Iib Axiodilaton And Pq String Charge

Let $M$ be an oriented smooth ten-dimensional manifold and $\mathcal{W}\mathrm{\ne }\mathrm{\varnothing }$ an oriented $C^2$ embedded two-dimensional source locus, closed as a subset of $M$. Write $U:=M\setminus \mathcal{W}$, an open smooth manifold. Apply the named supergravity supplier's spacetime conventions to $U$: $\mathrm{\hslash }=c=1$ and a smooth Lorentzian Einstein-frame metric $G^{(E)}$ on $U$ of signature $(-,+,\dots ,+)$. Take smooth maps $C_0,\mathrm{\Phi }:U\to \mathbb{R}$, where $C_0$ is the dimensionless type-IIB RR axion and $\mathrm{\Phi }$ the dimensionless dilaton, and define $G^{(s)}=e^{\mathrm{\Phi }\mathrm{/}2}{G}^{(E)}$ on $U$. The idealized source worldsheet $\mathcal{W}$ labels the excluded string core; neither metric is required to extend across it, and no induced physical metric or pullback of the exterior potentials to $\mathcal{W}$ is assumed. Thus this convention includes singular string exteriors without asserting a regular core or a distributional solution of the coupled source equations. Closed as a subset does not require the worldsheet to be boundaryless.

Specify an end chart $(x^0,x^1,y^2,\dots ,y^9)$ identifying an open end $E\subset M\setminus \mathcal{W}$ with ${\mathbb{R}}^{1,1}\times \{y\in {\mathbb{R}}^8:\mathrm{\mid }y\mathrm{\mid }>R_0\}$ for some $R_0>0$, and put $r:=\mathrm{\mid }y\mathrm{\mid }$. Write $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$ in these coordinates and ${\mathbb{S}}^7:=\{z\in {\mathbb{R}}^8:\mathrm{\mid }z\mathrm{\mid }=1\}$. In this item, “flat asymptotic” means that $G_{\mu \nu }^{(E)}-{\eta }_{\mu \nu }$ and all its first and second coordinate derivatives tend to zero as $r\to \mathrm{\infty }$, uniformly over $y\mathrm{/}r\in {\mathbb{S}}^7$ and locally uniformly for $(x^0,x^1)$ in compact subsets of ${\mathbb{R}}^{1,1}$. Assume also that $C_0$ and $\mathrm{\Phi }$ tend to constants $C_{0,\mathrm{\infty }}\in \mathbb{R}$ and ${\mathrm{\Phi }}_{\mathrm{\infty }}\in \mathbb{R}$ in the same uniform and local-uniform sense. Define the asymptotic axiodilaton ${\tau }_{\mathrm{\infty }}:=C_{0,\mathrm{\infty }}+i{e}^{-{\mathrm{\Phi }}_{\mathrm{\infty }}}$ and the upper half-plane $\mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$; then ${\tau }_{\mathrm{\infty }}\in \mathbb{H}$.

Let $B_2^{(1)}$ and $B_2^{(2)}$ be the real NS–NS and RR two-form gauge potentials, respectively, smooth locally on $M\setminus \mathcal{W}$, with three-form field strengths $H_3^{(j)}:=d{B}_2^{(j)}$. A source's charge labels are the integer column $Q:=(p,q{)}^{\mathsf{T}}\in {\mathbb{Z}}^2\setminus \{0\}$ in this ordered basis: $p$ is its NS–NS charge and $q$ its RR charge, normalized so the unit F-string and D-string have labels $(1,0)$ and $(0,1)$. The formal source-charge pairing is the exterior two-form $Q^{\mathsf{T}}{B}_2$, where $B_2:=(B_2^{(1)},B_2^{(2)}{)}^{\mathsf{T}}$. Call $Q$ primitive when $\gcd (\mathrm{\mid }p\mathrm{\mid },\mathrm{\mid }q\mathrm{\mid })=1$. A charge vector with a larger gcd is interpreted in the source family as the corresponding multiple-string threshold sector, not as a new elementary primitive string.

For $\mathrm{\Lambda }=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{S}\mathrm{L}(2,\mathbb{Z})$, where $a,b,c,d\in \mathbb{Z}$ and $ad-bc=1$, fix the convention ${\tau }_{\mathrm{\infty }}^{\mathrm{\prime }}=(a{\tau }_{\mathrm{\infty }}+b)\mathrm{/}(c{\tau }_{\mathrm{\infty }}+d)$ and $Q^{\mathrm{\prime }}=\mathrm{\Lambda }Q$. In the same source basis the two-form column transforms as $B_2^{\mathrm{\prime }}=({\mathrm{\Lambda }}^{\mathsf{T}}{)}^{-1}{B}_2$, so the source pairing $Q^{\mathsf{T}}{B}_2$ is unchanged. These transformation rules define the classical charge/background covariance convention; they do not assert an exact equivalence of the full quantum theory.

Let $T_F>0$ denote the fixed string-frame reference tension of the unit $(1,0)$ F-string, with dimension energy per length (length${}^{-2}$ in the adopted units). Define the dimensionless normalized central-charge factor ${\mathcal{Z}}_{p,q}({\tau }_{\mathrm{\infty }}):=(p-q{\tau }_{\mathrm{\infty }})\mathrm{/}\sqrt{\mathrm{Im}{\tau }_{\mathrm{\infty }}}$, using the positive real square root. This algebraic factor is defined for every stated charge and asymptotic modulus; it assigns no physical energy to the arbitrary exterior data above.

The physical tension terminology refers only to the cited classical infinite straight-string family: take $M={\mathbb{R}}^{1,1}\times {\mathbb{R}}^8$ and $\mathcal{W}={\mathbb{R}}^{1,1}\times \{0\}$, with exterior fields independent of $x^0,x^1$, satisfying the exterior field equations of Schwarz's action (6), and with the self-dual five-form field strength $F_5$ and all fermionic fields identically zero. Here $T_{p,q}^{(E)}$ is the source-reported asymptotic Einstein-frame energy per longitudinal length: an interval of asymptotic proper length $L$ has energy $T_{p,q}^{(E)}L$ in this stationary translation-invariant idealization. Both energy and length use the asymptotic normalization $G^{(E)}\to \eta$; this definition requires no metric on the singular core and assigns no finite total energy to the infinite string. “BPS” means saturation of the adopted tension-charge bound $T_{p,q}^{(E)}\ge T_F\mathrm{\mid }{\mathcal{Z}}_{p,q}\mathrm{\mid }$, namely equality. Neither the bound nor the existence and energy of these classical solutions are derived here: the source-reported value $T_{p,q}^{(E)}=T_F\mathrm{\mid }{\mathcal{Z}}_{p,q}\mathrm{\mid }$ is a physical premise of the consuming theorem, restricted to this family (and its stated multiple-string threshold sectors), rather than a consequence of the preceding kinematic exterior conventions.

Supergravity Conventions And Normalizations

Schwarz, §§2–4

Uranga, §14.3.1 and Appendix B.2

The source's fields, charge vector, and transformation law fix one sign convention. Uranga's lecture notes use a different two-form-doublet convention; their discussion supports the distinction between classical continuous covariance and conjectural discrete quantum duality, but does not override the explicit Schwarz charge convention used here.