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Ω-Background Fixed-Point Expansion

Physical scope

Conditional equality of a source-specific, formally regulated N=2* U(N) Ω-background instanton series for N≥2 with the positive-FI ADHM fixed-point sum. Its adopted S4 application is U(2). No continuum path integral, full S4 partition function, contour integral, regulator removal, or analytic convergence is claimed.

Statement

Fix $N \geq 2$ , $k \geq 0$ , and $ζ > 0$ . Let $X_{k, N}^{ζ}$ be the positive-FI stable complex ADHM quotient with rotation and dual-framing torus $T = (S^{1})^{2} \times (S^{1})^{N}$ acting by $(t_{1}, t_{2}, h) \cdot (B_{1}, B_{2}, I, J) = (t_{1} B_{1}, t_{2} B_{2}, I h^{- 1}, t_{1} t_{2} h J)$ , where $W = \sum_{α = 1}^{N} e_{α}^{- 1}$ . Write $ϵ_{i} = c_{1}^{T} (t_{i})$ and $a_{α} = c_{1}^{T} (e_{α})$ , and assume these weights are algebraically independent. Let $q$ be dimensionless and formal. For an $N$ -tuple of partitions $\vec{Y} = (Y_{1}, \dots, Y_{N})$ , each $Y_{α}$ a finite downward-closed subset of $Z_{\geq 0}^{2}$ , put $∣ \vec{Y} ∣ = \sum_{α} ∣ Y_{α} ∣$ , $V_{\vec{Y}} = \sum_{α = 1}^{N} e_{α} \sum_{(i, j) \in Y_{α}} t_{1}^{i} t_{2}^{j}$ , and $W^{\lor} = \sum_{α} e_{α}$ . Assume the complexified torus acts algebraically on a smooth separated finite-type variety of complex dimension $2 k N$ , with a finite invariant affine atlas and finite proper fixed locus. Assume the fixed points and tangent characters are those in the positive-FI stable ADHM partition-fixed-point and tangent-character result. Extend $T$ by a flavor circle acting trivially on the quotient, with defining character $χ_{m}$ and additive weight $\tilde{m} = c_{1} (L_{χ_{m}})$ . Let $K_{m} = K \times S_{m}^{1}$ use the product right Milnor model, and set $X_{k, N, K_{m}}^{ζ} = E K_{m} \times_{K_{m}} X_{k, N}^{ζ}$ with the diagonal Borel quotient relation. Equivariant cohomology is $H^{*} (X_{k, N, K_{m}}^{ζ}; Q)$ ; its Chern classes are the rationalized ordinary Chern classes of the descended equivariant bundles, defined by the projective-bundle relation of the preceding Definition. At a fixed point with tangent characters of additive weights $w_{χ}$ , the top class $c_{2 k N}^{T_{m}} (T X_{k, N}^{ζ} \otimes L_{- 1})$ restricts to $\prod_{χ} (w_{χ} - \tilde{m})$ : the Borel fiber is the sum of character lines tensored by the inverse flavor line, and top Chern equals the Euler class of the complex-oriented real bundle, multiplicative over the ordered sum. Thus $E_{\tilde{m}} = c_{2 k N}^{T_{m}} (T X_{k, N}^{ζ} \otimes L_{- 1}) \in H^{4 k N} (X_{k, N, K_{m}}^{ζ}; Q)$ . Assume its localized Borel–Moore pushforward is defined by the proper-fixed-locus construction and satisfies the noncompact localization formula. For the source-reported $N = 2^{*}$ convention, set $m = i m_{E}$ for real Euclidean mass $m_{E}$ and $\tilde{m} = m + (ϵ_{1} + ϵ_{2}) / 2$ .

