The Wess–Zumino–Witten–Novikov Model, Taub-NUT Spaces, and M-Theory

M5-branes wrapped on multi-centered Taub-NUT spaces are central to deriving the Standard Model of physics from F-theory through ADE-Tate-Hitchin brane-webs in light of the M/F-theory duality. Paradigmatic instances of such systems are asymptotically locally flat hyper-Kähler 4-folds locally asymptotic at infinity to ${{\mathbb{R}}^{3}}\times {{S}^{1}}$ . A central task is to establish that the M5-brane wrapped on cycles inherit the chiral modes of the heterotic string worldsheet in a three-torus compactification via a Kaluza-Kleining of the M5-brane on a K3 surface, in both, the Abelian and non-Abelian cases. Ensuing is a derivation of such in terms of a Wess–Zumino–Witten–Novikov model for a multi-centered Taub-NUT space directly arising from M-theory. Let’s recall that Dp-brane solutions preserving half SUSY have the following form:

$\displaystyle ds_{{Dp}}^{2}={{\Omega }^{{-1}}}\left[ {d{{t}^{2}}-ds_{p}^{2}} \right]-\Omega dx_{{9-p}}^{2}$

$\displaystyle {{e}^{{2\phi }}}={{\Omega }^{{1-p}}}$

$\displaystyle F_{{01...pm}}^{A}={{\partial }_{m}}{{H}^{A}}$

$\displaystyle {{M}_{{Kah}}}^{{AB}}={{\text{I}}_{{\text{Re}{{\text{p}}_{G}}}}}^{{AB}}+2{{\Omega }^{{-1}}}{{H}^{A}}{{H}^{B}}$

with the M5-brane action in a D = 11 SUGRA background is given as such:

$\displaystyle \begin{array}{l}S=2\int_{{{{M}_{6}}}}{{d{{x}^{6}}}}\left[ {\sqrt{{-\det \left( {{{g}_{{\mu \nu }}}+i{{{\tilde{H}}}_{{\mu \nu }}}} \right)}}} \right.\\+\frac{{\sqrt{{-g}}}}{{4{{{\left( {\partial a} \right)}}^{2}}}}{{\partial }_{\lambda }}a{{{\tilde{H}}}^{{\lambda \mu \nu }}}\left. {{{H}_{{\mu \nu \rho }}}{{\partial }^{\rho }}_{a}} \right]-\int_{{{{M}_{6}}}}{{{{C}_{6}}}}+{{H}_{3}}\wedge {{C}_{3}}\end{array}$

where:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{{\tilde{H}}}^{{\rho \mu \nu }}}\equiv \frac{1}{{6\sqrt{{-g}}}}{{\epsilon }^{{\rho \mu \nu \lambda \sigma \tau }}}{{H}_{{\lambda \sigma \tau }}}} \\ {{{{\tilde{H}}}_{{\lambda \sigma \tau }}}\equiv \frac{{{{\partial }^{\rho }}a}}{{\sqrt{{{{{\left( {\partial a} \right)}}^{2}}}}}}{{{\tilde{H}}}_{{\rho \mu \nu }}}} \\ {g=\det {{g}_{{\mu \nu }}}} \\ {{{\epsilon }^{{0...5}}}=-{{\epsilon }_{{0...5}}}=1} \end{array}} \right.$

and the $M5$ -brane action takes the following form:

$\displaystyle S=\int_{{{{{\hat{M}}}_{6}}}}{{{{d}^{6}}x}}\left( {-\frac{{\sqrt{{-g}}}}{6}{{{\tilde{\mathcal{L}}}}_{{M5}}}^{{F,G}}\left( {\tilde{F}} \right)} \right)-\int_{{{{{\hat{M}}}_{6}}}}{{\left( {{{C}_{6}}+H\wedge {{C}_{3}}} \right)}}$

where:

$\displaystyle {{\widetilde{\mathcal{L}}}_{{M5}}}^{{F,G}}\left( {\tilde{F}} \right)\doteq \left( {{{{\tilde{G}}}^{{\mu \nu \rho }}}{{G}_{{\mu \nu \rho }}}+3{{{\tilde{F}}}^{{\mu \nu \rho }}}{{F}_{{\mu \nu \rho }}}} \right)+2{{\mathcal{L}}_{{M5}}}\left( {F,G} \right)$

with:

$\displaystyle \begin{array}{l}{{\mathcal{L}}_{{M5}}}=-\frac{1}{{36\left( {1+{{G}^{2}}} \right)}}{{\epsilon }^{{{{\mu }_{1}}}}}{{^{{{{\mu }_{2}}}}}^{{{{\mu }_{3}}}}}{{^{{{{\mu }_{4}}}}}^{{{{\mu }_{5}}}}}^{{{{\mu }_{6}}}}{{G}_{{{{{^{{{{\mu }_{1}}}}}}^{{{{\mu }_{2}}}}}^{{{{\mu }_{3}}}}}}}{{F}_{{{{\mu }_{4}}\nu \lambda }}}{{F}_{{{{\mu }_{5}}}}}^{{\lambda \kappa }}{{F}_{{{{\mu }_{6}}\kappa }}}^{\nu }\\+\frac{1}{{1+{{G}^{2}}}}\sqrt{{-\det \left( {{{g}_{{\mu \nu }}}+\frac{1}{2}{{{\left( {F+G} \right)}}_{{\mu \rho \sigma }}}{{{\left( {F+G} \right)}}_{{{{\nu }^{{\rho \nu }}}}}}} \right)}}\end{array}$

and where the Hamiltonian metaplectic action in the Heisenberg representation yields:

