Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $K$-theoretic stable envelopes and our elliptic stable envelopes. We show that the $K$-theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $\frak{gl}_2$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $K$-theoretic stable envelopes.Comment: Latex, 37 pages; v.2: Appendix and Figure 1 added; v.3: missing shift in Theorem 2.9 added and a proof of Theorem 2.9 adde

A Mathematical Theory of the Topological Vertex

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.Comment: 66 pages, 10 figures; notation simplified, references adde

Open-closed Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks

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: 42 pages, 7 figure

Phase space polarization and the topological string: a case study

We review and elaborate on our discussion in hep-th/0606112 on the interplay between the target space and the worldsheet description of the open topological string partition function, for the example of the conifold. We discuss the appropriate phase space and canonical form for the system. We find a map between choices of polarization and the worldsheet description, based on which we study the behavior of the partition function under canonical transformations.Comment: 18 pages, invited review for MPL

Chern-Simons Theory and Topological Strings

We review the relation between Chern-Simons gauge theory and topological string theory on noncompact Calabi-Yau spaces. This relation has made possible to give an exact solution of topological string theory on these spaces to all orders in the string coupling constant. We focus on the construction of this solution, which is encoded in the topological vertex, and we emphasize the implications of the physics of string/gauge theory duality for knot theory and for the geometry of Calabi-Yau manifolds.Comment: 46 pages, RMP style, 25 figures, minor corrections, references adde

Geometric Transitions and Open String Instantons

We investigate the physical and mathematical structure of a new class of geometric transitions proposed by Aganagic and Vafa. The distinctive aspect of these transitions is the presence of open string instanton corrections to Chern-Simons theory. We find a precise match between open and closed string topological amplitudes applying a beautiful idea proposed by Witten some time ago. The closed string amplitudes are reproduced from an open string perspective as a result of a fascinating interplay of enumerative techniques and Chern-Simons computations.Comment: 24 pages, 4 figures, published versio