THE QUANTUM CONNECTION


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1 THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 GromovWitten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X, ) denote the moduli space (stack) of genus 0 npointed stable maps µ : C X with µ ([C]) = Recall that stability means that every contracted irreducible component has at least 3 special points, where a special point is either a marked point or a point of intersection with the rest of the curve The stack M 0,n (X, ) is known to be proper by an analogue of DeligneMumford stable reduction for maps Assume that X is convex, ie for all µ : P X, the obstruction space Obs(µ) := H (C, µ T X ) vanishes Then M 0,n (X, ) is a smooth and connected stack of dimension () c (T X ) + dim X + n 3 In particular, for X convex, M 0,n (X, ) has an ordinary fundamental class [M 0,n (X, )] in this homology dimension, and one can do intersection theory Examples of convex varieties include homogeneous varieties G/P, where G is a reductive algebraic group and P a parabolic subgroup Let ev i : M 0,n (X, ) X, i =,, n, be the ith evaluation morphism (C, p,, p n, µ) µ(p i ) Let γ,, γ n H (X, C) Then the npoint genus 0 GromovWitten invariant of X of class is defined as γ,, γ n := ev (γ ) ev n(γ n ) [M 0,n (X,)] In this setting, the GromovWitten invariants have an enumerative interpretation Assume that the γ i are homogeneous classes, that Γ i X are subvarieties in general position of class γ i, and that codim Γ i = dim M 0,n (X, ), so that the above integral is potentially nonzero Then γ,, γ n is the number of npointed maps µ : P X of class with µ(p i ) Γ i We use the notation γ k to denote γ,, γ where the class γ appears k times Example Let [pt] be the class of a point in P 2 Then [pt] 5 2 = since there is a unique conic through 5 points in general position in P 2
2 We will need the divisor equation, which describes GromovWitten invariants for which one of the classes is a divisor class Let γ,, γ n H (X, C) and γ H 2 (X) The divisor equation is ( ) (2) γ,, γ n, γ = γ γ,, γ n This is immediate from the enumerative interpretation above, since γ is the number of choices for the image of p n+ 2 Small quantum cohomology Let X be as in the previous section The Kähler cone of X, denoted K(X), is defined as the intersection of H, (X) with the cone in H 2 (X, R) generated by ample divisor classes in H 2 (X, Z) Define the complexified Kähler cone K C (X) = {ω H 2 (X, C) : Re(ω) K(X)} and the Kähler moduli space K C (X) = K C (X)/H 2 (X, 2πiZ) For simplicity, assume that h 2,0 (X) = 0, so that H, (X) = H 2 (X), and assume that X has no odddimensional cohomology Then we have a map H 2 (X, C) H 2 (X, C)/H 2 (X, 2πiZ) = H 2 (X, C ) defined in coordinates by t i q i = e t i, which sends K C (X) to K C (X) Example 2 For X = P n, the ample divisor classes in H, (X, Z) = H 2 (X, Z) = Z are the positive integers, so K C (X) = {z : Re z > 0} and is mapped by the exponential to K C (X) = D, the punctured unit disc We can now define the (small) quantum product ω on H (XC) for each ω H 2 (X, C) and study its convergence Let = T 0, T,, T m H (X, C) be a homogeneous basis, and let T,, T p denote the classes in H 2 (X, C) Let T 0,, T m H (X, C) be the dual basis defined by (T i, T j ) = δ ij, where (, ) is the intersection pairing Let ω = t i T i be a class in H 2 (X, C) Then we define the quantum product with respect to ω by T i ω T j = T i T j + e ω T i, T j, T k T k >0 k Note that this expression exists as an element of H (X, C) only if the infinite sum in each coefficient converges For X Fano, each coefficient is in fact a finite sum Indeed, in order for the GromovWitten T i, T j, T k to be nonzero, one must have an equality of dimensions (see ()) i + j + k = c (T X ) + dim X; since the LHS is bounded above and c (T X ) is ample, this equality can hold for only finitely many 2
