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Stony Brook University SSStttooonnnyyy BBBrrrooooookkk UUUnnniiivvveeerrrsssiiitttyyy The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University. ©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Two-Point Gromov-Witten Formulas for Symplectic Toric Manifolds A Dissertation Presented by Alexandra Mihaela Popa to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Stony Brook University August 2012 Copyright by Alexandra Mihaela Popa 2012 Stony Brook University The Graduate School Alexandra Mihaela Popa We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Aleksey Zinger- Dissertation Advisor Associate Professor, Department of Mathematics Jason Starr- Chairperson of Defense Associate Professor, Department of Mathematics Radu Laza Assistant Professor, Department of Mathematics Martin Rocek Professor, C.N. Yang Institute for Theoretical Physics This dissertation is accepted by the Graduate School. Charles Taber Interim Dean of the Graduate School ii Abstract of the Dissertation Two-Point Gromov-Witten Formulas for Symplectic Toric Manifolds by Alexandra Mihaela Popa Doctor of Philosophy in Mathematics Stony Brook University 2012 We show that the standard generating functions for genus 0 two-point twisted Gromov- Witten invariants arising from concavex vector bundles over symplectic toric manifolds are explicit transforms of the corresponding one-point generating functions. The latter are, in turn, transforms of Givental’s J-function. We obtain closed formulas for them and, in partic- ular, for two-point Gromov-Witten invariants of non-negative toric complete intersections. Such two-point formulas should play a key role in the computation of genus 1 Gromov-Witten invariants (closed, open, and unoriented) of toric complete intersections as they indeed do in the case of the projective complete intersections. iii Contents Index of notation v Acknowledgements vii 1 Introduction 1 1.1 Someresults.................................... 2 1.2 Outlineofthedissertation . .... 5 2 Overview of symplectic toric manifolds 7 2.1 Definition,charts,andK¨ahlerclasses . ........ 7 2.2 Cohomology, K¨ahler cone, and Picard group . ....... 18 2.3 Torusactionandequivariantnotation. ....... 21 2.4 Examples ..................................... 24 3 Explicit Gromov-Witten formulas 27 3.1 Notation and construction of explicit power series . .......... 27 3.2 Statements..................................... 34 4 Equivariant theorems 38 4.1 Constructionofequivariantpowerseries . ........ 39 4.2 Equivariantstatements . ... 43 5 Proofs 48 5.1 Outline....................................... 48 5.2 Notationforfixedpointsandcurves. .... 51 5.3 Recursivity, polynomiality, and admissible transforms............. 52 5.4 Torus action on the moduli space of stable maps . ....... 56 5.5 RecursivityfortheGWpowerseries . .... 59 5.6 MPCfortheGWpowerseries. 61 5.7 Recursivity and MPC for the explicit power series . ......... 66 Bibliography 72 A Derivation of (5.2) from [LLY3] 75 iv Index of notation Symbol Description Tk, TN C˚ k, C˚ N e TN -equivariantp q p Eulerq class N 1, 2,...,N r s t u M, MJ k N integer matrix, k J submatrix of M with columns ˆ indexed byˆ|J | N ; Section 2.1 τ Ďr s XM symplectic toric manifold given by a minimal toric pair M,τ ; (2.2), Definition 2.1, Proposition 2.2 p p1 pk p q k k ˚ τ t t1 ...tk if p p1,...,pk Z , t t1,...,tk T / t HTN XM M1 “p MN qP k “p qP NP p q t z t z1,...,t zN if t T , z z1,...,zN C ; (2.1) ¨ p q P “p N NqP α α1,...,αN weights of the standard action of T on C ; Section 2.3 “ p q M0,m X, A the moduli space of stable maps from genus 0 curves p q with m marked points into X respresenting A; [MirSym, Chapter 24] evi :M0,m X, A X the evaluation map at the i-th marked point; p qÝÑ [MirSym, Chapter 24] E E` E´ X concavex vector bundle; Chapter 