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The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti [arXiv:0709.1453, arXiv:0807.0597] relates all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifolds/3-orbifolds to the Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. In this paper, we present a proof of the Remodeling Conjecture for open-closed orbifold Gromov-Witten invariants of an arbitrary affine toric Calabi-Yau 3-orbifold relative to a framed Aganagic-Vafa Lagrangian brane. This can be viewed as an all genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds.

A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits.(i) We define the stacky GKM graph of a smooth GKM stack, under the mild assumption that any one-dimensional $T$-orbit closure contains at least one $T$ fixed point. The stacky GKM graph is a decorated graph which contains enough information to reconstruct the $T$-equivariant formal neighborhood of the $1$-skeleton (union of zero-dimensional and one-dimensional $T$-orbits) as a formal smooth DM stack equipped with a $T$-action.(ii) We axiomize the definition of a stacky GKM graph and introduce abstract stacky GKM graphs which are more general than stacky GKM graphs of honest smooth GKM stacks. From an abstract GKM graph we construct a formal smooth GKM stack.(iii) We define equivariant orbifold Gromov–Witten invariants of smooth GKM stacks, as well as formal equivariant orbifold Gromov–Witten invariants of formal smooth GKM stacks. These invariants can be computed by virtual localization and depend only the stacky GKM graph or the abstract stacky GKM graph. Formal equivariant orbifold Gromov–Witten invariants of the stacky GKM graph of a smooth GKM stack $\mathcal{X}$ are refinements of equivariant orbifold Gromov–Witten invariants of $\mathcal{X}$.

The mixed spin P-fields (MSP for short) theory sets up a geometric platform to relate Gromov-Witten invariants of the quintic three-fold and Fan-Jarvis-Ruan-Witten invariants of the quintic polynomial in five variables. It starts with Wittens vision and the P-fields treatment of GW invariants and FJRW invariants. Then it brie y discusses the master space technique and its application to the set-up of the MSP moduli. Some key results in MSP theory are explained and some examples are provided.

For any finite abelian group G, the equivariant Gromov-Witten invariants of [Cr/G] can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng’s orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves. This expression will play a crucial role in the proof of the remodeling conjecture for affine toric Calabi-Yau 3-orbifolds in [19].

An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold., Section 3.1-3.4 (review of Gromov-Witten theory) are similar to Section 3.1-3.4 of arXiv:1107.4712. Section 4 generalizes the toric case in Section 5 of arXiv:1107.4712 to the GKM case

We extend our previous work [19] on coherent-constructible correspondence for toric varieties to include toric Deligne-Mumford (DM) stacks. Following Borisov-Chen-Smith [9], a toric DM stack XΣ is described by a “stacky fan” Σ = (N,Σ, β), where N is a finitely generated abelian group and Σ is a simplicial fan in NR = N ⊗Z R. From Σ we define a conical Lagrangian ΛΣ inside the cotangent T ∗MR of the dual vector space MR of NR, such that torus-equivariant, coherent sheaves on XΣ are equivalent to constructible sheaves on MR with singular support in ΛΣ. The microlocalization theorem of Nadler and the last author [40, 38] relates constructible sheaves on MR to a Fukaya category on the cotangent T ∗MR, giving a version of homological mirror symmetry for toric DM stacks.

Let X Σ be a complete toric variety. The coherent-constructible correspondence κ of Fang et al. (2011) [14] equates P erf T ( X Σ ) with a subcategory Sh c c ( M R ; Λ Σ ) of constructible sheaves on a vector space M R . The microlocalization equivalence μ of Nadler and Zaslow (2009) [27] and Nadler (2009) [25] relates these sheaves to a subcategory Fuk ( T ⁎ M R ; Λ Σ ) of the Fukaya category of the cotangent T ⁎ M R . When X Σ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, DCoh T ( X Σ ) ≅ DFuk ( T ⁎ M R ; Λ Σ ) , which is an equivalence of triangulated tensor categories. The nonequivariant coherent-constructible correspondence κ ¯ of Treumann (preprint) [33] embeds P erf ( X Σ ) into a subcategory S h c ( T R ∨ ; Λ ¯ Σ ) of constructible sheaves on a compact torus T R ∨ . When X Σ is nonsingular, the composition of κ ¯ and microlocalization yields a version of homological mirror symmetry, DCoh ( X Σ ) ↪ DFuk ( T ⁎ T R ; Λ ¯ Σ ) , which is a full embedding of triangulated tensor categories. When X Σ is nonsingular and projective, the composition τ = μ ∘ κ is compatible with T-duality, in the following sense. An equivariant ample line bundle L has a hermitian metric invariant under the real torus, whose connection defines a family of flat line bundles over the real torus orbits. This data produces a T-dual Lagrangian brane L on the universal cover T ⁎ M R of the dual real torus fibration. We prove L ≅ τ ( L ) in Fuk ( T ⁎ M R ; Λ Σ ) . Thus, equivariant homological mirror symmetry is determined by T-duality.

