Hyper-Kähler Theory, Batalin-Vilkoviski Analysis, TQFT, and SuSy-Yang-Mills Theory

In part 1 of this series of posts, I started a Kähler-Poincaré holomorphic BF action cohomology-analysis of the \delta-symmetry, which fully determines a ring of topological ‘observables’, and thus derived from the symmetries of the action

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

given

    \[\left\{ {\begin{array}{*{20}{c}}{Q{A_m} = {\Psi _n} + {{\not D}_m}c}\\{Qc = - \frac{1}{2}\left[ {c,c} \right]}\\{Q{B_{mn}} = - \left[ {c,{B_{mn}}} \right]}\end{array}} \right.\]

    \[\left\{ {\begin{array}{*{20}{c}}{Q{\Psi _m} = - \left[ {c,{\Psi _m}} \right]}\\{Q{A_{\overline m }} = {{\not D}_{\overline m }}c}\end{array}} \right.\]

in this post, part 2, I will analyze, among other relations, the following BRST-topological quantum field theory action

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

and set the stage for Calabi-Yau 3-folding analysis by showing that on a hyper-Kähler manifold, we can identify, via holonomy-group-Kähler-algebraic twisting, the gauge-fixed action with N = 1, D = 4 Yang–Mills action. Let me choose a BV gauge function

    \[\not Z' = {\kappa ^{\overline m \overline n }}{B_{^{\overline m \overline n }}} + \Phi {\not D^{\overline m }}{\Psi _{\overline m }}\]

in order to gauge–fix the fermionic action – see below – to derive the N = 1, D = 4 chiral multiplet action

    \[{S_{SYM}} = \int\limits_{{M_4}} {{d^4}} x\sqrt g {\rm{Tr}}\left( {\overline \Phi {{\not D}^\mu }{{\not D}_\mu }\Phi \Psi {\gamma ^\mu }{{\not D}_\mu }\psi } \right)\]

To characterize a quantum theory: a path integral, I need to gauge-fix the topological symmetry of the BF system in a way consistent with faithfulness to the BRST symmetry associated to this symmetry

Quantum Holomorphy, Kähler Manifolds and SuperString Theory

The highest form of pure thought is in mathematics ~ Plato

That the Poincaré supersymmetry is a phase of a deeper symmetry is evidenced by the fact that it is analytically related to a class of topological symmetries and that the field-spectrum of dimensionally reduced N = 1 D = 11 supergravity is completely determined in the context of an 8-dimensional gravitational topological quantum field theory (TQFT). Let me study the Kähler-holomorphic nature of such connections by first analyzing the holomorphic BF action

    \[{I_{n - BF}} = \int\limits_{{M_{2n}}} {{\rm{Tr}}} \left( {{B_{n,n - 2}} \wedge {F_{0,2}}} \right)\]

with holomorphic sectorial localization constraints

    \[\left\{ {\begin{array}{*{20}{c}}{Q{A_m} = {\Psi _m} + {D_m}c}\\{Qc = - \frac{1}{2}\left[ {c,c} \right]}\end{array}} \right.\]

    \[\left\{ {\begin{array}{*{20}{c}}{Q{\Psi _m} = - \left[ {c,{\Psi _m}} \right]}\\{Q{A_{\overline m }} = {D_{\overline m }}c}\end{array}} \right.\]

The aim of this series of posts is ultimately the Kähler-Poincaré analysis of the action

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