index of a family of Dirac operators in $K^1$ Suppose I have a family of Dirac operators over a compact base space B. From the paper of Atiyah and Singer about skew adjoint Fredholm operators we know that it has an index in $K^1(B)$.
Suppose furthermore I know "a lot" about these Dirac operators (like their spectrum, eigenspaces etc.) and $B$ is a simple space like e.g. a torus where I know everything about $K^1$ and the cohomology.
What methods are there to give an explicit description of the index in $K^1(B)$ or its image in the odd-dimensional cohomology?
Any suggestions or references welcome.
 A: Whether the following is useful might depend on your concrete example. Because you are mentioning $K^1 $ instead of $K^0$, I assume that your Dirac operator is ungraded (if it is graded, the index should be in $K^0$). The graded case is the ordinary Atiyah-Singer family index theorem. There is a version for the ungraded case, with index in $K^1$, which is expressed by the same formula, except that the symbol class is in $K^1 $ and its definition is slightly more complicated than in the $K^0$-case. This is explained in Atiyah-Patodi-Singer, "Spectral asymmetry and Riemann geometry III" (you do not have to read parts I and II, but only one section of part III, nevertheless, these are beautiful papers). 
There is a cohomological formula that can be derived from the K-theory formula, much in the same way as in the ordinary $K^0$-case.
In other words, the family index in $K^1$ is as computable as the ordinary one. By the way: if you take a graded operator, you get - forget the grading - an ungraded operator. Its $K^1$-index is always zero.
If your base space is a circle and if you really know the spectrum of the operator at any point of $B$, there is a nice spectral-theoretic way for computing the index, by the so-called "spectral flow". The spectral flow of a map $f: S^1 \to Fred_{sa}$ is the geometric intersection number of $f$ with the sub"manifold" of noninvertible operators in the space of selfadjoint Fredholms. Sounds complicated, but what you have to do is to count how often an eigenvalue crosses zero. A nice explanation of the spectral flow is in Booss-Woichechowski's book "Boundary value problems for Dirac-operators". 
If $B$ is a $2$-torus, then $K^1 (B)$ should be $Z^2$, the factor being detected on the two circles, and this reduces the problem to the case $B=S^1$.
A last comment: if some miracle happens and the dimension of the kernel is constant, then your computation is VERY easy. The family index in $K^1 (B)$ will be zero in that case.
I wrote down a simple proof of this fact in http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.4719v3.pdf, Theorem 4.2.1.
A: Two years ago my student  Daniel Cibotaru wrote a dissertation   entitled
Localization formulae in odd  K-theory
in which  he answers precisely this question in great generality. 
More precisely, given a smooth family $(T_b)_{b\in B}$ of complex, Fredholm selfadjoint  operators  parametrized by a  compact, connected, oriented smooth manifold  $B$   he describes  explicitely  a  (stratified) cycle in $B$  that is Poincare dual to the odd  Chern character of this family. This cycle   is non-homogeneous, i.e., it is a sum of cycles of various codimensions.  The codimension $1$-part is  the so called Maslov class or spectral flow class and it is more or less known.   Daniel has a very nice description of the codimension $3$-part.  For example if $B$ is a $3$-manifold, then the family  defines a cohomology class in $H^3(B,\mathbb{Z})\cong \mathbb{Z}$ and  Daniel  explains how to compute this as a signed count of points in $B$.    
More precisely, for a generic family $(T_b)_{b\in B}$, the locus   of points $b$  where $\ker T_b\neq 0$ is  an oriented  surface $S\subset B$  and  the family of vector spaces
$$ S\ni s\mapsto  \ker T_s $$
is a complex line bundle  over $S$.The above integer is none other than the degree of this complex line bundle.
The higher codimension parts have explicit but  rather complicated descriptions.
The philosophy of his dissertation can be  easily described: he constructs a smooth model  for  the classifying  space of $K^1$. This is an infinite dimensional Grassmanian-like object $\mathcal{X}$ equipped with a Schubert-like  stratification by strata of finite  codimensions. He then shows that the  closures  of  the Schubert strata determine    cohomology classes forming an integral  basis of the cohomology of $\mathcal{X}$.  The whole thing has a  strong symplectic flavor.
