Pure $N=2$ SU(2) Coulomb-branch and period data

Use natural units $c=\mathrm{\hslash }=1$, spacetime ${\mathbb{R}}^{1,3}$ with metric $\eta =\mathrm{diag}(-1,1,1,1)$, and a selected pure four-dimensional rigid $\mathcal{N}=2$ Yang--Mills model whose global gauge group is $SU(2)$. The microscopic matter-hypermultiplet index set is empty, as explicitly adopted in the independent pure-model postulate. In $\mathcal{N}=1$ language the gauge sector is one vector superfield and one adjoint chiral superfield with scalar $\varphi \in {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, connection normalization ${\varphi }_{\mathrm{c}\mathrm{a}\mathrm{n}}=\varphi \mathrm{/}g$, positive dimensionless gauge coupling $g$, zero chiral superpotential and no ultraviolet matter mass. The low-energy charged monopole fields introduced later are emergent infrared fields, not an ultraviolet hypermultiplet. This is the selected physical datum, not a specialization inferred from the different one-hypermultiplet postulate.

A classical Coulomb-branch configuration is a constant adjoint scalar configuration satisfying the model's zero auxiliary-field conditions, taken modulo the compact $SU(2)$ gauge action. This uses the compact-group orbit quotient in the preceding vacuum-moduli definition; it does not replace that quotient by a complexified-group stability construction. In the weak-coupling semiclassical frame, let $a_{\mathrm{c}\mathrm{l}}$ be the eigenvalue coordinate fixed by $\varphi =\frac{1}{2}\mathrm{diag}(a_{\mathrm{c}\mathrm{l}},-a_{\mathrm{c}\mathrm{l}})$, of mass dimension one. When a quantum theory with a vacuum-moduli set has actually been constructed, the quantum Coulomb-branch coordinate $u$ denotes the chosen gauge-invariant chiral quadratic-operator expectation value in the source trace normalization for which the classical relation is $u_{\mathrm{c}\mathrm{l}}=a_{\mathrm{c}\mathrm{l}}^2\mathrm{/}2$. Here $a_{\mathrm{c}\mathrm{l}}$ denotes the preceding classical eigenvalue coordinate; the quantum $a(u)$ below is a local period, with weak-coupling normalization $a(u{)}^2\mathrm{/}(2u)\to 1$ as $\mathrm{\mid }u\mathrm{\mid }\mathrm{/}{\mathrm{\Lambda }}^2\to \mathrm{\infty }$ on a chosen electric branch, where $\mathrm{\Lambda }>0$ is the fixed dynamical scale of mass dimension one. This is an asymptotic relation, not an identity at finite quantum $u$. The mass dimension of $u$ is two. This definition does not assert that $u$ globally parameterizes the quantum vacua.

Equip the formal electric/magnetic charge lattice ${\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}={\mathbb{Z}}^2$ with the alternating integer-valued form ${\omega }_{\mathrm{\Gamma }}((n_e,n_m),(k_e,k_m))=n_m{k}_e-n_e{k}_m$ in its reference coordinates. This form fixes an orientation convention for frames; it does not specify the normalization of a physical Dirac pairing. An electric/magnetic frame is an ordered lattice basis $({\gamma }_m,{\gamma }_e)$ satisfying ${\omega }_{\mathrm{\Gamma }}({\gamma }_m,{\gamma }_e)=1$, called a symplectic basis here. When realized by homology cycles, its intersection convention is ${\gamma }_m\cdot {\gamma }_e=1$.

On a sufficiently small open regular patch $U$ of the $u$-plane, supply holomorphic period functions $a_D,a:U\to \mathbb{C}$ for these magnetic and electric basis elements, both of mass dimension one. Require $a$ to be a holomorphic coordinate on $U$, with $da\mathrm{/}du\mathrm{\ne }0$, and require the local coupling below to have positive imaginary part. The local two-derivative abelian gauge action is encoded by a holomorphic prepotential ${\mathcal{F}}_U:a(U)\to \mathbb{C}$ of mass dimension two, defined in this coordinate by

$a_D=\frac{\mathrm{\partial }{\mathcal{F}}_U}{\mathrm{\partial }a},\tau =\frac{\mathrm{\partial }{a}_D}{\mathrm{\partial }a}=\frac{{\mathrm{\partial }}^2{\mathcal{F}}_U}{\mathrm{\partial }{a}^2},\mathrm{Im}\tau >0.$

The dimensionless $\tau$ is the local complexified gauge coupling. A frame change on a connected overlap has a constant matrix $M=\left(\begin{array}{cc}p& q\\ r& s\end{array}\right)\in SL(2,\mathbb{Z})$, meaning $p,q,r,s\in \mathbb{Z}$ and $ps-qr=1$, and acts by

$\left(\begin{array}{c}{\tilde{\gamma }}_m\\ {\tilde{\gamma }}_e\end{array}\right)=M\left(\begin{array}{c}{\gamma }_m\\ {\gamma }_e\end{array}\right),\left(\begin{array}{c}{\tilde{a}}_D\\ \tilde{a}\end{array}\right)=M\left(\begin{array}{c}{a}_D\\ a\end{array}\right),\tilde{\tau }=\frac{p\tau +q}{r\tau +s}\mathrm{.}$

After shrinking the overlap if necessary, use a local prepotential in the new electric coordinate $\tilde{a}$. No determinant-minus-one change is a frame change in this convention. Neither period is asserted to be a single-valued global function on a punctured branch.

In the selected frame, the formal electric/magnetic charge label $(n_e,n_m)\in {\mathbb{Z}}^2$ represents $\gamma =n_e{\gamma }_e+n_m{\gamma }_m$, with period pairing ${\mathcal{Z}}_{\mathrm{S}\mathrm{W}}(u;n_e,n_m)=n_{e}a(u)+n_m{a}_D(u)$. The pair labels the central-charge formula; the lattice definition does not assert that every label is an allowed line operator, a dynamical particle, a stable state, or a state in any constructed Hilbert space. Those are separate model data. For the same $\gamma$ in the changed frame the charge row transforms by $({\tilde{n}}_m,{\tilde{n}}_e)=(n_m,n_e)M^{-1}$, so the period pairing is unchanged.

The word exact in this page refers only to the adopted source-proposed Wilsonian two-derivative low-energy data on a regular patch of the selected pure SU(2) model. It does not mean a proof of a nonperturbative continuum path integral, a global single-valued prepotential, or an exact statement about arbitrary $\mathcal{N}=2$ theories.

Seiberg and Witten, §§2.2–2.4, §3.1, §5.1, and §6, equations (6.7)–(6.14).

Bilal, §2.2, §§3.2–3.4, and §6.1.

Tong, §4.4.1.

Classical and conditional quantum vacuum-moduli data: Classical and quantum supersymmetric vacuum-moduli data.

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