Jaffe–Leśniewski–Osterwalder cocycle - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D,\gamma)$ be an even theta-summable spectral triple. Its Jaffe–Leśniewski–Osterwalder cocycle is the sequence of even cochains

\Phi_{2n}(a_0,\ldots,a_{2n}) =\int_{\Delta_{2n}}\operatorname{Tr}\!\left( \gamma a_0e^{-t_0D^2}[D,a_1]e^{-t_1D^2}\cdots [D,a_{2n}]e^{-t_{2n}D^2}\right)\,dt,

where $\Delta_m=\{(t_0,\ldots,t_m):t_j\geq0,\ \sum_jt_j=1\}$ . Theta summability makes the integrand trace class and provides the growth estimates for an entire cochain. The identities $(b+B)\Phi=0$ make $\Phi$ an entire cyclic cocycle, whose class is the entire Chern character of the triple.

Why the formula converges

The heat factors distribute a total heat time of one among the commutators. Hölder inequalities for trace ideals, together with $\operatorname{Tr}(e^{-tD^2})<\infty$ , control the integrand near the faces of the simplex. The simplex volume supplies factorial decay in the cochain degree. These estimates establish the entire-growth condition, not merely termwise finiteness Jaffe–Leśniewski–Osterwalder, §§2–4.

Cohomological meaning

The JLO class is invariant under suitable differentiable deformations of the unbounded cycle. Under stronger summability hypotheses, it represents the same periodic cyclic-cohomology class as the Chern character of the associated bounded Fredholm module. The heat-kernel formula is especially useful when no finite degree alone captures the character.

Odd case and conventions

For an odd spectral triple there is no grading operator $\gamma$ ; the odd JLO character is expressed by odd-degree cochains, equivalently through a standard suspension construction. Normalizing constants and rescaling $D$ vary across sources, but they do not change the underlying cohomology class when the corresponding conventions are used consistently.

References

A. Jaffe, A. Leśniewski, and K. Osterwalder, “Quantum K-Theory. I. The Chern Character,” Communications in Mathematical Physics 118 (1988), 1–14. DOI record. Relevant: the heat-kernel cocycle, entire estimates, and Chern character.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-maintained text. Relevant: chapter IV on theta-summable Fredholm modules and the JLO entire cocycle.