Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc

By Sheldon Katz, Chiu-Chu Melissa Liu. 2001

263 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e8f47071fbdb2104f805d3a5c100f3aa4fbc4418

How researchers see this paper

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In Applied Mathematics

1. Marcos Mariño
2. C. Vafa
3. K. Fukaya
4. R. Pandharipande
5. A. Klemm
6. V. Bouchard
7. A. Okounkov
8. Sergei Gukov

In Geometry and Topology

1. Marcos Mariño
2. Sergei Gukov
3. C. Vafa
4. B. Eynard
5. A. Klemm
6. Mina Aganagic
7. V. Bouchard
8. Sara Pasquetti

In Mathematical Physics

1. Marcos Mariño
2. Sergei Gukov
3. C. Vafa
4. B. Eynard
5. Mina Aganagic
6. E. Witten
7. A. Klemm
8. R. Pandharipande

Citations in the corpus

• Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture (2012, background): “When n > 0, Katz and Liu [33] have constructed a virtual class and virtual cycle in Mg,n(X, L, β, ~w).”
• Hodge integrals and invariants of the unknot (2003, background): “Liu of the corresponding open string Gromov–Witten theory [10].”
• Disk Instantons, Mirror Symmetry and the Duality Web (2001, methodology): “The authors of [32] have recently checked and verified that the integer choice of ambiguity changes the topological string amplitudes exactly”
• Open Gromov-Witten disk invariants in the presence of an anti-symplectic involution (2013, unknown): “Katz and Liu [22, 24] define such invariants for an S1-equivariant pair (M,L).”
• Dirichlet Branes and Mirror Symmetry (2009, background): “There is yet to be a wholly satisfactory mathematical theory of open Gromov-Witten invariants, but in many interesting cases there is: see [297, 429].”
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions (2006, background): “Katz and Liu took a first step toward the mathematical definition of open invariants given the additional structure of an S(1) action on the pair (X,L) [10, 16].”
• Torus Knots and Mirror Symmetry (2011, methodology): “In the A-model, framing arises as an ambiguity associated to the choice of localization action in the moduli space of maps with boundaries [29], and the open string invariants depend naturally on an integer parametrizing this ambiguity.”
• Lagrangian Floer theory on compact toric manifolds II: bulk deformations (2008, methodology): “ing (1) - (6) of Proposition 6.5. The proof is the same as the proof of Lemma 10.7 [FOOO3] and so is omitted. Remark 6.9. The independence of open Gromov-Witten invariant such as c(β;p) was proved in [KL] by taking equivariant perturbations in the situation where an appropriate S1-action exists. LAG”
• Framed knots at large N (2001, methodology): “The dual geometry for the unknot is known [2], and some explicit computations of open Gromov-Witten invariants for this geometry have been done using localization techniques [9][10][11].”
• Chern-Simons theory and topological strings (2004, unknown): “It turns out that the leading term of Fw,g(p, t) as t → ∞ (which we will simply denote by Fw,g(p)) can be computed in open Gromov-Witten theory by using Hodge integrals (Katz and Liu, 2002; see also Li and Song, 2002).”
• All Loop Topological String Amplitudes from Chern-Simons Theory (2002, methodology): “The role of framing in topological string theory was discovered in [7] in a closely related context and studied subsequently in [8,41,42].”
• Opening Mirror Symmetry on the Quintic (2006, background): “four-dimensional analogue of the S(1) action used in [25].”
• The Topological Vertex (2003, background): “A-model localization calculations for all genus open string amplitudes with a single stack of D-branes have been introduced in [45,43].”
• Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds (2003, background): “dary values. Denote by A0(E,F) the sheaf of C∞ sections of Ewith boundary values in F, and denote by A(0,1)(E) the sheaf of C∞ E-valued (0,1)-forms. The following is easy to check (see section 3.4 of [KL]). Lemma 6.2. The sequence 0 →(E,F) →A0(E,F) →A∂ 0,1(E) →0 defines a fine resolution of (E,F). Fro”
• The Superpolynomial for Knot Homologies (2005, methodology): “We expect that things would work out similarly if we used a different field of coefficients; it is less clear what would happen if we tried to use coefficients in Z.”
• Nilpotent slices, Hilbert schemes, and the Jones polynomial (2004, unknown): “is a homotopy equivalence (see [8] or [20]).”