At a fixed point $\vec{Y}$ , the tangent character is ${ch}_{T} (T_{\vec{Y}} X_{k, N}^{ζ}) = V_{\vec{Y}}^{\lor} V_{\vec{Y}} (t_{1} + t_{2} - 1 - t_{1} t_{2}) + V_{\vec{Y}}^{\lor} W^{\lor} + t_{1} t_{2} W V_{\vec{Y}} .$ For each tangent character $χ$ , write $w_{χ} = c_{1}^{T} (χ)$ and let $E_{\vec{Y}} = \prod_{χ} w_{χ}$ and $N_{\vec{Y}} (\tilde{m}) = \prod_{χ} (w_{χ} - \tilde{m})$ , with multiplicities. Then the mass-inserted coefficient and formal series are $I_{k, N}^{a d j} (\tilde{m}) = \sum_{∣ \vec{Y} ∣ = k} \frac{N_{\vec{Y}} (\tilde{m})}{E_{\vec{Y}}}, Z_{N, a d j}^{l o c} (q; \tilde{m}) = \sum_{k \geq 0} q^{k} \sum_{∣ \vec{Y} ∣ = k} \frac{N_{\vec{Y}} (\tilde{m})}{E_{\vec{Y}}} \in Q (ϵ_{1}, ϵ_{2}, a_{1}, \dots, a_{N}, \tilde{m}) ⟨ ⁣ ⟨ q ⟩ ⁣ ⟩ .$

For the adopted round $S_{r}^{4}$ $N = 2^{*}$ model, specialize to $N = 2$ , $ϵ_{1} = ϵ_{2} = r^{- 1}$ and $\tilde{m} = i m_{E} + r^{- 1}$ , with $q = e^{2 π i τ}$ on the model's continued Coulomb contour. The source assumes, but does not construct, the supersymmetry-compatible noncommutative S4 regulator agreeing near each pole with the flat-space regulator. The displayed identity is a formal fixed-point calculation for the regulated local factor; it does not prove existence of that regulator, a continuum functional measure, the full S4 partition function, or convergence of the series.

Statement references

Mass-inserted ADHM series definition

Stable ADHM fixed-point and tangent-character core

Intrinsic localized Borel–Moore pushforward

Noncompact localization with a proper fixed locus

Source-qualified $N = 2^{*}$ U(2) S4 model

Pestun, §1 equations (1.3)–(1.4), §4.3 equations (4.47)–(4.50), and §5 equations (5.11)–(5.12)

Nekrasov, §1 equation (1.3) and §§3.3–3.6

Nakajima and Yoshioka, §3.1, PDF pp. 17–19

Nekrasov and Okounkov, §§2.2, 2.4, 3.1 and 6.1

Facts & Assumptions

Given: $N \geq 2$ , $k \geq 0$ , $ζ > 0$ , the positive-FI stable ADHM quotient, its compact rotation/framing torus action, and the localized mass-inserted Borel–Moore pushforward stipulated in the Statement.

[F1]

The current draft Regulated ADHM $N = 2^{*}$ Instanton Series defines equivariant Chern classes as rationalized ordinary Chern classes of Borel-descended bundles; its top class restricts at a fixed point to $\prod_{χ} (w_{χ} - \tilde{m})$ . Its coefficient is the localized proper-fixed-locus pushforward capped with the complex-oriented fundamental class, has field $Q (ϵ_{1}, ϵ_{2}, a_{1}, \dots, a_{N}, \tilde{m})$ , and the series is coefficientwise formal.

[F2]

The current draft Stable ADHM partition fixed points and tangent character has an authored local proof: for the positive-FI stable quotient it gives smooth complex dimension $2 k N$ , isolated fixed points bijective with $N$ -tuples of partitions of total size $k$ , and the stated tangent character. Its proof also supplies the stable quotient, finite invariant affine atlas, and finite proper fixed locus. It asserts no sheaf/ASD correspondence; the separate noncompact pushforward is supplied by [F4].

[F3]

Current draft Intrinsic localized pushforward for a proper fixed locus defines the localized trace as $p_{F, *} \circ (i_{*})^{- 1}$ on normalized equivariant Borel–Moore groups, with a complex-oriented fundamental class and cap-product insertions. Its stated interface matches the present use.

[F4]

Current draft Noncompact localization with a proper fixed locus has an authored conditional proof: for a smooth separated finite-type complex variety with a finite invariant affine cover and proper fixed locus, the localized trace of a homogeneous insertion is the sum of its restrictions divided by the normal Euler classes. Its Statement specifies the normalized Borel–Moore degree and finite-fixed-point reciprocal-Euler case; the stated trace interface matches this use.

