RESEARCH STATEMENT GUANGBO XU

Transcription

1 RESEARCH STATEMENT GUANGBO XU My research interest lies in symplectic geometry, where many problems originate from string theory. My work is based on the analysis of certain elliptic PDE, which combines the traditional holomorphic curve equation and equations in gauge theory. These are the symplectic vortex equation, and its generalization, called the gauged Witten equation. My current research consists of the following three categories. Gauged linear σ-model (GLSM). This project (joint with Gang Tian) intends (ultimately) to prove the so-called Landau-Ginzburg/Calabi-Yau correspondence. The LG/CY correspondence is a conjectural relation between Gromov-Witten theory of a Calabi-Yau defined by a quintic polynomial Q : C 5 C, and another quantum theory of the singularity defined by the same Q. The proof will be based on a mathematical construction of the gauged linear σ-model of Witten, where the core is to study the moduli space of the classical equation of motion. In [TX14a], Tian and I studied the basic analytical properties of this equation (the gauged Witten equation). In [TX14b], Tian and I outlined a construction of GLSM invariants, assuming the existence of a virtual cycle. The construction of a virtual cycle will appear in a forthcoming paper [TX]. We are also considering the adiabatic limits of gauged Witten equation, which connects our theory to Gromov-Witten theory. A separate but relevant result I obtained in [Xu12] generalizes a classical result of C. Taubes. Vortex equation and Hamiltonian Floer theory. A different perspective on Hamiltonian Floer theory is via the symplectic vortex equation instead of holomorphic curves. This new perspective has certain advantages. In [Xu13a], I defined the vortex Floer homology group for a Hamiltonian G-manifold. In [WXb], Wu and I applied the chain-level theory to give an alternative construction of the spectral invariants of Hamiltonian diffeomorphism group of symplectic quotients. In [SX14], Schecter and I studied a finite-dimensional model of the vortex Floer theory, and we obtained a new Morse-type chain complex. Vortex equation and Lagrangian Floer theory. Lagrangian Floer theory is another type of Floer theory where one uses holomorphic curves to study Lagrangian submanifolds. It plays a fundamental role in Kontsevich s Homological Mirror Symmetry conjecture. Similar to Hamiltonian Floer theory, Lagrangian Floer theory can also be approached via vortex equation. In [Xu13b], I proved a compactness theorem of the moduli space vortex equation over bordered Riemann surfaces. In a joint project with C. Woodward and D. Wang, we study the vortex equation over disks, towards a definition of a quantum deformation of the Kirwan map. Two results haven been obtained: in [WXa], Wang and I proved the compactness result of adiabatic limits of the disk vortices; in a forthcoming paper [Xub], I constructed the local model of affine vortices, which is a prerequisite of the nonlinear analysis in our project. In the remaining of this statement, I will first review the background of my research, and then give more details of my results in the above three categories. At the end I will brief two other ongoing projects. 1

2 2 GUANGBO XU 1. Background Quantum field theory has greatly transformed mathematics over the past few decades. The ill-defined path integral can be localized to moduli spaces of the classical equations of motion, which allows mathematicians to understand quantum field theory rigorously. Here we review three types of elliptic PDE coming from string theory. Their moduli spaces contain nontrivial information about the background geometries, which can be generally referred to as certain quantum theories. The connections among these theories are the main topics of my research J-holomorphic curves and Gromov-Witten theory. J-holomorphic curves are solutions to the Cauchy-Riemann equation ( ) u u s + J = 0 t for maps u from a Riemann surface Σ (with local coordinate z = s + it) to an almost Kähler manifold (X, ω, J). The most important breakthrough were made by Gromov and Floer and since then J-holomorphic curve technique has become the most fundamental tool in symplectic geometry. Astounding results have been obtained via the holomorphic curve technique, such as Gromov s nonsqueezing theorem [Gro85] and Floer s proof of Arnold conjecture [Flo89]. The quantum invariant of a symplectic manifold X defined by using