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

given

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

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

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

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

thus getting the crucial identity

Modulo gauge invariance, one gets two topological freedoms corresponding to the two components in ; moreover, in order to perform a metaplectic-gauge-fixing for the two-form and , one needs to inject two anti-commuting anti-ghosts and , and two Lagrange multipliers and

and

and given that holds, we have a Batalin-Vilkoviski (BV) system. Let me introduce BV anti-fields for and and their ghosts, antighosts and Lagrangian multipliers, with upper-let symbol labeling antifields. Note now that the antifield of a field with ghost number has ghost number and opposite statistics. Hence, for a -invariant BV action one has

Now, the property is equivalent to the Kähler-master equation

where indicate the derivatives from the left and from the right. So the BV action

implies the gauge-invariance of the semi-classical action as well as the nilpotency on all the fields, hence also entailing

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

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

we have

which implies that is eliminated in the path integral. And, after Gaussian-integration on , the gauge-fixed action becomes

Let us compare 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

even-odd , so that one can identify the topological ghost as a left-handed Weyl spinor and the topological anti–ghosts as a right-handed Weyl spinor and so the holonomy group of 4-dimensional Kahler manifold is locally given by

hence we can identify the forms and as and forms respectively. The Kähler-twist is now

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

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

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.

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