Single M5 worldvolume tensor action

Let $(M^{11},g_M)$ be the oriented Lorentzian CJS background of Eleven-Dimensional CJS Supergravity Bosonic Action. Let ${\mathrm{\Sigma }}_6$ be a smooth oriented six-manifold and let $\iota :{\mathrm{\Sigma }}_6\to M^{11}$ be a smooth embedding (an injective immersion that is a homeomorphism onto its image). In dimensionless worldvolume coordinates define the normalized induced metric by $g:={\mathrm{\ell }}_p^{-2}{\iota }^{\ast }{g}_M$, where ${\mathrm{\ell }}_p>0$ is the Planck length from the CJS postulate; require $g$, with components $g_{mn}$, to be nondegenerate Lorentzian with signature $(-+++++)$. For this local formula, restrict to embeddings whose image lies in an open set $V\subset M^{11}$ carrying smooth potentials $C^{(3)}\in {\mathrm{\Omega }}^3(V)$ and $C^{(6)}\in {\mathrm{\Omega }}^6(V)$; $C^{(3)}:={\mathrm{\ell }}_p^{-3}{C}_{3,\mathrm{C}\mathrm{J}\mathrm{S}}{\mathrm{\mid }}_V$ is the dimensionless normalization of the local CJS three-form, and $C^{(6)}$ is a supplied dimensionless local dual potential in the source convention. This rescaling is part of the adopted local field normalization; the CJS action does not itself supply $C^{(6)}$. Write $P={\iota }^{\ast }$ for pullback to ${\mathrm{\Sigma }}_6$. No transition data between potential patches or global Wess–Zumino functional is specified.

Use dimensionless worldvolume coordinates ${\xi }^m$, $m=0,\dots ,5$, and adopt the source-normalized dimensionless field convention for the displayed local functional, with $\mathrm{\hslash }=c=1$. In this convention $g_{mn}$, $A_{mn}$, $H_{mnl}$, $a$, and ${\tilde{H}}_{mn}$ are dimensionless; this normalization is an adopted convention and does not follow from setting $\mathrm{\hslash }=1$ alone. The overall physical tension is suppressed and fixed separately. Then $g=\det (g_{mn})<0$. Let $A_2\in {\mathrm{\Omega }}^2({\mathrm{\Sigma }}_6)$ be one real antisymmetric worldvolume two-form, written $A_2=\frac{1}{2}{A}_{mn}d{\xi }^m\wedge d{\xi }^n$, and let $a\in C^{\mathrm{\infty }}({\mathrm{\Sigma }}_6,\mathbb{R})$ be an auxiliary scalar. With the source's component convention set $F_{mnl}=2({\mathrm{\partial }}_l{A}_{mn}+{\mathrm{\partial }}_m{A}_{nl}+{\mathrm{\partial }}_n{A}_{lm}),F_3=\frac{1}{3!}{F}_{mnl}d{\xi }^m\wedge d{\xi }^n\wedge d{\xi }^l=2d{A}_2,H_{mnl}=F_{mnl}-P[C^{(3)}{]}_{mnl},H^{\ast mnl}=\frac{{\epsilon }^{mnlpqr}}{3!\sqrt{-g}}{H}_{pqr},$ where ${\epsilon }^{012345}=+1$ is the oriented alternating symbol and indices are raised with $g$. Restrict to the non-null auxiliary-scalar branch $(\mathrm{\partial }a{)}^2:=g^{mn}{\mathrm{\partial }}_{m}a{\mathrm{\partial }}_{n}a>0$, and set ${\tilde{H}}_{mn}=\frac{H_{mnl}^{\ast }{\mathrm{\partial }}^{l}a}{\sqrt{(\mathrm{\partial }a{)}^2}}\mathrm{.}$ For fixed $g$ and $a$ with spacelike $da$, at each $\xi \in {\mathrm{\Sigma }}_6$ let ${\mathcal{D}}_{\xi }$ be the connected component, in the usual finite-dimensional real vector-space topology, containing $0\in {\mathrm{\Lambda }}^3{T}_{\xi }^{\ast }{\mathrm{\Sigma }}_6$ of the set of $h\in {\mathrm{\Lambda }}^3{T}_{\xi }^{\ast }{\mathrm{\Sigma }}_6$ for which $\det \text{\hspace{0.17em}⁣}(g_{mn}(\xi )+i\frac{({\ast }_{g}h{)}_{mnl}{\mathrm{\partial }}^{l}a(\xi )}{\sqrt{(\mathrm{\partial }a{)}^2(\xi )}})<0,$ where $({\ast }_{g}h{)}^{mnl}:={\epsilon }^{mnlpqr}{h}_{pqr}\mathrm{/}(3!\sqrt{-g})$. Restrict the smooth field configurations to those with $H_3(\xi )\in {\mathcal{D}}_{\xi }$ for every $\xi$, and take the pointwise positive real square root $\sqrt{-\det (g+i\tilde{H})}$. On this domain define the source-normalized functional $S_{\mathrm{P}\mathrm{S}\mathrm{T}}={\int }_{{\mathrm{\Sigma }}_6}{d}^6\xi [\sqrt{-\det (g_{mn}+i{\tilde{H}}_{mn})}+\frac{\sqrt{-g}}{4(\mathrm{\partial }a{)}^2}{\mathrm{\partial }}_{m}a{H}^{\ast mnl}{H}_{nlp}{\mathrm{\partial }}^{p}a]+{\int }_{{\mathrm{\Sigma }}_6}(P[C^{(6)}]+\frac{1}{2}{F}_3\wedge P[C^{(3)}])\mathrm{.}$ The display first specifies a local six-form density $\mathcal{L}$. If ${\mathrm{\Sigma }}_6$ is compact, set $S={\int }_{{\mathrm{\Sigma }}_6}\mathcal{L}$. On a noncompact worldvolume use the local action in the following precise sense: for two admissible configurations $q,q_0$ agreeing outside a compact subset of the interior, define $S[q]-S[q_0]:={\int }_{{\mathrm{\Sigma }}_6}(\mathcal{L}[q]-\mathcal{L}[q_0])$. The difference density is smooth and compactly supported, so this integral is finite. This notation defines the difference directly and does not subtract divergent numbers; no finite total action on a noncompact worldvolume is asserted.

