In differential geometry, the Ooguri–Vafa metric is a four-dimensional Hyperkähler metric. The Ooguri–Vafa metric is named after Hirosi Ooguri and Cumrun Vafa, who first described it in 1996 using the Gibbons–Hawking ansatz. Another construction was given by Davide Gaiotto, Gregory Moore and Andrew Neitzke in 2008.
Definition The Ooguri–Vafa metric is defined on the four-dimensional total spaces of principal U(1)-bundles over open subsets of the three-dimensional euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} . In particular the whole space results in R 3 × S 1 {\displaystyle \mathbb {R} ^{3}\times S^{1}} . Define the elliptical fibers τ ( z ) = 1 2 π i log ( z ) {\displaystyle \tau (z)={\frac {1}{2\pi i}}\log(z)} with τ 1 = Re ( τ ( z ) ) {\displaystyle \tau _{1}=\operatorname {Re} (\tau (z))} and τ 2 = Im ( τ ( z ) ) {\displaystyle \tau _{2}=\operatorname {Im} (\tau (z))} and let λ {\displaystyle \lambda } be the string coupling constant. Further define the scaled spatial coordinate
y = ( x , z λ , z ¯ λ ) {\displaystyle \mathbf {y} =\left(x,{\frac {z}{\lambda }},{\frac {\bar {z}}{\lambda }}\right)} . The metric of Ooguri and Vafa has the form
d s 2 = λ 2 [ V − 1 ( d t − A ⋅ d y ) 2 + V d y 2 ] {\displaystyle ds^{2}=\lambda ^{2}[V^{-1}(dt-\mathbf {A} \cdot d\mathbf {y} )^{2}+Vd\mathbf {y} ^{2}]}
where A = ( A x , A z , A z ¯ ) {\displaystyle \mathbf {A} =(A_{x},A_{z},A_{\bar {z}})} and
A x = − τ 1 = i 4 π log ( z z ¯ ) , A z = 0 , A z ¯ = 0 {\displaystyle A_{x}=-\tau _{1}={\frac {i}{4\pi }}\log \left({\frac {z}{\bar {z}}}\right),\quad A_{z}=0,\quad A_{\bar {z}}=0}
and V {\displaystyle V} is a potential which gets modified from the form V = τ 2 = 1 4 π log ( 1 z z ¯ ) {\displaystyle V=\tau _{2}={\frac {1}{4\pi }}\log \left({\frac {1}{z{\bar {z}}}}\right)} .
Requirements for the potential There are 5 requirements for the potential V {\displaystyle V} :
V {\displaystyle V} should be a function of only x {\displaystyle x} and | z | {\displaystyle |z|} , i.e. V ( x , | z | ) {\displaystyle V(x,|z|)} for | z | = z z ¯ {\displaystyle |z|={\sqrt {z{\bar {z}}}}} . For the metric to be a hyperkähler metric, the following conditions must be met:
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