The Quantum Master Equation and Supersymmetric-Field-Cosmology

Supersymmetric-field theory remains the best new physics beyond the SM and also “builds a bridge between the low energy phenomenology and high-energy fundamental physics”. Moreover, I will show here that SuSy can incorporate aspects of loop quantum gravity to shed deep light on fundamental issues in quantum cosmology. Before getting deep, recall that the central action for supersymmetric field theory is:

    \[\begin{array}{l}{S_0} = \sum\limits_\nu {\int {d\phi } } \left[ {{{\tilde D}^2}} \right.\left\{ {\Omega _i^\dagger } \right.\left( \wp \right)\nabla _a^2{\Omega ^j}\left( \wp \right)\\{\left. {\left. { + \,\omega _i^a\left( \wp \right)\omega _a^i} \right\}} \right]_{\left| {_G} \right.}}\end{array}\]

with \left| {_G} \right. the Grassmannian variable and total action is given by:

    \[\begin{array}{c}{S_T} = {S_0} + \sum\limits_\nu {\int {d\phi } } \left[ {{{\tilde D}^2}} \right.\left\{ {{B_i}} \right.\left( \wp \right){{\tilde D}^a}\Gamma _a^i\left( \wp \right)\\ + \,{{\bar c}_i}\left( \wp \right){{\tilde D}^a}{\nabla _a}{c^i}{\left. {\left. {\left( \wp \right)} \right\}} \right]_{\left| {_G} \right.}}\end{array}\]

and the super-field/anti-super-field theory expansion in the Landau type gauge is given by:

    \[\begin{array}{l}{Z_L}\left[ 0 \right] = \int {\not D} M{e^{ - {W_L}\left[ {\Phi ,{\Phi ^ * }} \right]}} = \\\int {\not DM\exp \left[ { - \left( {{S_0}} \right.} \right.} + \sum\limits_\nu {\int {d\phi } } \left[ {\Gamma _{1i}^{a * }} \right.\\ \cdot {\nabla _a}{c^i} + c_{1i}^ * f_{kj}^i{c^k}{c^j} + \bar c_{1i}^ * \left. {\left. {{{\left. {{B^i}} \right]}_{\left| {_G} \right.}}} \right)} \right]\end{array}\]

with the super-covariant derivatives of

    \[\left\{ {\begin{array}{*{20}{c}}{{\Omega ^i}\left( \wp \right)}\\{{\Omega ^{i\dagger }}\left( \wp \right)}\end{array}} \right.\]

are defined by:

    \[{\nabla _a}{\Omega ^i}\left( \wp \right) = {\tilde D_a}{\Omega ^i}\left( \wp \right) - if_{kj}^i\Gamma _a^k\left( \wp \right){\Omega ^j}\left( \wp \right)\]


    \[{\nabla _a}{\Omega ^{i\dagger }}\left( \wp \right) = {\tilde D_a}{\Omega ^{i\dagger }}\left( \wp \right) - if_{kj}^i{\Omega ^{k\dagger }}\left( \wp \right)\Gamma _a^i\left( \wp \right)\]

and the super-derivative is:

    \[{\tilde D_a} = {\not \partial _a} + K_a^b{\theta _b}\]

where the Ashtekar-Barbero connection K_a^i, for a, b = 1, 2, 3, is given in terms of co-triads e_a^i:

    \[K_a^i(x) = {K_{ab}}(x){e^{bi}}(x)\]

and e_a^i satisfying

    \[E_i^a = \left| {\det \left( {e_a^i} \right)} \right|e_a^i(x)\]

and the extrinsic curvature {K_{ab}} being:

    \[{K_{ab}} = \frac{1}{{2{N_{lapse}}}}\left( {{{\dot h}_{ab}} - {\nabla _a}N{{_b^{shift}}^\prime } - {\nabla _b}N{{_a^{shift}}^ * }} \right)\]

For definitions of terms, see my ‘SuperSymmetric Field Theory and the Quantum Master Equation’ post. Some philosophical points are in order first.

T-Branes and F-Theory: α’- Corrections and the D-Term-Equations

I last introduced T-branes, which are non-Abelian deformation of intersecting D-brane systems in the corresponding compactification manifold. Then I showed that we have a Kähler-equivalence of the derivatives in the pull-back with the gauge-covariant ones, which gave us:

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

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

with \iota \Phi the inclusion of the complex Higgs field \Phi, and S representing the symmetrization over gauge indices.

Locally, the Higgs field is given by:

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

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

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


    \[\iota \Phi \gamma = 0\]

a Kähler coordiante expansion of \gamma and 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