This defines the single-abelian bosonic PST source convention on the stated potential patch; it does not define global patching or multiple-M5 dynamics.

Pasti–Sorokin–Tonin, §2, Eq. (17)

Pasti–Sorokin–Tonin define ${\mathcal{V}}^{mn}:=-2\sqrt{-\frac{(\mathrm{\partial }a{)}^2}{g}}\frac{\delta \sqrt{-\det (g_{pq}+i{\tilde{H}}_{pq})}}{\delta {\tilde{H}}_{mn}}\mathrm{.}$ Their Eq. (9) reports the $A_2$ equation ${\epsilon }^{lmnpqr}{\mathrm{\partial }}_n\text{\hspace{0.17em}⁣}(\frac{{\mathrm{\partial }}_{p}a}{(\mathrm{\partial }a{)}^2}(H_{qrs}{\mathrm{\partial }}^{s}a-{\mathcal{V}}_{qr}))=0,$ and Eq. (10) reports that, after an appropriate gauge fixing of $\delta A_{mn}=\frac{1}{2}{\mathrm{\partial }}_{[m}a{\phi }_{n]}$, its general solution reduces to the nonlinear self-duality equation $H_{qrs}{\mathrm{\partial }}^{s}a-{\mathcal{V}}_{qr}=0.$ The source cites earlier PST work for the gauge-fixing step and does not give the complete nonlinear variation proof.

Eq. (17) of the pinned PDF prints the auxiliary term with $H_{mnp}$, so the index $m$ occurs three times in ${\mathrm{\partial }}_{m}a{H}^{\ast mnl}{H}_{mnp}{\mathrm{\partial }}^{p}a$. This is not a scalar under the Einstein summation convention. The paragraph immediately preceding Eq. (17) explicitly instructs replacement of the field strength in Eq. (1) by Eq. (16). Applying that instruction to its correctly contracted auxiliary density gives ${\mathrm{\partial }}_{m}a{H}^{\ast mnl}{H}_{nlp}{\mathrm{\partial }}^{p}a$, with each dummy index occurring exactly twice. The local functional above is defined by this explicit substitution into Eq. (1), followed by the displayed Wess–Zumino addition, rather than by treating the ill-typed printed component of Eq. (17) as a scalar. No erratum is asserted.

The Born–Infeld density and auxiliary density are added, as in both source equations; they are not multiplied. At $H_3=0$ the first density is $\sqrt{-g}$ and the second vanishes. For real $g$ and $\tilde{H}$, transposition shows $\det (g+i\tilde{H})=\det (g-i\tilde{H})$, its complex conjugate, so this determinant is real. At zero field it is $g<0$; the stated negative-determinant component therefore supplies the real positive square-root branch. These checks establish a well-typed local definition in the source convention. The nonlinear field equation and PST gauge-reduction assertion above remain explicitly source-reported model data; this definition does not claim an independent proof of that variation or supply it as a mathematical theorem.