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F-Theory, the D-Term Equation and Representation Theory

Let us see how the Yukawa couplings among 4-D fermionic fields can be derived from the F-theory superpotential and relate them to the tree-level superpotential. This is of utmost importance since D7/D3-brane-phenomenology of 4-D F-theory can be promoted to M-theory in light of the F/M-theory duality and the compactness of Calabi-Yau 4-folds. Start with a Kähler coordinate expansion of \gamma which gives us, after inserting it in:

    \[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]

the following:

    \[\begin{array}{l}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = \\{\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}\end{array}\]

which is the exact 7-brane superpotential for F-theory and the integrand is independent of \lambda, entailing that the F-term conditions are purely topological and in no need for \alpha '-corrections.

However, the D-term in:

    \[{D^K} = \int_{\tilde S} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}} \]

 

is in need of \alpha '-corrections, since it is evaluable as:

    \[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\lambda P\left[ J \right]} \right.} \wedge F - \frac{{i\lambda }}{6}{\iota _\Phi }{\iota _\Phi }{J^3} + \\\frac{{i{\lambda ^3}}}{2}{\iota _\Phi }{\iota _\Phi }J \wedge F \wedge F - {\rm{P}}\left[ {J \wedge B} \right] \wedge F\\\left. { + i{\lambda ^2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge B} \right) \wedge \frac{{i\lambda }}{2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge {B^2}} \right)} \right\}\end{array}\]

and the non-Abelian D-term has the form:

    \[D = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {\tilde A\left( {\tilde T} \right)/\tilde A\left( {\tilde N} \right)} \]

With {Y_4} our target Calabi-Yau 4-fold and Lie algebra G, for:

    \[\left[ {{{D'}_i}} \right] \in {H^2}\left( {{Y_4}} \right)\]

we have:

    \[\int_{{Y_4}} {\left[ {{{D'}_i}} \right]} \wedge \left[ {{{D'}_j}} \right] \wedge \tilde \omega = - {C_{ij}}\int_S {\tilde \omega } \]

with \tilde \omega \in {H^2}\left( {{{\hat B}_3}} \right) and {C_{ij}} the Cartan matrix of G, effectively reflecting the F/M-theory duality.

In the local patch on the C-manifold, one takes the flat-space-Kähler-form as having the form:

    \[J = \underbrace {\frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar x + \frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar y}_{ = :\omega } + 2i{\rm{d}}z \wedge {\rm{d}}\bar z\]

Then, we decompose the Kähler-background B-field as:

    \[B \equiv B\left| {_{\tilde S}} \right. + {B_{z\overline z }}{\rm{d}}z \wedge {\rm{d}}\bar z\]

with:

    \[\tilde F = \lambda F - B\left| {_{\tilde S}} \right.\]

thus giving us:

    \[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\rm{P}} \right.} \left[ J \right] \wedge \tilde F + \frac{{i\lambda }}{2}\left( {{\iota _\Phi }{\iota _\Phi }J} \right)\\ - \left( {{{\tilde F}^2} - {\omega ^2}} \right) - i\lambda \left( {{\iota _\Phi }{\iota _\Phi }B} \right)\omega \wedge \tilde F\\ - \omega \wedge {\rm{P}}\left[ {{B_{z\overline z }}{\rm{d}}z \wedge {\rm{d}}\bar z} \right]\end{array}\]

with the Abelian pull-back \omega to {S_4} determined by:

    \[J = \underbrace {\frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar x + \frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar y}_{ = :\omega } + 2i{\rm{d}}z \wedge {\rm{d}}\bar z\]

where locally the Higgs field is given by:

    \[\Phi \equiv \phi \frac{\partial }{{\bar \partial z}} + \bar \phi \frac{{\bar \partial }}{{\partial \bar z}}\]

with \phi a matrix in the complexified adjoint representation of G and \bar \phi its Hermitian conjugate. Thus, I could derive:

    \[\gamma \equiv z{\rm{d}}x \wedge {\rm{d}}y\]

with:

    \[\iota \Phi \gamma = 0\]

