Mathematical reason itself does not err ~ Kurt Gödel
Continuing from my last two posts, I finally derived, via Kähler-twisting relations
![\[\left\{ {\begin{array}{*{20}{c}}{{\Psi _m} = {\lambda ^\alpha }{\sigma _{\mu \alpha \dot 1}}e_m^\mu }\\{{\kappa _{\overline m \overline n }} = {{\overline \lambda }_{\dot \alpha }}\overline \sigma _{\mu \nu \dot 2}^{\dot \alpha }e_{\overline m }^\mu \,e_{\overline n }^\nu }\\{\kappa = \delta _{\dot 2}^{\dot \alpha }{{\overline \lambda }_{\dot \alpha }}}\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-210d1ee198c8e48d020ae107049d9485_l3.png "Rendered by QuickLaTeX.com")
that on a hyper-Kähler manifold, by ‘constant spinor’ change-of-variables, we can identify
![\[\begin{array}{c}{S^{g \cdot f}} = \int\limits_{{M_4}} {{d^4}} x\sqrt g {\rm{Tr}}\left( {{F_{\overline m \overline n }}} \right.{F^{\overline m \overline n }} + \\\frac{1}{2}{\left| {{J_{m\overline n }}{F^{m\overline n }}} \right|^2} - 2{\varepsilon ^{\overline m \overline n pq}}{\kappa _{\overline m \overline n }}{{\not D}_p}{\Psi _q}\\ + \,i\kappa {J^{\overline m l}}{{\not D}_{\overline m }}\left. {{\Psi ^l}} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-ae328a1f5e1d77177eca64c26f6d3dda_l3.png "Rendered by QuickLaTeX.com")
with the N = 1, D = 4 Yang–Mills action
![\[\begin{array}{c}{S_{SYM}} = \int\limits_{{M_4}} {{d^4}} x\sqrt g \frac{1}{4}{\rm{Tr}}\left( {{F_{\mu \nu }}} \right.{F^{\mu \nu }}\\ + {F_{\mu \nu }}{\overline F ^{\mu \nu }} + \overline \lambda {\gamma ^\mu }{{\not D}_\mu }\left. \lambda \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-75a528887e8473c59ab81948870b200e_l3.png "Rendered by QuickLaTeX.com")
getting the following deep relation between N = 2 SYM theory, topological quantum field theory, and Kähler-theory
that since N = 2 SYM theory is a TQFT, and its Poincare supersymmetric version is obtained by coupling the N = 1 Yang–Mills multiplet to a chiral multiplet in the adjoint representation of the gauge group, we get an expression of the N = 1 scalar theory as a TQFT on a Kähler manifold.
In this post, I will derive a connection between the Batalin-Vilkoviski action, which I will eventually show is crucial to M-theory‘s braneworld cosmology, indeed, to any quantum-cosmological model consistent with a quantum gravity theory.
For any Calabi–Yau three–fold 
one can always use the holomorphic closed 
–form 
and define 
thus getting the transformed classical-BRST action
![\[{I_{cl}}\left( {A,{B_{0,1}}} \right) = \int\limits_{{M_6}} {{\Omega _{3,0}}} \wedge {\rm{Tr}}\left( {{B_{0,1}} \wedge F} \right)\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-da00098efc4186f06cdad702fc9f6695_l3.png "Rendered by QuickLaTeX.com")
with the corresponding BRST symmetry
![\[\left\{ {\begin{array}{*{20}{c}}{Q{A_m} = {\Psi _m} + {{\not D}_m}c}\\{Q{A_{\overline m }} = {{\not D}_{\overline m }}\,c}\\{Q{B_{\overline m }} = {{\not D}_{\overline m }}\,\chi - \left[ {c,{B_{\overline m }}} \right]}\\{Q\chi = - \left[ {c,\chi } \right]}\\{Qc = - \frac{1}{2}\left[ {c,c} \right]}\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-7ea2107d38b9fc341ab2ad59a138d573_l3.png "Rendered by QuickLaTeX.com")
whose invariance under the transformation of the 
field is guaranteed by part of the Bianchi identity
![\[{\not D_{\left[ {\overline m \,\,{F_{\overline n \,\overline l }}} \right]}} = 0\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-2b087ab5c5b4770373e40234d8edc7be_l3.png "Rendered by QuickLaTeX.com")
and closed-ness of 
and the fact that the BRST action
is invariant under the complexified gauge group 
.
The BRST symmetry of
is derivable from the Batalin–Vilkoviski action
![\[\begin{array}{c}{S_{BV}} = \int\limits_{{M_6}} {\sqrt g } {d^6}x{\rm{Tr}}\left( {{\varepsilon ^{\overline m \overline p \overline q }}} \right.{B_{\overline m }}{F_{\overline p \overline q }}\\{ + ^ * }{B^{\overline m }}\left( {{{\not D}_{\overline m }}\chi - \left[ {c,{B_{\overline m }}} \right]} \right) + \\^ * {A^m}\left( {{\Psi _m} + {{\not D}_{\overline m }}c} \right){ + ^ * }\,{A^{\overline m }}{{\not D}_{\overline m }}c \cdot \\^ * {\Psi ^m}\left[ {c,{\Psi _m}} \right] + \left( { - \frac{1}{2}} \right.\left. {\left. {\left[ {c,c} \right]} \right)} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-4bf5f1cdbb25ee1042a099d0fb2d6eec_l3.png "Rendered by QuickLaTeX.com")
Now we must quantize the model, and the best way is to
quantize the theory around a non-perturbative vacuum corresponding to a stable holomorphic vector bundle, since in that context, the BF model correspond to the twisted version of a supersymmetric Yang–Mills theory
since it allows the BRST symmetry to be treated as ordinary gauge ones and fixed with transversality conditions on the 
and 
![\[\left\{ {\begin{array}{*{20}{c}}{{{\not D}^{\overline m }}{A_{\overline m }} = 0}\\{{{\not D}^{\overline m }}{B_{\overline m }} = 0}\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-0522a453847c77cae9cfd2cb901af4e4_l3.png "Rendered by QuickLaTeX.com")
The corresponding BV fermion identity follows then
![\[\not Z = \overline \chi {\not D^{\overline m }} + \overline c {\not D^{\overline m }}{A_{\overline m }}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-77a35af87474ade7e360e1ef884ea250_l3.png "Rendered by QuickLaTeX.com")
So the shift symmetry on the 
part of the connection gives rise to three degrees of freedom, while the symmetry on the field to one and are collected into the ghost fields 
respectively. In the non–perturbative case, the gauge fixing conditions are
![\[\left\{ {\begin{array}{*{20}{c}}{{F_{mn}} = - \frac{4}{3}{\varepsilon _{mnp}}{B^p}}\\{{J^{\overline m \,n}} = 0}\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-05f2a74a0e618b14f5a0ce264d141324_l3.png "Rendered by QuickLaTeX.com")
Notice that 
reduces the complex gauge group to the unitary group 
and can be interpreted as a partial gauge-fixing for the complex gauge symmetry of
The BV fermion corresponding to
where 
are the antighosts associated to BV gauge–fixing conditions and whose BV action is given
![\[\begin{array}{c}{S_{{\rm{aux}}}} = \int\limits_{{M_6}} {\sqrt g } {d^6}x{\rm{Tr}}{\left( {^ * \overline \chi } \right._{\overline m \overline n }}{h^{\overline m \overline n }} + \\^ * \overline \eta h + \left. {^ * \overline {c\,} b} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-442c513be69331dcc4fc957fc8733390_l3.png "Rendered by QuickLaTeX.com")
Now, by eliminating the anti–fields by means of
![\[^ * \phi = \frac{{\delta Z}}{{\delta \phi }}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-966b50c3a56e77fe6bd93297dc201e77_l3.png "Rendered by QuickLaTeX.com")
and implementing the gauge–fixing conditions
and by integration on the Lagrangian multipliers, we get from
and
the (g.f.)-action
![\[\begin{array}{c}{S^{g.f.}} = \int\limits_{{M_6}} {{d^6}} x\sqrt g {\rm{Tr}}\left( { - \frac{3}{2}} \right.{F^{\overline m \overline n }} + \\\left| {{J^{m\overline n }}{F_{m\overline n }}} \right|\left. {^2} \right) + {\overline \chi ^{\overline m \overline n }}{{\not D}_{\left[ {\overline m \,{\Psi _{\overline n }}} \right]}} + \\2i\overline \eta {J^{\overline m n}}{J^{m\overline n }}{{\not D}_{\overline m }}{\Psi _n} + \frac{4}{3}{\varepsilon _{\overline m \overline n \overline p }} \cdot \\{\overline \chi ^{\overline m \overline n }}{{\not D}^{\overline p }}\left. {\overline \chi } \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-9bedbefcf655726ccf9184572cadf021_l3.png "Rendered by QuickLaTeX.com")
and by the identity
![\[\begin{array}{c} - \frac{1}{4}{\rm{Tr}}\left( {F \wedge * F} \right) + J \wedge {\rm{Tr}}\left( {F \wedge F} \right)\\ \cdot {\rm{Tr}}\left( { - \frac{3}{2}{F^{\overline m \overline n }}} \right.{F_{^{\overline m \overline n }}} + \left| {{J^{^{\overline m \overline n }}}{F_{^{\overline m \overline n }}}} \right|\left. {^2} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-85151d1fdb95c5c0d444103e99effb20_l3.png "Rendered by QuickLaTeX.com")
we can recognize in the first line of the (g.f.)-action the bosonic part of the N=1 D=6 SYM action, modulo the topological density 
, where 
is the Kähler two–form. Concerning the fermionic part, one can use of the mapping between chiral fermions and complex forms
![\[{S_ \pm } \otimes \mathbb{C} \sim \Omega _{{\rm{even}}}^{{\rm{odd}}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-b7b36a1bf4d436479bbdafdeff0a47c6_l3.png "Rendered by QuickLaTeX.com")
to map the topological ghosts into the right–handed spinor 
and the topological antighosts into the left–handed spinor 
. Hence, one can use the covariantly constant spinor 
of the Calabi–Yau three-fold to perform the mapping
![\[\left\{ {\begin{array}{*{20}{c}}{{\Psi _m} \to \overline \lambda {\Gamma _m}\zeta }\\{\chi \to \overline \chi \zeta }\\{{{\overline \chi }^{\overline m \overline n }} \to \zeta \,{\Gamma ^{\overline m \overline n }}\lambda }\\{\overline \eta \to {\varepsilon _{\overline m \overline n \overline p }}\zeta }\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-4090b6633816fdba36791ac8e4f49397_l3.png "Rendered by QuickLaTeX.com")
