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

In part 1 of this series of posts, I startedKähler-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

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


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

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

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. Note that the anti-self-duality condition in 4-dimensions is expressible in complex coordinates

    \[\left\{ {\begin{array}{*{20}{c}}{{F_{mn}} = 0}\\{{J_{m\overline n }}{F^{m\overline n }} = 0}\\{{F_{\overline m \overline n }} = 0}\end{array}} \right.\]

thus getting the crucial identity

    \[\begin{array}{c}{\rm{Tr}}\left( {{F_{\overline m \overline n }}{F^{\overline m \overline n }} + {{\left| {{J_{m\overline n }}{F^{m\overline n }}} \right|}^2}} \right) = \\\frac{1}{4}{\rm{Tr}}\left( {{F_{\mu \nu }}{F^{\mu \nu }} + {F_{\mu \nu }}{{\widetilde F}^{\mu \nu }}} \right)\end{array}\]

Modulo gauge invariance, one gets two topological freedoms corresponding to the two components in {\Psi _m}; moreover, in order to perform a metaplectic-gauge-fixing for the two-form {B_{2,0}} and {A_{1,0}}, one needs to inject two anti-commuting anti-ghosts {\kappa ^{mn}} and \kappa, and two Lagrange multipliers {b^{mn}} and b

    \[\left\{ {\begin{array}{*{20}{c}}{Q{\kappa ^{mn}} = {b^{mn}}}\\{Q\kappa = b}\end{array}} \right.\]


    \[\left\{ {\begin{array}{*{20}{c}}{Q{b^{mn}} = 0}\\{Qb = 0}\end{array}} \right.\]

and given that {Q^2} = 0 holds, we have a Batalin-Vilkoviski (BV) system. Let me introduce BV anti-fields for A and B and their ghosts, antighosts and Lagrangian multipliers, with upper-let symbol * labeling antifields. Note now that the antifield ^ * \varphi of a field with ghost number g has ghost number - g - 1 and opposite statistics. Hence, for a Q-invariant BV action S one has

    \[\left\{ {\begin{array}{*{20}{c}}{Q\phi = \frac{{{{\not \partial }_g}S}}{{{{\not \partial }^*}\phi }}}\\{{Q^*}\phi = - \frac{{{{\not \partial }_l}S}}{{\not \partial \phi }}}\end{array}} \right.\]

Now, the property {Q^2} = 0 is equivalent to the Kähler-master equation

    \[\frac{{{{\not \partial }_\tau }S{{\not \partial }_l}S}}{{\not \partial \phi \,{{\not \partial }^ * }\phi }} = 0\]

where \left( {{{\not \partial }_\tau },{{\not \partial }_l}} \right) indicate the derivatives from the left and from the right. So the BV action

    \[\begin{array}{c}S = \int\limits_{{M_4}} {{d^4}} x\sqrt g {\rm{Tr}}\left( {\frac{1}{4}} \right.{\varepsilon ^{mn\overline m \overline n }}{B_{mn}}{F_{\overline m \overline n }}\\ + {\,^ * }{A^m}\left( {{\Psi _m} + {{\not D}_m}c} \right) + {\,^ * }{A^{\overline m }}\left( {{{\not D}_{\overline m }}c} \right)\\ - {\,^ * }{B^{mn}}\left[ {c,{B_{mn}}} \right] - {\,^ * }\Psi \left[ {c,{\Psi _m}} \right] - \\\frac{1}{2}{\,^ * }c\left[ {c,c} \right] + {\,^ * }{\kappa _{mn}}{b^{mn}} + \left. {^ * \kappa b} \right)\end{array}\]

implies the gauge-invariance of the semi-classical action as well as the nilpotency {Q^2} = 0 on all the fields, hence also entailing

    \[{Q^ * }{B^{mn}} = {\varepsilon ^{mn\overline m \overline n }}{F_{\overline m \overline n }} - \left[ {{c^ * },{B^{mn}}} \right]\]

