4 Thermostats from Vortices

4.1 The vortex equations

Let \((M,g)\) be a closed oriented Riemannian \(2\)-manifold of negative Euler characteristic and \(\nu : L \to M\) a complex line bundle of positive degree. For a triple consisting of a Hermitian bundle metric \(\mathcal{h}\) on \(L\), a del-bar operator \(\overline{\partial}_L\) on \(L\), and a \((1,\! 0)\)-form \(\varphi\) on \(M\) with values in \(L\), we consider the following pair of equations \[R(\mathrm{D})+\frac{1}{2}\varphi\wedge\varphi^*+\mathrm{i}\ell\Omega_g=0 \qquad \text{and} \qquad \overline{\partial}_L\varphi=0.\] Here we write \(\ell:=\deg(L)/|\chi(M)|\), \(\mathrm{D}\) denotes the Chern connection on \(L\) with respect to \((\mathcal{h},\overline{\partial}_L)\), \(R(\mathrm{D})\) its curvature, \(\Omega_g\) the area form of \(g\) and the \(1\)-form \(\varphi^*\) with values in the dual \(L^{-1}\) of \(L\) is defined by \[\varphi^*(v)(\xi):=\mathcal{h}(\xi,\varphi(v))\] for all \(x \in M\), \(v \in TM\) and \(\xi \in \nu^{-1}(\{x\})\). We assume \(\mathcal{h}\) to be conjugate linear in the second variable, so that \(\varphi^*\) is an \(L^{-1}\)-valued \((0,\! 1)\)-form. We extend the wedge-product to bundle-valued forms in the standard way, so that for \(\varphi \in \Omega^{1}(L)\) and \(\varrho \in \Omega^1(L^{-1})\), we have \[(\varphi\wedge\varrho)(v,w)=\varrho(w)\varphi(v)-\varrho(v)\varphi(w)\] for all \(x \in M\) and \(v,w \in T_xM\). In particular, we obtain \[\begin{aligned} \left(\varphi\wedge\varphi^*\right)(v,w)&=\mathcal{h}(\varphi(v),\varphi(w))-\mathcal{h}(\varphi(w),\varphi(v))\\ &=\mathcal{h}(\varphi(v),\varphi(w))-\overline{\mathcal{h}(\varphi(v),\varphi(w))}=2\mathrm{i}\operatorname{Im}\mathcal{h}(\varphi(v),\varphi(w))\end{aligned}\] so that \(\varphi\wedge\varphi^*\) is a purely imaginary \((1,\! 1)\)-form on \(M\).

The complex gauge group \(\mathrm{G}_{\mathbb{C}}\) of \(L\) is the group of automorphisms of \(L\) (covering the identity on \(M\)) and the gauge group \(\mathrm{G}\) of \((L,\mathcal{h})\) consists of the automorphisms of \(L\) that are unitary with respect to \(\mathcal{h}\). Since an automorphism of a one-dimensional complex vector space is just a non-vanishing complex number, we have \(\mathrm{G}_{\mathbb{C}}\simeq C^{\infty}(M,\mathbb{C}^*)\) and \(\mathrm{G}\simeq C^{\infty}(M,\mathrm{U}(1))\), the smooth functions on \(M\) with values in the one-dimensional unitary group \(\mathrm{U}(1)\). An element \(\tau \in \mathrm{G}_{\mathbb{C}}\) acts on a Hermitian bundle metric \(\mathcal{h}\) on \(L\) by the rule \[\tag{4.1} \tau\cdot \mathcal{h}=|\tau|^2\mathcal{h}\] and on \(\varphi \in \Omega^{p,q}(L)\) by the rule \[\tag{4.2} \tau\cdot \varphi=\tau^{-1}\varphi.\] We define an action on the space of del-bar operators on \(L\) by \[\tag{4.3} \tau\cdot \overline{\partial}_L=\overline{\partial}_L+\tau^{-1}\overline{\partial}\tau.\] Writing \(\mathrm{D}_{\mathcal{h},\overline{\partial}_L}\) for the Chern connection on \(L\) determined by the Hermitian metric \(\mathcal{h}\) and del-bar operator \(\overline{\partial}_L\), we obtain:

We now have:

