH S 1( LX) should carry the module structure over the algebra D of differential operators generated by q i = e ti, and p j = / t j. A semi-classical limit 0 should give rise to the subalgebra in Hq (X) generated by the Kahler classes p i and q i. In particular relations between them should describe a lagrangian variety with respect to the Poisson bracket {p i, q j } = q i δ ij the characteristic variety of the D-module. 5. Realization: Toric manifolds. Holomorphic maps CP 1 X to a toric manifold X = C N //T k can be described as equivariant maps C 2 C N. This compactifies the map spaces up to toric manifolds X d = C N+D //T k. Here the homology class d of maps can be identified with an integral point in LieT k such that j = 1,..., N, D j = i m ijd i 0, and D = D j. We can interpret the map spaces as approximations of LX by algebraic loops (SC 1 CP 1 ), define FH S 1(X) as a certain limit (see [10]) of H S (X 1 d ) and using the explicit toric description of X d, compute the D-module. The algebra H = H (X) is generated by the integral Kahler classes P 1,..., P k (See [3]). Denote (, ) intersection pairing on H, Ω : H H the automorphism generated by P i P i. Introduce notations: U j = Σm ij P i, D j = Σm ij d i, t i = log q i, j = Σm ij / t i, r j = j( j )...( j (r 1) ) and for any M Z put M[x]! = M M m= (x + m)/ 0 m= (x + m). Theorem 1.[10] Suppose c 1 (X) > 0. Then 1. FH S 1 D/J where the left D-ideal J is generated by all operators r r N N q m l l N N with r j 0, l j 0 and r j l j = D j. 2. The kernel of this linear differential system is generated by the components of the following vector-function of t with values in the cohomology algebra H: f (t) = k N e Pt Σ d Z ke dt /D 1 [U 1 ]!...D N [U N ]!. 3. d edτ X d e p(t τ)/ = N k ( f (t), Ωf (τ)), where p i are Duistermaat - Heckman forms ω i + H i on each X d corresponding to our basis in Lie T k H 2 (X d ) and the S 1 -action. 4. Suitable limits to = 0 give rise to the algebra Hq (X) and a generating series for symplectic volumes of X d. Example. For k = 1, N = n and (m ij ) = (1,..., 1) we get X = CP n 1. Then P n = 0 and components of f = e Pt d=0 edt /[(P +1)...(P +d)] n nd give all n solutions of ( d/dt) n I = e t I, and the first one (P = 0) is q d /(d!) n nd. 6. Toric complete intersections. Given T k -invariant homogeneous polynomials on C N, one can plug components of a rational curve C 2 C N into them and 6

7 equate to zero identically. In the spaces X d the solutions form the zero locus of a PSL 2 (C)-invariant holomorphic section of a suitable equivariant vector bundle. If such sections were transverse to the zero section the loci would represent equivariant Euler classes of these bundles. One may hope to reconstruct the D-modules and quantum cohomology algebras of complete intersections X X from such classes, substituting them for fundamental cycles of the map spaces X d. For simplicity let us consider the case of Calabi Yau complete intersections in X = CP n 1. Let l 1,..., l r > 0 be Chern numbers of r line bundles with l l r = n. Introduce the algebra H = C[P]/(P n r ) with the intersection pairing (P n r 1, 1) = l 1...l r (the image of H (X) H (X )) and denote Ed l(p, ) the S1 -equivariant Euler class of that suitable vector bundle over X d. Theorem 2. [10] Σ d=0 edτ X d e p(t τ)/ Ed l(p, ) = ( 1)n 1 1+r n ( g l (t), Ω g l (τ)) where g l = e Pt d=0 edt (l 1 d)[p]!...(l r d)[p]!/(d[p]!) n. The n r components of g l in H provide a complete solution to the differential equation (D = d/dt): D n r I = l 1...l r e t (l 1 D + 1)...(l 1 D + l 1 1)...(l r D + 1)...(l r D + l r 1)I. This is exactly the equation that was found in [5] as the Picard Fuchs equation for mirrors of projective Calabi Yau complete intersections, satisfied by the hypergeometric series d l 1!...l r!q d /(d!) n. In particular we have obtained the equation D 4 I = 5e t (5D+1)...(5D+4)I entirely in topological terms of map spaces and not as a Picard Fuchs equation of a Landau Ginzburg model. Furthermore, its solution e P t q d (5P + 1)...(5P + 5d)/(P + 1) 5...(P + d) 5, rewritten as e Pt (G 1 + G 2 P + G 3 P 2 + G 4 P 3 ) = G 1 (q) + P(G 1 (q)log q + G 2 (q)) +..., yields I 1 as G 1 and Ĩ as G 2. Thus each coefficient of G 1,..., G 4 should hide some enumerative information about rational curves in CP 4 relative to (one or many) quintics: what appeared meaningless in the Picard-Fuchs equation due to accidental choice of the coordinate λ in the family of quintic-mirrors, turns out related directly to the exterior geometry of quintics in CP 4. Problem: Recover this information. 7. Integral representations. It is not surprizing that toric geometry provides integral formulas for some hypergeometric series. However they will illuminate possible nature of mirror manifolds. Definition. A function F : E C on the fibered space π : E B generates the lagrangian variety L = {(p, t) T B x π 1 (t) : df x = π (p)}. Lemma. Theorem 1 with k = N and m ij = δ ij formally gives P = 0 and f = e dt /d 1!...d N! N d = exp( e t j / ). d Z N + Theorem 3 [10]. Let X = C N //T k be a compact toric manifold with c 1 > 0. 7

