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 Incoming links: Cauchy-Riemann equations

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 3B / b / Solution Created 2026-09-24 Updated 2026-09-25

Differentiate the affine relation with respect to

x

and

y

:

a u_{x} + b v_{x} = 0, a u_{y} + b v_{y} = 0.

Using the Cauchy-Riemann equations, the second equation becomes

b u_{x} - a v_{x} = 0.

Thus

(a b b - a) (u_{x} v_{x}) = 0.

The determinant is

- (a^{2} + b^{2}) \neq = 0

, so

u_{x} = v_{x} = 0

and therefore

f^{'} = 0

. A holomorphic function with zero derivative on a connected open set is constant. Hence

f

is constant, as in analytic function with image in an affine real line.

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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

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 .

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}) .

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},

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

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

Thus

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

A radial vortex of winding

N > 0

has

Φ = f (r) e^{i Nθ}, A = a (r) d θ,

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}) .

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}) .

Since

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

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

satisfies the Liouville equation

Δ ψ = e^{2 ψ} .

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}} .

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}},

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}} .

For

N = 1

these reduce to

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

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 .

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

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 105 / 1 / b / Solution Created 2026-09-24 Updated 2026-09-25

Suppose a

C^{1}

classical solution existed and set

w = u + i v

. The system is precisely

u_{x} = v_{t}, u_{t} = - v_{x},

which are the Cauchy-Riemann equations for

w

as a function of the complex number

z = x + i t

. Hence

w

is holomorphic near

z = 0

. Every holomorphic function is real analytic, so its restriction

w (x) = w (0, x) = f (x) + i g (x)

to the real axis is real analytic near zero. Its real part and imaginary part show that both

f

and

g

must be real analytic, contradicting the hypothesis. Therefore no such

C^{1}

solution exists.

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Perturbative gauge-coupling independence of the Wilsonian superpotential 2026-10-06

The holomorphic coupling is

τ = ϑ / (2 π) + 4 πi / g^{2}

. Perturbative coefficients are independent of the topological angle

ϑ

. A holomorphic function invariant under continuous shifts of

Re τ

is constant by the Cauchy-Riemann equations, hence a perturbative local superpotential coefficient cannot depend on

τ

. Terms in

e^{2 πi τ}

can evade this argument nonperturbatively. This does not forbid renormalization of the separate gauge kinetic F-term, or wave-function renormalization from the Kähler potential.

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