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The Rank-One One-Instanton Mass Insertion

Physical scope

A finite formal mass-inserted coefficient on the explicit rank-one, charge-one stable ADHM quotient. It is not a specialization of the adopted U(2) S4 model and makes no continuum path-integral claim.

Example

For the explicit $k = N = 1$ stable scalar ADHM quotient and generic rotation weights with $ϵ_{1} ϵ_{2} \neq 0$ , extend the rotation torus by a flavor circle. Use natural units $c = ℏ = 1$ and assign mass dimension one to the formal rotation weights $ϵ_{1}, ϵ_{2}$ and additive flavor mass $\tilde{m}$ . Let $q$ be a dimensionless formal variable marking instanton charge. The unique fixed point has tangent weights $ϵ_{1}, ϵ_{2}$ , so the charge-one contribution of the top-Chern-class insertion for the inverse flavor line is $q I_{1, 1}^{a d j} (\tilde{m}) = q \frac{(ϵ_{1} - \tilde{m}) (ϵ_{2} - \tilde{m})}{ϵ_{1} ϵ_{2}} .$ The diagram shows the unique one-box partition, with the first coordinate horizontal for $B_{1}$ and the second vertical for $B_{2}$ . This is a finite formal rank-one calculation only; the adopted S4 postulate in this batch is specifically $U (2)$ .

Example references

Explicit quotient and mass-inserted Borel–Moore coefficient

Facts & Assumptions

Given: The explicit scalar quotient with $k = N = 1$ , rotation weights $ϵ_{1}, ϵ_{2}$ , and formal flavor weight $\tilde{m}$ .

[F1]

The current draft Explicit Rank-One One-Instanton ADHM Quotient has a complete authored local proof: its stable-locus map satisfies the classical geometric quotient conditions, the quotient is $C^{2}$ with one fixed point and tangent weights $ϵ_{1}, ϵ_{2}$ , and the Borel-descended top Chern insertion has the stated mass-shifted weight product and charge-one coefficient.

Verification

1.1givenF1

The unique fixed point and its two tangent weights are the rank-one quotient data in [F1].

2.1givenF1step 1.1algebra∎

The inverse flavor line shifts each tangent weight by $- \tilde{m}$ , so the insertion numerator is $(ϵ_{1} - \tilde{m}) (ϵ_{2} - \tilde{m})$ , while the tangent Euler denominator is $ϵ_{1} ϵ_{2}$ . Dividing gives the displayed coefficient, and multiplication by $q$ records charge one.

Depends on

Explicit Rank-One One-Instanton ADHM Quotient

Dependency tree · 2 levels

Sources

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