the moduli space of J- holomorphic curves is the so-called Gromov-Witten invariant. It corresponds to the correlation function of 2D nonlinear σ-model (a quantum field theory), which are multilinear functions X : ( H (X; Q) ) n Q. (1.1) Among all symplectic target, the GW invariants of Calabi-Yau manifolds are of special importance, because Calabi-Yau manifolds are of the form of the extra dimensions in superstring theory. These invariants are the core of the A-model geometry in view of mirror symmetry Vortex equation and gauged Gromov-Witten theory. J-holomorphic curves can be coupled with gauge fields. Suppose G is a compact Lie group with Lie algebra g. Suppose (X, ω) has a Hamiltonian G-action with a moment map µ : X g. The triple (X, ω, µ) is called a Hamiltonian G-manifold. Choose a G-invariant almost complex structure J. The symplectic vortex equation over a Riemann surface Σ is the following equation for a pair (A, u) A u = 0, F A + µ(u) = 0. (1.2) Here A is a connection on a G-bundle P Σ, u is a section of the associated bundle P G X. A u is the covariant Cauchy-Riemann operator with respect to the connection A, F A Ω 2 (Σ, adp ) is the curvature of A, and µ(u) is a section of adp. Solutions to (1.2) will be called vortices. (1.2) was firstly introduced by I. Mundet ([Mun99]) and Cieliebak-Gaio-Salamon ([CGS00]). One of the applications of this theory is to define a quantum invariant of a Hamiltonian G- manifold (X, ω, µ), which is called the gauged (or Hamiltonian) GW invariant: G X : ( H G(X) ) n Q. (1.3)

3 RESEARCH STATEMENT Witten equation and Landau-Ginzburg A-model. Potential energy is another important feature in field theory. A quantum field theory involving a complex manifold with a holomorphic function is usually called a Landau-Ginzburg model. If Q : C N C is a quasi-homogeneous polynomial, then Witten [Wit93a] proposed to study the following equation for maps u : C C N. u i z + iq(u) = 0, i = 1,..., N. (1.4) More generally, one can consider equations over a general Riemann surface Σ. The moduli space of of the Witten equation contains geometric information about the hypersurface singularity defined by Q. Recently Fan-Jarvis-Ruan ([FJR08, FJR13, FJR11]) carried out the construction based on Witten s idea, and they defined the correlation function Q : ( H Q ) n Q. (1.5) Here H Q is some cohomology group canonically associated to the singularity. When Q defines a Calabi-Yau hypersurface X Q CP N 1, H Q is isomorphic to the cohomology of X Q Relations among these theories. GW and gauged GW theories are related by the adiabatic limit. Rescaling the metric on the Riemann surface Σ by a factor λ 2, (1.2) becomes A u = 0, F A + λ 2 µ(u) = 0. (1.6) If λ >> 0 and (A, u) is a solution, then u will be close to µ 1 (0) in L 2 -sense. In the limiting situation, solutions will approximate holomorphic curves in the Marsden-Weinstein quotient X = µ 1 (0)/G. Using this principle, [GS05] proved a correspondence between the gauged GW invariants of X and the GW invariants of X, under certain restrictions. Moreover, Salamon conjectured that there exists a deformation of the Kirwan map κ X : HG (X) H (X), called the quantum Kirwan map κ Q X, which intertwines with the two correlators (1.1) and (1.3), i.e., X κq X = G X. (1.7) The definition of κ G X should be based on the moduli space of affine vortices, i.e., solutions to the vortex equation over the noncompact Riemann surface C. This is because affine vortices are extra instantons appeared in the adiabatic limits, which should correct the classical Kirwan map. The rigorous construction of κ Q X is the project of F. Ziltener (see [Zil14]). Another conjectural relation exists between the GW theory and the LG A-model theory. In 1990s, string theorists discovered a correspondence between the nonlinear σ-model of a Calabi- Yau hypersurface in CP 4 defined by a quintic polynomial Q : C 5 C, and the Landau-Ginzburg model of Q. This is the so-called LG/CY correspondence. The mathematical theories on the LG side and the CY side are the LG A-model theory and the GW theory we just reviewed. However, the correspondence physicists discovered remains mysteriously to mathematicians and there haven t been even a clearly stated mathematical conjecture. Witten s observation [Wit93b] gives a more geometric understanding of LG/CY correspondence. His theory is now called the gauged linear σ-model (GLSM), which is the main source of ideas of the project I am going to introduce.