$H = {\dot \psi ^{2\pi ik}}{\Im _i} + V_t^{p + 1}\not K + \oint_{p + i} {\delta _k^{{\rm{susy}}}} \left| {_{{B_{{\rm{Bos}}}}}} \right.d\,\Omega {({\phi _{si}})^{p + 1}}{\not H_i} + \lambda {\not H^i}$

where:

$\left\{ {\begin{array}{*{20}{c}}{{\Im _i} = - \not \partial {\phi _{si}}{T_{Dp}}d\,\Omega {{({\phi _{si}})}^{2\pi ik}}}\\{\not K = - {{\not \partial }_i}{{\widetilde E}^i} + {{( - 1)}^{p + 1}}{T_{Dp}}{S^{{\rm{Fer}}}}}\\{{{\not H}_i} = \widetilde P{\alpha _i}\widetilde E_i^\alpha {{\not \partial }_i}{\phi _{si}} + \widetilde E{{\not F}_{ij}}}\\{H = \frac{1}{{2\pi ik}}\left[ {{{\widetilde P}^2} + {{\widetilde E}^i}{{\widetilde E}^j}{G_{ij}} + T_{Dp}^2{e^{ - 2{\phi _{si}}}}{\rm{det}}\left( {{G_{ij}} + {{\not F}_{ij}}} \right)} \right]}\end{array}} \right.$

with:

$S = * {\left( {{{\not R}_{\mu \nu }}{\varepsilon ^{{\rm{Fer}}}}} \right)_p}$

and:

$E_i^\alpha = \delta \int {d\not E_m^\alpha } {\not \partial _i}{\dot X^m}$

where the Ramond-Ramond gauge-coupling sector is given by the following action:

$\displaystyle \mathcal{L}_{G}^{{Loc}}=\sum\limits_{{b=1}}^{{N-1}}{{\frac{1}{{2g_{b}^{2}}}}}{{\int{\text{d}}}^{2}}\theta {{W}^{\alpha }}{{W}_{\alpha }}{{\delta }^{2}}\left( {\left( {1-{{e}^{{ib\phi }}}} \right)z} \right)$

and the action of a $Dp$ -brane is given by a Dirac-Born-Infeld part coupled to a Chern-Simons WZ part:

$\displaystyle S_{{DBI}}^{{cs}}=-{{T}_{p}}\int_{{{\mathcal{W}}'}}{{{{d}^{{p+1}}}}}\xi {{e}^{{-\Phi }}}\Xi -{{T}_{p}}\int_{{{\mathcal{W}}'}}{{C_{F}^{B}}}$

with:

$\displaystyle \Xi \doteq \sqrt{{{{{\det }}_{{\left[ {a,b} \right]}}}\left( {{{g}_{{ab}}}+{{B}_{b}}+2\pi {\alpha }'{{F}_{{ab}}}} \right)}}$

$\displaystyle C_{F}^{B}\doteq \text{TrP}\left[ {C\wedge {{e}^{{-B}}}} \right]\wedge {{e}^{{2\pi {\alpha }'F}}}$

where $P$ is the worldvolume pullback with $p$ -orientifold action:

$\displaystyle {{S}_{{Op}}}={{2}^{{p-4}}}-{{T}_{p}}\int\limits_{{{{\mathcal{W}}_{\parallel }}}}{{{{d}^{{p+1}}}}}\xi {{e}^{{-\phi }}}\left( \Pi \right)-\Psi$

with:

$\displaystyle \Pi \doteq \sqrt{{-\det P\left[ {{{g}_{{\mu \nu }}}} \right]}}-{{2}^{{2-4}}}$

and

$\displaystyle \Psi \doteq -{{T}_{p}}\int_{{{{\mathcal{W}}_{\parallel }}}}{{P\left[ {{{C}_{{p+1}}}} \right]}}$

where the pullback to the $Dp$ -worldvolume yields the 10-D SYM action:

$\displaystyle {{S}_{{YM}}}=\frac{1}{{4g_{{_{{YM}}}}^{2}}}\int{{{{d}^{{10}}}}}x\left[ {\text{Tr}\left( {{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}} \right)+2\text{iTr}\left( {\bar{\psi }\,{{\Gamma }^{\mu }}{{D}_{\mu }}\psi } \right)} \right]$

with string coupling:

$\displaystyle \frac{1}{{{{g}_{s}}}}={{e}^{{-\Phi }}}$

and the 10-D SUGRA dimensionally-reduced Type-IIB action is given as such:

$\displaystyle S_{{DBI}}^{{c{s}'}}=-\frac{{{{T}_{p}}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{4{{g}_{s}}}}\int_{{{\mathcal{W}}'}}{{{{d}^{{p+1}}}\left( {\hat{F}+\hat{X}} \right)-\frac{{{{T}_{p}}}}{{{{g}_{s}}}}{{V}_{\vartheta }}}}$

with:

$\displaystyle \hat{F}\equiv {{F}_{{ab}}}{{F}^{{ab}}}$

$\displaystyle \hat{X}\equiv \frac{2}{{{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\partial }_{a}}{{X}^{m}}{{\partial }^{a}}{{X}_{m}}$

