[Paper Review] Discrete connections on the triangulated manifolds and difference linear equations

S. P. Novikov|ArXiv.org|Mar 13, 2003

Molecular spectroscopy and chirality4 citations

TL;DR

This paper develops a discrete differential geometry framework for $GL_n$-connections on triangulated $n$-manifolds using a nonstandard discretization based on first-order triangle difference equations. It establishes that the framed abelian holonomy representation—encoded by gauge-invariant coefficients $\rho_{ij}^{TT'}$ and holonomy invariants $\mu(\gamma_k)$—completely determines the discrete connection up to abelian gauge transformations, with a complete reconstruction procedure provided for $n \geq 2$. The key contribution is a necessary and sufficient set of conditions for reconstructing discrete $GL_n$-connections from topological and holonomy data.

ABSTRACT

Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the interpretation of DG Connection as linear first order (''triangle'') difference equation acting on the scalar functions of vertices in any simplicial manifold. This theory appeared as a by-product of the new type of discretization of the special Completely Integrable Systems, such as the famous 2D Toda Lattice and corresponding 2D stationary Schrodinger operators. A nonstandard discretization of the 2D Complex Analysis based on these ideas was developed in our recent work closely connected with this one. A complete classification theory is constructed here for the Discrete DG Connections based on the mixture of the abelian and nonabelian features.

Motivation & Objective

• To develop a discrete analog of differential-geometric $GL_n$-connections on triangulated $n$-manifolds using a nonstandard discretization based on first-order difference equations.
• To classify discrete DG connections by combining abelian and nonabelian features through gauge-invariant coefficients $\rho_{ij}^{TT'}$.
• To solve the inverse problem: determine whether the full discrete connection can be reconstructed from a minimal set of gauge-invariant data.
• To establish a complete and necessary set of invariants—$\rho_{ij}^{TT'}$ and $\mu(\gamma_k)$—that fully characterize the discrete $GL_n$-connection up to abelian gauge transformations.
• To generalize the reconstruction procedure from 2D and 3D to arbitrary $n \geq 2$ manifolds using homological algebra and cochain complex techniques.

Proposed method

• Define discrete DG connections via coefficients $b_{T:P}$ assigned to vertex-simplex pairs, inducing a first-order triangle difference operator $Q$ acting on vertex functions.
• Introduce gauge-invariant quantities $\rho_{ij}^{TT'} = \mu_{ij}^T \mu_{ji}^{T'}$ for adjacent $n$-simplices sharing an edge $[ij]$, forming a 1-cocycle in the Poincaré dual cell complex.
• Use the star of each edge $ij$, $St(ij)$, to solve for local connection coefficients $\mu_{ij}^T$ up to a constant $\delta_{ij}$, leveraging $H^1(St(ij), \mathbb{C}^*) = 1$ for solvability.
• Construct a 2-cochain $\tilde{\mu}[\Delta]$ on 2-simplices $\Delta = [ijl]$ from the $\rho_{ij}^{TT'}$ data, proving it forms a closed multiplicative cocycle.
• Prove the 2-cochain $\tilde{\mu}[\Delta]$ is exact using homological relations derived from closed paths in the dual complex, enabling global reconstruction.
• Reconstruct the original connection via $\mu_{ij}^T = \delta_{ji} \tilde{\mu}_{ij}^T$, where $\delta$ adjusts for abelian gauge freedom and satisfies $d\delta = \tilde{\mu}[\Delta]$.

Experimental results

Research questions

• RQ1 Can the full discrete $GL_n$-connection be reconstructed from the minimal set of gauge-invariant coefficients $\rho_{ij}^{TT'}$ associated with adjacent $n$-simplices sharing an $(n-1)$-face?
• RQ2 What additional topological invariants—beyond $\rho_{ij}^{TT'}$—are required to fully determine the discrete connection, and how are they encoded?
• RQ3 Is the framed abelian holonomy representation, defined via products $\mu(\gamma) = \prod \mu_{l,l+1}^{T_l}$ along closed paths $\gamma$, sufficient to reconstruct the discrete connection up to abelian gauge transformations?
• RQ4 How do the relations among $\rho_{ij}^{TT'}$ and holonomy invariants $\mu(\gamma_k)$ reflect the topology of the triangulated manifold, particularly its homology groups?
• RQ5 Can the reconstruction procedure be generalized from $n=2,3$ to arbitrary $n \geq 2$ using cohomological arguments?

Key findings

• The set of gauge-invariant coefficients $\rho_{ij}^{TT'}$ and holonomy invariants $\mu(\gamma_k)$ forms a complete and necessary set of invariants for reconstructing the discrete $GL_n$-connection on any triangulated $n$-manifold with $n \geq 2$.
• The 2-cochain $\tilde{\mu}[\Delta]$ defined on 2-simplices is a closed multiplicative cocycle, and its exactness ensures global consistency of the reconstruction process.
• The reconstruction of $\mu_{ij}^T$ is unique up to abelian gauge transformations, as the freedom in the solution is parameterized by closed 1-cocycles.
• For compact oriented 2-manifolds, the only nontrivial relation is $\prod_{[ij] \in M} \rho_{ij}^{TT'} = 1$, reflecting the global orientation and boundary structure.
• For $n \geq 3$, the reconstruction relies on the solvability of the system $\rho_{ij}^{TT'} = \tilde{\mu}_{ij}^T \tilde{\mu}_{ji}^{T'}$ in the star of each edge, guaranteed by $H^1(St(ij), \mathbb{C}^*) = 1$.
• The Uniqueness Theorem holds: the framed abelian holonomy representation determines the discrete $GL_n$-connection uniquely up to abelian gauge transformations, with the full set of conditions being both necessary and sufficient.