1 “ ‘ ÝÑ ˚ ` VE` M0,m X, A VE` π˚ev E ; (1.1) ÝÑ p q ” 1 ˚ ´ VE´ M0,m X, A VE´ R π ev E ; (1.1) ÝÑ p q ” ˚ VE M0,m X, A VE VE´ VE` ; (1.1) İ İİ İ ÝÑ p q ” ‘ ˚ ` ˚ ´ VE, VE M0,m X, A see (1.7), Chapter 4; e VE ev e E e VE ev e E , İİ 1 1 ÝÑ p q ˚ p ` q p q“˚ p ´ q p q e VE ev2 e E e VE ev2 e E 2 p q p q“ p q p q ψi H M0,m X, A the first Chern class of the universal cotangent line bundle P p p qq for the i-th marked point 2 ψi HTN M0,m X, A and its equivariant version; Chapters 1, 4 P p p Y qq p1 p2 ψ η1,ψ η2 A two-point GW invariant of Y ; (1.3) τ τ KM the closed K¨ahler cone of XM ; Proposition 2.16 @ D τ τ Λ d H2 X ; Z : ω, d 0 ω K ; (1.6), Footnote 2 P p M q ě @ P M d R Λ R ring, R Λ@ D adQ : ad (R d Λ , rr ss rr ss” dPΛ P @ P 1 " 1 * Qd Qd Qřd`d ; Chapter 1 R ~ R ring, R ~¨ ”R ~´1 R ~ ; Section 3.1 ” rr ss ` r s A A1,..., Ak formal variables; Section 3.1 ”p V W q V W R A p the space of homogeneous polynomials of degree p in A r s v with coefficients in the ring R; Section 3.1 k N supp v j : vj 0 if v R (or v C ) τ p q t τ ‰ u P P XM J z XM : supp z J ; Remark 2.10 V pτ q tr sP p qĎ u N τ M set of k-tuples in N indexing the T -fixed points of XM ; (2.4),r (2.13),s Corollary 2.20(a) 1, if j J, τ V τ J z1,...,zN XM , zj P with J M ;(2.13) r s r sP ” #0, otherwise, P ˚ τ ˚ J , :H N X H N restriction induced by the point J ; (2.28) ¨p q ¨ J T p M qÝÑ T r s φJ the equivariant Poincar´edual of the point J ; (2.35) ˇ τ τ ´1 p r s k ˇ Lp X Lp X C , where z,c t z,t c , t T ; (2.14) ÝÑ M ” M ˆ {„ p q „ p ¨ q @ P γi Lei ; (2.16) ` ´ τ Li ,Li XM non-negativer and non-trivial line bundle, negative line bundle ÝÑ a b ` ` ´ ´ such that E Li , E Li ; (1.5) ” i“1 ” i“1 ℓ˘ ˚ℓ˘ ˘ ˘ ˚ 1i Àki À ℓri Z Li γ1 ... γk ; (3.5), Proposition 2.17 P “ b b k ˘ 2 τ ˘ ˘ λi HTN XM e Li ℓrixr; (4.4) P p q p q“ r“1 2 τ Uj H XM ; Z c1 L´řMj ; (2.16) P 2p τ q p ˚ q Hi H XM ; Z c1 γi ; (2.16) P 2p τ q p q N uj HTN XM c1 Dj where Dj ET j L´Mj ; (2.27), (2.24) P 2 p τ q pr sq˚ r s” N ˆ xi H N X c1 γ where γ ET triv γi; (2.27), (2.23) P T p M q p i q i ” ˆ τ τ EM J N : Dj , Dj z XM : zj 0 ; (2.18) # Ďr s jPJ “H+ ”tr sP “ u Dj d Ş Uj, d ; (3.3) ˘p q ˘ Li d c1 Li , d ; 3.3 p q N @a p qD pb q ` ´ νE d Dj d @ Li d D Li d ; (3.3) p q j“1 p q´i“1 p q`i“1 p q τ 1 IJ XM I řJ , the P řpassing throughř I and J ; (5.8) pτ Y q V τ r s r s Ij XM I j if I M and j N I; Section 5.2 p Yt uqV τ P Pr s´ v I, j the element of M so that v I, j I and v I, j I j p q V τ p q‰ p qĂ Yt u whenever I M and j N j ; Section 5.2 f the coefficient of qPd in f wheneverPr s´tf uR Λ ; Section 3.1 q;d P rr ss f A;p the degree p homogeneous part of f R A ; Section 3.1 p p1 pk P rr ss J K d d d ě0 k A ~ q A1 ~ q1 ... Ak ~ qk p p1,...,pk Z ; `J K dq ` dq1 ` dqk @ “p qPp q ! ) ! ) ! Remarks) 3.2, 4.2 vi Ackowledgements I am indebted to Aleksey Zinger for teaching me localization techniques and for suggesting all the questions that I have worked on so far. I thank him for his thorough comments on and corrections of countless drafts of my papers, in particular this dissertation. He spent a very generous amount of time on my mathematical guidance and shared his Gromov-Witten expertise and talent in many discussions with me; I am very grateful. I thank Melissa Liu for explaining parts of Givental’s work needed for this dissertation, for her answers to my questions on toric manifolds, and for an inspiring discussion about a preliminary stage of the results in this dissertation. I am indebted to Iulic˘aPopa, my teacher, for over twenty years of wonderful lessons of mathematics and guidance. In particular, I thank him for the computations without paper and pen and the discovery-type problems in my childhood, and for his teaching me in high school the proofs of the fundamentals ahead of the time they were taught in class.
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