### 3.4 Reduced density matrix and entropy

The density matrix defined by tracing over -modes is given by the Euclidean path
integral over field configurations on the complete instanton with a cut along the axis
where the field in the path integral takes the values and below and above
the cut respectively. The trace is obtained by equating the fields across the cut and doing the
unrestricted Euclidean path integral on the complete Euclidean instanton . Analogously, is given
by the path integral over field configurations defined on the n-fold cover of the complete instanton.
This space is described by the metric (50) where angular coordinate is periodic with period
. It has a conical singularity on the surface so that in a small vicinity of the
total space is a direct product of and a two-dimensional cone with angle deficit
. Due to the abelian isometry generated by the Killing vector this construction can be
analytically continued to arbitrary (non-integer) . So that one can define a partition function
by the path integral over field configurations over , the -fold cover of the instanton . For a
bosonic field described by the field operator one has that . Defining the effective
action as , the entanglement entropy is still given by formula (16), i.e., by
differentiating the effective action with respect to the angle deficit. Clearly, only the term linear in
contributes to the entropy. Thus, the problem reduces to the calculation of this term in the
effective action.