Hence we have:

    \[\left\{ {\begin{array}{*{20}{c}}{{\iota _\Phi }{\iota _\Phi }J = 2i\left[ {\phi ,\bar \phi } \right]}\\{{\iota _\Phi }{\iota _\Phi }{J^3} = 6i\left[ {\phi ,\bar \phi } \right]{\omega ^2}}\end{array}} \right.\]

Now: realize that 2i\left[ {\phi ,\bar \phi } \right] is a zero-form and 6i\left[ {\phi ,\bar \phi } \right]{\omega ^2} does not have transverse-legs to \tilde S, and thus the pull-back {\rm{P}} has a trivial action. So, after solving:

    \[{\rm{P}}\left[ J \right] = \omega + 2i{\lambda ^2}\left( {{\rm{D}}\phi } \right) \wedge \left( {{\rm{\bar D}}\bar \phi } \right)\]

the D-term equation amounts to D = 0 with:

    \[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\omega \wedge \tilde F} \right.} + {\lambda ^2}{\rm{D}}\phi \wedge \overline {{\rm{D}}\phi } \\ \wedge \left( {2i\tilde F - {B_{z\overline z }}\omega } \right) + \lambda \left[ {\phi ,\bar \phi } \right]\\\left. {\left( {{\omega ^2} - {{\tilde F}^2} - i{B_{z\overline z }}\omega \wedge \tilde F} \right)} \right\}\end{array}\]

and with the B-field vanishing on the sheave of the C-manifold, one gets a reduction to:

    \[\begin{array}{l}D = \lambda \int_{\tilde S} {S\left\{ {\omega \wedge F + 2i{\lambda ^2}} \right.} {\rm{D}}\phi \wedge \overline {{\rm{D}}\phi } \wedge F\\ + \left[ {\phi ,\bar \phi } \right]\left. {\left( {{\omega ^2} - {\lambda ^2}{F^2}} \right)} \right\}\end{array}\]

which yields a non-Abelian \alpha '-corrected Chern-Simons action of a stack of D7-branes with both terms at leading order in \lambda, entailing that for matrix algebras:

    \[{\Im ^G} \subset GL\left( {n,\mathbb{C}} \right)\]

they are the matrix products in the fundamental representation of {\Im ^G}

and so the \alpha '-corrections on D-terms with the gauge flux F diagonalization yields

the D-term equations

    \[\begin{array}{l}D = \lambda {\int_{\tilde S} {\rm{P}} _{ab}}\left[ J \right] \wedge F = \\\lambda \int_{\tilde S} {\left( {\omega + 2i{\lambda ^2}\partial \phi \wedge \overline {\partial \phi } } \right)} \wedge F\end{array}\]

Deep upshot: the \alpha '-corrections are given entirely by the abelian pull-back of the Kähler-form J to \tilde S

    \[{{\rm{P}}_{ab}}\left[ J \right] \equiv \left( {\omega + 2i{\lambda ^2}\partial \phi \wedge \overline {\partial \phi } } \right)\]

And this has a deep physical interpretation which can be extracted from the energy-momentum tensor and D-term of Q-clouds.

In the special case that is of interest, the Yukawa couplings among 4-d matter fields can be derived from the superpotential:

    \[W = m_ * ^4\int_S {{\rm{tr}}} \left( {F \wedge \Phi } \right)\]

with {m_ * } the F-theory characteristic scale, and F = dA - iA \wedge A with dynamical dependence on the D-term:

    \[D = \int_S {\omega \wedge F} + \frac{1}{2}\left[ {\Phi ,\bar \Phi } \right]\]

Our equations of motion that follow from the superpotential and the D-term are given by:

    \[{{\bar \partial }_A}\Phi = \bar \partial \Phi - i\left[ {{A_{0,1}},\Phi } \right] = 0\]

    \[{F_{0,2}} = 0\]

which are the F-term equations, and the following holds for our fundamental \omega form on S:

    \[\omega \wedge F + \frac{1}{2}\left[ {\Phi ,\bar \Phi } \right] = 0\]

which is the D-term equation.

