 
Summary: qinvariants and determinant lines
Xianzhe Dais)
Department of Mathematics, University of Southern California, Los Angeles,
California 90089
Daniel S. Freedb)
Department of Mathematics, Universiiy of Texas at Austin, Austin, Texas 78712
(Received 30 April 1994; accepted for publication 17 May 1994)
The pinvariant of an odd dimensional manifold with boundary is investigated. The
natural boundary condition for this problem requires a trivialization of the kernel of
the Dirac operator on the boundary. The dependence of the Tinvariant on this
trivialization is best encoded by the statement that the exponential of the
qinvariant lives in the determinant line of the boundary. Our main results are a
variational formula and a gluing law for this invariant. These results are applied to
reprove the formula for the holonomy of the natural connection on the determinant
line bundle of a family of Dirac operators, also known as the "global anomaly
formula." The ideas developed here fit naturally with recent work in topological
quantum field theory, in which gluing (which is a characteristic formal property of
the path integral and the classical action) is used to compute global invariants on
closed manifolds from local invariants on manifolds with boundary.
The qinvariant was introduced by Atiyah, Patodi, and Singer (APS)' in a series of papers
