Charge-one BPST instanton on the round four-sphere

Assume full Axiom of Choice. Let $B^5=\{z\in {\mathbb{R}}^5:\mathrm{\mid }z\mathrm{\mid }\le 1\}$ and $S^4=\mathrm{\partial }{B}^5=\{z\in {\mathbb{R}}^5:\mathrm{\mid }z\mathrm{\mid }=1\}$ have the Euclidean induced unit round metric $g_X$. Put $p_{\mathrm{\infty }}=(0,0,0,0,-1)$ and $p_0=(0,0,0,0,1)$, and set $U_N=S^4\setminus \{p_{\mathrm{\infty }}\}$ and $U_S=S^4\setminus \{p_0\}$. Define the north chart by

${\kappa }_N:{\mathbb{R}}^4⟶S^4\setminus \{p_{\mathrm{\infty }}\},{\kappa }_N(x)=(\frac{2x}{1+\mathrm{\mid }x{\mathrm{\mid }}^2},\frac{1-\mathrm{\mid }x{\mathrm{\mid }}^2}{1+\mathrm{\mid }x{\mathrm{\mid }}^2}),$

and the south chart by

${\kappa }_S:{\mathbb{R}}^4⟶S^4\setminus \{p_0\},{\kappa }_S(y)=(\frac{2y}{1+\mathrm{\mid }y{\mathrm{\mid }}^2},\frac{\mathrm{\mid }y{\mathrm{\mid }}^2-1}{1+\mathrm{\mid }y{\mathrm{\mid }}^2})\mathrm{.}$

Write $x=(x^1,x^2,x^3,x^4)$ and $y=(y^1,y^2,y^3,y^4)$ for the inverse chart coordinates, so ${\kappa }_N(0)=p_0$, $r^2=\mathrm{\mid }x{\mathrm{\mid }}^2={\sum }_{\mu =1}^4(x^{\mu }{)}^2$, and on the overlap $y=x\mathrm{/}{r}^2$. The north coordinates have $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4>0$. The metrics in these charts are respectively $4(1+r^2{)}^{-2}{\sum }_{\mu =1}^4(d{x}^{\mu }{)}^2$ and $4(1+\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-2}{\sum }_{\mu =1}^4(d{y}^{\mu }{)}^2$.

Define $SU(2)=\{U\in M_2(\mathbb{C}):U^{\ast }U=I,\det U=1\}$. Write ${ϵ}_{123}=1$ and let

${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),T_a=-\frac{i}{2}{\sigma }_a\mathrm{.}$

The $T_a$ are traceless anti-Hermitian generators. Set $B(X,Y)=-{\mathrm{tr}}_2(XY)$. Define antisymmetric symbols by ${\eta }_{aij}={ϵ}_{aij}$, ${\eta }_{ai4}={\delta }_{ai}$, ${\bar{\eta }}_{aij}=-{ϵ}_{aij}$, and ${\bar{\eta }}_{ai4}={\delta }_{ai}$ for spatial indices $i,j\in \{1,2,3\}$, where ${\delta }_{ai}$ is the Kronecker delta; extend them antisymmetrically in the last two indices. For a scale parameter $\rho >0$, define $u(x)=(x^{4}I-i{x}^j{\sigma }_j)\mathrm{/}r\in SU(2)$ on $U_N\cap U_S$. Form the quotient of the two trivial right $SU(2)$-bundles by identifying $(x,h{)}_S\sim (x,u(x{)}^{-1}h{)}_N$ for each $x\in U_N\cap U_S$ and $h\in SU(2)$, so its local sections satisfy $s_S=s_N{u}^{-1}$. Take the set of admitted bundle sectors to be the singleton containing the isomorphism class of this glued bundle, and take the representation family to be the singleton defining representation on ${\mathbb{C}}^2$. Define the local connection potentials by

$A_N=\frac{2{\eta }_{a\mu \nu }{x}^{\nu }}{r^2+{\rho }^2}{T}_{a}d{x}^{\mu },A_S=-\frac{2{\rho }^2{\bar{\eta }}_{a\mu \nu }{y}^{\nu }}{1+{\rho }^2\mathrm{\mid }y{\mathrm{\mid }}^2}{T}_{a}d{y}^{\mu }\mathrm{.}$

On the overlap these obey $A_S={\mathrm{Ad}}_u{A}_N+ud(u^{-1})$, so they define a smooth principal connection on the glued bundle. Let $F$ be its global curvature, with chart representatives $F_N$ and $F_S$. In the north chart its curvature, written $F_N=\frac{1}{2}{F}_{\mu \nu }^{N}d{x}^{\mu }\wedge d{x}^{\nu }$, has components

$F_{\mu \nu }^N=-\frac{4{\rho }^2}{(r^2+{\rho }^2{)}^2}{\eta }_{a\mu \nu }{T}_a\mathrm{.}$

Let ${\ast }_{g_X}$ be the Hodge star characterized on two-forms by $\alpha \wedge {\ast }_{g_X}\beta =⟨\alpha ,\beta {⟩}_{g_X}d{\mathrm{vol}}_X$, acting componentwise on the Lie-algebra coefficients. This curvature is self-dual, meaning $F_N={\ast }_{g_X}{F}_N$. With $B$ as fixed above, define the charge by $q_B=(8{\pi }^2{)}^{-1}{\int }_{S^4}B(F\wedge F)$ and, for any real $g>0$, define the action by $S_B(g)=g^{-2}{\int }_{S^4}B(F\wedge {\ast }_{g_X}F)$. Then $q_B=1$, $S_B(g)=8{\pi }^2\mathrm{/}{g}^2<\mathrm{\infty }$, and the action saturates the self-dual bound. This is one explicit Belavin–Polyakov–Schwartz–Tyupkin (BPST) charge-one connection; no uniqueness or moduli classification is asserted.

Explicit charge-one BPST connection on the round four-sphere