4.2 The vortex equations on a root of \(SM\)

Since \(L\) has positive degree and \(\chi(M)<0\), there exist unique positive coprime integers \((m,n)\) so that we have an isomorphism \(L^n\simeq K^m\) of complex line bundles. We fix an \(n\)-th root \(SM^{1/n}\) of the unit tangent bundle \(SM\) of \((M,g)\) and let \(K^{1/n}\) denote the corresponding \(n\)-th root of \(K\), so that we have an isomorphism \(\mathcal{Z} : L\to K^{m/n}\) of complex line bundles. Note that such a root exists since \(n\) divides \(\chi(M)\). We equip \(SM^{1/n}\) with the generalised Riemannian structure \((X,H,\mathbb{V})\) as in Example 2.3. We may write \(\mathcal{h}=e^{2f}\mathcal{h}_0\) for a unique smooth real-valued function \(f\) on \(M\). Abusing notation, we also use the letter \(f\) to denote the pullback of \(f\) to \(SM^{1/n}\). Recall that the space of del-bar operators on a line bundle \(L \to M\) is an affine space modelled on \(\Omega^{0,1}\). Therefore, without loosing generality, we can assume that there exists a \(1\)-form \(\theta\) on \(M\) so that \[\tag{4.4} \overline{\partial}_L=\overline{\partial}_{K^{m/n}}-\ell\,\theta^{0,1},\] where \(\theta^{0,1}=\frac{1}{2}(\theta-\mathrm{i}\star_g\theta) \in \Omega^{0,1}\) denotes the \((0,\! 1)\)-part of \(\theta\) and \(\star_g\) the Hodge-star with respect to \(g\). We may also think of \(\theta\) as a real-valued function on \(SM\) and abusing notation, we also write \(\theta\) to denote its pullback to \(SM^{1/n}\). Note that the function \(\theta\) on \(SM^{1/n}\) satisfies \(\mathbb{V}\mathbb{V}\theta=-\theta\). The pullback of \(\theta^{0,1}\) to \(SM^{1/n}\) can be expressed as \(\frac{1}{2}(\theta+\mathrm{i}\mathbb{V}\theta)\overline{\omega}\), where we write \(\omega=\omega_1+\mathrm{i}\omega_2\) and \(\overline{\omega}=\omega_1-\mathrm{i}\omega_2\). Therefore, the connection form \(\zeta\) on \(SM^{1/n}\) of the Chern connection \(\mathrm{D}\) of \((L,\overline{\partial}_L,\mathcal{h})\) can be written as \[\zeta=-\mathrm{i}\ell\psi+w\omega-\frac{\ell}{2}(\theta+\mathrm{i}\mathbb{V}\theta)\overline{\omega}\] for some unique complex-valued function \(w\) on \(SM^{1/n}\). On \(SM^{1/n}\), the condition that \(\mathrm{D}\) preserves \(\mathcal{h}=e^{2f}\mathcal{h}_0\) translates to \[\mathrm{d}\left(e^{2f}\boldsymbol{B}_1\overline{\boldsymbol{B}_2}\right)=e^{2f}\Big((\mathrm{d}\boldsymbol{B}_1+\zeta\boldsymbol{B}_1)\overline{\boldsymbol{B}_2}+\boldsymbol{B}_1(\mathrm{d}\overline{\boldsymbol{B}_2}+\overline{\zeta}\overline{\boldsymbol{B}_2})\Big)\] where \(\boldsymbol{B}_1,\boldsymbol{B}_2\) represent arbitrary smooth sections of \(L\). A straighforward calculation yields \[\zeta=-\mathrm{i}\ell\psi+\left(\frac{\ell}{2}(\theta-\mathrm{i}\mathbb{V}\theta)+Xf-\mathrm{i}Hf\right)\omega-\frac{\ell}{2}(\theta+\mathrm{i}\mathbb{V}\theta)\overline{\omega}.\] The \((1,\! 0)\)-form \(\varphi\) with values in \(L\) is a section of \(K\otimes L\simeq K^{(n+m)/n}\), so that on \(SM^{1/n}\) the form \(\varphi\) is represented by a complex-valued \(1\)-form \(\boldsymbol{\varphi}\), which we may write as \[\boldsymbol{\varphi}=\ell\left(\frac{\mathbb{V}a}{1+\ell}+\mathrm{i}a\right)\omega,\] where the real-valued function \(a\) satisfies \(\mathbb{V}\mathbb{V}a=-(1+\ell)^2 a\), since \(\ell=m/n\).

