The $U(2)$ $\mathcal{N}=2^{\ast }$ Round-$S^4$ Matrix Integral and Its One-Instanton Term

Use the adopted round-$S_r^4$ $\mathcal{N}=2^{\ast }$ $U(2)$ model with one adjoint hypermultiplet of real Euclidean mass $m_E$, dimensionless coupling $g_{\mathrm{Y}\mathrm{M}}>0$, theta angle $\theta$, and natural units $c=\mathrm{\hslash }=1$. Thus $[r]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^{-1}$, while $m_E$, Cartan parameters, and ${ϵ}_i$ have mass dimension one; $\tau =4\pi i\mathrm{/}{g}_{\mathrm{Y}\mathrm{M}}^2+\theta \mathrm{/}(2\pi )$ and $q=e^{2\pi i\tau }$ are dimensionless.

The source continues its algebraic mass as $m=i{m}_E$ for real Euclidean $m_E$. Put

${ϵ}_1={ϵ}_2=ϵ=r^{-1},\tilde{m}=i{m}_E+\frac{{ϵ}_1+{ϵ}_2}{2}\mathrm{.}$

For the algebraic fixed-point calculation, work in $K_m=\mathbb{Q}({ϵ}_1,{ϵ}_2,a_1^{\mathrm{P}},a_2^{\mathrm{P}},\mu )$, the rational function field in algebraically independent rotation, Coulomb, and additive flavor variables; the formal flavor weight has mass dimension one. Sum the fixed-point terms as rational functions before specializing $\mu$ to $\tilde{m}$ and imposing the physical round-sphere values stated above.

For $U(2)$ the nonzero roots are $\pm {\alpha }_{12}$, and the Weyl group is $S_2$. The source's finite-dimensional integration contour is the real vector space of skew-Hermitian matrices $\mathfrak{u}(2)$. Its Cartan coordinates are real, with Weyl factor $\mathrm{\Delta }(a_E)={\prod }_{\alpha \in \mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{s}(U(2))}\alpha \cdot a_E$ in a Cartan reduction. The full Lie-algebra formula below has no separate Weyl factor; its measure and gauge-volume normalization are specified there.

Write the real Cartan coordinates as $a_E=(a_1^E,a_2^E)\in {\mathbb{R}}^2$. They parametrize the skew-Hermitian Cartan matrix $\mathsf{a}=i\mathrm{diag}(a_1^E,a_2^E)\in \mathfrak{u}(2)$ on the integration contour; $\mathsf{a}$ is the zero mode ${\mathrm{\Phi }}_0^E$ of the continued scalar. The continued scalar coordinates used by the determinants are not the matrix $\mathsf{a}$: the one-loop factor has argument $i{a}_E$, and the Nekrasov factor has complexified Coulomb weights $a_{\alpha }^{\mathrm{P}}=i{a}_{\alpha }^E$. Thus ${\alpha }_{12}(a_E)=a_1^E-a_2^E$ in the one-loop product and ${\alpha }_{12}(a^{\mathrm{P}})=i(a_1^E-a_2^E)$ in the instanton series.

The source's unregularized one-loop product is

$Z_{1\text{-loop}}^{\mathcal{N}=2^{\ast }}(i{a}_E;m_E)={\prod }_{\alpha \in \mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{s}(U(2))}{\prod }_{n=1}^{\mathrm{\infty }}{\left[\frac{((\alpha \cdot a_E{)}^2+{ϵ}^2{n}^2{)}^2}{((\alpha \cdot a_E+m_E{)}^2+{ϵ}^2{n}^2)((\alpha \cdot a_E-m_E{)}^2+{ϵ}^2{n}^2)}\right]}^{n\mathrm{/}2}\mathrm{.}$

For its Barnes-$G$ regularization, define

$G(1+z)=(2\pi {)}^{z\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-\frac{(1+\gamma )z^2+z}{2}){\prod }_{n=1}^{\mathrm{\infty }}{(1+\frac{z}{n})}^n\exp \text{\hspace{0.17em}⁣}(\frac{z^2}{2n}-z),H(z)=G(1+z)G(1-z),$

where $\gamma$ is Euler's constant. Pestun's Eq. (4.50) drops the complete $a_E$-independent divergent exponential and fixes the remaining factor to

$Z_{1\text{-loop}}^{\mathcal{N}=2^{\ast }}(i{a}_E;m_E)={\prod }_{\alpha \in \mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{s}(U(2))}\frac{H(i\alpha \cdot a_E\mathrm{/}ϵ)}{{[H((i\alpha \cdot a_E+i{m}_E)\mathrm{/}ϵ)H((i\alpha \cdot a_E-i{m}_E)\mathrm{/}ϵ)]}^{1\mathrm{/}2}}\mathrm{.}$

On the stated real contour every argument of $H$ is imaginary, $H(iy)=\mathrm{\mid }G(1+iy){\mathrm{\mid }}^2>0$, and the square root is the positive one. No additional $a_E$-independent finite factor is included. The Weyl factor is not part of this one-loop root product.