$\displaystyle {{V}_{\vartheta }}\equiv V_{{p+1}}^{{WV}}+\vartheta \left( {{{F}^{4}}} \right)$

In the string-frame, the type-IIB SUGRA action is given by:

$\displaystyle {{S}_{{IIB}}}={{S}_{{NS}}}+{{S}_{R}}+{{S}_{{CS}}}$

with:

$\displaystyle {{S}_{{NS}}}=\frac{1}{{2k_{{10}}^{2}}}\int{{{{d}^{{10}}}}}x{{\left( {-{{G}_{{10}}}} \right)}^{{1/2}}}{{e}^{{-2\Phi }}}\left[ {{{R}_{{10}}}+4\left( {{{\partial }^{\mu }}\Phi } \right)\left( {{{\partial }_{\mu }}\Phi } \right)-\frac{1}{2}{{{\left| {{{H}_{{\left( 3 \right)}}}} \right|}}^{2}}} \right]$

$\displaystyle {{S}_{R}}=-\frac{1}{{4k_{{10}}^{2}}}\int{{{{d}^{{10}}}}}x{{\left( {-{{G}_{{10}}}} \right)}^{{1/2}}}\left[ {{{{\left| {{{F}_{1}}} \right|}}^{2}}+{{{\left| {{{{\tilde{F}}}_{{\left( 3 \right)}}}} \right|}}^{2}}+\frac{1}{2}{{{\left| {{{{\tilde{F}}}_{{\left( 5 \right)}}}} \right|}}^{2}}} \right]$

$\displaystyle {{S}_{{CS}}}=-\frac{1}{{4k_{{10}}^{2}}}\int{{{{C}_{{\left( 4 \right)}}}\wedge {{H}_{{\left( 3 \right)}}}\wedge {{F}_{{\left( 3 \right)}}}}}$

where the superpotential is given by:

$\displaystyle W=\int_{Y}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {S,U} \right){{e}^{{-{{a}_{i}}{{T}_{i}}}}}$

where:

$\displaystyle \int_{Y}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}$

is the Gukov-Vafa-Witten superpotential stabilization complex term, as well as the axio-dilaton field:

$\displaystyle S={{e}^{{-\phi }}}+i{{C}_{0}}$

Given the presence of $E3$ -brane instantons, ${{T}_{i}}$ are of Kähler moduli Type-IIB-orbifold class:

$\displaystyle {{T}_{i}}={{e}^{{-\phi }}}{{\tau }_{1}}+i{{\rho }_{i}}$

with ${{\tau }_{i}}$ being the volume of the divisor ${{D}_{i}}$ and ${{\rho }_{i}}$ the 4-form Ramond-Ramond axion field corresponding to:

$\displaystyle {{\tau }_{i}}=\frac{1}{2}\int_{{{{D}_{i}}}}{{J\wedge J}}=\frac{1}{2}{{k}_{{ijk}}}+{{\,}^{j}}{{t}^{k}}$

and:

$\displaystyle {{\rho }_{i}}=\int_{{{{D}_{i}}}}{{{{C}_{4}}}}$

where $J$ is the Kähler form:

$\displaystyle J=\sum\limits_{i}{{{{t}_{i}}}}{{\eta }_{i}}$

and:

$\displaystyle \left\{ {{{\eta }_{i}}} \right\}\in {{H}^{{1,1}}}\left( {Y,\mathbb{Z}} \right)$

an integral-form basis and ${{k}_{{ijk}}}$ the associated intersection coefficients. Hence, the desired Kähler potential is given as such:

$\displaystyle K=-2\text{In}\left( {\tilde{\mathcal{V}}+\frac{\xi }{{2g_{s}^{{3/2}}}}} \right)-\text{In}\left( {S+\tilde{S}} \right)-\text{In}\left( {-i\int_{Y}{{\Omega \wedge \tilde{\Omega }}}} \right)$

with ${\tilde{\mathcal{V}}}$ the Calabi-Yau volume, and in the Einstein frame, has the following form:

$\displaystyle \mathcal{V}=\frac{1}{{3!}}\int_{Y}{{J\wedge J\wedge J=}}\frac{1}{6}{{k}_{{ijk}}}{{t}^{i}}{{t}^{j}}{{t}^{k}}$

The $F$ -term is given by:

$\displaystyle {{V}_{F}}={{e}^{K}}\left( {\sum\limits_{{i={{T}_{i}};j={{S}_{j}}}}{{{{K}^{{ij}}}}}{{D}_{i}}W{{D}_{j}}\tilde{W}-3{{{\left| {{{W}_{i}}} \right|}}^{2}}} \right)$

with the large volume scenario $D$ -term given as such:

$\displaystyle {{V}_{D}}=\sum\limits_{{i=1}}^{N}{{\frac{1}{{\operatorname{Re}\left( {{{f}_{i}}} \right)}}}}{{\left( {\sum\limits_{j}{{Q_{j}^{{\left( i \right)}}{{{\left| {{{\phi }_{j}}} \right|}}^{2}}-{{{\hat{\xi }}}_{i}}}}} \right)}^{2}}$

with:

$\displaystyle \operatorname{Re}\left( {{{f}_{i}}} \right)\doteq {{e}^{{-\phi }}}\frac{1}{2}\int_{{{{D}_{i}}}}{{J\wedge J-}}{{e}^{{-\phi }}}\int_{{{{D}_{i}}}}{{\text{c}{{\text{h}}_{2}}}}\left( {{{\mathcal{L}}_{i}}-B} \right)$

and the Fayet-Illopoulos terms being:

