SuperSymmetric Field Theory and the Quantum Master Equation

By George Shiber Friday, July 8, 2016Saturday, August 17, 2019 In Grand Unification Physics, Loop Quantum Gravity, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry Physics, String/M-Theory

I will show that there is a quantum description of supersymmetric field theory that maps, via the quantum master equation, the two generating functionals, one linear, the other non-linear, of the extended actions of the superfield/super-antifield BRST formalism. It is key to note that local supersymmetric fields analytically entail that space-time has Sasaki-Einstein-curvature. Here is the central action for the supersymmetric field theory:

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:

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

and

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

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 $e_a^i$ thusly:

$K_a^i(x) = {K_{ab}}(x){e^{bi}}(x)$

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:

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

with the total action given by:

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

with $\left| {_G} \right.$ the Grassmannian variable describing a space with supersymmetric degrees of freedom at ${\theta _a} = 0$ and mini-superspace variables $\wp : = \left( {\nu ,\phi ,\theta } \right)$ and ${\nabla _a}$ the mini-superspace covariant derivative with the 4-D metric being:

${h^{ab}} = {\delta ^{ij}}e_i^ae_j^b = e_i^ae_i^b$

where the field-strength for a matrix valued spinor field $\left( {\Gamma _a^i} \right)$ is given by

$\omega _a^i\left( \wp \right){\nabla ^b}{\nabla _a}\Gamma _b^i\left( \wp \right)$

Thus, the action ${S_0}$ is invariant under the following gauge transformations

$\delta \,{\Omega ^i}\left( \wp \right) = if_{kj}^i{\Lambda ^k}\left( \wp \right){\Omega ^j}\left( \wp \right)$

$\delta \,{\Omega ^{i\dagger }}\left( \wp \right) = if_{kj}^i{\Omega ^{k\dagger }}\left( \wp \right){\Lambda ^j}\left( \wp \right)$

$\delta \,\Gamma _a^i\left( \wp \right) = {\nabla _a}{\Lambda ^i}\left( \wp \right)$

where ${\Lambda ^i}$ is the infinitesimal bosonic transformation.

Here is a serious problem: this gauge symmetry, in the path-integral formalism, implies that there exists infinitely many

$^{\left( \Lambda \right)}\Gamma _a^i$

that are physically equivalent to $\Gamma _a^i$ , and hence, divergences in the functional integral. To quantize such a theory, we must eliminate redundant gauge degrees of freedom and work with the Faddeev-Popov gauge condition:

${\hat G^{FP}}\left[ {\Gamma _a^i\left( \wp \right)} \right] = 0$

which yields, given the linearised gauge-fixing action corresponding to the above Faddeev-Popov gauge condition together with the induced ghost term:

with ${B^i}\left( {\nu ,\phi ,\theta } \right)$ being the Nakanishi-Lautrup external superfield, ${c^i}\left( \wp \right)$ and ${\bar c^i}\left( \wp \right)$ the ghost and antighost superfields in that order and ${s_b}$ the Slavnov-variation. In such a gauge, the total effective action for supersymmetric field theory is:

${S_T} = {S_0} + {S_{gh}} + {S_{gf}}$

and in order to be an integrable system, the Landau gauge

${\hat G^{{L^i}}} = \tilde D\Gamma _a^i\left( \wp \right) = 0$

consequently must be imposed and therefore, the total action becomes, as required:

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

Now, the Curci-Ferrari non-linear gauge forces us to perform a shift in auxiliary superfield:

${B^i}\left( {\nu ,\phi ,\theta } \right) \to {B^i}\left( \wp \right) - \frac{1}{2}f_{jk}^i{\bar c^j}\left( \wp \right){c^k}\left( \wp \right)$

giving us the total effective action corresponding to the above non-linear gauge:

and both are are invariant under the third-quantized infinitesimal BRST transformations:

$\left\{ {\begin{array}{*{20}{c}}{{\delta _b}{\Omega ^i}\left( \wp \right) = if_{kj}^i{c^k}\left( \wp \right){\Omega ^j}\left( {\wp \delta \lambda } \right)}\\{{\delta _b}{\Omega ^{i\dagger }}\left( \wp \right) = - if_{kj}^i{\Omega ^{\dagger k}}\left( \wp \right){c^j}\left( \wp \right)\delta \lambda }\\{{\delta _b}{c^i}\left( \wp \right) = \frac{1}{2}f_{kj}^i{c^k}\left( \wp \right){c^j}\left( \wp \right)\delta \lambda }\\{{\delta _b}\Gamma _a^i\left( \wp \right) = {\nabla _a}{c^i}\left( \wp \right)\delta \lambda }\\{{\delta _b}{{\bar c}^i}\left( \wp \right) = {B^i}\left( \wp \right)\delta \lambda }\\{{\delta _b}{B^i}\left( \wp \right) = 0}\end{array}} \right.$

and crucial, $\delta \lambda$ being an infinitesimal anticommuting space-time independent parameter. Hence, we have nilpotency of order two. By using the BRST transformation, the sum of gauge-fixing as well as ghost parts of:

we get a reduction to

${S_{gf + gh}} = \sum\limits_\nu {\int {d\phi {s_b}} } \Psi$

with

$\Psi \equiv {\tilde D^2}\left\{ {{{\bar c}_i}\left( \wp \right){{\hat G}^{FP}}^i\left[ {\Gamma _a^i\left( \wp \right)} \right]} \right\}$

Let us transition to the superfield/super-antifield formalism

Here, we find that the functional for the supersymmetric field theory in 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 ${W_L}$ the extended supersymmetric quantum action.