3 In general, the sum is infinite and may converge for ω K C (X), since then ω > 0 and thus the coefficients e ω decrease as increases As ω + in K C (X), the coefficients approach 0, so convergence becomes more likely For X CalabiYau, the sum is conjecturally convergent for ω sufficiently large We will assume X is Fano from now on, so that the quantum product is welldefined for all ω H 2 (X, C) The quantum product is supercommutative by definition As described in Barbara s lecture, linear equivalences between certain boundary divisors on M 0,n (X, ) (obtained by pulling back the linear equivalences of the 3 boundary divisors on M 0,4 ) yield that the quantum product is associative 2 The quantum connection Let X be Fano We have a welldefined quantum product ω for each ω H 2 (X, C) Each quantum product is a generating function for the same Gromov Witten invariants (namely the 3point ones), but with different weights depending on ω We would like to study all of these quantum products simultaneously The idea to do so is very simple: let H be the trivial vector bundle on H 2 (X, C) with fiber H (X, C) Let i = / t i Vect(H 2 (X, C)), i =,, p Then we can view T i as a constant section of the trivial bundle H We will define a connection on H such that i (T j ) = T i T j, where T i T j denotes the section of H whose value at the fiber over ω is T i ω T j Definition 2 The quantum connection of X is the connection on H defined by i = t i T i This defines a connection on the trivial bundle H = O m+ over A p Remarks 22 () By definition i (T j ) = T i T j (2) More invariantly, = d p i= A idt i, where A i Mat m+ (C) is the matrix whose jth column is the coefficients of T i T j (3) The function e ω on H 2 (X, C) descends to the quotient H 2 (X, C)/H 2 (X, 2πiZ) = H 2 (X, C ) Thus, so do the quantum product and the quantum connection Example 23 Let X = P m, and set T i = H i H (X, C), where H is the hyperplane class As was shown in Barbara s lecture, T T 0 = T T T m = T m T T m = e t T 0 So is the connection on the trivial bundle H = O m+ over A defined by = d A dt, 3
4 where A = 0 e t Mat m+ (C) Proposition 24 is flat Proof The curvature of a connection on a locally free sheaf E over a space X is the map 2 : E E O Ω 2 X defined, for any v, w Vect(X), by 2 v,w = v w w v [v,w] We have [ i, j ] = 0 since partial derivatives on a vector space commute Using the definition of, for i, j p we have: 2 i, j (T k ) = i j (T k ) j i (T k ) = i (T j T k ) + j (T i T k ) = t i (T j T k ) + t j (T i T k ) + T i (T j T k ) T j (T i T k ) The sum of the last two terms is 0 by associativity and supercommutativity of As for the first two terms, note that by the divisor equation (2) we have: T j T k = T j T k + ( ) e t T e tp Tp T j T k, T l >0 l Hence t i (T j T k ) = T j T k >0 l ( e t T e tp Tp T i ) ( T j ) T k, T l This expression is symmetric in i and j, so the sum of the first two terms is 0 as well We next view as a connection on H 2 (X, C ) = (C ) p using Remark 22(3), and compute its monodromy around 0 in each factor We will need the following general lemma (see eg [], III, 4) Lemma 25 Let d + Λ dx x 4
5 be a connection on O n over a disk of radius ɛ, where Λ Mat n (O) Suppose Res x=0 ( ) := Λ(0) is a nonresonant matrix, ie no two eigenvalues differ by a nonzero integer Then the monodromy matrix M is conjugate to e 2πiΛ(0) Example 26 For λ C, consider the connection d + λ dx x on O over A This has a flat section x λ with monodromy e 2πiλ Proposition 27 The monodromy of around 0 in the jth factor of H 2 (X, C ) = (C ) p is conjugate to e 2πiT j Proof Set t i constant for i j Let q j = e t j, so that dt j = d log(q j ) = dq j /q j So we can write = d A j dt j = d + A j dq j q j The kth column of A j is the coefficients of T j T k = T j T k + e t j T j, >0 where e t j T j = q T j j The monodromy is independent of the choice of neighborhood of 0, so we may assume that ω lies in the punctured neighborhood K C (X) of 0 We may additionally normalize T,, T p to lie in H 2 (X, Z) Then T j is a positive integer since is effective and T j is Kähler So q T j j is holomorphic in q j, and A j (0) = T j Since T j raises degree, A j (0) is nilpotent, hence nonresonant The preceding lemma thus implies that the monodromy is conjugate to e 2πiT j Example 28 For X = P, set q = e t The quantum connection is ( ) 0 e t = d dt ( ) 0 q dq = d + q Then and Res q=0 ( ) = M( ) ( 0 0 ( 2πi ) = H ) 5