Writing \[\boldsymbol{A}:=\frac{\mathbb{V}a}{1+\ell}+\mathrm{i}a,\] we obtain:

4.3 Fractional differentials

Note that we may think of \(\varphi/\ell\) as a section of \(K\otimes L\simeq K^{(n+m)/n}\) which we denote by \(A\). Thus, we may interpret \(A\) as a differential of fractional degree \((n+m)/n=1+\ell\). Recall that the choice of an \(n\)-th root \(SM^{1/n}\) of \(SM\) equips \(K^{(n+m)/n}\) with a Hermitian bundle metric which we denote by \(\mathcal{h}_0\). Defining \(|A|^2_g:=\mathcal{h}_0(A,A)\), the pullback of the function \(|A|^2_g\) to \(SM^{1/n}\) is \(|\boldsymbol{A}|^2\). Moreover, the co-differential \(\delta_g\theta\) of \(\theta\) with respect to \(g\) pulls-back to \(SM^{1/n}\) to become \(-X\theta-H\mathbb{V}\theta\) and the Laplacian \(\Delta_g f\) of \(f\) with respect to \(g\) pulls-back to \(SM^{1/n}\) to become \(XXf + HHf\). Using this notation, the equation (4.6) can be written as \[K_g-\delta_g\theta=-1+\ell e^{2f}|A|^2_g-\frac{1}{\ell}\Delta_g f.\] Observe also that since \(\overline{\partial}_L\varphi=0\), the equation (4.4) implies \[\overline{\partial}_{K^{1+\ell}}A=\ell\,\theta^{0,1}\otimes A.\]

4.4 The thermostat

In order to associate a thermostat on \(SM^{1/n}\) to a solution of the vortex equation, we first consider as a motivating example the case \(L=K^2\). In this case \(n=1\) and \(m=2\) so that no choice of a root of \(SM\) is necessary. We may take \(\overline{\partial}_L\) to be the del-bar operator on \(K^2\) induced by the metric \(g\), that is, we choose \(\theta\) to vanish identically. Furthermore we choose \(\mathcal{h}\) to be \(\mathcal{h}_0\) so that \(f\) vanishes identically as well. Thinking of \(\varphi\) as a section of \(K\otimes L\simeq K^3\), we obtain a cubic differential \(A\), and the vortex equations become \[K_g=-1+2|A|^2_g\qquad\text{and}\qquad \overline{\partial}_{K^3} A=0.\] In particular, the cubic differential \(A\) is holomorphic with respect to the standard holomorphic line bundle structure on \(K^3\). Now observe that \(L\) admits a square root \(L^{1/2}\simeq K\) and hence we may interpret \(\varphi/2\) as a section of \(K\otimes \mathrm{Hom}(L^{-1/2},L^{1/2})\). Using the Hermitian metric induced by \(\mathcal{h}_0\) on \(L^{1/2}\simeq K\), we may identify \(L^{1/2}\simeq \overline{L^{-1/2}}\). As a real vector bundle \(\overline{L^{-1/2}}\) is isomorphic to \(L^{-1/2}\). Therefore, we may interpret \(\varphi/2\) as a \(1\)-form on \(M\) with values in the endomorphisms of \(L^{-1/2}\), thought of as a real vector bundle. Identifying \(\mathbb{C}\simeq \mathbb{R}^2\) in the usual way, multiplication with the complex number \(z\), thought of as a linear map \(\mathbb{R}^2 \to \mathbb{R}^2\), has matrix representation \[\begin{pmatrix} \operatorname{Re}z & - \operatorname{Im}z \\ \operatorname{Im}z & \operatorname{Re}z \end{pmatrix}\] with respect to the standard basis of \(\mathbb{R}^2\). Taking into account the identification \(L^{1/2}\simeq \overline{L^{-1/2}}\), which just amounts to complex conjugation, the \(1\)-form \(\varphi/2\) is thus represented by \[\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} \operatorname{Re}\boldsymbol{\varphi} & -\operatorname{Im}\boldsymbol{\varphi} \\ \operatorname{Im}\boldsymbol{\varphi} & \operatorname{Re}\boldsymbol{\varphi}\end{pmatrix}=\frac{1}{2}\begin{pmatrix} \operatorname{Re}\boldsymbol{\varphi} & -\operatorname{Im}\boldsymbol{\varphi}\\ -\operatorname{Im}\boldsymbol{\varphi} & -\operatorname{Re}\boldsymbol{\varphi}\end{pmatrix}.\] The Chern connection on \(L\) induces a connection on \(L^{-1/2}\) whose connection form is \(-(1/2)\zeta\). Adding \(\varphi/2\) to this connection, thought of as a connection on the real vector bundle \(L^{-1/2}\), we obtain a connection \(\nabla\) with connection form \[\Upsilon=(\Upsilon^i_j)=-\frac{1}{2}\begin{pmatrix} \operatorname{Re}(\zeta-\boldsymbol{\varphi}) & -\operatorname{Im}(\zeta-\boldsymbol{\varphi}) \\ \operatorname{Im}(\zeta+\boldsymbol{\varphi}) & \operatorname{Re}(\zeta+\boldsymbol{\varphi}) \end{pmatrix}\] Since \(L^{-1/2}\simeq K^{-1}\), the vector bundle \(L^{-1/2}\), as a real vector bundle, is isomorphic to the tangent bundle of \(M\). Thus \(\Upsilon\) defines a connection \(\nabla\) on \(TM\) and in [26] it is shown that the orbits of the thermostat \(\phi\) on \(SM\) defined by the condition \(F \hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\Upsilon^2_1=0\) project to \(M\) to become the geodesics of \(\nabla\), when ignoring the parametrisation.

