Quantum Holomorphy, Kähler Manifolds and SuperString Theory

By George Shiber Thursday, February 18, 2016Saturday, August 17, 2019 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry holonomy, Kähler Manifolds

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

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

Note first that from the supersymmetry algebra, one can derive that the irreducible quotient ${V_0}$ is isomorphic to the Fock space of the fermionic operators

$\left\{ {\begin{array}{*{20}{c}}{\Psi = \frac{1}{2}\left( {{\Psi _1} + iw{\Psi _2}} \right)}\\{{\Psi ^\dagger } = \frac{1}{2}\left( {{\Psi ^\dagger } - iw\Psi _2^\dagger } \right)}\end{array}} \right.$

And hence, by unitary, we see that half of the supersymmetry is broken, with Kac-Moody-spanners

$\left\{ {\begin{array}{*{20}{c}}{\left( {\Psi _1^\dagger + iw\Psi _2^\dagger } \right)\left| 0 \right\rangle }\\{{\Psi _1}\Psi _2^\dagger \left| 0 \right\rangle }\end{array}} \right.$

The BRST operator defining a topological symmetry is a scalar operator which can be defined in any curved space, while supersymmetry is an extremely delicate concept in curved space. Hence, topological symmetry is a more fundamental concept than Poincaré supersymmetry. Yet, to perform the twisting-operation that relates Poincaré supersymmetry and topological symmetry, you need to use manifolds with special holonomy.
The standard construction of a topological quantum field theory leads to models with N = 2 supersymmetry

to see that, consider topological Yang–Mills theory in four and eight dimensions. The crucial and relevant BRST transformations are

$\left\{ {\begin{array}{*{20}{c}}{\delta {A_\mu } = {\Psi _\mu } + {{\not D}_\mu }c}\\{\delta c = - \left[ {c,\Phi } \right]}\end{array}} \right.$

and

$\left\{ {\begin{array}{*{20}{c}}{\delta \,{\Psi _\mu } = \not D\Phi - \left[ {c,{\Psi _\mu }} \right]}\\{\delta \,\Phi = - \left[ {c,\Phi } \right]}\end{array}} \right.$

together representing the geometrical identity

$\begin{array}{c}\left( {\delta + d} \right)\left( {A + c} \right) + \frac{1}{2}\left[ {A + c,A + c} \right]\\ = F + \Psi + \Phi \end{array}$

The antighosts are an anti-commuting anti-self dual 2-form ${\kappa _{\mu \nu }}$ and an anticommuting scalar $\eta$ , and for each, there is an associated Lagrange multiplier field, and their BRST equations are

$\left\{ {\begin{array}{*{20}{c}}{{\delta _{{\kappa _{\mu \nu }}}} = {b_{\mu \nu }} - \left[ {c,{\kappa _{\mu \nu }}} \right]}\\{\delta \overline \Phi = \eta - \left[ {c,\overline \Phi } \right]}\end{array}} \right.$

and

$\left\{ {\begin{array}{*{20}{c}}{\delta {b_{\mu \nu }} = - \left[ {c,{b_{\mu \nu }}} \right]}\\{\delta \eta = \left[ {c,\eta } \right]}\end{array}} \right.$

Thus, the twist operation – which has different geometrical interpretation in 4 and 8 dimensions – is a mapping from these ghost and antighost fermionic degrees of freedom on a pair of spinors, which leads one to reconstruct the spinor-spectrum of N = 2 supersymmetry, both in 4 and 8 dimensions. Using self-duality equations as gauge-functions allows us to build a $\delta$ -exact action providing twisted supersymmetric theories with a $\delta$ -exact energy-momentum tensor. The cohomology of the $\delta$ –symmetry fully determines a ring of topological ‘observables’, a subsector of the familiar set of ‘observables’ for the gauge particles. The latter is selected from the cohomology of the ordinary gauge invariance. So, from

$\left\{ {\begin{array}{*{20}{c}}{\delta {A_\mu } = {\Psi _\mu } + {{\not D}_\mu }c}\\{\delta c = - \left[ {c,\Phi } \right]}\end{array}} \right.$

and

$\left\{ {\begin{array}{*{20}{c}}{\delta \,{\Psi _\mu } = \not D\Phi - \left[ {c,{\Psi _\mu }} \right]}\\{\delta \,\Phi = - \left[ {c,\Phi } \right]}\end{array}} \right.$

as well as

and

$\left\{ {\begin{array}{*{20}{c}}{\delta {b_{\mu \nu }} = - \left[ {c,{b_{\mu \nu }}} \right]}\\{\delta \eta = \left[ {c,\eta } \right]}\end{array}} \right.$

It follows that in these TQFT one has twice as many fermionic degrees of freedom than bosonic ones, and to solve this conundrum, one works in models with a milder topological symmetry, the best being the Chern–Simons type model, characterized by metric-independent classical actions, leading us to models that are twisted N = 1 supersymmetric theories. In this series of posts, I will work with the following holomorphic BF action

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

definable on any complex manifold $M$ of complex dimension $n$ and we can define a theory that is classically invariant under the following topological symmetry, which is localized in the holomorphic sector

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

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

One can recover such a heterotic symmetry as a symmetry associated to the action

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

and counting the ghost degrees of freedom, we have 2-components for ${\Psi _m}$ and one for $c$ , and then the number of fermion degrees of freedom will fit with those of a single Majorana spinor and we can reach N = 1 supersymmetry (for the next post).

Let me now construct the action for a BF system on a Kähler manifold. On such a manifold, one can define a complex structure

$\left\{ {\begin{array}{*{20}{c}}{{J^{mn}} = 0}\\{{J^{m\overline n }} - i{g^{m\overline n }}}\\{{J^{\overline m \overline n }} = 0}\end{array}} \right.$

allowing the introduction of complex coordinates ${z^m}$ and ${z^{\overline m }}$ and $1 \le m$ , $\overline m \le N$ in $2N$ -dimensions by

$\left\{ {\begin{array}{*{20}{c}}{J_n^m{z^n} = i{z^m}}\\{J_{\overline n }^{\overline m }{z^{\overline n }} = - i{x^{\overline m }}}\end{array}} \right.$

So, in four dimensions, the action

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

becomes

where

$F = dA + A \wedge A$

is the curvature of the Yang–Mills field

$A$

The equations of motion are

$\left\{ {\begin{array}{*{20}{c}}{{F_{\overline m \overline n }} = 0}\\{{\varepsilon ^{mn\overline m \overline n }}{{\not D}_{\overline n }}{B_{mn}} = 0}\end{array}} \right.$

Classically, ${A_m}$ is undetermined, ${A_{\overline m }}$ is a pure gauge and ${B_{mn}}$ is holomorphic. Altogether, there are two gauge invariant degrees of freedom that are not specified at the classical level. Modulo gauge invariance, there is a mixed propagation between $A$ and $B$ . The symmetries of the action

become

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

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

where the charged 2-form can be interpreted as a Hodge dual to the Yang–Mills field. In the next post, part 2 of this series, I will explain the BRST quantization of the action