The conformal factor and a central extension of a formal loop group with values in \(\text{PSL}(2,\mathbb{R})\) (Q1310528)

For a stationary and axially symmetric space-time we can take coordinates \(x_ 0\), \(x_ 1\), \(z\), \(\rho\) such that a metric is: \(ds^ 2 = \sum h_{pq} dx^ p dx^ q - \lambda^ 2 (dz^ 2 + d\rho^ 2)\) where \(h_{pq}\) and \(\lambda(>0)\) are functions of \(z\), \(\rho\) (\(\lambda\), and also \(\tau = \lambda \sqrt{h_{00}}\), is called the conformal factor). These functions satisfy the Einstein vacuum field equations. Let \({\mathcal S}E\) be the set of formal solutions of the equation for the matrix \(h\). This set is equivalent to a solution space \({\mathcal S} P\) lying in \(AN(F)\) where \(AN\) means the connected lower triangular subgroup of \(\text{SL}(2,\mathbb{R})\) and \(F = \mathbb{R}[[z,\rho]]\). For any element of \({\mathcal S}P\) there exists a unique \(\tau \in F\) (up to a positive multiplicative constant). A procedure of a linearization leads to the formal loop group \({\mathcal F}{\mathcal G}\) and a potential space \({\mathcal S}{\mathcal P}\). They are defined as follows. Endow \(F\) with the two-sided filtration \(\{F_ l = \rho^{| l|} F,\;l \in \mathbb{Z}\}\). Then \({\mathcal F}{\mathcal G}\) is the set \(\{g(t) = \sum g_ l t^ l\), \(l \in \mathbb{Z}\), \(g_ l \in {\mathfrak gl}(2,F_ l)\), \(\text{det }g(t) = 1\}/\pm I_ 2\), and \({\mathcal S}{\mathcal P}\) is a subset of the subgroup \(\{g_ l = 0\) for \(l < 0\}\) of \({\mathcal F}{\mathcal G}\) given by some linear equations. The spaces \({\mathcal S}{\mathcal P}\) and \({\mathcal S}P\) are linked by the projection \(p(t) \mapsto p_ 0\) (this map is not surjective). The group \({\mathcal G}^{(\infty)} = \text{PSL}(2, \mathbb{R} [[s]])\) is called the Hauser group. It is embedded into \({\mathcal F}{\mathcal G}\) by \(s = \rho(t^{-1} - t) + 2z\). Let us consider a symmetric space (an analogue of the Lobachevsky plane) \({\mathcal F}{\mathcal G}/{\mathcal F}{\mathcal K}\) where \({\mathcal F}{\mathcal K}\) is the fixed point subgroup of the ``Cartan involution'' \(\theta^{(\infty)}\): \(g(t) \to g(-1/t)^{\prime-1}\) (the prime denotes matrix transposition). The action of \({\mathcal F}{\mathcal G}\) on this symmetric space gives rise to a transitive action of the Hauser group on \({\mathcal S}{\mathcal P}\). A similar construction works for central extensions. A central extension of \({\mathcal F}{\mathcal G}\) by \(F\) is given by a 2-cocycle \(\Xi\) with values in \(F\). The centrally extended potential space \(\Gamma({\mathcal S}{\mathcal P})\) consists of pairs \((p(t),\tau^{-1})\) where \(\tau \in F\) corresponds to \(p(t) \in {\mathcal S}{\mathcal P}\). The main result of the paper claims that the Hauser group acts on \(\Gamma({\mathcal S}{\mathcal P})\) transitively. It implies that \(\tau\) is expressed as the value of the cocycle at the corresponding potential: \(\tau = \text{exp }{1\over 2} \Xi (\theta^{(\infty)} (p(t)^{-1}), p(t))\).