Independent pure $N=2$ SU(2) ultraviolet vector-multiplet datum

Adopt independently a rigid four-dimensional $\mathcal{N}=2$ pure Yang–Mills model on oriented and time-oriented $M={\mathbb{R}}^{1,3}$, $\eta =\mathrm{diag}(-1,1,1,1)$, with increasing $x^0$ time orientation and natural units $c=\mathrm{\hslash }=1$. The global group is $SU(2)$, with one supplied smooth right principal bundle $P\to M$ and smooth equivariant local trivializations. The connection and its curvature are $A$ and $F_A=dA+\frac{1}{2}[A\wedge A]$, in the right-action convention of the Yang–Mills model supplier. Use anti-Hermitian $T_a=-i{\sigma }_a\mathrm{/}2$ and $\kappa (X,Y)=-2{\mathrm{tr}}_{\mathbf{2}}(XY)$; let ${\kappa }_{\mathbb{C}}$ be its complex-bilinear extension and ${\kappa }_H(\bar{X},Y)={\kappa }_{\mathbb{C}}(\bar{X},Y)$. Write $g=g_{\mathrm{Y}\mathrm{M}}>0$ for the supplied vector/connection normalization, so $A_{\mu }=-ig{B}_{\mu }$ for its canonical Hermitian vector component $B_{\mu }$. Every microscopic prepotential exponential, curvature/component dictionary and $1\mathrm{/}{g}^2$ action coefficient below uses this same fixed dimensionless coupling. Fix theta angle zero. A bundle sector means its smooth equivariant isomorphism class over $M$; only the supplied sector is used.

The microscopic matter-hypermultiplet index set is empty. The vector multiplet comprises the connection, one complex adjoint scalar $\varphi$, and two adjoint left-Weyl fermions. In $\mathcal{N}=1$ packaging it is a real adjoint vector superfield $V$ and an adjoint chiral superfield $\mathrm{\Phi }$, not a matter hypermultiplet. Use the supplied superspace derivatives, chirality, ordered Berezin measures and vector-supergauge convention, including

${\mathcal{C}}_{\alpha ,R}=-\frac{1}{4}{\bar{D}}^2(e^{-2g{V}_R}{D}_{\alpha }{e}^{2g{V}_R}),[X{]}_D=-\frac{1}{4}{I}_D(X),[Y{]}_F=\frac{1}{2}{I}_F(Y)\mathrm{.}$

Here ${\mathcal{C}}_{\alpha ,R}$ is the represented matrix curvature, not the abstract Lie-algebra curvature used in the action. With Hermitian $t_{a,R}=i{\rho }_{R\ast }(T_a)$, let $W_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a$ denote the canonically normalized component field-strength superfields, so ${\mathcal{C}}_{\alpha ,R}=2g{t}_{a,R}{W}_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a$. Define the abstract connection-normalized curvature by

${\mathcal{W}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n},\alpha }=g{T}_a{W}_{\mathrm{c}\mathrm{a}\mathrm{n},\alpha }^a,{\rho }_{R\ast }({\mathcal{W}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n},\alpha })=\frac{{\mathcal{C}}_{\alpha ,R}}{2i}\mathrm{.}$

In the fundamental representation, ${\kappa }_{\mathbb{C}}=-2{\mathrm{tr}}_{\mathbf{2}}$ therefore gives ${\kappa }_{\mathbb{C}}({\mathcal{W}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}^{\alpha },{\mathcal{W}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n},\alpha })=\frac{1}{2}{\mathrm{tr}}_{\mathbf{2}}({\mathcal{C}}^{\alpha }{\mathcal{C}}_{\alpha })$, with the same ordered spinor product on both sides. This is a classical representation/component normalization, not a quantum field-rescaling or threshold-matching theorem.

Adopt the connection-normalized microscopic action

$S_{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}={\int }_M{d}^{4}x\{\frac{1}{g^2}[{\kappa }_H(\bar{\mathrm{\Phi }},e^{2g{V}_{\mathrm{a}\mathrm{d}}}\mathrm{\Phi }){]}_D+({[\frac{1}{4g^2}{\kappa }_{\mathbb{C}}({\mathcal{W}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}^{\alpha },{\mathcal{W}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n},\alpha })]}_F+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.})\}\mathrm{.}$

Here $I_D,I_F$ are the raw ordered mathematical integrals, and the physical action brackets and chiral components use Physical superspace action brackets and canonical chiral components. The Hermitian-conjugate operation applies only to the chiral F term; the D term is real. Equivalently, the fundamental matrix F integrand is ${\mathrm{tr}}_{\mathbf{2}}({\mathcal{C}}^{\alpha }{\mathcal{C}}_{\alpha })\mathrm{/}(8g^2)$, plus its Hermitian conjugate. Its bosonic gauge density is fixed to $-\kappa (F_A\wedge {\ast }_{\eta }{F}_A)\mathrm{/}(2g^2)$. There is no chiral superpotential, hypermultiplet, hypermultiplet mass, or FI parameter in this ultraviolet datum. The scalar is connection-normalized; its canonical version is ${\varphi }_{\mathrm{c}\mathrm{a}\mathrm{n}}=\varphi \mathrm{/}g$. In the same local Berezin normalization the quadratic microscopic prepotential is ${\mathcal{F}}_{\mathrm{U}\mathrm{V}}(\mathrm{\Phi })=i{\kappa }_{\mathbb{C}}(\mathrm{\Phi },\mathrm{\Phi })\mathrm{/}(2g^2)$, with positive imaginary Hessian $\kappa \mathrm{/}{g}^2$. The prepotential has mass dimension two. The dimensions are $[x]=-1$, $[\theta ]=-1\mathrm{/}2$, $[A]=[\varphi ]=1$, $[F_A]=2$, $[{\mathcal{W}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}]=[\mathcal{C}]=3\mathrm{/}2$, and $[g]=0$.

Eight real rigid supercharges, their $\mathcal{N}=2$ algebra and invariance of this adopted action are model premises, not a locally proved invariance theorem. Variations are smooth and compactly supported; actions are on relatively compact regions or where integrable. A small $\mathcal{N}=1$ adjoint-mass perturbation and the Seiberg–Witten solution are separate subsequent inputs. In particular an emergent infrared monopole hypermultiplet is not microscopic matter added to this pure model. This postulate defines classical ultraviolet field/action data; it constructs no quantum measure, regulator limit, interacting continuum theory, or empirical verification.

Physical superspace action brackets and canonical chiral components

Seiberg and Witten, §§2.1–2.3.

Four-dimensional real extended super-Poincare algebra

Four-dimensional $N=1$ superspace Grassmann algebra and Berezin integration

Four-dimensional $N=1$ chiral superfield on flat superspace

Four-dimensional $N=1$ non-Abelian vector superfield in connection normalization

Classical pure SU(2) Yang–Mills model