In the bosonic case, to derive the equations of motion, define:

    \[\left\{ {\begin{array}{*{20}{c}}{{\Phi _{xy}} = \left\langle {{\Phi _{xy}}} \right\rangle + {\varphi _{xy}}}\\{{A_{\bar m}} = \left\langle {{A_{\bar m}}} \right\rangle + {a_{\bar m}}}\end{array}} \right.\]

and expand the F-term equations and the D-term equation to first order in the fluctuations \left( {\varphi ,a} \right). Thus we find:

    \[{{\bar \partial }_{\left\langle A \right\rangle }}\varphi + i\left[ {\left\langle \Phi \right\rangle ,a} \right] = 0\]

    \[{{\bar \partial }_{\left\langle A \right\rangle }}\varphi = 0\]

    \[\omega \wedge {{\bar \partial }_{\left\langle A \right\rangle }}a - \frac{1}{2}\left[ {\left\langle {\bar \Phi } \right\rangle ,\varphi } \right] = 0\]

with the following relations:

    \[\left\{ {\begin{array}{*{20}{c}}{a = {a_{\bar x}}d\bar x + {a_{\bar y}}d\bar y}\\{\varphi = {\varphi _{xy}}dx \wedge dy}\end{array}} \right.\]

and locally, we have the Kähler form:

    \[\omega = \frac{i}{2}\left( {dx \wedge d\bar x + dy \wedge d\bar y} \right)\]

Hence, our equations:

    \[{{\bar \partial }_{\left\langle A \right\rangle }}\varphi + i\left[ {\left\langle \Phi \right\rangle ,a} \right] = 0\]

    \[{{\bar \partial }_{\left\langle A \right\rangle }}\varphi = 0\]

    \[\omega \wedge {{\bar \partial }_{\left\langle A \right\rangle }}a - \frac{1}{2}\left[ {\left\langle {\bar \Phi } \right\rangle ,\varphi } \right] = 0\]

admit zero mode solutions that are localized on fermionic curves which are determined by the background of \Phi which in the absence of fluxes depends holomorphically on the complex coordinates of S. So, a nontrivial VEV \left\langle \Phi \right\rangle with the property that its rank changes at curves {\Sigma _i} implies that instead of a single S there are intersecting surfaces:

    \[\left\{ {\begin{array}{*{20}{c}}{{S_{{\rm{GUT}}}}}\\{{S_i}}\end{array}} \right.\]

Now, at any point on S\left\langle \Phi \right\rangle splits {G_p} to {G_{{\rm{GUT}}}} times U(1) due to the 7-branes wrapping the {S_i}, and at {\Sigma _i} there are additional commuting generators whose associated fluctuations give rise to matter localized on {\Sigma _i} as implied by solving the equations of motion. At point p where the matter curves intersect, there is bi-uplifts to {G_p}. Locally, a worldvolume flux \left\langle F \right\rangle is included, entailing that a hypercharge generator exists that breaks {G_{{\rm{GUT}}}} to the SM group.

A sketch of the proof:

Take

    \[{G_p} = U{(N)^N}\]

such that:

    \[\left\langle {{\Phi _{xy}}} \right\rangle = \frac{1}{n}{m^2}{\rm{diag}}\left( { - 2x + y,x + y,x - 2y} \right)\]

with m the mass parameter; thus we have a VEV breaking of U(N) to U{(1)^N} at generic points in S and so the group is enhanced to:

    \[SU(2) \times U{(1)^N}\]

at curves:

    \[\left\{ {\begin{array}{*{20}{c}}{{\Sigma _a} = \left\{ {x = 0} \right\}}\\\begin{array}{l}{\Sigma _b} = \left\{ {y = 0} \right\}\\{..._{{._{{._{{._.}}}}}}}\end{array}\\{{\Sigma _n} = \left\{ {x = y} \right\}}\end{array}} \right.\]