For a partition $Y$, use a finite downward-closed subset of ${\mathbb{Z}}_{\ge 0}^2$ and write $\mathrm{\mid }Y\mathrm{\mid }$ for its box count. The source labels its boxes from $1$; translate $(i,j)$ to $(i-1,j-1)$ for the ADHM theorem, which does not change arm or leg lengths. If $s=(i,j)$ in the source's convention, let ${\nu }_i(Y)$ be the height of column $i$ and ${\nu }_j^{\mathrm{\prime }}(Y)$ the length of row $j$, and define $v_Y(s)={\nu }_i(Y)-j$ and $h_Y(s)={\nu }_j^{\mathrm{\prime }}(Y)-i$; these relative vertical and horizontal lengths may be negative when $s$ is outside $Y$. Let $\vec{Y}=(Y_1,Y_2)$, $\mathrm{\mid }\vec{Y}\mathrm{\mid }=\mathrm{\mid }{Y}_1\mathrm{\mid }+\mathrm{\mid }{Y}_2\mathrm{\mid }$, and let $a_{\alpha }^{\mathrm{A}}=c_1^T(e_{\alpha })$ be the additive dual-framing weights in the ADHM convention. Let $a_{\alpha }^{\mathrm{P}}$ be Pestun's Coulomb weights in the Nekrasov factor.

The exact convention match used here is $a_{\alpha }^{\mathrm{A}}=-a_{\alpha }^{\mathrm{P}}$: the ADHM framing representation has character $W={\sum }_{\alpha }{e}_{\alpha }^{-1}$, whereas Pestun's Cartan element acts in the fundamental representation.

In the source convention define

$E_{\alpha \beta }(s)=-h_{Y_{\beta }}(s){ϵ}_1+(v_{Y_{\alpha }}(s)+1){ϵ}_2+a_{\beta }^{\mathrm{P}}-a_{\alpha }^{\mathrm{P}}\mathrm{.}$

The source's formal $\mathcal{N}=2^{\ast }$ instanton factor is

$Z_{\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}}^{\mathcal{N}=2^{\ast }}(a_1^{\mathrm{P}},a_2^{\mathrm{P}},\tilde{m};{ϵ}_1,{ϵ}_2,q)={\sum }_{\vec{Y}}{q}^{\mathrm{\mid }\vec{Y}\mathrm{\mid }}{\prod }_{\alpha ,\beta =1}^2{\prod }_{s\in Y_{\alpha }}\frac{(E_{\alpha \beta }(s)-\tilde{m})({ϵ}_1+{ϵ}_2-E_{\alpha \beta }(s)-\tilde{m})}{E_{\alpha \beta }(s)({ϵ}_1+{ϵ}_2-E_{\alpha \beta }(s))}\mathrm{.}$

For the finite-dimensional matrix integral, let $d{\mu }_0(\mathsf{a})$ be Lebesgue measure on the real vector space $\mathfrak{u}(2)$ induced by $(X,Y)=-{\mathrm{tr}}_{\mathbf{2}}(XY)$, and let ${\mathrm{vol}}_0(U(2))$ be Haar volume for the bi-invariant Riemannian metric induced by the same form. This fixes the normalization of both measures; since $[\mathsf{a}]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$, $d{\mu }_0(\mathsf{a})$ has mass dimension four. The full Lie-algebra integral has no separate Weyl factor. If it is rewritten on the Cartan, use the source's $\mathrm{\Delta }(a_E)={\prod }_{\alpha \in \mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{s}(U(2))}\alpha \cdot a_E$ and the Weyl-group quotient $\mathrm{\mid }{S}_2\mathrm{\mid }=2$ exactly once. In the gauge-fixing normalization of Pestun's Eq. (4.3), the group volume is instead written ${\mathrm{vol}}_{g_{\mathrm{Y}\mathrm{M}}}(U(2))=g_{\mathrm{Y}\mathrm{M}}^{\dim U(2)}{\mathrm{vol}}_0(U(2))=g_{\mathrm{Y}\mathrm{M}}^4{\mathrm{vol}}_0(U(2))$; Eq. (1.4)'s unscaled $\mathrm{vol}(U(2))$ is ${\mathrm{vol}}_0(U(2))$.