$\displaystyle {{\hat{\xi }}_{i}}=-\text{Im}\left( {\frac{1}{{\tilde{\mathcal{V}}}}\int_{Y}{{{{e}^{{-\left( {B+iJ} \right)}}}}}{{\Gamma }_{i}}} \right)$

where ${{\Gamma }_{i}}$ are the $D7$ -brane charge-vectors. Then we saw that the Chern-Simons orientifold action gets a Calabi-Yau curvature correction in the form of:

$\displaystyle {{S}_{{{{O}_{p}},CS}}}=-{{2}^{{p-4}}}-{{T}_{p}}\int\limits_{{{{\mathcal{W}}^{\prime }}}}{{P\left[ C \right]}}\wedge \Theta$

with

$\displaystyle \Theta \doteq \sqrt{{\frac{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'T{{\mathcal{W}}^{\prime }}} \right)}}{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'N{{\mathcal{W}}^{\prime }}} \right)}}}}$

due to the Gauss–Codazzi equations:

$\displaystyle {{S}_{{{{O}_{p}}}}}={{2}^{{p-4}}}-{{T}_{p}}{{\int_{{{\mathcal{W}}'}}{\text{d}}}^{{p+1}}}\xi {{e}^{{-\phi }}}\sqrt{{-\det \left( {P\left[ {{{g}_{{\mu \nu }}}} \right]} \right)}}-{{2}^{{2-4}}}-{{T}_{p}}\int_{{{\mathcal{W}}'}}{{P\left[ {{{C}_{{p+1}}}} \right]}}$

and the Ramond-Ramond term being:

$\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{Tr}}}\left( {{{e}^{{F/2\pi }}}} \right)$

which yields the Type-IIB Calabi-Yau three-fold superpotential:

$\displaystyle {{V}_{W}}=\int\limits_{X}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {\left( {{{e}^{{-\phi }}}+i{{C}_{0}}} \right),U} \right){{e}^{{-a\left( {{{e}^{{-\phi }}}{{\tau }_{i}}+i{{\rho }_{i}}} \right)}}}$

with the topologically mixed Yang-Mills action taking the following form:

$\displaystyle {{\mathcal{L}}_{{TYM}}}\equiv -\frac{1}{4}e{{\tilde{F}}_{{\mu \nu }}}^{M}{{\tilde{F}}^{{\mu \nu N}}}{{\hat{M}}_{{MN}}}+\kappa {{\mathcal{L}}_{{CS}}}$

with the corresponding Chern-Simons action:

$\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{ch}}}\left( {\tilde{F}} \right)\wedge \sqrt{{\frac{{\hat{A}\left( {{{R}_{T}}} \right)}}{{\hat{A}\left( {{{R}_{N}}} \right)}}}}$

It follows that the equation of motion for an M5-brane, in the abelian case, is given as such:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\nabla }^{2}}{{\phi }^{\alpha }}_{\beta }=0} \\ {i{{\Gamma }^{m}}{{\nabla }_{m}}{{\psi }^{\alpha }}=0} \\ {{{H}_{{mnp}}}=\frac{1}{{3!}}{{\epsilon }_{{mnpqrs}}}{{H}^{{qrs}}}} \end{array}} \right.$

with:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{H}_{{mnp}}}=3{{\partial }_{{\left[ m \right.}}}{{B}_{{\left. {np} \right]}}}} \\ {{{\epsilon }^{{012345}}}=1} \\ {\alpha ,\beta =1,...,4:\quad \text{ind}\left[ {\text{irre}{{\text{p}}^{{fun}}}USp(4)/\text{R-sym}} \right]} \end{array}} \right.$

and are clearly invariant under SUSY transformations:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\delta {{\phi }^{{\alpha \beta }}}=-i{{{\bar{\epsilon }}}^{{\left[ \alpha \right.}}}{{\psi }^{{\left. \beta \right]}}}} \\ {\delta {{B}_{{mn}}}=-i{{{\bar{\epsilon }}}^{\alpha }}{{\Gamma }_{{mn}}}{{\psi }_{\alpha }}} \\ {\delta {{\psi }^{\alpha }}={{\nabla }_{m}}{{\phi }^{\alpha }}_{\beta }{{\Gamma }^{m}}{{\epsilon }^{\beta }}+\frac{1}{{2\cdot 3!}}{{\Gamma }^{{mnp}}}{{H}_{{mnp}}}{{\epsilon }^{\alpha }}} \end{array}} \right.$

with ${{\epsilon }^{\alpha }}$ a chiral Killing spinor on the M5-brane worldvolume. Since the M5-brane worldvolume is isomorphic to a multi-centered Taub-NUT space ${{\text{R}}^{{1,1}}}\times {{M}_{{mTN}}}$ with:

$\displaystyle ds_{{mTN}}^{2}={{H}^{{-1}}}{{\left( {d{{x}^{5}}+\theta } \right)}^{2}}+Hd\vec{x}\cdot d\vec{x}$

$\displaystyle H=1+\sum\limits_{{I=1}}^{n}{{{{h}_{I}}}}$

$\displaystyle \theta =\sum\limits_{{I=1}}^{n}{{{{\theta }_{I}}}}$

$\displaystyle {{h}_{I}}=\frac{R}{2}\frac{{{{N}_{I}}}}{{\left| {\vec{x}-{{{\vec{x}}}_{I}}} \right|}}$

