4.1. Dirac monopoles on a punctured torus [02H5]

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4.1. Dirac monopoles on a punctured torus

Let $\mathbb{T}=\mathbb{R}^{3}/\Lambda$ be a $3$ –torus for some lattice $\Lambda\simeq\mathbb{Z}^{3}$ . Endow $\mathbb{T}$ with a flat metric $g_{\mathbb{T}}$ .

Let $\tau\colon\thinspace\mathbb{T}\rightarrow\mathbb{T}$ be the standard involution $x\mapsto-x$ on $\mathbb{T}$ and denote by $q_{1},\dots,q_{8}$ its fixed points. For each $j=1,\dots,8$ choose a non-negative integer $m_{j}$ .

Fix a $\tau$ –symmetric configuration of further $2n$ distinct points $p_{1},\tau(p_{1}),\dots,p_{n},\tau(p_{n})$ . Sometimes we will use the notation $-p_{i}$ for $\tau(p_{i})$ . Denote by $\mathbb{T}^{\ast}$ the punctured torus

\mathbb{T}^{\ast}=\mathbb{T}\setminus\{q_{1},\dots,q_{8},p_{1},\tau(p_{1}),\dots,p_{n},\tau(p_{n})\}.

Finally choose integer weights $k_{1},\dots,k_{n}>0$ and assume the following balancing condition holds:

(4.1)

\sum_{j=1}^{8}{m_{j}}+\sum_{i=1}^{n}{k_{i}}=16.

In particular, $n\leq\sum_{i=1}^{n}{k_{i}}\leq 16$ .

For each $j=1,\dots,8$ let $\rho_{j}$ denote the distance function from the point $q_{j}$ with respect to $g_{\mathbb{T}}$ . Similarly, by abuse of notation we let $\rho_{i}$ denote the distance function from $\pm p_{i}$ in $\mathbb{T}/\tau$ . By restricting the branched double cover $\mathbb{T}\rightarrow\mathbb{T}/\tau$ to a sufficiently small ball centred at $\pm p_{i}$ in $\mathbb{T}/\tau$ we will also regard $\rho_{i}$ as the distance function on $\mathbb{T}$ from the point $p_{i}$ or $\tau(p_{i})$ .

We look for a Dirac monopole $(h,\theta)$ on $\mathbb{T}^{\ast}$ with the following singular behaviour: $h$ is a harmonic function on $\mathbb{T}^{\ast}$ with prescribed singularities at the punctures

(4.2)

h\sim\frac{2m_{j}-4}{2\rho_{j}}\mbox{ as }\rho_{j}\rightarrow 0,\qquad h\sim\frac{k_{i}}{2\rho_{i}}\mbox{ as }\rho_{i}\rightarrow 0.

Proposition 4.3.

Assume the balancing condition (4.1) is satisfied.

(i)

There exists a harmonic function $h$ on $\mathbb{T}^{\ast}$ with prescribed singular behaviour (4.2) such that $\ast_{g_{\mathbb{T}}}dh$ is the curvature $d\theta$ of a connection $\theta$ on some principal $U(1)$ –bundle $P\rightarrow\mathbb{T}^{\ast}$ .
(ii)

The moduli space of Dirac monopoles $(h,\theta)$ on $P$ is isomorphic to $\mathbb{R}\times\hat{\mathbb{T}}$ , where $\hat{\mathbb{T}}$ is the dual torus parametrising flat $U(1)$ –connections on $\mathbb{T}$ .
(iii)

The involution $\tau$ lifts to the involution $\tilde{\tau}$ of the $U(1)$ –bundle $P$ which acts simultaneously as $\tau$ on $\mathbb{T}^{\ast}$ and as the standard involution on the circle fibres.

Proof.

The necessary and sufficient condition for the existence of the harmonic function $h$ is

(4.4)

\sum_{j=1}^{8}{2m_{j}-4}+2\sum_{i=1}^{n}{k_{i}}=\frac{1}{2\pi}\int_{\partial\mathbb{T}_{\sigma}}{\ast dh}=0,

where $\mathbb{T}_{\sigma}$ denotes the complement of the union of small balls of radius $\sigma$ centred at the punctures. Thus if (4.1) is satisfied, a harmonic function $h$ with the singular behaviour (4.2) does indeed exists and is unique up to the addition of a constant.

By Lefschetz–Poincaré duality $H_{2}(\mathbb{T}^{\ast})\simeq H^{1}_{c}(\mathbb{T}^{\ast})$ . The latter group sits in a long exact sequence

0\rightarrow H^{0}(\mathbb{T})\rightarrow\mathbb{Z}^{2n+8}\rightarrow H^{1}_{c}(\mathbb{T}^{\ast})\rightarrow H^{1}(\mathbb{T})\rightarrow 0,

where $\mathbb{Z}^{2n+8}$ is generated by the $2n+8$ punctures. Thus $H_{2}(\mathbb{T}^{\ast})$ is $(2n+10)$ –dimensional and maps onto $H_{2}(\mathbb{T})$ with kernel spanned by the classes of $2n+8$ spheres centred at the punctures. Note that the sum of these $2n+8$ homology classes vanishes.

Because of (4.2), $2n+7$ of the $2n+10$ integrality constraints on $\tfrac{i}{2\pi}\ast dh$ to represent the first Chern class of a line bundle are automatically satisfied since we chose $2m_{j}-4,k_{i}\in\mathbb{Z}$ . The remaining $3$ constraints can be reinterpreted in terms of the position of the punctures following the arguments in the proof of [9, Proposition 3.5]:

\sum_{j=1}^{8}{(2m_{j}-8)\,q_{j}}+\sum_{i=1}^{n}{k_{i}\,\big(p_{i}+\tau(p_{i})\big)}\in\Lambda.

Since the points $q_{j}$ belong to the half-lattice $\tfrac{1}{2}\Lambda$ this condition is automatically satisfied.

We have therefore proved the existence of a principal $U(1)$ bundle $P\rightarrow\mathbb{T}^{\ast}$ endowed with a connection $\theta$ with curvature $\ast dh$ . Since $\mathbb{T}$ is not simply connected $\theta$ is uniquely determined up to a flat connection, i.e. a point of the dual torus $\hat{\mathbb{T}}$ . This concludes the proof of (i) and (ii).

By uniqueness up to the addition of a constant the harmonic function $h$ is $\tau$ –invariant and therefore can be thought of as defined on $\mathbb{T}^{\ast}/\tau$ . Since $\tau^{\ast}(\ast dh)=-\ast dh$ , we can lift $\tau$ (uniquely up to gauge transformations) to an involution $\tilde{\tau}$ of the circle bundle $P$ by requiring that $\tilde{\tau}$ acts simultaneously as $\tau$ on $\mathbb{T}^{\ast}$ and as the standard involution on the circle fibres. ∎