4 4 GUANGBO XU 2. Gauged Witten equation and gauged linear σ-model This project is joint with my PhD advisor Gang Tian. The main idea comes from Witten s gauged linear σ-model (GLSM) [Wit93b], which explains the geometry behind the so-called Landau-Ginzburg/Calabi-Yau correspondence. In spite of the lack of precise mathematical statement, the LG/CY correspondence is roughly a conjectural relation between the correlation function (1.5) for a quintic polynomial Q : C N C, and the correlation function (1.1) for the quintic hypersurface X = X Q in CP 4. Witten s idea is: consider the C -invariant Lagrange multiplier W = pq : C 5+1 C, where the C -action on C 5+1 has weight (1, 1, 1, 1, 1, 5). One can consider a U(1)-gauge theory with a superpotential W (mathematically, it should be a combination of U(1)-gauged GW theory and the LG A-model theory). The U(1)-action on C 6 has a moment map µ(x, p) = i 2( x 2 5 p 2 τ ). If τ > 0, the corresponding GIT quotient/symplectic quotient of C 6 is O CP 4( 5). W descends to a function W on the symplectic quotient whose critical locus is the quintic hypersurface X Q. For τ < 0, the quotient is C 5 /Z 5 and the critical locus of the descended function is the origin. This explains the LG/CY correspondence in the classical level. Moreover, in the quantum level, the LG and CY theories can be embedded in a unifying theory where τ is a parameter. The LG/CY correspondence is actually a phase transition as τ crosses zero The setup of gauged Witten equation. In order to give a mathematical construction of GLSM, first we have to set up a good elliptic PDE. In [TX14a] we introduce this equation, which we call the gauged Witten equation. It is a generalization of both the symplectic vortex equation (1.2) and Witten equation (1.4) introduced above. We can work in a general situation but for simplicity, I only describe the setup in the case of homogeneous polynomials. Suppose Q : C N C is a degree r homogeneous polynomial. First we need an r-spin structure over the Riemann surface Σ. This is a holomorphic line bundle L Σ together with an isomorphism φ : L r K Σ. On the other hand, choose an arbitrary line bundle L Σ and denote E = (LL ) (LL ) (L ) ( r) (of rank N + 1). Then the function W = pq can be lifted to a section W Γ ( E, π K Σ ). (2.1) Any Hermitian metric H on (L, L ) induces its gradient H W Γ ( E, π (E, T (0,1) Σ) ). On the other hand, the S 1 S 1 -action on C N+1 has a moment map µ(x, p) = (µ 0 (x), µ 1 (x, p)) = ( i 2 ( x 2 τ 1 ), i ) 2 ( x 2 r p 2 τ 2 ). (2.2) A Hermitian metric H induces a lift µ H : E ir of µ. Then the gauged Witten equation is u + H W(u) = 0, F H + µ H (u) = 0. (2.3) Here the variables of the equation are (u, H), where u is a smooth section of E and H is a Hermitian metric. We also allow the variation of holomorphic structure of L.