8 Then 1. The quantum cohomology algebra Hq (X) is the algebra of functions on the lagrangian variety generated by F = u u N : C N C with π : u q given by q i = u m ij j. 2. Take the holomorphic volume form ω q on π 1 (q) equal ( j du j /u j )/( i dq i /q i ). Then integrals I(log q) = ω q e (u u N )/ Γ π 1 (q) over cycles Γ corresponding to dimh q (X) critical points of F π 1 (q) provide a complete set of solutions to the differential system of Theorem 1.1. Notice that dimπ 1 (q) = N k = dim C X. According to our formulation of the mirror problem we should call the Landau Ginzburg data (E B, F, ω) a family mirror-symmetric to the toric manifold X. Furthermore, put X q = F 1 (1) π 1 (q), ω q = ω q /d(f π 1 (q)). Theorem 4 [5, 10] All solutions of the differential equation of Theorem 2 with r = 1 are integrals ω q over compact cycles Γ X q. In order to obtain the same result for r > 1 one should split u u N into r sums of length l 1,..., l r and consider the sums as equations of a complete intersection X in the fibres of π. Theorem 4 matches well to remarkable Batyrev s construction [4] of geometrical mirror pairs of toric hypersurfaces: fibers π 1 (q) are the complex tori which, when suitably compactified into a toric variety, meet F 1 (1) along Batyrev s Calabi Yau hypersurface, and ω extends to its holomorphic volume form. Thus Batyrev s mirrors of toric hypersurfaces are hypersurfaces in the mirrors of their ambient toric manifolds. Example. Replace the homogeneous equation q 1/5 (x x 5 5) = x 1...x 5 with the affine equation in Z 4 5-invariant coordinates u i = q 1/5 x 5 i/x 1...x 5. Then u 1...u 5 = q, and the equation is u u 5 = 1. This corresponds to our matrix (m ij ) = (1, 1, 1, 1, 1). 8. Homological geometry. Along with the differential equations of Theorems 1,2 the above integral formulas have intrinsic cohomological meaning in toric geometry. In fact the description of H q (X) by N + k generators and relations is a q-deformation of the following similar description of H (X). A symplectic quotient C N //T k can be identified with the free quotient C N /T k C of some domain in CN, and H (C N //T k ) with the equivariant cohomology H T k ( C N ). One begins with H T N (C N ) = H (BT N ) = C[u], then computes H T N ( C N ) which causes factorisation by some multiplicative ideal, and finally reduces the group T N to T k which imposes the additive relations. Therefore our (u 1,..., u N ) are in fact characteristic classes of T N, and the function F = u j is the universal 1-st Chern class c 1 (of all toric manifolds quotients of C N ). 8

9 Our solutions to the quantum cohomology differential systems are given by integrals in SpecH (BT N ) over Morse theoretic cycles of the function Re(c 1 ). It suggests that in general mirror manifolds should live in some cohomologies of each other. The integral formulas can be recovered from our D-module approach: one should first compute equivariant Floer cohomology which are equivariant also with respect to the maximal torus T N /T k of Aut(X), and then get rid of this extra-structure. The first step adds variables, the second expresses f as a De Rham class in excessive variables. References [1] V.I. Arnold. Critical points of smooth functions. In: Proceedings of ICM 74, v. 1, Vancouver, B.C., 1974, [2] P. Aspinwall, D. Morrison. Topological field theory and rational curves. Preprint, Oxford, [3] M.Audin. The topology of torus actions on symplectic manifolds. Birkhauser, Basel, [4] V.V. Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. Preprint, Essen Univ., [5] V. Batyrev, D. van Straten. Generalized hypergeometric functions and rational curves on Calabi - Yau complete intersections in toric varieties. Preprint, Essen Univ., [6] P. Candelas, X.C. de la Ossa, P.S. Green, L. Parkes. A pair of Calabi - Yau manifolds as an exactly soluble superconformal field theory. Nucl. Phys. B359 (1991), [7] B. Dubrovin. Integrable systems and classification of 2-dimensional topological field theories. Preprint, hep-th [8] A. Floer. Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120 (1989), [9] A.Givental. A symplectic fixed point theorem for toric manifolds. Preprint. Berkeley, 1992 (to appear in Progr. in Math., Floer s memorial volume). [10] A.Givental. Homological geometry. I. Projective hypersurfaces. Preprint, II. Integral representations. In preparation. [11] A. Givental, B.Kim. Quantum cohomology of flag manifolds and Toda lattices. Preprint, hep-th (to appear in Commun. Math. Phys.). [12] M. Kontsevich. Enumeration of rational curves via torus actions. Preprint, [13] M. Kontsevich, Yu. Manin. Gromov-Witten invariants, quantum cohomology, and enumerative geometry. Preprint, Max-Plank-Institut, [14] D. McDuff, D. Salamon. J-holomorphic curves and quantum cohomology. AMS, Providence, R.I., [15] Y. Ruan, G. Tian. A mathematical theory of quantum cohomology. Preprint, [16] K. Saito. On the periods of primitive integrals I. RIMS 1982,

10 [17] A.N. Varchenko, A.B. Givental. Period mappings and intersection forms. Funct. Anal. Appl. 16:2 (1982), [18] E.Witten. Two-dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom. 1 (1991),

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