$\displaystyle d{{\theta }_{I}}={{*}^{H}}_{3}d{{h}_{I}}$

the worldvolume metric is hence given by:

$\displaystyle ds_{6}^{2}=-{{\left( {d{{x}^{0}}} \right)}^{2}}+{{\left( {d{{x}^{1}}} \right)}^{2}}+ds_{{mTN}}^{2}$

and asymptotes as such:

$\displaystyle \begin{array}{l}ds_{{mTN}}^{2}={{\left( {1+\frac{{{{N}_{{D6}}}R}}{{2r}}} \right)}^{{-1}}}{{\left( {d{{x}^{5}}+\frac{1}{2}{{N}_{{D6}}}R\cos \theta d\phi } \right)}^{2}}\\+\left( {1+\frac{{{{N}_{{D6}}}R}}{{2r}}} \right)\left( {d{{r}^{2}}+{{r}^{2}}d{{\theta }^{2}}+{{r}^{2}}{{{\sin }}^{2}}\theta d{{\phi }^{2}}} \right)\end{array}$

Now, in light of the self-duality of the curvature form, a Killing spinor ${{\epsilon }^{\alpha }}$ satisfying:

$\displaystyle {{\Gamma }_{{01}}}{{\epsilon }^{\alpha }}=-{{\epsilon }^{\alpha }}$

exists. Since the bosonic solutions to the Euler-Lagrange equations preserve eight supersymmetries, we have the following relation:

$\displaystyle {{\partial }_{m}}{{\phi }^{\alpha }}_{\beta }=0$

In light-cone gauge, we hence have:

$\displaystyle \begin{array}{c}\delta {{\psi }^{\alpha }}=\frac{1}{4}{{\Gamma }^{{-ij}}}{{H}_{{-ij}}}{{\epsilon }^{\alpha }}+\frac{1}{4}{{\Gamma }^{{+ij}}}{{H}_{{+ij}}}{{\epsilon }^{\alpha }}+\frac{1}{2}{{\Gamma }^{{+-i}}}{{H}_{{+-i}}}\\+\frac{1}{{3!}}{{\Gamma }^{{ijk}}}{{H}_{{ijk}}}{{\epsilon }^{\alpha }}=0\end{array}$

with:

$\displaystyle {{H}_{{-ij}}}={{H}_{{ijk}}}={{H}_{{+-i}}}=0$

Thus, the Euler-Lagrange equations satisfying eight supersymmetries are:

$\displaystyle H=\sum\limits_{{I=1}}^{n}{{\upsilon _{+}^{I}}}d{{x}^{+}}\wedge {{\omega }_{I}}$

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

$\displaystyle {{\omega }_{I}}=\frac{1}{{4{{\pi }^{2}}R}}d{{\xi }_{I}}$

with:

$\displaystyle {{\xi }_{I}}={{H}^{{-1}}}{{h}_{I}}\left( {d{{x}^{5}}+\theta } \right)-{{\theta }_{I}}$

$\displaystyle \int{{{{\omega }_{I}}}}\wedge {{\omega }_{J}}=\int{{{{\omega }_{I}}}}\wedge {{*}^{H}}{{\omega }_{J}}=\frac{{{{N}_{I}}}}{{4{{\pi }^{2}}}}{{\delta }_{{IJ}}}$

Hence, the fermionic Euler-Lagrange equations are given by:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\Gamma }^{i}}{{\nabla }_{i}}{{\psi }^{\alpha }}=0} \\ {{{\partial }_{-}}{{\psi }^{\alpha }}=0} \end{array}} \right.$

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

$\displaystyle {{T}_{{mn}}}=\frac{\pi }{2}\sqrt{{-g}}{{H}_{{mpq}}}{{H}_{n}}^{{pq}}$

with the super-Poincaré operator given by:

$\displaystyle {{\mathcal{P}}_{+}}=\int{{d{{x}^{5}}}}{{T}_{{++}}}=\frac{1}{{4\pi }}\sum{{{{N}_{I}}}}\int{{d{{x}^{+}}\upsilon _{+}^{I}\left( {{{x}^{+}}} \right)}}\,\upsilon _{+}^{I}\left( {{{x}^{+}}} \right)$

Hence, the abelian conserved current takes the following form:

$\displaystyle {{J}_{m}}\left( \Lambda \right)=2\pi \sqrt{{-g}}{{H}_{{mpq}}}{{\partial }^{n}}{{\Lambda }^{p}}$

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

$\displaystyle Q\left( \Lambda \right)=2\pi \oint_{{\mathbb{R}\times {{S}^{1}}\times S_{\infty }^{2}}}{{{{H}_{{+r\mu }}}}}{{\Lambda }^{\mu }}{{r}^{2}}d\tilde{\Omega }_{2}^{{{{J}_{+}}\left( \Lambda \right)}}d{{x}^{+}}$

Restricting to ${{\Lambda }_{5}}$ , asymptotically we get:

