Research output: Contribution to journal › Article › peer-review
Local index theorem for orbifold Riemann surfaces. / Takhtajan, L.A.; Zograf, Peter .
In: Letters in Mathematical Physics, Vol. 109, No. 5, 01.05.2019, p. 1119-1143.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Local index theorem for orbifold Riemann surfaces
AU - Takhtajan, L.A.
AU - Zograf, Peter
PY - 2019/5/1
Y1 - 2019/5/1
N2 - We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.
AB - We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.
KW - Determinant line bundles
KW - Fuchsian groups
KW - Local index theorems
KW - Quillen’s metric
KW - \ Fuchsian groups
KW - Quillen's metric
UR - http://www.scopus.com/inward/record.url?scp=85058374048&partnerID=8YFLogxK
UR - http://www.mendeley.com/research/local-index-theorem-orbifold-riemann-surfaces
U2 - 10.1007/s11005-018-01144-w
DO - 10.1007/s11005-018-01144-w
M3 - Article
VL - 109
SP - 1119
EP - 1143
JO - Letters in Mathematical Physics
JF - Letters in Mathematical Physics
SN - 0377-9017
IS - 5
ER -
ID: 41877343