The generators {E_{{a^ \pm }}} determined by:

    \[{E_{{a^ + }}} = \left( {\begin{array}{*{20}{c}}0&1&0&{...}\\0&0&0&{...}\\0&0&0&{...}\\0&0&0&0\end{array}} \right)\]

with:

    \[{E_{{a^ - }}} = E_{{a^ + }}^\dagger \]

commute with \left\langle \Phi \right\rangle when x = 0. Inducing chirality involves including the flux:

    \[\left\langle F \right\rangle = - \frac{i}{n}M\left( {dx \wedge d\bar x - dy \wedge d\bar y} \right){\rm{diag}}\left( {1, - 2,1} \right)\]

Under the holomorphic gauge \left\langle {{A_{0,1}}} \right\rangle = 0 such that:

    \[{\left\langle {{A_{1,0}}} \right\rangle ^{{\rm{hol}}}} = \frac{i}{n}M\left( {\bar xdx - \bar ydy} \right){\rm{diag}}\left( {1, - 2,1} \right)\]

solutions to the equations of motion are derived by gauge transformations, noting that equations:

    \[\left\langle {{\Phi _{xy}}} \right\rangle = \frac{1}{n}{m^2}{\rm{diag}}\left( { - 2x + y,x + y,x - 2y} \right)\]

and:

    \[\left\langle F \right\rangle = - \frac{i}{n}M\left( {dx \wedge d\bar x - dy \wedge d\bar y} \right){\rm{diag}}\left( {1, - 2,1} \right)\]

satisfy:

    \[{{\bar \partial }_A}\Phi = \bar \partial \Phi - i\left[ {{A_{0,1}},\Phi } \right] = 0\]

    \[{F_{0,2}} = 0\]

Hence, the following equations of motion:

    \[{{\bar \partial }_{\left\langle A \right\rangle }}\varphi + i\left[ {\left\langle \Phi \right\rangle ,a} \right] = 0\]

    \[{{\bar \partial }_{\left\langle A \right\rangle }}\varphi = 0\]

    \[\omega \wedge {{\bar \partial }_{\left\langle A \right\rangle }}a - \frac{1}{2}\left[ {\left\langle {\bar \Phi } \right\rangle ,\varphi } \right] = 0\]

admit an F-theory zero-mode local model and the Yukawa couplings tree-level superpotential:

    \[W = m_ * ^4\int_S {{\rm{tr}}} \left( {F \wedge \Phi } \right)\]

includes the trilinear term:

    \[{W_{\rm{Y}}} = - im_ * ^4\int_S {{\rm{tr}}} \left( {A \wedge A \wedge \Phi } \right)\]

leading to 4-d couplings – given by an integral of the zero mode wavefunctions – among the zero modes of A and \Phi.

Solving the D-term equation:

    \[\omega \wedge F + \frac{1}{2}\left[ {\Phi ,\bar \Phi } \right] = 0\]

we get a description of an F-theory-GUT model in the vicinity of a single point by computing the down-like Yukawa couplings:

    \[{W_\parallel } = m_*^4\int_S {{\rm{tr}}} \left( {F \wedge \Phi } \right)\]

or the up-like Yukawa couplings:

    \[{W^\dagger } = m_*^4\int_S {{\rm{tr}}} \left( {F \wedge \Phi } \right)\]

The proof, and the structure of the argument, generalizes to F-theory models with {G_{{\rm{GUT}}}} = SU(5) with differing {G_p}, the most interesting cases being:

    \[{G_p} = SO(N),N \ge 10\]

    \[SP(N),N \ge 12\]

    \[{{G_p} = {E_8}}\]

and

    \[SP(8) \times SO(16)\]

all of which explicitly reflect the ‘no-two-time’ property of F-theory.