Hadjicostas's formula: Difference between revisions

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{{Orphan|date=December 2012}}
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{{unreferenced|date=November 2013}}
 
In mathematics, the '''Bismut connection <math>\nabla</math>''' is the unique [[connection (mathematics)|connection]] on a complex manifold that satisfies the following conditions,
# It preserves the metric <math>\nabla g =0</math>
# It preserves the complex structure <math>\nabla J=0</math>
# The [[torsion (algebra)|torsion]] <math>T(X,Y)</math> contracted with the metric, i.e. <math>T(X,Y,Z)=g(T(X,Y),Z)</math>, is totally [[skew-symmetric]].
[[Bismut]] has used this connection when proving a local index formula for the Dolbeault operator on non-[[Kähler manifold]]s.  Bismut connection has applications in type II and heterotic string theory.
 
The explicit construction goes as follows. Let <math>\langle-,-\rangle</math> denote the pairng of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. <math>\langle X,JY\rangle=-\langle JX,Y\rangle</math>. Further let <math>\nabla</math> be the Levi-Civita connection. Define first a tensor <math>T</math> such that <math>T(Z,X,Y)=-\frac12\langle Z,(\nabla_{X}J)Y\rangle </math>. It is easy to see that this tensor is anti-symmetric in the first and last entry, i.e. the new connection <math>\nabla+T</math> still preserves the metric. In concrete terms, the new connection is given by <math>\Gamma^{\alpha}_{\beta\gamma}-\frac12 J^{\alpha}_{~\delta}\nabla_{\beta}J^{\delta}_{~\gamma}</math> with <math>\Gamma^{\alpha}_{\beta\gamma}</math> being the Levi-Civita connection. It is also easy to see that the new connection preserves the complex structure. However, the tensor <math>T</math> is not yet totall anti-symmetric, in fact the anti-symmetrization will lead to the [[Nijenhuis tensor]]. Denote the anti-symmetrization as <math>T(Z,X,Y)+\textrm{cyc~in~}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)</math>, with <math>S</math> given explicitly as
 
:<math>S(Z,X,Y)=-\frac12\langle X,J(\nabla_{Y}J)Z\rangle-\frac12\langle Y,J(\nabla_{Z}J)X\rangle.</math>  
We show that <math>S</math> still preserves the complex structure (that it preserves the metric is easy to see), i.e. <math>S(Z,X,JY)=-S(JZ,X,Y)</math>.
 
:<math>\begin{align}
S(Z,X,JY)+S(JZ,X,Y)&=-\frac12\langle JX, \big(-(\nabla_{JY}J)Z-(J\nabla_ZJ)Y+(J\nabla_YJ)Z+(\nabla_{JZ}J)Y\big)\rangle\\
&=-\frac12\langle JX, Re\big((1-iJ)[(1+iJ)Y,(1+iJ)Z]\big)\rangle.\end{align}</math>
So if <math>J</math> is integrable, then above term vanishes, and the connection
 
:<math>\Gamma^{\alpha}_{\beta\gamma}+T^{\alpha}_{~\beta\gamma}+S^{\alpha}_{~\beta\gamma}.</math>
gives the Bismut connection.
 
[[Category:Complex manifolds]]
 
 
{{differential-geometry-stub}}

Latest revision as of 17:38, 11 July 2014

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