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The $U (2)$ $N = 2^{*}$ One-Instanton Factor

Physical scope

The finite, formal charge-one coefficient for the conditional regulated U(2) N=2* ADHM series; no contour evaluation or continuum path-integral claim.

Example

Take $N = 2$ and charge $k = 1$ in the mass-inserted formal ADHM series. Use its dimensionless formal charge variable $q$ ; the equivariant weights $ϵ_{i}, a_{α}, \tilde{m}$ have mass dimension one. Let $s = ϵ_{1} + ϵ_{2}$ , $d = a_{2} - a_{1}$ , and $\tilde{m} = i m_{E} + s / 2$ in the source's Euclidean mass convention. Exactly one box lies in either color, so the fixed-point tuples are $(□, \emptyset)$ and $(\emptyset, □)$ . Their tangent characters and Euler classes are ${ch}_{T} (T_{(□, \emptyset)} X) = t_{1} + t_{2} + \frac{e_{2}}{e_{1}} + t_{1} t_{2} \frac{e_{1}}{e_{2}}$ $E_{1} = ϵ_{1} ϵ_{2} d (s - d), N_{1} = (ϵ_{1} - \tilde{m}) (ϵ_{2} - \tilde{m}) (d - \tilde{m}) (s - d - \tilde{m})$ ${ch}_{T} (T_{(\emptyset, □)} X) = t_{1} + t_{2} + \frac{e_{1}}{e_{2}} + t_{1} t_{2} \frac{e_{2}}{e_{1}}$ $E_{2} = - ϵ_{1} ϵ_{2} d (s + d), N_{2} = (ϵ_{1} - \tilde{m}) (ϵ_{2} - \tilde{m}) (- d - \tilde{m}) (s + d - \tilde{m}) .$ For generic weights satisfying $ϵ_{1} ϵ_{2} d (s - d) (s + d) \neq 0$ , the coefficient of $q$ is $\frac{N_{1}}{E_{1}} + \frac{N_{2}}{E_{2}} = \frac{2 (ϵ_{1} - \tilde{m}) (ϵ_{2} - \tilde{m}) ({\tilde{m}}^{2} - \tilde{m} s + s^{2} - d^{2})}{ϵ_{1} ϵ_{2} (s^{2} - d^{2})} .$ For the adopted $S_{r}^{4}$ model, set $ϵ_{1} = ϵ_{2} = r^{- 1}$ and $\tilde{m} = i m_{E} + r^{- 1}$ . At $m_{E} = 0$ the displayed one-instanton coefficient vanishes because its first two numerator factors are zero; this is only the degree-one check of the $N = 4$ massless limit.

The diagram places the single box in each of the two possible framing colors. The first coordinate is horizontal and the second vertical.

Example references

Mass-inserted series and coefficient convention

Conditional fixed-point expansion

Adopted U(2) S4 model and mass specialization

Pestun, §5, equations (5.11)–(5.12)

Facts & Assumptions

Given: $N = 2$ , $k = 1$ , the ADHM character convention, generic rotation, framing and flavor weights, and the conditional localized mass-inserted series.

[F1]

The current draft Ω-Background Fixed-Point Expansion has an authored conditional proof: for each partition tuple the fixed-point contribution is the product of tangent weights shifted by $- \tilde{m}$ , divided by the tangent Euler product; the fixed points are the partition tuples of total size $k$ .

[F2]

Current draft Regulated ADHM $N = 2^{*}$ Instanton Series: the coefficient is the proper-fixed-locus localized pushforward of the rational Borel top Chern class of the tangent bundle tensored with the inverse flavor line, capped with the complex-oriented fundamental class; its restriction at a fixed point of tangent weights $w_{χ}$ is $\prod_{χ} (w_{χ} - \tilde{m})$ .

[F3]

Current draft Adopted $N = 2^{*}$ U(2) Model on the Round Four-Sphere adopts $N = 2^{*}$ $U (2)$ with real Euclidean mass $m_{E}$ and equal pole rotations $ϵ_{1} = ϵ_{2} = r^{- 1}$ ; its noncommutative S4 regulator remains a source assumption, not a proven existence claim.

[A1]

Assume the theorem's finite-proper-fixed-locus pushforward conditions and the source-reported mass-insertion interpretation hold for the displayed generic-parameter calculation. The source model's regulator remains assumed, and the formula is not an evaluation of the continued Coulomb contour.

Verification

1.1givenF1algebra

The only total-size-one tuples are $(□, \emptyset)$ and $(\emptyset, □)$ . Substitution of $V = e_{1}$ and $V = e_{2}$ into the ADHM tangent character gives respectively $t_{1} + t_{2} + e_{2} / e_{1} + t_{1} t_{2} e_{1} / e_{2}$ and $t_{1} + t_{2} + e_{1} / e_{2} + t_{1} t_{2} e_{2} / e_{1}$ .

2.1givenF1step 1.1algebra

Taking first Chern classes of those four weights gives $(ϵ_{1}, ϵ_{2}, d, s - d)$ and $(ϵ_{1}, ϵ_{2}, - d, s + d)$ . Multiplication yields $E_{1} = ϵ_{1} ϵ_{2} d (s - d)$ and $E_{2} = - ϵ_{1} ϵ_{2} d (s + d)$ ; shifting each weight by $- \tilde{m}$ gives exactly $N_{1}, N_{2}$ stated above.

3.1givenstep 2.1algebra

Put $A = (ϵ_{1} - \tilde{m}) (ϵ_{2} - \tilde{m})$ and $C = {\tilde{m}}^{2} - \tilde{m} s - d^{2}$ . The two shifted cross-color products are $(d - \tilde{m}) (s - d - \tilde{m}) = C + s d$ and $(- d - \tilde{m}) (s + d - \tilde{m}) = C - s d$ . Therefore $\frac{N_{1}}{E_{1}} + \frac{N_{2}}{E_{2}} = \frac{A}{ϵ_{1} ϵ_{2} d} (\frac{C + s d}{s - d} - \frac{C - s d}{s + d}) = \frac{A}{ϵ_{1} ϵ_{2} d} \frac{(C + s d) (s + d) - (C - s d) (s - d)}{s^{2} - d^{2}} .$ The numerator is $2 C d + 2 s^{2} d = 2 d (C + s^{2})$ , so cancellation of $d$ in the generic rational function field gives the displayed coefficient.

4.1givenF2F3A1step 3.1algebra∎

Under [F2] and [F3], the source S4 specialization has $ϵ_{1} = ϵ_{2} = r^{- 1}$ and $\tilde{m} = i m_{E} + r^{- 1}$ . Setting $m_{E} = 0$ makes $A = 0$ , so the degree-one coefficient vanishes; this checks only this coefficient and assigns no value at the excluded generic Euler-denominator poles.

Depends on

Dependency tree · 2 levels

Sources

Vasily Pestun, Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

The U(2)U(2) N=2∗N=2∗ One-Instanton Factor