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

By George Shiber Saturday, April 1, 2017Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematics, Physics Calabi-Yau 4-folds, F-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: