Batalin-Vilkoviski Equation, Calabi–Yau 4–Fold, D = 8 SYM-Action and Gauge-Invariance

By George Shiber Saturday, March 5, 2016Thursday, August 15, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry Calabi–Yau 4–Fold, Kähler

Reality is complicated. There is no justification for all of the hasty conclusions ~ Hideki Yukawa

Continuing from my last few posts dealing with Calabi–Yau N-fold holomorphic TQFT analysis, recall that I showed that by integration on the Lagrangian multipliers, we get, from the Batalin–Vilkoviski action

$\begin{array}{c}{S_{BV}} = \int\limits_{{M_6}} {\sqrt g } {d^6}x{\rm{Tr}}\left( {{\varepsilon ^{\overline m \overline p \overline q }}} \right.{B_{\overline m }}{F_{\overline p \overline q }}\\{ + ^ * }{B^{\overline m }}\left( {{{\not D}_{\overline m }}\chi - \left[ {c,{B_{\overline m }}} \right]} \right) + \\^ * {A^m}\left( {{\Psi _m} + {{\not D}_{\overline m }}c} \right){ + ^ * }\,{A^{\overline m }}{{\not D}_{\overline m }}c \cdot \\^ * {\Psi ^m}\left[ {c,{\Psi _m}} \right] + \left( { - \frac{1}{2}} \right.\left. {\left. {\left[ {c,c} \right]} \right)} \right)\end{array}$

and

$\begin{array}{c}{S_{{\rm{aux}}}} = \int\limits_{{M_6}} {\sqrt g } {d^6}x{\rm{Tr}}{\left( {^ * \overline \chi } \right._{\overline m \overline n }}{h^{\overline m \overline n }} + \\^ * \overline \eta h + \left. {^ * \overline {c\,} b} \right)\end{array}$

and by the identity

$\begin{array}{c} - \frac{1}{4}{\rm{Tr}}\left( {F \wedge * F} \right) + J \wedge {\rm{Tr}}\left( {F \wedge F} \right)\\ \cdot {\rm{Tr}}\left( { - \frac{3}{2}{F^{\overline m \overline n }}} \right.{F_{^{\overline m \overline n }}} + \left| {{J^{^{\overline m \overline n }}}{F_{^{\overline m \overline n }}}} \right|\left. {^2} \right)\end{array}$

to see in the first line of the (g.f.)-action the bosonic part of the N=1 D=6 SYM action, modulo the topological density $J \wedge {\rm{Tr}}\left( {F \wedge F} \right)$ , where $J$ is the Kähler two–form. Concerning the fermionic part, I showed how the mapping between chiral fermions and complex forms

${S_ \pm } \otimes \mathbb{C} \sim \Omega _{{\rm{even}}}^{{\rm{odd}}}$

map the topological ghosts $\left( {{\Psi _m},\chi } \right)$ into the right–handed spinor $\overline \lambda$ and the topological antighosts into the left–handed spinor $\lambda$ . Hence, one can use the covariantly constant spinor $\zeta$ of the Calabi–Yau three-fold to perform the mapping

$\left\{ {\begin{array}{*{20}{c}}{{\Psi _m} \to \overline \lambda {\Gamma _m}\zeta }\\{\chi \to \overline \chi \zeta }\\{{{\overline \chi }^{\overline m \overline n }} \to \zeta \,{\Gamma ^{\overline m \overline n }}\lambda }\\{\overline \eta \to {\varepsilon _{\overline m \overline n \overline p }}\zeta }\end{array}} \right.$

and in this way, one can see in the (g.f.)-action the twisted version of the N = 1 D = 6 Super Yang–Mills action, where the (g.f.)-action is

$\begin{array}{c}{S^{g.f.}} = \int\limits_{{M_6}} {{d^6}} x\sqrt g {\rm{Tr}}\left( { - \frac{3}{2}} \right.{F^{\overline m \overline n }} + \\\left| {{J^{m\overline n }}{F_{m\overline n }}} \right|\left. {^2} \right) + {\overline \chi ^{\overline m \overline n }}{{\not D}_{\left[ {\overline m \,{\Psi _{\overline n }}} \right]}} + \\2i\overline \eta {J^{\overline m n}}{J^{m\overline n }}{{\not D}_{\overline m }}{\Psi _n} + \frac{4}{3}{\varepsilon _{\overline m \overline n \overline p }} \cdot \\{\overline \chi ^{\overline m \overline n }}{{\not D}^{\overline p }}\left. {\overline \chi } \right)\end{array}$

Let us get deeper. On a Calabi–Yau four–fold one can write a generalization of the action

$\begin{array}{c}{I_{cl}}\left( {A,B} \right)\int\limits_{{M_4}} {{\rm{Tr}}} {B_{2,0}} \wedge {F_{0,2}} = \\\int\limits_{{M_4}} {{d^4}} x\sqrt g {\rm{Tr}}\left( {{\varepsilon ^{mn\overline m \overline n }}{B_{mn}}{F_{m\overline {} }}} \right)\end{array}$

${I_{cl}}\left( {A,{B_{0,2}}} \right)\int\limits_{{M_8}} {{\Omega _{4,0}}} \wedge {\rm{Tr}}\left( {{B_{0,2}} \wedge {F_{0,2}}} \right)$

where ${\Omega _{4,0}}$ is the holomorphic covariantly closed $\left( {4,0} \right)$ -form. After normalization of $\Omega$ such that $\Omega \wedge \overline \Omega$ is the volume element on ${M_8}$ , the action

${I_{cl}}\left( {A,{B_{0,2}}} \right)\int\limits_{{M_8}} {{\Omega _{4,0}}} \wedge {\rm{Tr}}\left( {{B_{0,2}} \wedge {F_{0,2}}} \right)$

displays the symmetry

$\left\{ {\begin{array}{*{20}{c}}{Q{A_M} = {\Psi _M} + {{\not D}_M}c}\\{Q{A_{\widehat M}} = {{\not D}_{\widehat M}}c}\end{array}} \right.$

$\begin{array}{c}Q{B_{\widehat M\widehat N}} = {{\not D}_{\left[ {\widehat M\chi \widehat N} \right]}} - \left[ {c,{B_{\widehat M\widehat N}}} \right] - \\\frac{1}{4}{\varepsilon _{\widehat M\widehat N\overline P \overline Q }}\left[ {^ * {B^{\overline P \overline Q }},\phi } \right]\end{array}$

and the keep is that $c$ is the complexified Faddeev–Popov ghost. So, the BV action above corresponding to

${I_{cl}}\left( {A,{B_{0,2}}} \right)\int\limits_{{M_8}} {{\Omega _{4,0}}} \wedge {\rm{Tr}}\left( {{B_{0,2}} \wedge {F_{0,2}}} \right)$

$\begin{array}{c}{S^{FP}} = \int\limits_{{M_8}} {\sqrt g } {d^8}x{\rm{Tr}}\left( {{\varepsilon ^{\widehat M\widehat N\overline P \overline Q }}} \right.{B_{\widehat M\widehat N}}{F_{\overline P \overline Q }}\\{ + ^ * }{B^{\widehat M\widehat N}}\left( {{{\not D}_{\left[ {\widehat M\chi \widehat N} \right]}}} \right. - \left[ {c,{B_{\widehat M\widehat N}}} \right] - \\\frac{1}{4}{\varepsilon _{\widehat M\widehat N\overline P \overline Q }}\left. {\left[ {^ * {B^{_{\overline P \overline Q }}}} \right]} \right) + \\^ * {A^M}\left( {{\Psi _M} + {{\not D}_{\widehat M}}c} \right){ + ^ * }{A^{\widehat M}} \cdot \\{{\not D}_{\widehat M}}c{ + ^ * }{\chi ^{\widehat N}}\left( {{{\not D}_{\widehat N}}\phi - \left[ {z,{\chi _{\widehat N}}} \right]} \right)\\{ + ^ * }{\overline \chi _{\widehat M\widehat N}}{h^{\widehat M\widehat N}}{ + ^ * }\overline \chi h{ + ^ * }\overline c \,b + \\^ * \overline \phi \eta { + ^ * }c\left( { - \frac{1}{2}} \right.\left[ {c,c} \right]{ - ^ * }\phi \cdot \\\left. {\left[ {c,\phi } \right]} \right))\end{array}$

thus allowing us to gauge–fix by imposing six complex conditions for ${B_{\widehat M\widehat N}}$