[L1]

Current draft Adopted $N = 2^{*}$ U(2) Model on the Round Four-Sphere adopts a round S4, gauge group $U (2)$ , one adjoint hypermultiplet with real Euclidean mass $m_{E}$ , $q = e^{2 π i τ}$ , equal pole rotations $ϵ_{1} = ϵ_{2} = r^{- 1}$ , and a source-assumed noncommutative S4 regulator. Its exact source-qualified interface is used conditionally, not as a proven physical fact.

[A1]

Assume the source-reported $N = 2^{*}$ instanton factor uses the mass-inserted tangent Euler class $\prod_{χ} (w_{χ} - \tilde{m})$ over the vector tangent Euler class. The source's §5 equation (5.11) gives this shifted-weight numerator and $\tilde{m} = m + (ϵ_{1} + ϵ_{2}) / 2$ ; this is a formal regulator convention, not a proof of a continuum determinant or path-integral equality.

[A2]

Apply the fixed-point, tangent-character, smoothness, finite-atlas, and proper-fixed-locus conclusions of [F2] under its exact positive-FI hypotheses. This proof uses only that stable quotient result; it makes no ADHM-to-ASD or ADHM-to-Gieseker identification and takes the separate pushforward from [F4].

[A3]

Assume the localized Borel–Moore construction [F3] and formula [F4] are available for this quotient and for the flavor-extended torus, with the smooth finite-type action, finite invariant affine atlas, and proper finite fixed locus required by [F4]. The required geometric hypotheses follow from [F2] under [A2]; the added flavor torus acts trivially on the base, so it preserves the same affine cover and fixed locus.

[A4]

Assume the exact source regulator input [L1] identifies the formal local factor of the adopted $N = 2^{*}$ U(2) S4 model with the mass-inserted series for $N = 2$ under the stated Euclidean mass shift. No full S4 integral or regulator construction is inferred.

Proof

1.1givenA2F2

By [A2] and [F2], the $T$ -fixed points of $X_{k, N}^{ζ}$ are isolated and correspond bijectively to the $N$ -tuples $\vec{Y}$ with $∣ \vec{Y} ∣ = k$ . The tangent character at each point is the displayed character, so its additive weights, with multiplicity, have Euler product $E_{\vec{Y}}$ .

2.1givenF1F2step 1.1algebra

Extend $T$ by the flavor circle acting trivially on the base. This does not change the fixed-point set, and at $\vec{Y}$ the top Chern insertion restricts to $N_{\vec{Y}} (\tilde{m}) = \prod_{χ} (w_{χ} - \tilde{m})$ by [F1]. Each nontrivial tangent character class $w_{χ}$ is inverted in the character-localized coefficient ring; no invertibility of the shifted numerator factors is needed.

3.1F1F3F4A3step 1.1step 2.1

Apply the noncompact Borel–Moore localization formula [F4] under [A3] to $E_{\tilde{m}} \cap [X_{k, N}^{ζ}]$ . Each fixed component is a proper point, its proper trace is evaluation, and its normal bundle is the tangent representation; hence the coefficient is the finite sum $\sum_{∣ \vec{Y} ∣ = k} N_{\vec{Y}} (\tilde{m}) / E_{\vec{Y}}$ . Its Borel–Moore degree is zero because $c_{2 k N}$ has degree $4 k N$ , equal to the degree of the fundamental class.

4.1F1F2step 3.1algebra

By the definition [F1], the pushforward in step 3.1 is $I_{k, N}^{a d j} (\tilde{m})$ . For finite $N$ and fixed $k$ there are finitely many partition tuples, so each coefficient is a finite rational function in the stated field. The coefficient sequence therefore defines the displayed formal power series; at $k = 0$ , the unique empty tuple has empty numerator and denominator products equal to $1$ .

5.1givenA1A4L1step 4.1∎

Under the source-reported insertion convention [A1] and the exact model input [A4], the $N = 2$ specialization of step 4.1 is the formal local $N = 2^{*}$ instanton factor with $ϵ_{1} = ϵ_{2} = r^{- 1}$ , $\tilde{m} = i m_{E} + r^{- 1}$ , and the model's $q$ and continued Coulomb contour. The argument proves only this conditional fixed-point formula; it establishes neither the assumed S4 regulator nor a continuum functional integral.

Ω-Background Fixed-Point Expansion

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources