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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.

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) relates the A-model open and closed topological string amplitudes (open and closed Gromov-Witten invariants) of a symplectic toric Calabi-Yau 3-fold to Eynard-Orantin invariants of its mirror curve. The Remodeling Conjecture can be viewed as a version of all genus open-closed mirror symmetry. The SYZ conjecture explains mirror symmetry as $T$-duality. After a brief review on SYZ mirror symmetry and mirrors of symplectic toric Calabi-Yau 3-orbifolds, we give a non-technical exposition of our results on the Remodeling Conjecture for symplectic toric Calabi-Yau 3-orbifolds. In the end, we apply SYZ mirror symmetry to obtain the descendent version of the all genus mirror symmetry for toric Calabi-Yau 3-orbifolds., Comment: 18 pages. Exposition of arXiv:1604.07123

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 study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford (DM) stacks (with possibly non-trivial generic stabilizers and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack with boundaries mapped into a Aganagic-Vafa brane. All genus open-closed Gromov-Witten invariants are defined by torus localization and depend on the choice of a framing which is an integer. We also provide another definition of all genus open-closed Gromov-Witten invariants based on algebraic relative orbifold Gromov-Witten theory; this generalizes the definition in Li-Liu-Liu-Zhou [arXiv:math/0408426] for smooth toric Calabi-Yau 3-folds. When the toric DM stack a toric Calabi-Yau 3-orbifold (i.e. when the generic stabilizer is trivial), we define generating functions of open-closed Gromov-Witten invariants or arbitrary genus $g$ and number $h$ of boundary circles; it takes values in the Chen-Ruan orbifold cohomology of the classifying space of a finite cyclic group of order $m$. We prove an open mirror theorem which relates the generating function of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of the toric Calabi-Yau 3-orbifold. This generalizes a conjecture by Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] (proved in full generality by the first and the second authors in [arXiv:1103.0693]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold., Comment: 44 pages, 7 figures

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.

We present a proof of the mirror conjecture of Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] on disk enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary framing. In particular, we recover previous results on the conjecture for (i) an inner brane at zero framing in the total space of the canonical line bundle of the projective plane (Graber-Zaslow [arXiv:hep-th/0109075]), (ii) an outer brane at arbitrary framing in the resolved conifold (Zhou [arXiv:1001.0447]), and (iii) an outer brane at zero framing in the total space of the canonical line bundle of the projective plane (Brini [arXiv:1102.0281, Section 5.3])., Comment: 39 pages, 11 figures

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

Confocal Scanning Laser Microscopy: Applications for Imaging Dynamic Processes in Skin In Vivo

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

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

Motivated by the important work of Brown adn York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface., 4 pages; final published version

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