$\displaystyle \begin{array}{l}Q\left( {{{\Lambda }_{5}}\left( \infty \right)} \right)=\frac{1}{{2\pi R}}\text{Tr}\sum\limits_{I}{{\oint_{{\mathbb{R}\times {{S}^{1}}\times S_{\infty }^{2}}}{{d{{\Omega }_{2}}}}}}d{{x}^{+}}\left[ {H{{\partial }_{r}}} \right.\left( {\frac{{{{h}_{I}}}}{H}} \right)+\\{{\varepsilon }^{{rjk}}}{{\theta }_{j}}{{\partial }_{k}}\left. {\left( {\frac{{{{h}_{I}}}}{H}} \right)} \right]\upsilon _{+}^{I}{{\Lambda }_{5}}\left( \infty \right)\end{array}$

and where the D4-brane $U(1)$ gauge field is given as such:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{A}_{\mu }}=4{{\pi }^{2}}R{{B}_{{\mu 5}}}} \\ {U(1):\quad {{A}_{\mu }}\to {{A}_{\mu }}+4{{\pi }^{2}}R{{\partial }_{\mu }}{{\Lambda }_{5}}} \end{array}} \right.$

In the non-abelian case, the M5-brane compactified on a circle of radius R yields at low energy a 5D M-SYM. An elliptic multi-centered Taub-NUT reduction on ${{x}^{5}}$ imposes the following metric:

$\displaystyle ds_{5}^{2}=-{{\left( {d{{x}^{0}}} \right)}^{2}}+{{\left( {d{{x}^{1}}} \right)}^{2}}+Hd\vec{x}\cdot d\vec{x}$

with Yang-Mills action:

$\displaystyle {{S}_{F}}=\frac{1}{{8{{\pi }^{2}}R}}\int{{{{d}^{5}}}}x\sqrt{H}\text{tr}\left( {F\wedge {{*}^{H}}F} \right)+\theta \wedge \text{tr}\left( {F\wedge F} \right)$

and the scalar action:

$\displaystyle \begin{array}{l}{{S}_{s}}^{\phi }=-\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\int{{{{d}^{5}}}}x\sqrt{{-g}}\left( {\sqrt{H}} \right.{{D}_{\mu }}{{\phi }_{{\alpha \beta }}}{{D}^{\mu }}{{\phi }^{{\alpha \beta }}}+\\\frac{1}{4}\frac{1}{{{{H}^{{5/2}}}}}{{\partial }_{i}}H{{\partial }_{i}}H{{\phi }_{{\alpha \beta }}}{{\phi }^{{\alpha \beta }}}-\sqrt{H}\left[ {{{\phi }^{{\alpha \beta }}},\left. {{{\phi }_{\beta }}^{\delta }} \right]} \right.\left. {\left[ {{{\phi }_{{\delta \gamma }}},{{\phi }^{\gamma }}_{\alpha }} \right]} \right)\end{array}$

and our Chern-Simons term is hence given by:

$\displaystyle CS=\text{tr}\left( {{{A}_{\mu }}{{\partial }_{\nu }}{{A}_{\lambda }}+\frac{2}{3}{{A}_{\mu }}{{A}_{\nu }}{{A}_{\lambda }}} \right)d{{x}^{\mu }}\wedge d{{x}^{\nu }}\wedge d{{x}^{\lambda }}$

We 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 as such:

$\displaystyle i{{\gamma }_{{234}}}{{\varepsilon }^{\alpha }}={{\varepsilon }^{\alpha }}$

with SUSY-variation:

$\displaystyle \begin{array}{l}\delta {{\psi }^{\alpha }}=\frac{1}{2}{{F}_{{\mu \nu }}}{{\gamma }^{{\mu \nu }}}{{\varepsilon }^{\alpha }}+2i\sqrt{H}{{M}_{{\beta \gamma }}}{{D}_{\mu }}\left( {\frac{1}{{\sqrt{H}}}{{\phi }^{{\alpha \beta }}}} \right){{\gamma }^{\mu }}{{\varepsilon }^{\gamma }}\\-\frac{1}{{\sqrt{H}}}{{M}_{{\beta \gamma }}}{{\phi }^{{\alpha \beta }}}{{{\tilde{F}}}_{{\mu \nu }}}{{\gamma }^{{\mu \nu }}}{{\varepsilon }^{\gamma }}+2{{M}_{{\beta \gamma }}}{{M}_{{\delta \lambda }}}\left[ {{{\phi }^{{\alpha \beta }}},{{\phi }^{{\gamma \delta }}}} \right]{{\varepsilon }^{\gamma }}\end{array}$

and field strength:

$\displaystyle {{F}_{{\mu \nu }}}={{\nabla }_{\mu }}{{A}_{\nu }}-{{\nabla }_{\nu }}{{A}_{\mu }}+{{\left[ {{{A}_{\mu }},{{A}_{\nu }}} \right]}_{\mathfrak{g}}}$

The covariant derivative is hence given in terms of a field transforming in the adjoint representation of the gauge group:

$\displaystyle {{D}_{\mu }}\chi ={{\nabla }_{\mu }}\chi +{{\left[ {{{A}_{\mu }},\chi } \right]}_{\mathfrak{g}}}$

and the BPS conditions are given by:

$\displaystyle {{F}_{{ij}}}={{F}_{{+-}}}={{F}_{{-+}}}=0$

Hence, we can derive:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{D}_{i}}\left( {\sqrt{H}{{\phi }^{\alpha }}_{\beta }} \right)={{D}_{-}}{{\phi }^{\alpha }}_{\beta }=0} \\ {{{{\left[ {{{\phi }^{\alpha }}_{\beta },{{\phi }^{\beta }}_{\gamma }} \right]}}_{\mathfrak{g}}}=0} \end{array}} \right.$

from which the Euler-Lagrange equations follow as such:

