
where the M5-brane action in a D = 11 SUGRA background is given by:
with:
and the -brane action takes the following form:
where:
with:
where the Hamiltonian metaplectic action in the Heisenberg representation on the Dp+1 dimensional worldvolume gives us:
where:
with:
and:
where the Ramond-Ramond gauge-coupling sector is given by the action:
and generally, the action of a -brane is given by a Dirac-Born-Infeld part coupled to a Chern-Simons WZ part:
with:
where is the worldvolume pullback with
-orientifold action:
with:
and
where the pullback to the -worldvolume yields the 10-D SYM action:
with string coupling:
and the 10-D SUGRA dimensionally reduced Type-IIB action is:
with:
and in the string-frame, the type-IIB SUGRA action is given by:
with:
where the Calabi-Yau superpotential is:
where:
is the Gukov-Vafa-Witten superpotential stabilization complex term, as well as the axio-dilaton field:
Given the presence of -brane instantons,
are of Kähler moduli Type-IIB-orbifold class:
with being the volume of the divisor
and
the 4-form Ramond-Ramond axion field corresponding to:
and:
where is the Kähler form:
and:
an integral-form basis and the associated intersection coefficients. Hence, the Kähler potential is given by:
with the Calabi-Yau volume, and in the Einstein frame, is given by:
The -term is given by:
with the large volume scenario -term given as such:
with:
and the Fayet-Illopoulos terms being:
where are the
-brane charge-vectors. Then we saw that the Chern-Simons orientifold action gets a Calabi-Yau curvature correction in the form of:
with
due to the Gauss–Codazzi equations:
and the Ramond-Ramond term being:
which yields the Type-IIB Calabi-Yau three-fold superpotential:
and where the topologically mixed Yang-Mills action is given by:
with the corresponding Chern-Simons action:
From the topologically mixed Yang-Mills action, it follows that the Euler-Lagrange EOM for an M5-brane, in the abelian tensor multiplet case, is:

where the following holds:
![Rendered by QuickLaTeX.com \displaystyle \left\{ {\begin{array}{*{20}{c}} {{{H}_{{mnp}}}=3{{\partial }_{{\left[ m \right.}}}{{B}_{{\left. {np} \right]}}}} \\ {{{\epsilon }^{{012345}}}=1} \\ {\alpha ,\beta =1,...,4:\quad \text{ind}\left[ {\text{irre}{{\text{p}}^{{fun}}}USp(4)/\text{R-sym}} \right]} \end{array}} \right.](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-268e2455875144b107733e5e2897994a_l3.png)
and are invariant under SUSY transformations:
![Rendered by QuickLaTeX.com \displaystyle \left\{ {\begin{array}{*{20}{c}} {\delta {{\phi }^{{\alpha \beta }}}=-i{{{\bar{\epsilon }}}^{{\left[ \alpha \right.}}}{{\psi }^{{\left. \beta \right]}}}} \\ {\delta {{B}_{{mn}}}=-i{{{\bar{\epsilon }}}^{\alpha }}{{\Gamma }_{{mn}}}{{\psi }_{\alpha }}} \\ {\delta {{\psi }^{\alpha }}={{\nabla }_{m}}{{\phi }^{\alpha }}_{\beta }{{\Gamma }^{m}}{{\epsilon }^{\beta }}+\frac{1}{{2\cdot 3!}}{{\Gamma }^{{mnp}}}{{H}_{{mnp}}}{{\epsilon }^{\alpha }}} \end{array}} \right.](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-f22be97caaacf8a5ab4f3d5074c76fa6_l3.png)
with a chiral Killing spinor on the M5-brane worldvolume. Since the M5-brane worldvolume is a multi-centered Taub-NUT space
with:





our worldvolume metric is given as such:

and asymptotically takes the form:

and since such a class of manifolds have self-dual curvature, it follows that there is a Killing spinor satisfying:

Since the bosonic solutions to the E(s)OM preserve 8 supersymmetries, we have the following relation:

and in light-cone gauge, we thus have:


Hence, the E(s)OM satisfying 8 supersymmetries are:

On the multi-centered Taub-NUT space, our 2-form is given as such:

with:


Hence, the fermionic E(s)OM are given by:

The energy-momentum tensor for scalars and fermions is given as such:

with the super-Poincaré operator given by:

Hence, the abelian conserved current takes the following form:

for all 1-forms derived from the gauge symmetry corresponding to the Hermitian Yang-Mills equations, with total charge:

Restricting to , asymptotically we get:
![Rendered by QuickLaTeX.com \displaystyle \begin{array}{l}Q\left( {{{\Lambda }_{5}}\left( \infty \right)} \right)=\frac{1}{{2\pi R}}\text{Tr}\sum\limits_{I}{{\oint_{{\mathbb{R}\times {{S}^{1}}\times S_{\infty }^{2}}}{{d{{\Omega }_{2}}}}}}d{{x}^{+}}\left[ {H{{\partial }_{r}}} \right.\left( {\frac{{{{h}_{I}}}}{H}} \right)+\\{{\varepsilon }^{{rjk}}}{{\theta }_{j}}{{\partial }_{k}}\left. {\left( {\frac{{{{h}_{I}}}}{H}} \right)} \right]\upsilon _{+}^{I}{{\Lambda }_{5}}\left( \infty \right)\end{array}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-3341bd214a5cf26faf8f53a58abcc3e2_l3.png)
and where the D4-brane gauge field is given as such:

In the non-abelian case, the M5-brane compactified on a circle of radius R yields at low energy, a 5D M-SYM. A fibrational multi-centered Taub-NUT reduction on imposes the following metric:

with Yang-Mills action:

and the scalar action:
![Rendered by QuickLaTeX.com \displaystyle \begin{array}{l}{{S}_{s}}^{\phi }=-\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\int{{{{d}^{5}}}}x\sqrt{{-g}}\left( {\sqrt{H}} \right.{{D}_{\mu }}{{\phi }_{{\alpha \beta }}}{{D}^{\mu }}{{\phi }^{{\alpha \beta }}}+\\\frac{1}{4}\frac{1}{{{{H}^{{5/2}}}}}{{\partial }_{i}}H{{\partial }_{i}}H{{\phi }_{{\alpha \beta }}}{{\phi }^{{\alpha \beta }}}-\sqrt{H}\left[ {{{\phi }^{{\alpha \beta }}},\left. {{{\phi }_{\beta }}^{\delta }} \right]} \right.\left. {\left[ {{{\phi }_{{\delta \gamma }}},{{\phi }^{\gamma }}_{\alpha }} \right]} \right)\end{array}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-454b7a969ebf35a2e16ca01028a7062a_l3.png)
and our Chern-Simons term is given as such:

One can now analyze the M5-brane from the perspective of the D3/D5-brane configuration system. The Killing spinor equation in such a system is given by:

and where the supersymmetry variation is given as such:
![Rendered by QuickLaTeX.com \displaystyle \begin{array}{l}\delta {{\psi }^{\alpha }}=\frac{1}{2}{{F}_{{\mu \nu }}}{{\gamma }^{{\mu \nu }}}{{\varepsilon }^{\alpha }}+2i\sqrt{H}{{M}_{{\beta \gamma }}}{{D}_{\mu }}\left( {\frac{1}{{\sqrt{H}}}{{\phi }^{{\alpha \beta }}}} \right){{\gamma }^{\mu }}{{\varepsilon }^{\gamma }}\\-\frac{1}{{\sqrt{H}}}{{M}_{{\beta \gamma }}}{{\phi }^{{\alpha \beta }}}{{{\tilde{F}}}_{{\mu \nu }}}{{\gamma }^{{\mu \nu }}}{{\varepsilon }^{\gamma }}+2{{M}_{{\beta \gamma }}}{{M}_{{\delta \lambda }}}\left[ {{{\phi }^{{\alpha \beta }}},{{\phi }^{{\gamma \delta }}}} \right]{{\varepsilon }^{\gamma }}\end{array}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-1f65ecec3c413acb33f0dcef16ee2d23_l3.png)
with the field strength:
![Rendered by QuickLaTeX.com \displaystyle {{F}_{{\mu \nu }}}={{\nabla }_{\mu }}{{A}_{\nu }}-{{\nabla }_{\nu }}{{A}_{\mu }}+{{\left[ {{{A}_{\mu }},{{A}_{\nu }}} \right]}_{\mathfrak{g}}}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-2455b73e1f42bcdd882bb67ed46b3107_l3.png)
and the covariant derivative is hence given as such in terms of a field transforming in the adjoint representation of the gauge group:
![Rendered by QuickLaTeX.com \displaystyle {{D}_{\mu }}\chi ={{\nabla }_{\mu }}\chi +{{\left[ {{{A}_{\mu }},\chi } \right]}_{\mathfrak{g}}}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-f25fb7743497002c5d7b818b1b53b92a_l3.png)
and where our BPS conditions are given as such:

and so we can derive:
![Rendered by QuickLaTeX.com \displaystyle \left\{ {\begin{array}{*{20}{c}} {{{D}_{i}}\left( {\sqrt{H}{{\phi }^{\alpha }}_{\beta }} \right)={{D}_{-}}{{\phi }^{\alpha }}_{\beta }=0} \\ {{{{\left[ {{{\phi }^{\alpha }}_{\beta },{{\phi }^{\beta }}_{\gamma }} \right]}}_{\mathfrak{g}}}=0} \end{array}} \right.](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-7dd43891557f8a5af3c02c6193c82685_l3.png)
from which the equations of motions follow as:

In light of the BPS conditions:


it follows that up to metaplecticomorphism, solutions to the BPS equation are gauge transformations of configuration type:

Hence, the E(s)OM are given as such:

with:

with general Yang-Mills form:

Thus, becomes an element of the full M5-brane gauge algebra in light of the identity:

The fermionic EOM is given as such:

and has a bi-chirality solitonic split as such:


the second being the Dirac equation for:

and the D-term in the first equation is the only source of the non-abelian gauge field, and group-theoretically, we have in terms of a representation h.w. term member , the following equation:

and consequently, the following can be derived:

The energy-momentum tensor is thus given as such:
![Rendered by QuickLaTeX.com \displaystyle \begin{array}{l}{{T}_{{\mu \nu }}}=\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\left[ {2\sqrt{H}} \right.{{D}_{\mu }}{{\phi }_{{\alpha \beta }}}{{D}_{\nu }}{{\phi }^{{\alpha \beta }}}+\\\frac{1}{{2{{H}^{{3/2}}}}}{{\partial }_{\mu }}H{{\partial }_{\nu }}H{{\phi }_{{\alpha \beta }}}{{\phi }^{{\alpha \beta }}}+2\sqrt{H}{{F}_{{\mu \rho }}}{{F}_{\nu }}^{\rho }\\-{{g}_{{\mu \nu }}}\left( {2\sqrt{H}{{D}_{\rho }}{{\phi }_{{\alpha \beta }}}{{D}^{\rho }}{{\phi }^{{\alpha \beta }}}+} \right.\frac{1}{4}\frac{1}{{{{H}^{{5/2}}}}}{{\partial }_{i}}H{{\partial }_{i}}H{{\phi }_{{\alpha \beta }}}{{\phi }^{{\alpha \beta }}}\\+\frac{{\sqrt{H}}}{2}{{F}_{{\rho \sigma }}}{{F}^{{\rho \sigma }}}-\sqrt{H}\left[ {{{\phi }^{{\alpha \beta }}},{{\phi }_{{\beta \lambda }}}} \right]\left. {\left. {\left[ {{{\phi }^{{\lambda \rho }}},{{\phi }_{{\rho \alpha }}}} \right]} \right)} \right]\end{array}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-b78f9fcf432bd6d015989b35fc46bf30_l3.png)
which implies:

and so the energy-momentum tensor reduces to the following form:

yielding the spacial integral relation:

which corresponds to copies of multi-centered Taub-NUT spaces of a WZWN model for each
, and crucially, the non-abelian gauge charges for our M5-brane action have the following form:
![Rendered by QuickLaTeX.com \displaystyle {{J}^{\sigma }}\left( \Lambda \right)=\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\left[ {-2\sqrt{{-g}}\sqrt{H}{{F}^{{\sigma \lambda }}}{{D}_{\lambda }}\Lambda +{{\varepsilon }^{{\mu \nu \rho \sigma \lambda }}}{{\theta }_{\mu }}{{F}_{{\nu \rho }}}{{D}_{\lambda }}\Lambda } \right]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-678dac81038bc273a8500cc33a40834b_l3.png)
with corresponding charges:
![Rendered by QuickLaTeX.com \displaystyle \begin{array}{c}Q\left( {\Lambda \left( \infty \right)} \right)=\frac{1}{{4{{\pi }^{2}}R}}\text{tr}\sum\limits_{I}{{\oint{{d{{\Omega }_{2}}}}}}d{{x}^{+}}\left[ {H{{\partial }_{r}}} \right.{{K}_{I}}+\\{{\varepsilon }^{{rjk}}}\left. {{{\theta }_{j}}{{\partial }_{k}}{{K}_{I}}} \right]\upsilon _{+}^{I}\Lambda \left( \infty \right)=-\frac{1}{{2\pi }}\text{tr}\sum\limits_{I}{{{{N}_{I}}}}\int{{d{{x}^{+}}}}\upsilon _{+}^{I}\Lambda \left( \infty \right)\end{array}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-167cc1730b944161e3c9994027137c59_l3.png)
Hence, the action of our BPS sector yields our desired result, a WZWN model for a multi-centred Taub-NUT space in the context of M-theory:
![Rendered by QuickLaTeX.com \displaystyle \begin{array}{l}{{S}_{{BPS}}}=-\sum\limits_{I}{{\frac{{{{N}_{I}}}}{{4\pi }}}}\text{tr}\int{{d{{x}^{+}}}}d{{x}^{-}}g\left( {{{{\vec{x}}}_{I}}} \right){{\partial }_{+}}{{g}^{{-1}}}\left( {{{{\vec{x}}}_{I}}} \right)g\left( {{{{\vec{x}}}_{I}}} \right){{\partial }_{-}}{{g}^{{-1}}}\left( {{{{\vec{x}}}_{I}}} \right)\\+\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\int{{{{d}^{5}}}}x{{\partial }_{i}}H\left[ {{{g}^{{-1}}}{{\partial }_{-}},{{g}^{{-1}}}{{\partial }_{+}}} \right]{{g}^{{-1}}}{{\partial }_{i}}g\end{array}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-867b0edce5a44599e565d1d430f5eab6_l3.png)
The central question that naturally arises is if Calabi-Gray compactifications exist with Dp-branes, T-branes, orientifolds, and fluxes defined by M5-branes wrapping internal Tate-Hitchin divisors that admit Taub-NUT elliptic fibrations with admissible F-theory interpretations in terms of principal Higgs bundles definable over D7-brane’s 8D worldvolume, a topic for another post.