5 RESEARCH STATEMENT Compactness. The next task is to study the analytical properties of solutions to the gauged Witten equation, such as the asymptotic behavior near punctures of Σ, and Fredholm theory. Most importantly, we need to prove the compactness of the moduli space. These are all accomplished in [TX14a]. The proof of compactness relies on an a priori energy bound and a C 0 -bound of the section u since the target is noncompact. The energy is given by the following generalization of the Yang-Mills-Higgs functional E(H, u) = 1 ( H u 2 + F 2 L 2 H 2 + µ H (u) ) 2 + H W(u) 2. (2.4) L 2 L 2 L 2 We give an upper bound on this energy which only depends on the homology class of the solution u. On the other hand, the uniform C 0 -bound is the most difficult part of [TX14a]. The proof is based on a study on the a priori behavior of solutions near infinity. With energy and the C 0 -bound, the compactness follows from the standard analytic technique Correlation functions. The correlation function in our theory can be viewed as a variant of Gromov-Witten invariants. In the above setting, the space of supersymmetric ground states is the same H Q as the Landau-Ginzburg A-model and if degq = N, it is isomorphic to H (X Q ). The correlation function is a collection of a multilinear maps d g,n : ( H Q ) n Q, d Z, g 0, n 0. (2.5) Based on the analytical work in [TX14a] and the on-going effort of constructing virtual cycles ([TX]), in [TX14b] we give the formal definition of the correlation function on a single curve Adiabatic limit. Currently we are considering the relation between the GLSM correlation function and the GW invariant. Consider the linear situation, where Q : C N C is a homogeneous polynomial of degree r. Then we can construct the GLSM for the superpotential W = pq : C N+1 C. We choose the parameter in the moment map (2.2) to be τ 0 = τ 1 > 0. Then adding the scale parameter λ to this theory, the GLSM energy functional (2.4) becomes E λ (H, u) = 1 ( H u 2 + λ 2 FH 2 + λ 2 2 L 2 L µ H (u) ) L H W(u) 2. (2.6) 2 L 2 Then, similar to the case of Gaio-Salamon [GS05], as λ +, we see that solutions will approximate (in L 2 -sense) to µ 1 (0). Moreover, if in the limiting process, we turn off the perturbation, then solutions will also approximate (in L 2 -sense) to Q 1 (0). The limiting objects (modulo bubbling) will be holomorphic curves in the projective hypersurface X Q CP N 1. The relation between GLSM and GW invariants of X Q is expected to be revealed in this way Classification of affine vortices in C N. A classical result by Taubes [Tau80] is its classification of affine vortices with target C. His theorem can be stated as: for every nonzero polynomial f(z), there exists a unique solution h : C R to the Kazdan-Warner equation 2 h + ( e 2h f 2 1 ) = 0. Indeed, the pair (f, h) gives an affine vortex (d + h h, e h f) in the target C. Therefore, the moduli space of affine vortices in C with vortex number d is isomorphic to the space of degree d polynomials modulo C, which is C d.

6 6 GUANGBO XU In the adiabatic limit of gauged Witten equation, affine vortices in the target C N will appear as bubbles. In [Xu12], I proved the following generalization of Taubes result (proved independently by Venugopalan-Woodward [VW12]). Theorem 2.1. For N polynomials f 1,..., f N with max i degf i = d, there exists a unique solution h : C R with nice behavior at infinity to the Kazdan-Warner equation 2 h + (e N ) 2h f i 2 1 = 0. i=1 The associated affine vortices are the first set of examples of affine vortices beyond Taubes. 3. Gauged Hamiltonian Floer theory An important application of the holomorphic curve technique is the Hamiltonian Floer theory. A gauged version of Floer theory, was proposed in [CGS00]. Suppose (M, ω, µ) is a Hamiltonian G-manifold, and H = (H t ) is an S 1 -family of G-invariant Hamiltonians on X, then one can define the equivariant action functional A H roughly as ( ) A H : Loop(X g) R, A H (x, η) = u ω + µ(x t )η t H t (x t ) dt. (3.1) Its negative gradient flow equation is u ( u ) s + J t + X Ψ(u) X Ht (u) D = 0, S 1 Ψ s + µ(u) = 0, (3.2) where the variable is (u, Ψ) : R S 1 X g. This is actually a perturbed version of (1.6) where the connection is A = d + Ψdt. Based on the idea of [CGS00], I defined the gauged Hamiltonian Floer homology groups in [Xu13a]. As a theorem, it can be formally stated as Theorem 3.1. The gauged Hamiltonian