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety. Specifically, let $X$ be a proper toric variety of dimension $n$ and let $M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n$ be the Lie algebra of the compact dual (real) torus $T_\bR^\vee\cong U(1)^n$. Then there is a corresponding conical Lagrangian $\Lambda \subset T^*M_\bR$ and an equivalence of triangulated dg categories $\Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda),$ where $\Perf_T(X)$ is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and $\Sh_{cc}(M_\bR;\Lambda)$ is the triangulated dg category of complex of sheaves on $M_\bR$ with compactly supported, constructible cohomology whose singular support lies in $\Lambda$. This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on $M_\bR$., Comment: 20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v4

For an equivariant Morse stratification that contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal G-bundle over a connected, closed, nonorientable surface Σ is locally equivariantly \({\mathbb{Q}}\)-antiperfect when G = U(2), SU(2), U(3), SU(3); we propose that the Morse stratification is actually equivariantly \({\mathbb{Q}}\)-antiperfect in these cases. Our proposal yields formulas of Poincare series \({P_t^G({\rm Hom}(\pi_1(\Sigma),G);\mathbb{Q})}\) when G = U(2), SU(2), U(3), SU(3). Our U(2), SU(2) formulas agree with formulas proved by T. Baird, who also verified our conjectural U(3) formula.

The first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface Σ. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of Σ. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when Σ is not homeomorphic to the Klein bottle, and for n ≤ 3 when Σ is the Klein bottle. We also derive similar orientability results when the structure group is SU(n).

The Remodeling Conjecture proposed by Bouchard-Klemm-Mari\~{n}o-Pasquetti (BKMP) [arXiv:0709.1453, arXiv:0807.0597] relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semi-projective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera., Comment: 85 pages, 4 figures; final version

We study the web of dualities relating various enumerative invariants, notably Gromov-Witten invariants and invariants that arise in topological gauge theory. In particular, we study Donaldson-Thomas gauge theory and its reductions to D=4 and D=2 which are relevant to the local theory of surfaces in Calabi-Yau threefolds., Comment: 38 pages, To Appear in Adv. Theor. Math. Phys. (2017)

JMg,n where tpi = ci(L?) is the first Chern class of L?, and Xj = Cj(E) is the j-th Chern class of the Hodge bundle. The study of Hodge integrals is an important part of intersection theory on Mg,n> Hodge integrals also naturally arise when one computes Gromov-Witten invariants by localization techniques. For example, the following generating series of Hodge integrals arises when one computes local invariants of a toric Fano surface in a Calabi-Yau 3-fold by virtual localization [33]

We derive some Hodge integral identities by taking various limits of the Mariño-Vafa formula using the cut-and-join equation. These identities include the formula of general λ g \lambda _g -integrals, the formula of λ g − 1 \lambda _{g-1} -integrals on M ¯ g , 1 {\overline {\mathcal {M}}}_{g,1} , the formula of cubic λ \lambda integrals on M ¯ g {\overline {\mathcal {M}}}_g , and the ELSV formula relating Hurwitz numbers and Hodge integrals. In particular, our proof of the MV formula by the cut-and-join equation leads to a new and simple proof of the λ g \lambda _g conjecture. We also present a proof of the ELSV formula completely parallel to our proof of the Mariño-Vafa formula.

A spacetime is a four-manifold with a pseudo-metric of signature (+,+,-+-,?). A hypersurface or a 2-surface in a spacetime is spacelike if the induced metric is pos itive definite. A quasi-local energy-momentum vector is a vector in R3'1 associated to a spacelike 2-surface which depends on its first and second fundamental forms and the connection to its normal bundle in the spacetime. The time component of the four-vector is called quasi-local energy (mass). Similar to [10, 8], we require the quasi-local energy-momentum vector to satisfy the following properties. (1) It should be zero for the flat spacetime. (2) The quasi-local mass should be equivalent to the standard definition if the spacetime is spherically symmetric and the quasi-local mass is evaluated on the spheres [7]. (We say that two masses mi and m2 are equivalent if there is a universal constant c > 0 such that c~lm\ /Area(E)/167r. (6) The quasi-local energy-momentum vector should be nonspacelike and the quasi-local mass should be nonnegative.