In general \(L\) will not admit a square root, but we may nonetheless formally carry out the same construction, except that now the identification \(L^{1/2}\simeq \overline{L^{-1/2}}\) needs to amount for the metric \(e^f\mathcal{h}_0\) induced by \(\mathcal{h}\) on the formal root \(L^{1/2}\). We may thus define \[\tag{4.7} \begin{aligned} \Upsilon=(\Upsilon^i_j)&=-\frac{1}{2}\begin{pmatrix} \operatorname{Re}\zeta & -\operatorname{Im}\zeta \\ \operatorname{Im}\zeta & \operatorname{Re}\zeta\end{pmatrix}+\frac{1}{2}\begin{pmatrix} e^f & 0 \\ 0 & -e^f\end{pmatrix}\begin{pmatrix} \operatorname{Re}\boldsymbol{\varphi} & -\operatorname{Im}\boldsymbol{\varphi} \\ \operatorname{Im}\boldsymbol{\varphi} & \operatorname{Re}\boldsymbol{\varphi}\end{pmatrix}\\ &=-\frac{1}{2}\begin{pmatrix} \operatorname{Re}(\zeta-e^f\boldsymbol{\varphi}) & -\operatorname{Im}(\zeta-e^f\boldsymbol{\varphi}) \\ \operatorname{Im}(\zeta+e^f\boldsymbol{\varphi}) & \operatorname{Re}(\zeta+e^f\boldsymbol{\varphi}) \end{pmatrix}. \end{aligned}\] Note that the vortex equations can be written as \[\tag{4.8} \mathrm{d}\zeta=\frac{\ell}{2}\omega\wedge\overline{\omega}-\frac{1}{2}e^{2f}\boldsymbol\varphi\wedge\overline{\boldsymbol{\varphi}}\qquad\text{and}\qquad \mathrm{d}\boldsymbol{\varphi}=-\zeta\wedge\boldsymbol{\varphi}.\] We also obtain \[\mathrm{d}\omega=\left(\zeta/\ell+\frac{1}{2}(\theta+\mathrm{i}\mathbb{V}\theta)\overline{\omega}\right)\wedge\omega.\] From (4.8) we easily conclude \[\mathrm{d}\Upsilon+\Upsilon\wedge\Upsilon=\frac{\mathrm{i}}{4}\begin{pmatrix} 0 & -\ell \\ \ell & 0 \end{pmatrix}\omega\wedge\overline{\omega}.\] Again in formal analogy to the case \(L=K^2\), we obtain a thermostat \(\phi\) on \(SM^{1/n}\) by requiring that \(F \hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\Upsilon^2_1=0\). Using the notation above, we have \[\lambda=e^f a-\mathbb{V}\theta-\frac{1}{\ell}Hf.\]

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