The \(\text{KO}\)-valued spectral flow for skew-adjoint Fredholm operators. (English) Zbl 1496.19002
The classical spectral flow is defined for a path of self-adjoint Fredholm operators with invertible end points and counts the number of eigenvalues that cross zero along the path. This article studies the spectral flow for paths of Fredholm operators on real Hilbert spaces with Clifford algebra symmetries, taking values in the \(\text{KO}\)-theory of a point instead of the integers. The \(K\)-theory for real \(C^*\)-algebras has received new attention recently because of its application in the study of symmetry-protected topological phases. Certain symmetries of physical systems such as time reversal are implemented by anti-unitary operators on a Hilbert space, and replace complex by real \(C^*\)-algebras. The spectral flow has already been used for such applications in certain situations, which creates the motivation to develop the theory more systematically.
The article is based on classical results describing the \(\text{KO}\)-theory of a point using modules over Clifford algebras and spaces of Fredholm operators that satisfy appropriate commutation relations with respect to a Clifford algebra representation. More precisely, the authors use a pair of complex structures \(J_0\) and \(J_1\) that anticommute with the Clifford generators and that satisfy \(\|J_0 - J_1\| <2\). They define an index for such a pair and give an alternative formula for it. Then they define the spectral flow for a continuous path of skew-adjoint Fredholm operators with invertible end points, anticommuting with Clifford generators. They compare their definition with previous definitions and treat some examples related to topological phases. They carry the definition over to paths of unbounded operators and carefully treat the continuity of paths of such operators.
Given various approaches to defining the spectral flow, an important observation in the article is that all reasonable definitions of it agree. Here “reasonable” means that the definition should be homotopy invariant, additive for concatenation of paths, and be normalised suitably. This allows to identify the spectral flow with other constructions, by proving that these have the relevant properties as well. This idea is used to identify the spectral flow with the index of a certain Fredholm operator built out of the path, generalising a formula by Robbin and Salamon. The spectral flow is also identified with a Kasparov product in bivariant \(K\)-theory.
The article is based on classical results describing the \(\text{KO}\)-theory of a point using modules over Clifford algebras and spaces of Fredholm operators that satisfy appropriate commutation relations with respect to a Clifford algebra representation. More precisely, the authors use a pair of complex structures \(J_0\) and \(J_1\) that anticommute with the Clifford generators and that satisfy \(\|J_0 - J_1\| <2\). They define an index for such a pair and give an alternative formula for it. Then they define the spectral flow for a continuous path of skew-adjoint Fredholm operators with invertible end points, anticommuting with Clifford generators. They compare their definition with previous definitions and treat some examples related to topological phases. They carry the definition over to paths of unbounded operators and carefully treat the continuity of paths of such operators.
Given various approaches to defining the spectral flow, an important observation in the article is that all reasonable definitions of it agree. Here “reasonable” means that the definition should be homotopy invariant, additive for concatenation of paths, and be normalised suitably. This allows to identify the spectral flow with other constructions, by proving that these have the relevant properties as well. This idea is used to identify the spectral flow with the index of a certain Fredholm operator built out of the path, generalising a formula by Robbin and Salamon. The spectral flow is also identified with a Kasparov product in bivariant \(K\)-theory.
Reviewer: Ralf Meyer (Göttingen)
MSC:
| 19K56 | Index theory |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 81T75 | Noncommutative geometry methods in quantum field theory |
Keywords:
Fredholm index; spectral flow; Clifford algebra; \(\text{KO}\)-theory; \(\text{KK}\)-theory; Fredholm pairReferences:
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