1
$\begingroup$

I'm trying to couple matter to $\mathcal{N}=(0,2)$ SYM in 2d using superfield formalism. There are some paper (this on Sec. 6, or this on Sec. 3 [whose notation will be used here]) that construct what I need for gauge group $U(1)$. The major difficulty I'm facing with is to define the super gauge transformations: in the abelian case one has

$$ \Psi \rightarrow \Psi + i(\Lambda - \bar{\Lambda})\;, \qquad\qquad\qquad V \rightarrow V +\partial_-(\Lambda + \bar{\Lambda})\;, $$

where $\Lambda$ is a chiral superfield. These give the correct transformation for fields. The matter lagrangian density is easy:

$$ \bar{\Phi}e^{-\Psi}(\partial_- -V)e^{-\Psi}\Phi\;,\tag{1} $$

where I used $\bar{D}_-\Phi=0$ so that $\bar{\mathcal{D}}_-(e^{-\Psi}\Phi)=0$. Taking for $U(1)$ gauge transformation (see also this reference, which has some typos)

$$ \Phi \rightarrow e^{-2i\bar{\Lambda}}\Phi\;, \qquad\qquad\qquad \bar{\Phi} \rightarrow \bar{\Phi}e^{2i\Lambda}\;, $$

it is easy to check the gauge invariance of $(1)$:

$$ (\partial_- -V) = e^{i(\Lambda+\bar{\Lambda})}(\partial_--V')e^{-i(\Lambda+\bar{\Lambda})}\;,\tag{2} $$

which defines $V$ as a (super-)connection.

QUESTION

How these results can be generalized for non-Abelian gauge group? Is there any reference?

ATTEMPT

I tried to generalize the gauge transformation for a non-abelian group (which is important, once fixed WZ gauge, to find the compensating transformation to ripristinate it, after a SUSY variation). As far as we remain in WZ gauge (in this case $V^2=0$) it seems sensible to me to take

$$ e^{-\Psi} \rightarrow e^{i\bar{\Lambda}}e^{-\Psi}e^{-i\Lambda}\;, $$

which, at infinitesimal level, implies thanks to BCH formula

$$ \Psi \rightarrow + i(\Lambda - \bar{\Lambda}) + \tfrac{i}{2}[\Psi,\Lambda+\bar{\Lambda}] + o(\Lambda,\bar{\Lambda})\;,\tag{3} $$

also I was thinking that

$$ V \rightarrow V + \partial_-(\Lambda + \bar{\Lambda}) +\tfrac{i}{1}[V,\Lambda+\bar{\Lambda}]\;.\tag{4} $$

In fact $(3)$ and $(4)$, give the correct transformation of the component field in the WZ gauge. However $(2)$ becomes

$$ (\partial_- - V)=e^{\Psi}e^{-i\bar{\Lambda}}e^{-\Psi}e^{-i\Lambda}(\partial_--V')e^{i\bar{\Lambda}}e^{-\Psi}e^{i\Lambda}e^{\Psi}\;, $$

which does not define any (super-)connection (the r.h.s. contains a non-unitary transformation). The fact that surprises me is that le lagrangian $(1)$ written in components (which is written in the paper I linked) is SYM even for gauge group other than $U(1)$.

$\endgroup$
  • $\begingroup$ Related: physics.stackexchange.com/q/164054/2451 $\endgroup$ – Qmechanic Jul 2 '18 at 3:10
  • $\begingroup$ I knew what you linked me. The point is that $(1)$ does not seem to be super-gauge invariant. Any suggestion? $\endgroup$ – MaPo Jul 2 '18 at 7:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.