TY - JOUR
A1 - Mickelsson, Jouko
A1 - Paycha, Sylvie
T1 - The logarithmic residue density of a generalized Laplacian
JF - Journal of the Australian Mathematical Society
N2 - We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyahâ€“Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbellâ€“Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.
KW - residue
KW - index
KW - Dirac operators
Y1 - 2011
U6 - http://dx.doi.org/10.1017/S144678871100108X
SN - 0263-6115
SN - 1446-8107
VL - 90
IS - 1
SP - 53
EP - 80
PB - Cambridge Univ. Press
CY - Cambridge
ER -
TY - JOUR
A1 - Levy, Cyril
A1 - Jimenez, Carolina Neira
A1 - Paycha, Sylvie
T1 - THE CANONICAL TRACE AND THE NONCOMMUTATIVE RESIDUE ON THE NONCOMMUTATIVE TORUS
JF - Transactions of the American Mathematical Society
N2 - Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp. any) noninteger order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace-class operators generalises Fathizadeh and Wong's characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the $ \zeta $-function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.
Y1 - 2016
U6 - http://dx.doi.org/10.1090/tran/6369
SN - 0002-9947
SN - 1088-6850
VL - 368
SP - 1051
EP - 1095
PB - American Mathematical Soc.
CY - Providence
ER -
TY - JOUR
A1 - Guo, Li
A1 - Paycha, Sylvie
A1 - Zhang, Bin
T1 - Conical zeta values and their double subdivision relations
JF - Advances in mathematics
N2 - We introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values. In order to achieve the closed cone subdivision relation, we also interpret linear relations among fractions as subdivisions of decorated closed cones. As a generalization of the double shuffle relation of multiple zeta values, we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values.
KW - Convex cones
KW - Conical zeta values
KW - Smooth cones
KW - Decorated cones
KW - Subdivisions
KW - Multiple zeta values
KW - Shuffles
KW - Quasi-shuffles
KW - Fractions with linear poles
KW - Shintani zeta values
Y1 - 2014
U6 - http://dx.doi.org/10.1016/j.aim.2013.10.022
SN - 0001-8708 (print)
SN - 1090-2082 (online)
VL - 252
SP - 343
EP - 381
PB - Elsevier
CY - San Diego
ER -