\(LT\)-equivariant index from the viewpoint of \(KK\)-theory. A global analysis on the infinite-dimensional Heisenberg group. (English) Zbl 1445.19004
Following up [J. Noncommut. Geom. 13, No. 2, 553–586 (2019; Zbl 1429.58019)], this paper continues the author’s research project about index theory on infinite dimensional manifolds using the methods of noncommutative geometry.
Here the goal is to develop an infinite-dimensional version of Kasparov’s index theory for complete Riemannian manifolds equipped with proper cocompact group actions (cf. [Sov. Math., Dokl. 27, 105–109 (1983; Zbl 0526.22007); translation from Dokl. Akad. Nauk SSSR 268, 533–537 (1983); Invent. Math. 91, No. 1, 147–201 (1988; Zbl 0647.46053); J. Noncommut. Geom. 10, No. 4, 1303–1378 (2016; Zbl 1358.19001)]).
More specifically, the infinite dimensional manifolds \(\mathcal{M}\) considered in this project carry proper actions of the loop group \(LT\mathrel{{:}{=}}C^\infty(S^1,T)\) of a circle group \(T\) and are assumed to be “even-dimensional and \(\mathrm{Spin}^c\)” in a certain sense. The index theory of interest is the one of the Dirac operator on \(\mathcal{L}\otimes\mathcal{S}\), where \(\mathcal{S}\to\mathcal{M}\) denotes the spinor bundle and \(\mathcal{L}\to\mathcal{M}\) an \(LT\)-equivariant line bundle which is \(\tau\)-twisted with respect to some \(U(1)\)-central extension \(LT^\tau\) of \(LT\).
At the core of the index theory should be a “function \(C^*\)-algebra” \(C_0(\mathcal{M})\), but as infinite dimensional manifolds are not locally compact Hausdorff spaces, the Gelfand–Naimark-duality dictates that it cannot be a commutative one. The question of how to define a non-commutative \(C^*\)-algebra as a replacement is left open in this paper. Instead, the author gives ad-hoc definitions of several objects which would normally be derived from this function algebra:
The author’s previous work [J. Noncommut. Geom. 13, No. 2, 553–586 (2019; Zbl 1429.58019)] already features the definition of a Hilbert space \(\underline{L^2(\mathcal{M},\mathcal{L}\otimes\mathcal{S})}\), a “Dirac operator” \(\mathcal{D}\) on this Hilbert space and a \(C^*\)-algebra \(\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}\), where the underline always tags substituting objects for the corresponding non-existing objects without underline. The “index element” \(\underline{\underline{[\mathcal{D}]}}\mathrel{{:}{=}}(\underline{L^2(\mathcal{M},\mathcal{L}\otimes\mathcal{S})},\mathcal{D})\) should be interpreted as some kind of Kasparov module over the yet-to-be-discovered function \(C^*\)-algebra \(C_0(\mathcal{M})\).
This paper continues by providing the construction of the “analytic index” \(\underline{\mathrm{ind}_{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}(\mathcal{D})}\) as a Kasparov module over \((\mathbb{C},\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}})\), a \(C^*\)-algebra \(\underline{LT\rtimes C_0(\mathcal{M})}\), a Kasparov module \(\underline{j_\tau^{LT}([\mathcal{D}])}\) over \((\underline{LT\rtimes C_0(\mathcal{M})},\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}))\) which plays the role of the image of \(\underline{\underline{[D]}}\) under a “partial descent homomorphism”, and a Kasparov module \(\underline{[c_{\mathcal{M}}]}\) over \((\mathbb{C},\underline{LT\rtimes C_0(\mathcal{M})})\) playing the role of the Mischenko line bundle for the action of \(LT\) on \(\mathcal{M}\). Now, the index theorem of this paper is the equality in \(KK(\mathbb{C},\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}})\) between the “image of the assembly map at the index element” \(\underline{\mu_\tau^{LT}([\mathcal{D}])}\mathrel{{:}{=}}\underline{[c_{\mathcal{M}}]}\otimes\underline{j_\tau^{LT}([\mathcal{D}])}\) and the analytic index \(\underline{\mathrm{ind}_{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}(\mathcal{D})}\).