With these conventions, Pestun's source-reported formal partition-function expression is

$Z_{S^4}^{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}=\frac{1}{{\mathrm{vol}}_0(U(2))}{\int }_{\mathfrak{u}(2)}d{\mu }_0(\mathsf{a})\exp \text{\hspace{0.17em}⁣}[-\frac{8{\pi }^2{r}^2}{g_{\mathrm{Y}\mathrm{M}}^2}(\mathsf{a},\mathsf{a})]Z_{1\text{-loop}}^{\mathcal{N}=2^{\ast }}(i{a}_E(\mathsf{a});m_E){\mid Z_{\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}}^{\mathcal{N}=2^{\ast }}(a^{\mathrm{P}},\tilde{m};{ϵ}_1=r^{-1},{ϵ}_2=r^{-1},q)\mid }^2,$

where $a_E(\mathsf{a})$ is the unordered real pair determined by unitary conjugation of $\mathsf{a}$ to $i\mathrm{diag}(a_1^E,a_2^E)$, $a_{\alpha }^{\mathrm{P}}=i{a}_{\alpha }^E$, and $(\mathsf{a},\mathsf{a})=-{\mathrm{tr}}_{\mathbf{2}}({\mathsf{a}}^2)={\sum }_{\alpha =1}^2(a_{\alpha }^E{)}^2$ on that Cartan. The source's $\mathrm{\mid }{Z}_{\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}}{\mathrm{\mid }}^2$ is the north-pole instanton factor times the south-pole anti-instanton factor on the physical real slice; after complex continuation these two factors are kept independent. This explicit finite-dimensional normalization does not construct the continuum functional measure.

In the positive-FI stable $U(2)$ ADHM quotient of the earlier item lem-qftx-stable-adhm-partition-fixed-points-and-tangent-character (a smooth quotient whose torus-fixed points are indexed by partition tuples and whose tangent character is specified there), the charge-one fixed points are the two pairs $((\mathrm{\square }),\mathrm{\varnothing })$ and $(\mathrm{\varnothing },(\mathrm{\square }))$. Give a fixed point with tangent weights $w_1,\dots ,w_4$ the mass-twisted weight ${\prod }_{j=1}^4(w_j-\mu )\mathrm{/}{\prod }_{j=1}^4{w}_j$, from the equivariant Euler insertion of the tangent bundle tensored with an inverse flavor-character line. After the specialization $\mu =\tilde{m}$, the sum of the two charge-one terms is

$C_1=\frac{({ϵ}_1-\tilde{m})({ϵ}_2-\tilde{m})(a_2^{\mathrm{P}}-a_1^{\mathrm{P}}+{ϵ}_1+{ϵ}_2-\tilde{m})(a_1^{\mathrm{P}}-a_2^{\mathrm{P}}-\tilde{m})}{{ϵ}_1{ϵ}_2(a_2^{\mathrm{P}}-a_1^{\mathrm{P}}+{ϵ}_1+{ϵ}_2)(a_1^{\mathrm{P}}-a_2^{\mathrm{P}})}+\frac{(a_1^{\mathrm{P}}-a_2^{\mathrm{P}}+{ϵ}_1+{ϵ}_2-\tilde{m})(a_2^{\mathrm{P}}-a_1^{\mathrm{P}}-\tilde{m})({ϵ}_1-\tilde{m})({ϵ}_2-\tilde{m})}{(a_1^{\mathrm{P}}-a_2^{\mathrm{P}}+{ϵ}_1+{ϵ}_2)(a_2^{\mathrm{P}}-a_1^{\mathrm{P}}){ϵ}_1{ϵ}_2}\mathrm{.}$

This is a source-reported structural prescription with a locally derived mathematical one-instanton coefficient for the stated tangent-bundle insertion, not a locally proved continuum path-integral identity.

Each square below is one box $(0,0)$ in the color named beneath it; an empty color has no box. The two cases exhaust total size one and the algebraic verification does not rely on the diagram.

Pestun, Eq. (1.4), §§4–5, and the noncommutative-$S^4$ qualification

Peelaers, Lecture 1 §1.2 and Lecture 2

Adopted $\mathcal{N}=2^{\ast }$ $U(2)$ $S^4$ model

Model-matched finite-regulator Ward identity

Stable ADHM fixed points and tangent character

Proper-fixed-locus localized Borel–Moore pushforward

Full Axiom of Choice