# How to rearrange the fermions in CohFT?

A simple question about notation of Moore Nekrasov and Shatashvili which makes me confused. Page 3, the authors rearranged the action into a novel form.

For D=3+1,5+1,9+1 respectively, the path integral is $$I_D=(\frac{\pi}{g})^{\frac{(N^2-1)(D-3)}{2}} \frac{1}{{\rm{Vol(G)}}}\int d^DXd^{\frac{D}{2}-1}\Psi e^{-S}$$ where $S=\frac{1}{g}(\frac{1}{4}\Sigma {\rm{Tr}}[X_\mu,X_\nu]^2+\frac{i}{2}\Sigma {\rm{Tr}}(\bar{\Psi}\Gamma^\mu[X_\mu,\Psi]))$

The authors rearrange the fields into the following form: $$\phi=X_D+iX_{D-1} \\ B_j=X_{2j-1}+iX_{2j} ~~~~~~(\rm{for ~~~j=1,2...D/2-1})\\ \Psi \rightarrow \Psi_a=(\psi_j,\psi_j^{\dagger}), \vec{\chi },\eta$$ $B_j$ are often denoted as $\mathbf{X}=\{X_a, a=1,....D-2\}$

My question is what the rearrangement of fermion field means? For D=4, a=2, j=1 ,$\chi$ has one component, before arrangement, we have a Dirac spinor $\Psi$, and what do we have after arrangement? are $\chi$ and $\eta$ Weyl spinors?

If we expand formula (2.4) using nilpotent symmetry (2.3), why no such terms like $\chi^{\dagger}[\bullet,\bullet]$?

Edit: From some related paper Kazakov Kostov and Nekrasov, the rearrangement is clear while there are some other puzzles. From KKN, first rewrite matrix model into complex fermion form, which is the formula (2.1) of KKN $$\frac{1}{\rm{Vol(G)}}\int dXd\lambda \exp(\frac{1}{2}\Sigma _{\mu<\nu}\rm{Tr}[X_\mu,X_\nu]^2+ )+\rm{Tr}\bar{\lambda}_\dot{\alpha}\sigma_\mu^{\alpha \dot{\alpha}}[X_\mu,\lambda_\alpha]$$ Two complex fermions $\lambda$ can be written as four real fermions $\chi$, $\eta$,$\psi_\alpha$, $\alpha=1,2$, $$\lambda_1=\frac{1}{2}(\eta-i\chi),\\ \lambda_2=\frac{1}{2}(\psi_1+\psi_2),\\ \bar{\lambda}_{\dot{a}} =\sigma_2^{a\dot{a}}\lambda_a^*,~~~s=[X_1,X_2]$$ Using the following nilpotent symmetry: $$\delta X_\alpha=\psi_\alpha,~~~\delta\psi_\alpha=[\phi,X_\alpha] ,\\ \delta\bar{\phi}=\eta,~~~\delta \eta [\phi,\bar{\phi}],\\ \delta\chi=H,~~~\delta H=[\phi,\chi],\\ \delta \phi=0$$

KKN claims that the action can be written as formula (2.5), $$S=\delta \left( -i\rm{Tr} \chi s -\frac{1}{2} \rm{Tr}\chi H - \Sigma_a\psi_a[X_a,\bar{\phi}] -\frac{1}{2} \eta[\phi,\bar{\phi}]\right)$$

I found that there is some inconsistence between formula (2.1) and formula (2.5). Look at fermionic part that consist $X_2$ in (2.5) which is proportional to $i\rm{Tr} \chi[\psi_1,X_2] + \rm{Tr}\psi_2[X_2,\eta]$. If we start from (2.1), fermionic part that consist $X_2$ should be proportional to $\bar{\lambda}_\dot{1}\sigma_2^{2\dot{1}}[X_2,\lambda_2] +\bar{\lambda}_\dot{2}\sigma_2^{1\dot{2}}[X_2,\lambda_1]$. Further using definition of $\bar{\lambda}_\dot{a}=\sigma_2^{a\dot{a}}\lambda_a^*$, we always get terms of form $\lambda_1^*[X_2,\lambda_1]$ and $\lambda_2^*[X_2,\lambda_2]$ which can not consist $i\rm{Tr} \chi[\psi_1,X_2] + \rm{Tr}\psi_2[X_2,\eta]$.

Anything wrong here? Some comments and references are welcoming.

• It's analogous to a decomposition of the spinor etc. under the $U(N)$ subgroup of $O(2N)$ - note that you paired the bosonic spacetime coordinates to complex coordinates. Try to figure out how the spinor decomposes in the three dimensionalities. – Luboš Motl Jul 14 '13 at 6:05
• I have revised the question. – thone Jul 14 '13 at 13:11