Also, the anti-fields must be replaced in the path integral by the BV formula

    \[^ * \phi = \frac{{\delta Z}}{{\delta \phi }}\]

and due to the path-integral anti-self-duality condition for

    \[\left\{ {\begin{array}{*{20}{c}}{{F_{mn}} = 0}\\{{J_{m\overline n }}{F^{m\overline n }} = 0}\\{{F_{\overline m \overline n }} = 0}\end{array}} \right.\]

we have

    \[\begin{array}{c}Z = {\kappa ^{mn}}\left( {{B_{mn}} - {\varepsilon _{mn\overline m \overline n }}{F^{\overline m \overline n }}} \right) + \\\kappa \left( {\frac{1}{2}b\_i\,{J_{m\overline n }}{F^{m\overline n }}} \right)\end{array}\]

which implies that {B_{2,0}} is eliminated in the path integral. And, after Gaussian-integration on b, the gauge-fixed action becomes

    \[\begin{array}{c}{S^{g \cdot f}} = \int\limits_{{M_4}} {{d^4}} x\sqrt g {\rm{Tr}}\left( {{F_{\overline m \overline n }}} \right.{F^{\overline m \overline n }} + \\\frac{1}{2}{\left| {{J_{m\overline n }}{F^{m\overline n }}} \right|^2} - 2{\varepsilon ^{\overline m \overline n pq}}{\kappa _{\overline m \overline n }}{{\not D}_p}{\Psi _q}\\ + \,i\kappa {J^{\overline m l}}{{\not D}_{\overline m }}\left. {{\Psi ^l}} \right)\end{array}\]

Let us compare {S^{g \cdot f}} with that of N = 1 SYM on a Kahler manifold. It is known that on a complex spin manifold the spinors can be identified with forms

    \[{S_ \pm } \otimes \mathbb{C} \sim \Omega _l^o\]

even-odd , so that one can identify the topological ghost {\Psi _m} as a left-handed Weyl spinor {\lambda _\alpha } and the topological anti–ghosts \left( {{\kappa _{\overline m \overline n }},\kappa } \right) as a right-handed Weyl spinor {\overline \lambda ^{\dot \alpha }} and so the holonomy group of 4-dimensional Kahler manifold is locally given by

    \[\begin{array}{c}U\left( 2 \right) \sim SU{\left( 1 \right)_R} \subset SU{\left( 2 \right)_L} \otimes \\SU{\left( 2 \right)_R}\end{array}\]

hence we can identify the forms {\sigma _{\mu \alpha \dot 1}}d{x^\mu } and {\sigma _{\mu \alpha \dot 2}}d{x^\mu } as \left( {1,0} \right) and \left( {0,1} \right) forms respectively. The Kähler-twist is now

    \[\left\{ {\begin{array}{*{20}{c}}{{\Psi _m} = {\lambda ^\alpha }{\sigma _{\mu \alpha \dot 1}}e_m^\mu }\\{{\kappa _{\overline m \overline n }} = {{\overline \lambda }_{\dot \alpha }}\overline \sigma _{\mu \nu \dot 2}^{\dot \alpha }e_{\overline m }^\mu \,e_{\overline n }^\nu }\\{\kappa = \delta _{\dot 2}^{\dot \alpha }{{\overline \lambda }_{\dot \alpha }}}\end{array}} \right.\]

On a hyper-Kähler manifold, such a twist, via ‘constant spinor’ change of variables, allows us to identify

    \[\begin{array}{c}{S^{g \cdot f}} = \int\limits_{{M_4}} {{d^4}} x\sqrt g {\rm{Tr}}\left( {{F_{\overline m \overline n }}} \right.{F^{\overline m \overline n }} + \\\frac{1}{2}{\left| {{J_{m\overline n }}{F^{m\overline n }}} \right|^2} - 2{\varepsilon ^{\overline m \overline n pq}}{\kappa _{\overline m \overline n }}{{\not D}_p}{\Psi _q}\\ + \,i\kappa {J^{\overline m l}}{{\not D}_{\overline m }}\left. {{\Psi ^l}} \right)\end{array}\]

with the N = 1, D = 4 Yang–Mills action

    \[\begin{array}{c}{S_{SYM}} = \int\limits_{{M_4}} {{d^4}} x\sqrt g \frac{1}{4}{\rm{Tr}}\left( {{F_{\mu \nu }}} \right.{F^{\mu \nu }}\\ + {F_{\mu \nu }}{\overline F ^{\mu \nu }} + \overline \lambda {\gamma ^\mu }{{\not D}_\mu }\left. \lambda \right)\end{array}\]

getting the deep relation between N = 2 SYM theory, topological quantum field theory, and Kähler-theory:

And 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 Kahler manifold.