Floer homology group V HF (X) of a Hamiltonian G- manifold (X, ω, µ) can be defined formally as the LG-equivariant Morse homology of the pair ( Loop(X g), AH ), where LG is the loop group of G. V HF (X) can be viewed as a substitute of the ordinary Hamiltonian Floer homology of the symplectic quotient X, because critical points of the action functional (3.1) corresponds to 1- periodic orbits of the induced Hamiltonian on X. V HF (X) has certain advantage, since in many cases we can avoid using the virtual technique. Moreover, it is conjectured that V HF (X) is isomorphic over integral coefficients to the singular homology of X, while one only has an isomorphism over the rationals via the ordinary Floer theory. Proving this isomorphism is the topic of the forthcoming paper [Xua]. The following two pieces of work are related to the gauged Hamiltonian Floer theory A finite dimensional model and adiabatic limits. In symplectic geometry one usually uses Morse homology theory as a finite dimensional model of Floer theory. In the context of gauged Floer theory, consider a pair of functions f, µ : X R on a compact manifold X and the Lagrange multiplier function F : X R R given by F(x, η) = f(x) + ηµ(x). Its critical points correspond to those of f X where X = µ 1 (0). In the joint paper [SX14] with S. Schecter, we studied the dynamics of the Morse flow of F, whose equation is the ODE ẋ(t) + f(x(t)) + η(t) µ(x(t)) = 0, η(t) + λ 2 µ(x(t)) = 0. (3.3)

7 RESEARCH STATEMENT 7 Here λ is the scale parameter, playing the similar role as the λ in (1.6). Via the λ + limit, one can see that the flow of (3.3) approximates to the negative gradient flow of f restricted to X. Therefore, the Morse homology of (X R, F) is isomorphic to the homology of X. Moreover, we discovered a new chain complex associated with the opposite limit of (3.3) as λ 0. In this limit, trajectories connecting two critical points of F are the fast-slow trajectories where each fast trajectory γ (i) F η i ; each slow trajectory γ (i) S ( X := γ (1) S, γ(1) F,..., γ(k) S, γ(k) F ) is a solution to ẋ(t) + (f + η iµ) (x(t)) = 0 for some constant is a path in the 1-dimensional critical submanifold { } C = (x, η) X R f(x) + η µ(x) = 0. The counting of those fast-slow trajectories results in a complex which we prove to be isomorphic to the Morse-Smale-Witten complex defined by (3.3) for small positive λ. Therefore, this new chain complex gives a different perspective of the homology of the hypersurface X. I believe that this piece of work will shed some light on understanding certain duality for vortex equations. For example, Witten s quantum cohomology = Verlinde algebra [Wit93c] is based on such a duality principle but is not yet well-understood by mathematicians Application to spectral invariants. The spectral invariants of Hamiltonian diffeomorphisms are invariants associated to the chain-level Floer theory. Let X be a symplectic manifold and Ham(X) be the group of Hamiltonian diffeomorphisms of X and Ham(X) be its universal cover. Each element of Ham(X) is represented by a Hamiltonian loop (H t ) t S 1, which gives the Floer chain complex CF (X, H). Since the homology HF (X, X) of CF (X, H) is isomorphic to the singular homology of X, each a H (X) can be represented by a cycle X = a x x CF (X, H). (3.4) The spectral invariant of X associates to each a H (X) and H Ham(X) the min-max value { c(a, H) = min max A H (x) X = } a x x, a x 0. (3.5) X CF (X,H), [X]=a The spectral invariant gives new information about symplectic topology. A remarkable application is the construction of partial symplectic quasi-states and Calabi quasimorphisms by Entov-Polterovich ([EP03, EP06, EP09]). On the other hand, the definition of spectral invariant depends on the foundational work of Floer theory, including the transversality of moduli space and the virtual technique. Gauged Hamiltonian Floer theory gives a way to avoid virtual technique in many interesting cases. Therefore we can simplify many constructions. This is the idea of my joint project with W. Wu. As the first step, we constructed the spectral invariant using the chain complex of the gauged Hamiltonian Floer theory I defined in [Xu13a]. We proved Theorem 3.2. ([WXb]) The spectral invariant defined using gauged Hamiltonian Floer theory of a Hamiltonian G-manifold X satisfies usual properties of the ordinary spectral invariant of its symplectic quotient X.

