[Paper Review] A closed formula for the evaluation of $\mathfrak{sl}_N$-foams

Louis‐Hadrien Robert, Emmanuel Wagner|arXiv (Cornell University)|Feb 14, 2017

Geometric and Algebraic Topology44 references17 citations

TL;DR

This paper presents a purely combinatorial closed formula for evaluating closed $\mathfrak{sl}_N$-foams, replacing the computationally impractical Kapustin-Li formula. The evaluation yields an integral polynomial and provides a diagrammatic, TQFT-style categorification of the MOY calculus, directly linking to equivariant cohomology of partial flag manifolds and enabling explicit computations in these cohomology rings.

ABSTRACT

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivariant cohomology rings of partial flag manifolds.

Motivation & Objective

• To provide a purely combinatorial, computation-friendly evaluation formula for closed $\mathfrak{sl}_N$-foams, replacing the matrix-factorization-based Kapustin-Li formula.
• To establish a direct connection between the evaluation of $\mathfrak{sl}_N$-foams and integral equivariant cohomology of partial flag manifolds.
• To offer a diagrammatic, TQFT-style categorification of the MOY calculus for $\mathfrak{sl}_N$ link polynomials.
• To enable reverse-engineering and explicit computations in the cohomology rings of partial flag varieties via the foam evaluation.
• To unify and generalize existing approaches to $\mathfrak{sl}_N$ link homology by providing a self-contained, combinatorial framework.

Proposed method

• Develops a symmetric evaluation map for closed $\mathfrak{sl}_N$-foams using a universal construction inspired by [BHMV95].
• Employs a combinatorial formula involving Schur polynomials, Littlewood-Richardson coefficients, and signed sums over tableaux.
• Introduces a sign rule based on the relative ordering of labels ($(-1)^{|A_2 < A_1|}$) to handle orientation and grading shifts.
• Uses the $ abla$-function to encode inner products and $ abla(A,B)$ to represent the pairing between labelings.
• Applies the $ riangle$-function to encode the contribution of the foam's 2-dimensional structure and its boundary labels.
• Relies on identities of Schur polynomials (Appendix A) and detailed tableau manipulations (Appendix B) to prove the main identity.

Experimental results

Research questions

• RQ1 Can a purely combinatorial, closed-form evaluation formula be derived for closed $\mathfrak{sl}_N$-foams, independent of matrix factorizations?
• RQ2 How does this evaluation formula relate to the equivariant cohomology of partial flag manifolds?
• RQ3 Can the MOY calculus for $\mathfrak{sl}_N$ link polynomials be categorified via a TQFT-style construction using this foam evaluation?
• RQ4 What is the precise combinatorial structure (involving tableaux, signs, and Littlewood-Richardson coefficients) that underlies the foam evaluation?
• RQ5 Can this framework be used to perform explicit computations in the cohomology rings of partial flag varieties?

Key findings

• The paper provides a closed-form evaluation formula for closed $\mathfrak{sl}_N$-foams as an integral polynomial, explicitly given by a signed sum over tableaux with Schur polynomials and Littlewood-Richardson coefficients.
• The evaluation is shown to be equivalent to the equivariant $\mathfrak{sl}_N$ link polynomial, establishing a direct link to the cohomology of partial flag manifolds.
• The main identity is proven through a detailed combinatorial manipulation of tableaux and signs, culminating in a symmetric expression involving $c^\nu_{\alpha_1\alpha_2}$, $c^{\tau}_{\alpha_1\gamma_1}$, and $c^{\beta}_{\alpha_2b\tau}$ coefficients.
• The formula is self-contained and combinatorial, allowing for reverse-engineering and explicit computation in cohomology rings of partial flag varieties.
• The evaluation respects the structure of the MOY calculus, providing a diagrammatic and computationally feasible alternative to the Kapustin-Li formula.
• The method successfully replaces the Kapustin-Li formula, which relies on matrix factorizations and is impractical for direct computation, with a purely algebraic and combinatorial approach.