[Paper Review] On the limiting procedure by which $SDiff(T^2)$ and $SU(\infty)$ are associated

J.D. Swain|ArXiv.org|May 1, 2004

Geometric and Algebraic Topology3 citations

TL;DR

This paper investigates the limiting relationship between the group of area-preserving diffeomorphisms on the 2-torus, $SDiff(T^2)$, and the infinite-dimensional unitary group $SU(\infty)$, showing that the standard large-$N$ limit of $SU(N)$ does not converge to $SDiff(T^2)$ due to divergent higher-order corrections in the commutation relations. The key result is that $SU(\infty)$ is fundamentally larger than $SDiff(T^2)$, challenging the conventional identification of the two groups.

ABSTRACT

There have been various attempts to identify groups of area-preserving diffeomorphisms of 2-dimensional manifolds with limits of SU(N) as $N o\infty$. We discuss the particularly simple case where the manifold concerned is the two-dimensional torus $T^2$ and argue that the limit, even in the basis commonly used, is ill-behaved and that the large-N limit of SU(N) is much larger than $SDiff(T^2)$.

Motivation & Objective

• To clarify the mathematical relationship between $SDiff(T^2)$ and $SU(\infty)$ via the large-$N$ limit of $SU(N)$.
• To assess whether the Lie algebra of $SU(N)$ converges to that of $SDiff(T^2)$ as $N \to \infty$.
• To identify the source of discrepancy in the standard limiting procedure used in physics literature.
• To demonstrate that $SU(\infty)$ is strictly larger than $SDiff(T^2)$ due to non-vanishing higher-order corrections.

Proposed method

• Construct the Lie algebra of $SDiff(T^2)$ using vector fields $L_{mn} = i e^{i(mx+ny)} (n\partial_x - m\partial_y)$, with commutation relations $[L_{mn}, L_{m'n'}] = (mn' - m'n) L_{m+m',n+n'}$.
• Define the $SU(N)$ Lie algebra generators $J'_{m,n} = iN/(2k\pi) \omega^{mn/2} g^m h^n$, where $g$ and $h$ satisfy $hg = \omega gh$ with $\omega^N = 1$.
• Derive the commutation relations for $SU(N)$: $[J'_{m,n}, J'_{m',n'}] = \frac{N}{k\pi} \sin\left(\frac{k\pi}{N}(mn' - m'n)\right) J'_{m+m',n+n'}$.
• Analyze the $N \to \infty$ limit by expanding the sine function and identifying the leading and next-order terms in $1/N^2$.
• Identify a specific class of generators with quantum numbers $(m,n) = (N/a, 0)$, $(m',n') = (0, N/b)$ to probe the behavior of the $1/N^2$ correction term.
• Show that the $1/N^2$ correction term scales as $N^4$, which diverges as $N \to \infty$, invalidating the naive limit.

Experimental results

Research questions

• RQ1 Does the Lie algebra of $SU(N)$ converge to that of $SDiff(T^2)$ in the $N \to \infty$ limit?
• RQ2 What is the behavior of the commutator structure constants in the large-$N$ limit of $SU(N)$?
• RQ3 Why does the standard identification of $SU(\infty)$ with $SDiff(T^2)$ fail in the case of the 2-torus?
• RQ4 Are there $SU(N)$ generators that do not map into $SDiff(T^2)$ in the large-$N$ limit?
• RQ5 What is the significance of the $1/N^2$ correction term in the context of the $SU(N) \to SDiff(T^2)$ correspondence?

Key findings

• The $1/N^2$ correction term in the $SU(N)$ commutation relations does not vanish in the large-$N$ limit for certain generator pairs.
• For generators with quantum numbers $(m,n) = (N/a, 0)$ and $(m',n') = (0, N/b)$, the $1/N^2$ correction term scales as $N^4$, which diverges as $N \to \infty$.
• This divergence implies that the large-$N$ limit of $SU(N)$ does not converge to $SDiff(T^2)$ in the standard basis.
• The Lie algebra of $SU(\infty)$ is strictly larger than that of $SDiff(T^2)$, indicating a fundamental mismatch in the groups.
• The result supports the idea that $SU(\infty)$ may describe a more general theory, possibly including topology change, rather than just area-preserving diffeomorphisms.
• The failure of the naive large-$N$ limit underscores the need for refined limiting procedures in relating finite-$N$ gauge groups to diffeomorphism groups.