In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed. The Quillen metric was used by Quillen to give a differential-geometric interpretation of the ample line bundle over the moduli space of vector bundles on a compact Riemann surface, known as the Quillen determinant line bundle. It can be seen as defining the Chern–Weil representative of the first Chern class of this ample line bundle. The Quillen metric construction and its generalizations were used by Bismut and Freed to compute the holonomy of certain determinant line bundles of Dirac operators, and this holonomy is associated to certain anomaly cancellations in Chern–Simons theory predicted by Edward Witten. The Quillen metric was also used by Simon Donaldson in 1987 in a new inductive proof of the Hitchin–Kobayashi correspondence for projective algebraic manifolds, published one year after the resolution of the correspondence by Shing-Tung Yau and Karen Uhlenbeck for arbitrary compact Kähler manifolds.
Determinant line bundle of a family of operators Suppose D t {\displaystyle D_{t}} are a family of Fredholm operators D t : V → W {\displaystyle D_{t}:V\to W} between Hilbert spaces, varying continuously with respect to t ∈ X {\displaystyle t\in X} for some topological space X {\displaystyle X} . Since each of these operators is Fredholm, the kernel and cokernel are finite-dimensional. Thus there are assignments
t ↦ ker D t , t ↦ coker D t {\displaystyle t\mapsto \ker D_{t},\quad t\mapsto {\text{coker}}D_{t}}
which define families of vector spaces over X {\displaystyle X} . Despite the assumption that the operators D t {\displaystyle D_{t}} vary continuously in t {\displaystyle t} , these assignments of vector spaces do not form vector bundles over the topological space X {\displaystyle X} , because the dimension of the kernel and cokernel may jump discontinuously for a family of differential operators. However, the index of a differential operator, the dimension of the kernel subtracted by the dimension of the cokernel, is an invariant up to continuous deformations. That is, the assignment
t ↦ ind ( D t ) := dim ker D t − dim coker D t {\displaystyle t\mapsto {\text{ind}}(D_{t}):=\dim \ker D_{t}-\dim {\text{coker}}D_{t}}
is a constant function on X {\displaystyle X} . Since it is not possible to take a difference of vector bundles, it is not possible to combine the families of kernels and cokernels of D t {\displaystyle D_{t}} into a vector bundle. However, in the K-theory of X {\displaystyle X} , formal differences of vector bundles may be taken, and associated to the family D t {\displaystyle D_{t}} is an element
ind ( D t ) = [ t ↦ ker D t − coker D t ] ∈ K ( X ) . {\displaystyle {\text{ind}}(D_{t})=[t\mapsto \ker D_{t}-{\text{coker}}D_{t}]\in K(X).}
This virtual index bundle contains information about the analytical properties of the family D t {\displaystyle D_{t}} , and its virtual rank, the difference of dimensions, may be computed using the Atiyah–Singer index theorem, provided the operators D t {\displaystyle D_{t}} are elliptic differential operators. Whilst the virtual index bundle is not a genuine vector bundle over the parameter space X {\displaystyle X} , it is possible to pass to a genuine line bundle constructed out of ind ( D t ) {\displaystyle {\text{ind}}(D_{t})} . For any t {\displaystyle t} , the determinant line of D t : V → W {\displaystyle D_{t}:V\to W} is defined as the one-dimensional vector space
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