6 We would now like to construct flat sections of These sections will be written in terms of a generalization of GromovWitten invariants known as descendant Gromov Witten invariants, or gravitational descendants 3 Psi classes 3 Descendant GromovWitten invariants Definition 3 For i n, let L i be the line bundle on M 0,n (X, ) whose fiber at [(C, p,, p n, µ)] is the cotangent line T p i (C) Let ψ i = c (L i ) Formally, let π : C M 0,n (X, ) be the universal curve with marked sections s,, s n Then L i = s i (T π ), where T π is the relative cotangent sheaf of π For A B = {,, n} and + 2 =, let D(A, B, 2 ) denote the closure of the set of maps (P, p i A) (P, p j B) X of class on the first component and 2 on the second This is a boundary divisor on M 0,n (X, ) For X a point, we denote the resulting boundary divisor on M 0,n by D(A B) Also recall that there exist forgetful morphisms M 0,n+ (X, ) M 0,n (X, ) and M 0,n (X, ) M 0,n defined by forgetting the last marked point (respectively, the map), then contracting unstable irreducible components Example 32 () We have M 0, (P, ) P [(C, p, µ)] µ(p) Under this isomorphism L = T P = O( 2), and ψ = 2H (2) We have M 0,4 = P obtained by fixing p, p 2, p 3 to 0,, and allowing p 4 to vary The 3 boundary components of M 0,4 arise from bubbling when p 4 collides with p, p 2, p 3 The cotangent line to p is only affected when p 4 collides with p It easily follows that ψ = D({, 4} {2, 3}) A second method to compute ψ is to consider the forgetful morphism ν : M 0,4 M 0,3 = pt which forgets p 4 Examining the number of marked points on irreducible components immediately yields the following facts If p 2 or p 3 lie on the same irreducible component C 0 as p, then C 0 is not contracted by ν Otherwise, C 0 is contracted It follows that ψ is the boundary divisor separating p from p 2, p 3, which again gives ψ = D({, 4} {2, 3}) The second method used in Example 32(2) can be generalized to prove the following: 6
7 Lemma 33 On M 0,n (X, ) with n 3, ψ = D({} A, {2, 3} B, 2 ) A B={4,,n} + 2 = 32 Descendant invariants Definition 34 For γ i H (X) and a i nonnegative integers, i =,, n, define an npoint descendant GromovWitten invariant by τ a (γ ),, τ n (γ n ) := ev (γ ) ψ a ev n(γ n ) ψn an [M 0,n (X,)] Note that τ 0 (γ ),, τ 0 (γ n ) = γ,, γ n Example 35 By Example 32(), τ () P = 2 The divisor equation (2) can be generalized to descendant invariants as follows: for γ H 2 (X), (3) ( ) τ a+ (γ ), γ 2,, γ n, γ = γ τ a+ (γ ), γ 2,, γ n + τ a (γ γ), γ 2,, γ n We will need another recursion among the descendant invariants called a topological recursion relation To justify it, recall the recursive structure of the boundary of M 0,n (X, ): the boundary divisor D(A, B, 2 ) can be expressed as the image of the morphism M 0,n +(X, ) X M 0,n2 +(X, 2 ) M 0,n (X, ), where n = n + n 2, = + 2, and the fiber product morphisms send a stable map to the image of the last marked point Geometrically, this morphism glues 2 maps together along their last marked points Lemma 33 expresses ψ as a sum of boundary divisors, and the above recursive structure expresses a boundary divisor as a fiber product of moduli spaces of lower dimension Informally, integration yields the following: (4) τ a+ (γ ), γ 2,, γ n = m τ a (γ ), T j T j, γ 2,, γ n 2 = + 2 j=0 This is called a topological recursion relation for descendant invariants, since it originates from the topological recursive structure of the boundary of M 0,n (X, ) 4 Flat sections 4 The quantum connection z We will use a modification of the quantum connection on H defined by p z := zd A i dt i, i= 7
8 where the A i are defined in Remark 22(2) and z is a formal variable This is not a connection, but is a λconnection with λ = z (recall that a λconnection : E E Ω X satisfies a deformed Leibniz rule (fσ) = f (σ) + λdf σ for f O X and σ E) Curvature of a λconnection is defined in the same way as for a usual connection, and one can prove flatness of z in the same way as in Lemma 24 A section of H is defined to be an expression m σ = f j T j with f j O(A p )[[z, z ]], and is said to be flat if z (σ) = 0 42 Flat sections We will now show that certain sections of H are flat for z j=0 Proposition 4 Let ω H 2 (X, C) Then for 0 a m, the sections of H defined by s a = e ω/z T a + e m ω z (n+) τ n (T a e ω/z ), T j T j >0 j=0 n=0 form a basis of flat sections for z Proof (sketch; see [2], Proposition 023 for details) We want to show that (5) z s a t i = T i s a for 0 a m Expand