The fermionic sector for the supersymmetric field theory in Landau-gauge becomes

${W_L} = {\tilde D^2}\left[ {{{\bar c}_i}\left( \wp \right){{\tilde D}^a}\Gamma _a^i\left( \wp \right)} \right]$

‘Solving’ for super-antifields for the corresponding Landau gauge yields:

$\left\{ {\begin{array}{*{20}{c}}{\Omega _1^{i * } = \frac{{\delta \,{\Psi _L}}}{{\delta \,{\Omega _i}}} = 0}\\{\Omega _1^{i\dagger * } = \frac{{\delta \,{\Psi _L}}}{{\delta \,\Omega _i^\dagger }} = 0}\end{array}} \right.$

$\left\{ {\begin{array}{*{20}{c}}{c_{1i}^ * = \frac{{\delta \,{\Psi _L}}}{{\delta \,{c^i}}} = 0}\\{\bar c_{1i}^ * = \frac{{\delta \,{\Psi _L}}}{{\delta \,{{\bar c}^i}}} = {{\tilde D}^2}{{\tilde D}_a}\Gamma _i^a}\end{array}} \right.$

and

$\Gamma _{1i}^{a * } = \frac{{\delta \,{\Psi _L}}}{{\delta \,\Gamma _a^i}} = - {\tilde D^2}{\tilde D^a}{\bar c_i}$

Thus the generating functional for supersymmetric field theory in non-linear gauge of superfields/super-antifields is:

and so the fermionic gauge-fixing for the non-linear gauge is:

and the corresponding super-antifields gauge-fixing fermionic system becomes:

$\left\{ {\begin{array}{*{20}{c}}{\Omega _2^{i * } = \frac{{\delta \,{\Psi _{NL}}}}{{\delta \,{\Omega _i}}} = 0}\\{\Omega _2^{i\dagger * } = \frac{{\delta \,{\Psi _{NL}}}}{{\delta \,\Omega _i^\dagger }} = 0}\end{array}} \right.$

$c_{2i}^ * = \frac{{\delta \,{\Psi _{NL}}}}{{\delta {c^i}}} = {\tilde D^2}\left( {\frac{1}{4}f_i^{jk}{{\bar c}_j}{{\bar c}_k}} \right)$

$\bar c_{2i}^ * = \frac{{\delta \,{\Psi _{NL}}}}{{\delta {{\bar c}^i}}} = {\tilde D^2}\left[ {{{\tilde D}_a}\Gamma _i^a - \frac{1}{2}f_i^{jk}{{\bar c}_j}{c_k}} \right]$

and

$\Gamma _{2i}^{a * } = \frac{{\delta \,{\Psi _{NL}}}}{{\delta \,\Gamma _a^i}} = - {\tilde D^2}{\tilde D^a}{\bar c_i}$

Now it becomes clear what the difference between the non-linear and linear extended quantum actions:

and we can now arrive at the structural form of solutions to the quantum master equation:

$\left\{ {\begin{array}{*{20}{c}}{\Delta {e^{i{W_\Psi }\left[ {\Phi ,{\Phi ^ * }} \right]}} = 0}\\{\Delta \equiv {{\left( { - 1} \right)}^\varepsilon }\frac{{{{\not \partial }_l}}}{{\not \partial \Phi }}\frac{{{{\not \partial }_l}}}{{\not \partial {\Psi ^ * }}}}\end{array}} \right.$

which has the form:

${W_\Psi }\left[ {\Phi ,{\Phi ^ * }} \right] \equiv \left( {{W_{NL}},{W_L}} \right)$

We are now in a position to establish a map between the two generating functionals corresponding to the above extended actions by use of the superfield/super-antifield dependent BRST transformations

The superfield/super-antifield dependent BRST transformation yields the following Jacobian functional measure:

$\begin{array}{l}{{Z'}_L}\left[ 0 \right] = \int {\not D} M\left( {s{\rm{Det}}J\left[ {\Phi ,{\Phi ^ * }} \right]} \right) \cdot \\\exp \left\{ { - {W_L}\left[ {\Phi ,{\Phi ^ * }} \right]} \right\} = \\\int {\not D} M{e^{ - \left( {{W_L}\left[ {\Phi ,{\Phi ^ * }} \right] - is{\rm{Tr}}\,{\rm{In}}J\left[ {\Phi ,{\Phi ^ * }} \right]} \right)}}\end{array}$

with ${Z'_L}$ referring to the Jacobian change of variables. The associated matrix for the superfield/super-antifield dependent BRST transformation is therefore:

By BRST-nilpotency, we get a reduction to:

$s{\rm{Det}}J\left[ {\Phi ,{\Phi ^ * }} \right] = \frac{1}{{1 + {s_b}}}\Lambda \left[ {\Phi ,{\Phi ^ * }} \right]$

thus simplifying our Jacobian functional measure:

$\begin{array}{c}{{Z'}_L}\left[ 0 \right] = \int {\not D\Phi \exp } \left( { - {W_L}} \right.\left[ {\Phi ,{\Phi ^ * }} \right]\\ - i{\rm{In}}\left. {\left( {1 + {s_b}\Lambda \left[ {\Phi ,{\Phi ^ * }} \right]} \right)} \right)\end{array}$

Hence, the Slavnov variation yields:

$\begin{array}{c}{s_b}\Lambda \left[ {\Phi ,{\Phi ^ * }} \right] = \sum\limits_\nu {\int {d\phi \exp \left( { - i} \right.} } \\\left[ {c_{2i}^ * \left( \wp \right)\left( {\frac{1}{2}f_{kj}^i{c^k}\left( \wp \right){c^j}\left( \wp \right)} \right)} \right. + \\\left( {\bar c_{2i}^ * \left( \wp \right) - \bar c_{1i}^ * \left( \wp \right)} \right){\left. {\left. {{B^i}\left( \wp \right)} \right]} \right)_{\left| {_G} \right.}} - 1\end{array}$