Subfactor Index, Sector Information, and Entropy

A finite-index inclusion $\mathcal N\subset\mathcal M$ measures a precise finite enlargement of one factor by another. For a faithful outer action of a finite group $G$, the fixed-point inclusion $\mathcal M^G\subset\mathcal M$ has index $|G|$, its canonical endomorphism records the finite sector content, and the averaging expectation bounds the relative entropy lost by forgetting the charged extension. None of these finite formulas survives automatically for an infinite group or infinite-index inclusion.

Required background. Conjugates, statistical dimension, and statistics operators provide the sector meaning of index; correctable subalgebras provide the information-preservation comparison. Helpful background. DHR reconstruction identifies gauge fixed points, braided sectors treat low-dimensional variants, and centers and edge information separate sector labels from distillable entanglement.

Index and the averaging expectation

Let $G$ be finite and act faithfully and outerly by automorphisms $\alpha_g$ on a factor $\mathcal M$. Put $\mathcal N=\mathcal M^G$. The group average

$$E:\mathcal M\longrightarrow\mathcal N, \qquad E(x)=\frac1{|G|}\sum_{g\in G}\alpha_g(x),$$

is a faithful normal conditional expectation. For $x\geq0$, the identity term gives

$$E(x)\geq\frac1{|G|}x.$$

The optimal Pimsner–Popa constant is $|G|^{-1}$ for an outer action, and the Jones index is

$$[\mathcal M:\mathcal N]=|G|.$$

Jones introduced the discrete index values and the basic construction for subfactors Jones 1983, §§2–4, pp. 4–16. Longo’s canonical-endomorphism formulation identifies index with squared statistical dimension and is the form suited to AQFT Longo 1989, §§2–3, pp. 221–232.

If $\phi$ is a normal state on $\mathcal M$, the operator inequality implies $\phi\circ E\geq|G|^{-1}\phi$. Monotonicity of the logarithm in the relative modular formulation then gives the finite information bound

$$S_{\mathcal M}(\phi\Vert\phi\circ E)\leq\log|G|.$$

It measures the distinguishability between a state and its $G$-averaged version. Equality is state-dependent; the index supplies an upper bound, not the assertion that every state loses exactly $\log|G|$.

The expectation also gives a simple correctability comparison. It fixes every neutral observable $n\in\mathcal N$ pointwise, $E(n)=n$, so the identity recovery protects the subalgebra $\mathcal N$. It need not recover $\mathcal M$: for a $\mathbb Z_2$-odd charged operator $x$ with $\alpha(x)=-x$, one has $E(x)=0$. Two states distinguished only through odd expectations can become indistinguishable after averaging. Finite index bounds this loss but does not invert the coarse graining on charged observables.

This example also separates the Jones index from a vector-space dimension. Both factors are infinite dimensional, yet their relative index is the finite number $|G|$. The number records the size of the canonical bimodule or endomorphism, not the count of basis elements in either local algebra.

Chiral-net fixed points and sectors

Let $\mathcal A$ be a completely rational chiral conformal net and let a finite group $G$ act faithfully by vacuum-preserving internal symmetries. For every interval $I$, assume the local action is outer. Then

$$\mathcal A^G(I)\subset\mathcal A(I)$$

has the averaging expectation above and index $|G|$. The dual canonical endomorphism of the inclusion decomposes into the sectors associated with irreducible representations of $G$, with multiplicities equal to their representation dimensions. Consequently

$$d(\theta)=\sum_{\pi\in\widehat G}(\dim\pi)^2=|G|.$$

This is the operator-algebraic reason finite orbifolding produces finite sector data. In a rational model, those sectors must also close under the appropriate fusion rules; the explicit conformal families and fusion coefficients belong to the minimal-model fusion analysis, while the present conclusion is only the fixed-point index and its canonical sector content.

As a concrete check, take $G=\mathbb Z_2$. The expectation is $E(x)=(x+\alpha(x))/2$, the index is $2$, and the two one-dimensional irreducible representations give $1^2+1^2=2$. For any normal $\phi$,

$$S(\phi\Vert\phi\circ E)\leq\log2.$$

If $\phi$ is already invariant, $\phi\circ E=\phi$ and the entropy is zero, independently confirming that the index bound need not be saturated.

Finite-index hypotheses cannot be dropped

Replace $G$ by an infinite compact group, for example $U(1)$. Haar averaging may still define a normal conditional expectation onto fixed points, but there is no positive finite constant $c$ with $E(x)\geq c x$ in the generic outer action. The fixed-point inclusion is then typically infinite index, the sum of squared representation dimensions diverges, and $\log|G|$ is not a number that can bound relative entropy.

This adversarial case separates three statements often conflated: an expectation may exist; an index may nevertheless be infinite; and a finite entropy bound requires a finite Pimsner–Popa constant. Complete rationality or a finite sector category is likewise a real hypothesis, not a consequence of conformal covariance alone.

Exercises

1. Verify the expectation. Prove that $E$ is idempotent and satisfies $E(n_1xn_2)=n_1E(x)n_2$ for $n_i\in\mathcal N$.

Solution

Applying a second group average counts each product $hg$ exactly $|G|$ times, so $E^2=E$. Fixed elements pass through every $\alpha_g$, giving the bimodule identity. Finite averaging also makes normality immediate.

2. Entropy extremes. Give one condition for zero entropy and one necessary ingredient for possible saturation of the $\log|G|$ bound.

Solution

If $\phi$ is $G$-invariant, then $\phi\circ E=\phi$ and the entropy is zero. Saturation requires a state that distinguishes all group-translated components as sharply as the inclusion permits; it is not implied by finite index and may fail in a given representation.

References

• Jones, Vaughan F. R. “Index for Subfactors.” Inventiones Mathematicae 72 (1983): 1–25. DOI.
• Longo, Roberto. “Index of Subfactors and Statistics of Quantum Fields. I.” Communications in Mathematical Physics 126 (1989): 217–247. DOI.