More specifically, the infinite dimensional manifolds \(\mathcal{M}\) considered in this project carry proper actions of the loop group \(LT\mathrel{{:}{=}}C^\infty(S^1,T)\) of a circle group \(T\) and are assumed to be “even-dimensional and \(\mathrm{Spin}^c\)” in a certain sense. The index theory of interest is the one of the Dirac operator on \(\mathcal{L}\otimes\mathcal{S}\), where \(\mathcal{S}\to\mathcal{M}\) denotes the spinor bundle and \(\mathcal{L}\to\mathcal{M}\) an \(LT\)-equivariant line bundle which is \(\tau\)-twisted with respect to some \(U(1)\)-central extension \(LT^\tau\) of \(LT\).
At the core of the index theory should be a “function \(C^*\)-algebra” \(C_0(\mathcal{M})\), but as infinite dimensional manifolds are not locally compact Hausdorff spaces, the Gelfand–Naimark-duality dictates that it cannot be a commutative one. The question of how to define a non-commutative \(C^*\)-algebra as a replacement is left open in this paper. Instead, the author gives ad-hoc definitions of several objects which would normally be derived from this function algebra:
The author’s previous work [J. Noncommut. Geom. 13, No. 2, 553–586 (2019; Zbl 1429.58019)] already features the definition of a Hilbert space \(\underline{L^2(\mathcal{M},\mathcal{L}\otimes\mathcal{S})}\), a “Dirac operator” \(\mathcal{D}\) on this Hilbert space and a \(C^*\)-algebra \(\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}\), where the underline always tags substituting objects for the corresponding non-existing objects without underline. The “index element” \(\underline{\underline{[\mathcal{D}]}}\mathrel{{:}{=}}(\underline{L^2(\mathcal{M},\mathcal{L}\otimes\mathcal{S})},\mathcal{D})\) should be interpreted as some kind of Kasparov module over the yet-to-be-discovered function \(C^*\)-algebra \(C_0(\mathcal{M})\).
This paper continues by providing the construction of the “analytic index” \(\underline{\mathrm{ind}_{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}(\mathcal{D})}\) as a Kasparov module over \((\mathbb{C},\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}})\), a \(C^*\)-algebra \(\underline{LT\rtimes C_0(\mathcal{M})}\), a Kasparov module \(\underline{j_\tau^{LT}([\mathcal{D}])}\) over \((\underline{LT\rtimes C_0(\mathcal{M})},\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}))\) which plays the role of the image of \(\underline{\underline{[D]}}\) under a “partial descent homomorphism”, and a Kasparov module \(\underline{[c_{\mathcal{M}}]}\) over \((\mathbb{C},\underline{LT\rtimes C_0(\mathcal{M})})\) playing the role of the Mischenko line bundle for the action of \(LT\) on \(\mathcal{M}\). Now, the index theorem of this paper is the equality in \(KK(\mathbb{C},\underline{LT\mathbin{{\rtimes}_\tau}\mathbb{C}})\) between the “image of the assembly map at the index element” \(\underline{\mu_\tau^{LT}([\mathcal{D}])}\mathrel{{:}{=}}\underline{[c_{\mathcal{M}}]}\otimes\underline{j_\tau^{LT}([\mathcal{D}])}\) and the analytic index \(\underline{\mathrm{ind}_{LT\mathbin{{\rtimes}_\tau}\mathbb{C}}(\mathcal{D})}\).
Reviewer: Christopher Wulff (Göttingen)
MSC:
| 19K56 | Index theory |
| 58B34 | Noncommutative geometry (à la Connes) |
| 58J22 | Exotic index theories on manifolds |
| 22E67 | Loop groups and related constructions, group-theoretic treatment |
| 19K35 | Kasparov theory (\(KK\)-theory) |
Keywords:
infinite-dimensional manifolds; loop groups; Dirac operators; assembly maps; \( K K\)-theory; index theoryReferences:
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