8 8 GUANGBO XU 4. Lagrangian Floer theory and vortex equation Another important application of holomorphic curve technique is Lagrangian Floer theory, which leads to Fukaya s A -category and Kontsevich s Homological Mirror Symmetry conjecture. It is natural to consider vortex equation with Lagrangian boundary conditions, i.e., (1.6) with boundary condition u( Σ) L, where L is a G-invariant Lagrangian of X contained in µ 1 (0). Such a Lagrangian descend to a usual Lagrangians L in the symplectic quotient X. My first result on this topic is the compactness of the moduli space of vortices with such a boundary condition, when G = S 1. It is a direct generalization of [MT09]. The theorem is Theorem 4.1 ([Xu13b]). The moduli space of vortices with uniformly bounded energy on bordered Riemann surfaces of given topological type with boundaries mapped to an S 1 -Lagrangian L is compact modulo reasonable degenerations. (We allow the complex structure of the bordered surface to vary and degenerate.) 4.1. Open quantum Kirwan map. An algebraic structure derived from the moduli space of holomorphic disks with boundary in a Lagrangian L is the Fukaya A -algebra of L. It can be used to define and deform Lagrangian Floer homology and then to detect displacibility of Lagrangian submanifolds. The A -multiplication is a sequence of multilinear maps m k : ( C(L, Λ) ) k C(L, Λ). (4.1) Here C(L, Λ) is certain cochain group of L with coefficients in a Novikov ring Λ, and m k is defined by counting holomorphic disks with k + 1 marked points. It takes a great amount of effort to define m k (see [FOOO09]) where one needs to take great care using the so-called virtual technique ; on the other hand, the vortex equation gives a way to lifting this problem to nice spaces where virtual technique can be avoided. For example, Woodward considered the quasi-map limit of the vortex equation in [Woo11], which he used to prove similar results about Lagrangian displacing as in [FOOO10, FOOO11]. By varying the scale parameter in the vortex equation, quasi-maps can be related to holomorphic disks in the symplectic quotients, via the adiabatic limit. Recall that the adiabatic limit in the closed case gives the conjectural quantum Kirwan map κ Q X (see 1.7). Woodward in [Woo11] conjectured that there should exist an open version of the quantum Kirwan map, which intertwines the A -multiplication of the quotient pair (X, L) and the quasi-map A -multiplication of (X, L), defined by counting vortices (A, u) over the upper half plane H, with boundary value u( H) L. D. Wang, C. Woodward and I have been working on this conjecture. As the first step, Wang and I proved the following compactness theorem. Theorem 4.2. ([WXa]) In adiabatic limit disk vortices with boundary in a G-Lagrangian L converges to certain stable objects, whose components consists of the following objects: holomorphic disks/spheres in (X, L), holomorphic disk/spheres in (X, L), affine vortices over C or H. To study the moduli space of affine vortices, a standard idea is to realize the moduli space as the zero set of a Fredholm section F : B E, where B is a Banach manifold and E is a Banach bundle. I proved the following result, which was missed in the foundational work of Ziltener [Zil14] on affine vortices. Theorem 4.3. ([Xub]) There is a Banach manifold B modelled on the weighted Sobolev space W 2,p δ (C) (resp. W 2,p δ (H)) weighted by the function (1+ z 2 ) δ/2 such that every affine vortex over

