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 Incoming links: Witten hyperbolic vortex

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 308 / 2 / Solution 2026-10-03

Away from a zero of

ϕ

, the first Bogomolny vortex equation gives

a_{\overset{z}{ˉ}} = - i \partial_{\overset{z}{ˉ}} lo g ϕ, f_{z \overset{z}{ˉ}} = - 2 i \partial_{z} \partial_{\overset{z}{ˉ}} lo g ∣ ϕ ∣.

(1)

Substitution into the second equation, with

\nabla^{2} = 4 \partial_{z} \partial_{\overset{z}{ˉ}}

, gives

- \frac{2}{Ω} \nabla^{2} lo g ∣ ϕ ∣ = 1 - ∣ ϕ ∣^{2} .

(2)

Now set

Ω = Ω∣ ϕ ∣^{2}

. The Gaussian curvature formula gives

K Ω = K Ω - \nabla^{2} lo g ∣ ϕ ∣.

(3)

Therefore

(K + 1/2) Ω = (K + 1/2) Ω

implies

- \nabla^{2} lo g ∣ ϕ ∣ = \frac{Ω}{2} (1 - ∣ ϕ ∣^{2}),

(4)

which is exactly the vortex equation.

The metric

d s_{n}^{2} = \frac{8 n ^{2} ∣ z ∣ ^{2 n - 2}}{( 1 - ∣ z ∣ ^{2 n} ) ^{2}} d z d \overset{z}{ˉ}

(5)

is the pullback of the

n = 1

Poincare disc model metric under

w = z^{n}

. It consequently has

K = - 1/2

away from the origin. Comparing it with

d s_{1}^{2}

gives the Witten hyperbolic vortex

∣ ϕ ∣ = \frac{n ∣ z ∣ ^{n - 1} ( 1 - ∣ z ∣ ^{2} )}{1 - ∣ z ∣ ^{2 n}} .

(6)

Putting

n = N + 1

gives winding number

N

. A gauge choice with positive radial factor is

ϕ = \frac{n z ^{N} ( 1 - ∣ z ∣ ^{2} )}{1 - ∣ z ∣ ^{2 n}},

(7)

and then

a_{\overset{z}{ˉ}} = i (\frac{z}{1 - ∣ z ∣ ^{2}} - \frac{n z ∣ z ∣ ^{2 n - 2}}{1 - ∣ z ∣ ^{2 n}}), a_{z} = \overline{a_{\overset{z}{ˉ}}} .

(8)

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 3 / Solution Created 2026-09-24 Updated 2026-09-25

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Write

F_{12} = \partial_{1} A_{2} - \partial_{2} A_{1}

. Up to a total derivative, the covariant-gradient identity and completion of the magnetic and potential terms give

V = \frac{1}{2} \int_{D} [e^{- 2 ρ} (F_{12} - \frac{e ^{2 ρ}}{2} (1 - ∣Φ ∣^{2}))^{2} + ∣ D_{1} Φ + i D_{2} Φ ∣^{2}] d^{2} x + \frac{1}{2} \int_{D} F_{12}, d^{2} x .

(1)

For positive flux, equality in the Bogomolny bound therefore gives the Bogomolny vortex equations

D_{1} Φ + i D_{2} Φ = 0, F_{12} = \frac{e ^{2 ρ}}{2} (1 - ∣Φ ∣^{2}) .

(2)

The opposite simultaneous signs describe negative winding.

For the orientation

d x^{1} \land d x^{2} > 0

, the Hodge star operator is

* 1 = e^{2 ρ} d x^{1} \land d x^{2},

(3)

* d x^{1} = d x^{2}, * d x^{2} = - d x^{1},

(4)

* (d x^{1} \land d x^{2}) = e^{- 2 ρ} .

(5)

Thus

B = * d A = e^{- 2 ρ} F_{12} = \frac{1}{2} (1 - ∣Φ ∣^{2}) .

(6)

A radial vortex of winding

N > 0

has

Φ = f (r) e^{i Nθ}, A = a (r) d θ,

(7)

with

f (0) = a (0) = 0

f (1) = 1

a (1) = N

, and equations

f^{'} = \frac{N - a}{r} f, \frac{a ^{'}}{r} = \frac{e ^{2 ρ}}{2} (1 - f^{2}) .

(8)

For

e^{2 ρ} = 8/ (1 - ∣ z ∣^{2})^{2}

, put

u = lo g ∣Φ∣

. Away from zeros, the first vortex equation gives

Δ u = - \frac{e ^{2 ρ}}{2} (1 - e^{2 u}) .

(9)

Since

Δ [- lo g (1 - z \overset{z}{ˉ})] = \frac{4}{( 1 - ∣ z ∣ ^{2} ) ^{2}},

(10)

the field

ψ = u - lo g (1 - z \overset{z}{ˉ}) + lo g 2

satisfies the Liouville equation

Δ ψ = e^{2 ψ} .

(11)

For holomorphic

g : D \to D

, direct use of the Cauchy-Riemann equations verifies

e^{2 ψ} = \frac{4∣ g ^{'} ( z ) ∣ ^{2}}{( 1 - ∣ g ( z ) ∣ ^{2} ) ^{2}} .

(12)

Choosing

g (z) = z^{N + 1}

gives the radial Witten hyperbolic vortex

Φ (z) = \frac{( N + 1 ) z ^{N} ( 1 - ∣ z ∣ ^{2} )}{1 - ∣ z ∣ ^{2 N + 2}},

(13)

A = a_{N} (r) d θ, a_{N} (r) = \frac{2 r ^{2}}{1 - r ^{2}} - \frac{( 2 N + 2 ) r ^{2 N + 2}}{1 - r ^{2 N + 2}} .

(14)

For

N = 1

these reduce to

f (r) = \frac{2 r}{1 + r ^{2}}, a_{1} (r) = \frac{2 r ^{2}}{1 + r ^{2}} .

(15)

Since

B d μ_{g} = d A = a_{1}^{'} (r) d r \land d θ

\int_{D} B d μ_{g} = 2 π \int_{0}^{1} a_{1}^{'} (r) d r = 2 π [a_{1} (1) - a_{1} (0)] = 2 π .

(16)

This directly verifies the Abelian Higgs vortex flux relation for unit winding.

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