out e ω and e ω/z, and use the divisor equation for descendant invariants (3) to rewrite s a as s a = T a + m z (n+) k! τ n(t a ), T j, ω (k) T j, >0 j=0 n,k=0 where ω (k) denotes that ω appears k times in the descendant invariant On the LHS of (5), writing ω = p i= t it i and expanding the above descendant invariant as a power series in the t i, one finds that differentiation with respect to t i adds a T i to the above descendant invariant Multiplication by z changes the index τ n to τ n+ On the RHS of (5), quantum multiplication gives an additional sum over ordinary 3point GromovWitten invariants Computing both sides of (5) and comparing the coefficients of T j, we are reduced to showing τ n+ (T a ), T j, T i, ω (k) = ( ) k τ n (T a ), T r, ω (k) T r, T j, T i, ω (k2) = k +k 2 =k This follows easily from the 3point topological recursion relation (4) and the divisor equation for descendants (3) Finally, note that the degree 0 term of s a (as a power series in the t i ) is T a So the s a are linearly independent since their degree 0 terms are 8 k
9 Example 42 As in Example 23, for X = P m we have 0 e t z = zd dt Therefore, a section m j=0 f j(t, z, z )T j of H is flat if and only if the f j satisfy the following system of st order linear PDEs (known as the quantum differential equations): z df m dt = f m (6) Define D = z df dt = f 0 z df 0 dt = e t f m ( z d dt) m+ e t By the general theory of differential equations (see eg [], III, ), Df = 0 if and only if f j := (zd/dt) j f, j = 0,, m, solve the system (6) We can solve the equation Df = 0 explicitly as follows Define S = d 0 e (H/z+d)t (H + z) m+ (H + 2z) m+ (H + dz) m+, and expand S = S m + S m H + + S 0 H m with S a C[[e t, z ]] Then it is easy to check that D(S a ) = 0 for a = 0,, m Consider the matrix M Mat m+ (C[[e t, z ]]) whose ath column is given by (zd/dt) j S a, j = 0,, m It is easy to check that M = id in degree 0 So by the above discussion, M defines a basis of solutions to the system (6) On the other hand, by Proposition 4, the matrix Ψ Mat m+ (C[[e t, z ]]) whose ath column is the components of s a also defines a basis of solutions to the system (6) As noted at the end of the proof of Proposition 4, Ψ = id in degree 0 So by the general theory of differential equations (see eg [], III, proof of 4), we conclude that Ψ = M Let us compute the last rows of these matrices explicitly in the case of P Since H 2 = 0, we have (H + kz) = ( 2H ) 2 k 2 z 2 kz 9
10 for k =,, d Writing S = S + S 0 H, we can calculate that S 0 = d 0 S = d 0 ( 2 ( + (d!) ) d (d!) 2 e dt z 2d t (d!) 2 ) e dt z (2d+) On the other side, write s 0 = A 0 + B 0 H and s = A + B H By Proposition 4, we have B 0 = tz + d>0 B = + d>0 e dt e dt n=0 n=0 z (n+) τ n ( thz ), d z (n+) τ n (H), d By (), dim M 0,2 (P, d) = 2d Thus, the only nonzero descendant invariants in B 0 occur for n = 2d or 2d, and the only nonzero descendant invariant in B occurs for n = 2d So the above expressions simplify to B 0 = tz + d>0 B = + d>0 τ 2d (H), d e dt z 2d ( τ 2d (), d t τ 2d (H), d ) e dt z (2d+) The equality of solutions Ψ = M to the quantum differential equations is ( ) ( A0 A z d = S dt 0 z d S ) dt B 0 B S 0 S Equating coefficients of powers of e t and z, the equalities B 0 = S 0 and B = S yield: τ 2d (), d = 2 ( + (d!) ) d τ 2d (H), d = (d!) 2 These 2point descendant invariants can also be computed directly by induction without any reference to the quantum connection (see [2], Example 03) The quantum connection organizes them as coefficients of its flat sections Finally, returning to the general case X = P m, the last rows of M and Ψ determine cohomology classes on P m which are usually called the Ifunction and Jfunction of 0
11 P m, respectively, in the literature: I P m := S = J P m := m S a H m a a=0 m ( a=0 P m s a ) H m a We have in particular shown that I P m = J P m, which is Givental s formulation of mirror symmetry for P m References [] A Borel et al Algebraic DModules, Academic Press (987) [2] D Cox and S Katz Mirror Symmetry and Algebraic Geometry, AMS (999) [3] M Guest From Quantum Cohomology to Integrable Systems, Oxford University Press (2008) [4] K Hori et al Mirror Symmetry, Clay Mathematics Monographs, v (2003)
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