9 RESEARCH STATEMENT 9 C (resp. H) is gauge equivalent to an element of B. The moduli space of affine vortices is the zero locus of a Fredholm section of a Banach bundle E B. 5. Other ongoing projects 5.1. Hitchin-Kobayashi correspondence for nodal curves. The Hitchin-Kobayashi correspondence builds a bridge between algebraic world and the analytical world. In the context of symplectic vortex equation, if the target space X is Kähler and the domain Σ is a smooth Riemann surface, vortices on Σ with target space X can be identified with (stable) holomorphic sections of P G C X. This is proved in [Mun00], which is a main tool of identifying moduli spaces of vortices and computing Hamiltonian Gromov-Witten invariants. Suppose there is a family of curves Σ t such that Σ t is smooth for t 0. On each smooth fibre Σ t, vortices corresponds to stable holomorphic objects by Mundet s theorem; for the singular fibre Σ 0, one can apply the correspondence for irreducible components of its normalization. However, by the compactness theorem of Mundet-Tian [MT09], it is not enough to compactify the moduli space, due to a further degeneration of vortices along long and thin necks. Mundet and I try to give algebraic descriptions of the singular solutions and the way that regular ones degenerate to singular ones. This project originated from our effort on understanding the product structure of the gauged GW invariants of projective spaces. An analogous result of pure Yang-Mills theory is [Pan96]. Moreover, as Mundet s stability condition depends on the area of the smooth curve, we speculate that our description also depends on the size of the curve, and it may also make contact with tropical geometry Analytic torsion in Landau-Ginzburg B-model. Classical mirror symmetry transforms the A-model symplectic geometry of a Calabi-Yau manifold X to the B-model complex geometry of its mirror Calabi-Yau X. Bershadsky-Cecotti-Ooguri-Vafa [BCOV94] proposed a B-model theory related to the Kodaira-Spencer theory of Calabi-Yau manifolds, and in particular, the genus one case makes contact with analytic torsion of the Dolbeault complex. Based on their proposal, Fang-Lu-Yoshikawa [FLY08] defined this genus one invariant (the BCOV invariant) of Calabi-Yau threefolds and proved a prediction in [BCOV94] (the corresponding A-model prediction is proved by Zinger [Zin09]). It is then a natural idea to consider an analogous construction in the Landau-Ginzburg model, in view of the LG/CY correspondence in a broad sense. More precisely, if X is a Kähler manifold and f : X C is a holomorphic function, then the object analogous to the -operator on compact Kähler manifold is the Schrödinger operator f = + df. For f behaving nicely at infinity (for example, polynomials on C N ), f shares similar properties with on compact manifolds. Moreover, the Bismut-Gillet-Soulé s theorem ([BGS88]), which plays important role in Fang-Lu-Yoshikawa s construction, should have an extension to the complex of f. Lastly, a variation of (X, f) (or rather the deformation of the singularity of f) induces a variation of Hodge structure similar to that induced from variation of complex structure. Therefore, I expect to obtain an equally interesting invariant, which could also be much simpler because, the computation of deformation of a function f is much easier than the computation of deformation of complex structures.

