Calculator Inputs
Formula used
This tool implements two widely used SU(2) conventions for a Seiberg–Witten curve. In hyperelliptic mode, the curve is written as:
y² = (x² − u)² − Λ^(4−Nf) Πᵢ (x + mᵢ)
The right-hand side becomes a quartic polynomial in x (for Nf ≤ 4). Branch points are approximated by solving RHS(x)=0, which finds the roots of that quartic.
In cubic pure mode, the curve is expressed as: y² = (x² − Λ⁴)(x − u), giving three simple branch points.
How to use
- Select a curve convention that matches your reference.
- Enter u and Λ; choose Nf and masses if needed.
- Set an x-range and sample count for the table.
- Press Compute to view coefficients, roots, and samples.
- Use the download buttons to export CSV or PDF.
Example data table
Example inputs: hyperelliptic mode, u=1.0, Λ=1.0, Nf=0, all masses 0.
| x | RHS = (x² − u)² − Λ⁴ | Interpretation |
|---|---|---|
| -2 | 8 | Positive y², real y possible. |
| -1 | -1 | Negative y², imaginary y region. |
| 0 | 0 | Branch point for these inputs. |
| 1 | -1 | Negative y², inside cut. |
| 2 | 8 | Positive y², outside cut. |
Professional notes on Seiberg–Witten curves
1) What the curve encodes
For SU(2) N=2 gauge theory, the Seiberg–Witten curve summarizes the low-energy description. Its branch structure determines periods of the Seiberg–Witten differential, which feed the effective coupling and BPS central charges. This calculator reports, in practice, the polynomial RHS(x) and where RHS(x)=0.
2) Why conventions matter
Different references rescale or shift x, y, u, and Λ, so “the same” curve can look different. The tool therefore offers two practical forms: a quartic hyperelliptic expression for Nf ≤ 4 and a cubic form often used for the pure theory. Compare outputs only after aligning conventions.
3) Quartic form implemented here
Hyperelliptic mode uses y² = (x² − u)² − Λ^(4−Nf) ∏(x + mᵢ). Expanding gives a quartic RHS(x)=a4x⁴+a3x³+a2x²+a1x+a0. The calculator prints a4…a0 so you can copy the polynomial into plotting, factoring, or period workflows.
4) Flavor count and mass dependence
Nf sets both the Λ power and how many mass factors appear in the product. Each mass shifts one factor (x + mᵢ), which can split symmetric root patterns and move branch points. When Λ is large, the flavor term can dominate, so small mass changes may shift roots noticeably.
5) Branch points and real-x intuition
Branch points are the solutions of RHS(x)=0 where y vanishes. The sample table evaluates y² along the real x-axis: y² > 0 allows real y, while y² < 0 implies imaginary y on that slice. This is a quick qualitative check for cut placement in a real-x sketch.
6) Numerical roots and near-singular tuning
Quartic branch points are found numerically and may be complex. Close to singular loci, roots can become nearly degenerate, making them sensitive to rounding and input perturbations. Treat displayed roots as accurate estimates for scanning, then validate with higher precision if your study needs it.
7) Picking a useful x-range
Choose x-min and x-max to cover the region where branch points are expected. If |u| is large compared with Λ, roots typically spread out and a wider range is helpful. If u is small, a tighter range reveals sign changes in y² and highlights nearby turning points.
8) Exports for analysis and reporting
CSV export captures inputs, coefficients, branch points, and sampled values for notebooks and plots. PDF export provides a compact snapshot for parameter scans and notes. Before drawing physics conclusions, confirm your reference’s normalization and verify that u, Λ, and masses match the same convention.
FAQs
1) What does u represent in this tool?
u is the Coulomb-branch modulus used in common SU(2) presentations. Changing u shifts the polynomial and typically moves branch points, altering the sign pattern of y² across the sampled x-range.
2) Why are there different curve forms?
Many texts use equivalent curves related by rescalings and shifts. The tool provides a quartic hyperelliptic option and a cubic pure-theory option so you can explore both without manually rewriting coefficients.
3) What are branch points here?
Branch points are solutions of RHS(x)=0 where y becomes zero. In hyperelliptic mode they are quartic roots (possibly complex). In cubic mode they occur at x=±Λ² and x=u.
4) Why do some roots appear complex?
For many parameter choices, the quartic has non-real roots. Complex branch points still matter for analytic continuation and period integrals, even if a real-x plot cannot directly display their locations.
5) How should I pick Nf and masses?
Choose Nf between 0 and 4 for this implementation, then enter up to four masses. Setting a mass to zero includes that flavor without shifting its factor; nonzero masses typically split symmetries and move roots.
6) Are the numbers exact?
Coefficients are exact from your inputs, but roots are numerical approximations. Near singular limits, roots can be ill-conditioned. Treat the displayed roots as estimates and verify with higher precision if needed.
7) What does the sample table show?
It evaluates y² = RHS(x) at evenly spaced x values. Positive entries indicate real y is possible; negative entries indicate imaginary y along the real x-line. It is useful for quick qualitative scans.
Use consistent conventions, and double-check all parameters before exporting.