$\left\{ {\begin{array}{*{20}{c}}{B_{\widehat M\widehat N}^ + = 0}\\{B_{\overline M \overline N }^ - = F_{\overline M \overline N }^ - }\end{array}} \right.$

and a gauge–fixing for ${\chi _{\overline M }}$

${\not D^{\widehat M}}{\chi _{\overline M }} = 0$

The projection on self-dual or anti-self-dual part $B_{0,2}^ \pm$ of the $\left( {0,2} \right)$ –forms can be done by using the anti-holomorphic $\left( {0,4} \right)$ form. Now, by enforcing

$\left\{ {\begin{array}{*{20}{c}}{B_{\widehat M\widehat N}^ + = 0}\\{B_{\overline M \overline N }^ - = F_{\overline M \overline N }^ - }\end{array}} \right.$

via the BRST doublets of complex antighosts and Lagrangian multipliers $\left( {{{\overline \chi }^{\overline M \overline N }},{h^{\overline M \overline N }}} \right)$ and $\left( {\overline \phi ,\overline \chi } \right)$ respectively. Hence, the following complex condition holds

${\not D^{\widehat Mc}}{A_{\overline M }} = 0$

whose real part is the Landau gauge condition and the imaginary part gives us

$\begin{array}{c}{\mathop{\rm Im}\nolimits} {{\not D}^{\widehat M}}{A_{\widehat M}} = 0 \Rightarrow \\{J^{M\widehat N}}{F_{^{M\widehat N}}} = 0\end{array}$

$\begin{array}{c}{\mathop{\rm Re}\nolimits} {{\not D}^{\widehat M}}{A_{\widehat M}} = 0 \Rightarrow \\{{\not \partial }^\mu }{A_\mu } = 0\end{array}$

And the corresponding gauge–fixing fermion to the gauge conditions is

$\begin{array}{c}Z = {\overline \chi ^{\overline M \overline N + }}{B_{\overline M \overline N }} + {\overline \chi ^{\overline M \overline N - }}\left( {{B_{\overline M \overline N }}} \right. - \\2\left. {{F_{\overline M \overline N }}} \right) + \overline \phi {{\not D}^{\overline M }}{\chi _{\overline M }} + \overline \chi \left( {i{J^{M\overline N }}} \right.\\{F_{M\overline N }} + \frac{1}{2}\left. h \right) + \overline c \left( {{{\not \partial }^\mu }{A_\mu } + \frac{1}{2}b} \right)\end{array}$

and by using the BV equation

$^ * \phi = \frac{{\delta Z}}{{\delta \phi }}$

and enforcing the gauge conditions, by integration on the Lagrangian multipliers,

one gets the desired action as a twisted form of the D = 8 supersymmetric Yang–Mills action whose gauge invariant part is

$\begin{array}{c}{S^{g.f.}} = \int\limits_{{M_8}} {{d^8}} x\sqrt g {\rm{Tr}}\left( {2{F^{\overline M \overline N - }}} \right.\\F_{\overline M \overline N }^ - + \frac{1}{2}{\left| {{J^{M\overline N }}{F_{M\overline N }}} \right|^2} + \overline \phi {{\not D}^{\overline M }}\\{{\not D}_{\overline M }}\phi + {\overline \chi ^{\overline M \overline N - }}{{\not D}_{\left[ {\overline M \chi \overline N } \right]}} + {\overline \chi ^{\overline M \overline N + }}\\{{\not D}_{\left[ {\overline M \Psi \overline N } \right]}} + \overline \chi {{\not D}^M}{\Psi _M} + \overline h \\{{\not D}^{\overline M }}\left. {{\chi _{\overline M }}} \right)\end{array}$