Ref. 1, page 15, equation (23) defines the $U(1)_V$ and $U(1)_A$ actions as $$e^{i\alpha F_V}: \Phi(x,\theta^{\pm},\bar{\theta}^{\pm}) \rightarrow e^{i\alpha q_V}: \Phi(x,e^{-i\alpha }\theta^{\pm},e^{i\alpha }\bar{\theta}^{\pm}) $$ The superfield can be written as $$\Phi(x,\theta^{\pm},\bar{\theta}^{\pm}) =x+\theta^+ \psi_+ +\theta^- \psi_- + \bar{\theta}^+ \bar{\psi}_+ + \bar{\theta}^- \bar{\psi}_- \ldots $$

The question is how to judge the $U(1)_V$ charge of $\psi_+$, $\psi_-$, $\bar{\psi}_+$ and $\bar{\psi}_-$ like table 2 in page 17. that the $U(1)_V$ charge of $\psi_\pm$ is -1 and the $U(1)_V$ charge of $\bar{\psi}_\pm$ is +1.


  1. A. Klemm, Introduction to topological string theory on Calabi-Yau manifolds, lecture notes, 2005. The pdf file is available here.

1 Answer 1


First let me refer you to the following review by Marcel Vonk containing a more detailed description of the world sheet N = [2, 2] supersymmetry in section 5.1.

The R-symmetry is a transformation acting on the spinor but not on the scalar components of the chiral superfields. It can also be assigned to the target manifold by setting arbitrary charges to the chiral superfields and their negatives to the corresponding complex conjugates.

The R-symmetry is a symmetry of the classical N = [2, 2] , theory, however, the axial R-symmetry is anomalous thus cannot be gauged.

The action of the R-symmetry is generated by the unit operator for the vector transformation and the chirality operator for the axial transformation.

In two dimensions one can choose the chirality operator to be diagonal (i.e., the Pauli matrix $\sigma_3$), please, see for example the following lecture notes by Rhys Davies. In this basis the Dirac operator becomes also diagonal (its components in flat space are just the holomorphic and antiholomorphic derivatives), and the Dirac Lagrangian decomposes to two independent Weyl components left and right moving.

The vector transformation does not depend on the chirality thus can be taken as $-1$ on both the right and left movers. Since the chirality operator is the Pauli matrix $\sigma_3$, the axial transformation can be taken as $+1$ on the left mover and $-1$ on the right mover. Complex conjugation reverses the charges of all vectors.


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