We study the equivariantly perturbed mirror Landau-Ginzburg model of the projective line. We show that the Eynard-Orantin recursion on this model encodes all genus all descendants equivariant Gromov-Witten invariants of the projective line. The non-equivariant limit of this result is the Norbury-Scott conjecture, while by taking large radius limit we recover the Bouchard-Marino conjecture on simple Hurwitz numbers., 34 pages, 5 figures; references added, typos corrected

In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to a well-defined algebro-geometric computation very similar to well-known localization techniques in Gromov-Witten theory. In particular, our computation of the invariants for multiple covers of a generic disc bounding a special Lagrangian submanifold in a Calabi-Yau threefold agrees completely with the original predictions of Ooguri and Vafa based on string duality. Our proposed invariants depend more generally on a discrete parameter which came to light in the work of Aganagic, Klemm, and Vafa which was also based on duality, and our more general calculations agree with theirs up to sign., Comment: This is the version published by Geometry & Topology Monographs on 22 April 2006

We describe some recent progress and open problems in Gromov-Witten theory of Calabi-Yau 3-folds, focusing on the quintic 3-fold and toric Calabi-Yau 3folds.

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are connected sets of real points inside a complex moduli variety endowed with a real structure; when the rank and the degree are coprime, they are in fact the connected components of the fixed-point set of the real structure. This presentation as a quotient enables us to generalize the methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincare polynomials of these moduli spaces in the coprime case. We also compute the mod 2 Poincare series of moduli stacks of all real and quaternionic vector bundles of a fixed topological type. As an application of our computations, we give new examples of maximal real algebraic varieties., Comment: Final version, 72 pages; formulae in the quaternionic, n>0 case corrected; proof of Theorem 1.3 revised; references added

In arXiv:math/0311139, as evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In arXiv:0911.4711, we showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on R^n. In this paper we investigate and reprove some of Kawamata's results from this perspective., Comment: 34 pages, 11 figures; dedicated to Loo-Keng Hua on the occasion of his 100th birthday

The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R^4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces., Comment: 38 pages; typos corrected, references added, minor changes (e.g. minor change of convention in Definition 5.13, 5.19, 6.5)

In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. We generalize their study to all closed, compact, connected, possibly nonorientable surfaces. We introduce the notion of "super central extension" of the fundamental group of a surface. It is the central extension when the surface is orientable. We establish a precise correspondence between Yang-Mills connections and representations of super central extension. Knowing this exact correspondence, we work mainly at the level of representation varieties which are finite dimensional instead of the level of strata which are infinite dimensional., 45 pages, 1 figure

Based on the duality between open-string theory on noncompact Calabi-Yau threefolds and Chern-Simons theory on three manifolds, M Marino and C Vafa conjectured a formula of one-partition Hodge integrals in term of invariants of the unknot (hep-th/0108064). Many Hodge integral identities, including the lambda_g conjecture and the ELSV formula, can be obtained by taking limits of the Marino-Vafa formula. Motivated by the Marino-Vafa formula and formula of Gromov-Witten invariants of local toric Calabi-Yau threefolds predicted by physicists, J Zhou conjectured a formula of two-partition Hodge integrals in terms of invariants of the Hopf link (math.AG/0310282) and used it to justify physicists' predictions (math.AG/0310283). In this expository article, we describe proofs and applications of these two formulae of Hodge integrals based on joint works of K Liu, J Zhou and the author (math.AG/0306257, math.AG/0306434, math.AG/0308015, math.AG/0310272). This is an expansion of the author's talk of the same title at the BIRS workshop: "The Interaction of Finite Type and Gromov-Witten Invariants", November 15--20, 2003., Comment: This is the version published by Geometry & Topology Monographs on 22 April 2006

We compute the local Gromov-Witten invariants of certain configurations of rational curves in a Calabi-Yau threefold. These configurations are connected subcurves of the `minimal trivalent configuration', which is a particular tree of P^1's with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov-Witten invariants of a blowup of P^3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov--Witten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex., This is the version published by Geometry & Topology on 7 March 2006

We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds., 66 pages, 10 figures; notation simplified, references added

We prove a remarkable formula for Hodge integrals conjectured by Marino and Vafa based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms., 36 pages; typos corrected; a reference added

We outline a proof of a remarkable formula for Hodge integrals conjectured by Marino and Vafa in hep-th/0108064 base on large N duality., 14 pages

We study the connectedness of the moduli space of gauge equivalence classes of flat G-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups including U(n) and Spin^C(n)., Comment: 24 pages