10 10 GUANGBO XU References [BCOV94] Bershadsky, Cecotti, Ooguri, and Vafa, Kodaira-Spencer theory of gravity and exact results of quantum string amplitudes, Comm. Math. Phys. 165 (1994), [BGS88] Jean-Michel Bismut, Henri Gillet, and Christophe Soulé, Analytic torsion and holomorphic determinant bundles I, II, III, Comm. Math. Phys. 115 (1988), 49 78, , [CGS00] Kai Cieliebak, Ana Gaio, and Dietmar Salamon, J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions, Int. Math. Res. Not. IMRN 16 (2000), [EP03] Michael Entov and Leonid Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. IMRN 2003 (2003), [EP06], Quasi-states and symplectic intersections, Comment. Math. Helv. 81 (2006), [EP09], Rigid subsets of symplectic manifolds, Compositio Math. 145 (2009), [FJR08] Huijun Fan, Tyler Jarvis, and Yongbin Ruan, Geometry and analysis of spin equations, Comm. Pure App. Math. 61 (2008), no. 6, [FJR11], The Witten equation and its virtual fundamental cycle, arxiv: , [FJR13], The Witten equation, mirror symmetry and quantum singularity theory, Ann. of Math. 178 (2013), [Flo89] Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), [FLY08] Hao Fang, Zhiqin Lu, and Ken-Ichi Yoshikawa, Analytic torsion for Calabi-Yau threefolds, J. Differential Geom. 80 (2008), [FOOO09] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Intersection Floer Theory: anomaly and obstruction, Studies in Advanced Mathematics, American Mathematical Society, [FOOO10], Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151 (2010), [FOOO11], Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Selecta Math. (N. S.) 17 (2011), [Gro85] Misha Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, [GS05] Ana Gaio and Dietmar Salamon, Gromov-Witten invariants of symplectic quotients and adiabatic limits, J. Symplectic Geom. 3 (2005), no. 1, [MT09] Ignasi Mundet i Riera and Gang Tian, A compactification of the moduli space of twisted holomorphic maps, Adv. Math. 222 (2009), [Mun99] Ignasi Mundet i Riera, Yang-Mills-Higgs theory for symplectic fibrations, Ph.D. thesis, Universidad Autonoma de Madrid, [Mun00], A Hitchin-Kobayashi correspondence for Kähler fibrations, J. Reine Angew. Math. 528 (2000), [Pan96] Rahul Pandharipande, A compactification over M g of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996), [SX14] Stephen Schecter and Guangbo Xu, Morse theory for Lagrange multipliers and adiabatic limits, J. Differential Equations 257 (2014), [Tau80] Clifford Taubes, Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys. 72 (1980), no. 3, [TX] Gang Tian and Guangbo Xu, Virtual fundamental cycles of gauged Witten equation, In preparation. [TX14a], Analysis of gauged Witten equation, submitted, [TX14b], Correlation functions in gauged linear σ-model, [VW12] Sushimita Venugopalan and Chris Woodward, Classification of vortices, , [Wit93a] Edward Witten, Algebraic geometry associated with matrix models of two dimensional gravity, Topological Methods in Modern Mathematics: A Symposium in Honor of John Milnor s Sixtieth Birthday (Lisa Goldberg and Anthony Phillips, eds.), Publish or Perish, Inc., [Wit93b], Phases of N = 2 theories in two dimensions, Nuclear Phys. B 403 (1993), [Wit93c], The Verlinde algebra and the cohomology of the Grassmannian, Geometry, Topology and Physics, International Press, Cambridge, MA, 1993.

11 RESEARCH STATEMENT 11 [Woo11] Chris Woodward, Gauged Floer theory for toric moment fibers, Geom. Funct. Anal. 21 (2011), [WXa] Dongning Wang and Guangbo Xu, Comactness of adaibatic limit of disk vortices, in preparation. [WXb] Weiwei Wu and Guangbo Xu, Vortex Floer theory and spectral invariants, In preparation. [Xua] Guangbo Xu, Gauged Hamiltonian Floer homology II: computation of the Floer homology groups, in preparation. [Xub], Local model of affine vortices, in preparation. [Xu12], U(1)-vortices and quantum Kirwan map, submitted, [Xu13a], Gauged Hamiltonian Floer homology I: definition of the Floer homology groups, submitted, [Xu13b], Moduli space of twisted holomorphic maps with Lagrangian boundary condition: compactness, Adv. Math. 242 (2013), [Zil14] Fabian Ziltener, A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane, vol. 230, Mem. Amer. Math. Soc., no. 1082, American Mathematical Society, [Zin09] Aleksey Zinger, The reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces, J. Amer. Math. Soc. 22 (2009),