$\displaystyle \sqrt{{-g}}{{D}_{\sigma }}\left( {\sqrt{H}{{F}^{{\sigma \lambda }}}} \right)+\frac{1}{4}{{\tilde{F}}_{{\mu \nu }}}{{\tilde{F}}_{{\rho \sigma }}}{{\epsilon }^{{\mu \nu \rho \sigma }}}=0$

In light of the BPS conditions:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{A}_{i}}=g{{\partial }_{i}}{{g}^{{-1}}}} \\ {{{A}_{-}}=g{{\partial }_{-}}{{g}^{{-1}}}} \end{array}} \right.$

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{A}_{+}}={g}'{{\partial }_{+}}{{{{g}'}}^{{-1}}}} \\ {{{A}_{-}}={g}'{{\partial }_{-}}{{{{g}'}}^{{-1}}}} \end{array}} \right.$

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

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{A}_{+}}=k{{\partial }_{+}}{{k}^{{-1}}}} \\ {{{A}_{-}}={{A}_{i}}=0} \\ {{{F}_{{i+}}}={{\partial }_{i}}{{A}_{+}}} \end{array}} \right.$

Hence, the Euler-Lagrange equations are given by:

$\displaystyle {{\partial }_{i}}{{\partial }_{i}}\tilde{K}+\frac{2}{H}{{\partial }_{i}}\tilde{K}{{\partial }_{i}}H=0$

with:

$\displaystyle {{A}_{+}}=\sum\limits_{{I=1}}^{n}{{{{{\tilde{K}}}_{I}}}}\left( {\vec{x}} \right)\upsilon _{+}^{I}\left( {{{x}^{+}}} \right)$

Our Yang-Mills form is thus given by:

$\displaystyle \begin{array}{l}F=\sum\limits_{I}{{\upsilon _{+}^{I}}}\left( {{{x}^{+}}} \right){{\partial }_{i}}{{{\tilde{K}}}_{I}}d{{x}^{+}}\wedge d{{x}^{i}}\\=4{{\pi }^{2}}R\sum\limits_{I}{{\upsilon _{+}^{I}}}\left( {{{x}^{+}}} \right)\omega _{{i5}}^{I}d{{x}^{+}}\wedge d{{x}^{i}}\end{array}$

Hence, $\upsilon _{+}^{I}$ becomes an element of the full M5-brane gauge algebra given that the following identity holds:

$\displaystyle {{F}_{{\mu \nu }}}=4{{\pi }^{2}}R{{H}_{{\mu \nu 5}}}$

The fermionic Euler-Lagrange equation is given by:

$\displaystyle i\sqrt{H}{{\gamma }^{\mu }}{{D}_{\mu }}{{\psi }^{\alpha }}-\frac{1}{8}{{\tilde{F}}_{{\mu \nu }}}{{\gamma }^{{\mu \nu }}}{{\psi }^{\alpha }}=0$

and has a bi-chirality solitonic split as such:

$\displaystyle -\sqrt{2}{{\gamma }_{0}}{{H}^{{1/2}}}{{D}_{+}}\psi _{+}^{\alpha }+\vec{\gamma }\cdot \vec{\nabla }\psi _{-}^{\alpha }+\frac{1}{4}{{H}^{{-1/2}}}\vec{\gamma }\cdot \vec{\nabla }H\psi _{-}^{\alpha }=0$

$\displaystyle \vec{\gamma }\cdot \vec{\nabla }\psi _{+}^{\alpha }-\frac{1}{4}{{H}^{{-1/2}}}\vec{\gamma }\cdot \vec{\nabla }H\psi _{+}^{\alpha }=0$

the second being the Dirac equation for:

$\displaystyle \hat{\psi }_{+}^{\alpha }={{e}^{{-\frac{1}{2}{{H}^{{1/2}}}}}}\psi _{+}^{\alpha }$

and the D-term in the first equation is the only source of the non-abelian gauge field. Hence we have, in terms of a representation highest weight term member $k$ , the following equation:

$\displaystyle {{k}^{{-1}}}=P\exp \left( {\sum\limits_{I}{{{{K}_{I}}\left( {\vec{x}} \right)\int_{0}^{{{{x}^{+}}}}{{\upsilon _{+}^{I}\left( {{{y}^{+}}} \right)d{{y}^{+}}}}}}} \right)$

Consequently, the following can be derived:

$\displaystyle {{A}_{+}}\left( {{{{\vec{x}}}_{I}}} \right)=k\left( {{{{\vec{x}}}_{I}}} \right){{\partial }_{+}}{{k}^{{-1}}}\left( {{{{\vec{x}}}_{I}}} \right)=\upsilon _{+}^{I}\left( {{{x}^{+}}} \right)$

The energy-momentum tensor is thus given as such:

$\displaystyle \begin{array}{l}{{T}_{{\mu \nu }}}=\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\left[ {2\sqrt{H}} \right.{{D}_{\mu }}{{\phi }_{{\alpha \beta }}}{{D}_{\nu }}{{\phi }^{{\alpha \beta }}}+\\\frac{1}{{2{{H}^{{3/2}}}}}{{\partial }_{\mu }}H{{\partial }_{\nu }}H{{\phi }_{{\alpha \beta }}}{{\phi }^{{\alpha \beta }}}+2\sqrt{H}{{F}_{{\mu \rho }}}{{F}_{\nu }}^{\rho }\\-{{g}_{{\mu \nu }}}\left( {2\sqrt{H}{{D}_{\rho }}{{\phi }_{{\alpha \beta }}}{{D}^{\rho }}{{\phi }^{{\alpha \beta }}}+} \right.\frac{1}{4}\frac{1}{{{{H}^{{5/2}}}}}{{\partial }_{i}}H{{\partial }_{i}}H{{\phi }_{{\alpha \beta }}}{{\phi }^{{\alpha \beta }}}\\+\frac{{\sqrt{H}}}{2}{{F}_{{\rho \sigma }}}{{F}^{{\rho \sigma }}}-\sqrt{H}\left[ {{{\phi }^{{\alpha \beta }}},{{\phi }_{{\beta \lambda }}}} \right]\left. {\left. {\left[ {{{\phi }^{{\lambda \rho }}},{{\phi }_{{\rho \alpha }}}} \right]} \right)} \right]\end{array}$

and entails:

$\displaystyle {{D}_{+}}{{\phi }^{\alpha }}_{\beta }=0$

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

$\displaystyle {{T}_{{++}}}=\frac{1}{{4{{\pi }^{2}}R}}\frac{1}{{\sqrt{H}}}\text{tr}\sum\limits_{{IJ}}{{{{\partial }_{i}}}}{{K}_{I}}{{\partial }_{i}}{{K}_{J}}{{\upsilon }^{I}}\left( {{{x}^{+}}} \right){{\upsilon }^{J}}\left( {{{x}^{+}}} \right)$

giving us the spacial integral relation:

$\displaystyle {{\mathcal{P}}_{+}}=\int{{{{d}^{3}}}}xd{{x}^{+}}\sqrt{{-g}}{{T}_{{++}}}=\frac{1}{{4\pi }}\sum\limits_{I}{{{{N}_{I}}}}\text{tr}\int{{d{{x}^{+}}{{\upsilon }^{I}}}}\left( {{{x}^{+}}} \right){{\upsilon }^{I}}\left( {{{x}^{+}}} \right)$

corresponding to $n$ copies of multi-centered Taub-NUT spaces of a Wess–Zumino–Witten–Novikov model for each ${{N}_{I}}$ , and crucially, the non-abelian gauge charges for our M5-brane action take the following form:

$\displaystyle {{J}^{\sigma }}\left( \Lambda \right)=\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\left[ {-2\sqrt{{-g}}\sqrt{H}{{F}^{{\sigma \lambda }}}{{D}_{\lambda }}\Lambda +{{\varepsilon }^{{\mu \nu \rho \sigma \lambda }}}{{\theta }_{\mu }}{{F}_{{\nu \rho }}}{{D}_{\lambda }}\Lambda } \right]$

with corresponding charges:

$\displaystyle \begin{array}{c}Q\left( {\Lambda \left( \infty \right)} \right)=\frac{1}{{4{{\pi }^{2}}R}}\text{tr}\sum\limits_{I}{{\oint{{d{{\Omega }_{2}}}}}}d{{x}^{+}}\left[ {H{{\partial }_{r}}} \right.{{K}_{I}}+\\{{\varepsilon }^{{rjk}}}\left. {{{\theta }_{j}}{{\partial }_{k}}{{K}_{I}}} \right]\upsilon _{+}^{I}\Lambda \left( \infty \right)=-\frac{1}{{2\pi }}\text{tr}\sum\limits_{I}{{{{N}_{I}}}}\int{{d{{x}^{+}}}}\upsilon _{+}^{I}\Lambda \left( \infty \right)\end{array}$

Thus, the action of our BPS sector yields the desired result: a Wess–Zumino–Witten–Novikov model for a multi-centered Taub-NUT space directly arising from M-theory:

$\displaystyle {{{S}_{{BPS}}}=-\sum\limits_{I}{{\frac{{{{N}_{I}}}}{{4\pi }}}}\text{tr}\int{{d{{x}^{+}}}}d{{x}^{-}}g\left( {{{{\vec{x}}}_{I}}} \right){{\partial }_{+}}{{g}^{{-1}}}\left( {{{{\vec{x}}}_{I}}} \right)g\left( {{{{\vec{x}}}_{I}}} \right){{\partial }_{-}}{{g}^{{-1}}}\left( {{{{\vec{x}}}_{I}}} \right)}$

$\displaystyle {+\frac{1}{{8{{\pi }^{2}}R}}\text{tr}\int{{{{d}^{5}}}}x{{\partial }_{i}}H\left[ {{{g}^{{-1}}}{{\partial }_{-}},{{g}^{{-1}}}{{\partial }_{+}}} \right]{{g}^{{-1}}}{{\partial }_{i}}g}$

The Wess–Zumino–Witten–Novikov Model, Taub-NUT Spaces, and M-Theory

George Shiber

M-Theory on Kovalev TCS G2-Manifolds, Yukawa Couplings, and Chirality

Exceptional Field Theory and M-Theory on Kovalev TCS G2-Manifolds

George Shiber

M-Theory on Kovalev TCS G2-Manifolds, Yukawa Couplings, and Chirality

Exceptional Field Theory and M-Theory on Kovalev TCS G2-Manifolds

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