Riemann Surfaces and Branch Cuts

A multivalued function like $\sqrt{z}$ or $\log z$ becomes single-valued once its domain is replaced by the right surface — a Riemann surface stitched from several sheets. This resolves the monodromy of analytic continuation geometrically, and it is the precise language for the branch cuts of scattering amplitudes that open at particle-production thresholds. This page builds on holomorphic.md and analytic-continuation.md.

Branch points and branch cuts

A branch point is a point around which a multivalued function fails to return to its starting value. For $f(z)=\sqrt{z}$, encircling $0$ once sends $\sqrt{z}\to -\sqrt{z}$; $0$ (and $\infty$) are its branch points. To work with a single-valued function one removes a branch cut — a curve joining branch points — across which the function is discontinuous. For $\sqrt{z}$ the standard cut is the negative real axis; the principal branch then has $\sqrt{z}>0$ for $z>0$.

The discontinuity across the cut is the physically meaningful object. For $\sqrt{z}$ just above vs. below the negative axis the values differ by a sign; in general the jump $$\mathrm{disc}f(x)=f(x+i0)-f(x-i0)$$ across a cut equals $2i\mathrm{Im}f$ on the cut (by the Schwarz reflection principle), and is the absorptive part that dispersion relations integrate.

Riemann surfaces

Rather than cut the plane, glue copies of it. The Riemann surface of a multivalued function is the surface on which it becomes single-valued and holomorphic, built by taking one sheet per branch and gluing sheets along the cuts:

• $\sqrt{z}$: two sheets, glued crosswise along the cut — topologically a sphere. Going around $0$ moves you from sheet 1 to sheet 2 and back after two loops (matching $\sqrt{z}$'s two values).
• $\log z$: infinitely many sheets in an endless spiral staircase (one per branch $+2\pi in$) — the universal cover of $\mathbb{C}\setminus \{0\}$, the same object as ${\pi }_1=\mathbb{Z}$ in de-rham.md.
• Algebraic functions $w$ solving $P(z,w)=0$: compact Riemann surfaces whose genus (number of handles) is a deep invariant (Riemann–Hurwitz), linking complex analysis to topology and algebraic geometry.

On its Riemann surface the function is honestly single-valued; the "multivaluedness" was an artifact of forcing it onto the plane.

The uniformization theorem

The Riemann mapping theorem classifies simply connected planar domains; its vast generalization classifies simply connected Riemann surfaces intrinsically:

Uniformization theorem. Every simply connected Riemann surface is conformally equivalent to exactly one of three models: the Riemann sphere $\hat{\mathbb{C}}$, the plane $\mathbb{C}$, or the disc $\mathbb{D}$.

Every Riemann surface is then a quotient of one of these by a group of deck transformations, giving a trichotomy — elliptic (sphere), parabolic (plane/torus), or hyperbolic (everything else, carrying the Poincaré metric). The type is tied to the genus: genus $0$ is the sphere, genus $1$ the tori $\mathbb{C}/\Lambda$, and genus $\ge 2$ hyperbolic. This is the complex-analytic face of the constant-curvature classification in geometry ($K>0,=0,<0$), and it explains why "most" Riemann surfaces are hyperbolic.

Physical branch cuts: thresholds

In QFT a scattering amplitude $\mathcal{M}(s)$, as a function of the Mandelstam energy-squared $s$, is holomorphic except for poles and cuts:

• poles on the real axis = stable particles / bound states (mass${}^2=s$ pole), from laurent-residues.md;
• branch cuts starting at each production threshold $s=(m_1+m_2+\cdots {)}^2$ = the energies at which new multiparticle states can be produced.

The branch point sits exactly at the threshold; above it the amplitude develops an imaginary part (the discontinuity), which by the optical theorem equals the total cross section into those states. The analytic structure — where the poles and cuts sit — thus encodes the entire particle content and is the backbone of the S-matrix program and dispersion relations (analytic-continuation.md). The second sheet of the amplitude's Riemann surface is where resonances / unstable particles live, as complex poles at $s=M^2-iM\Gamma$.

References

• Stein & Shakarchi, Complex Analysis, Ch. 8.
• Forster, Lectures on Riemann Surfaces.
• Eden, Landshoff, Olive & Polkinghorne, The Analytic S-Matrix.