New fields on super Riemann surfaces

doi:10.1088/0264-9381/12/10/018

IOPscience

Classical and Quantum Gravity

Quick search Find article

Quick search

All journals This journal only

Find article
Volume number: Issue number (if known): Article or page number:

Classical and Quantum Gravity Volume 12 Number 10 Create an alert RSS this journal

A Rogers and M Langer 1995 Class. Quantum Grav. 12 2619 doi:10.1088/0264-9381/12/10/018

New fields on super Riemann surfaces

FREE ARTICLE

A Rogers and M Langer

Tag this article Full text PDF (49 KB)

Abstract Cited By

CORRIGENDUM

This is a Corrigendum for the article 1994 Class. Quantum Grav. 11 2619

It has been pointed out by Rabin [1] that theorem 1 of our paper contains an error. This corrigendum corrects theorem 1. The main conclusion of the paper, that there exist function-like objects on super Riemann surfaces with the desired dimensionality properties, survives.

Our paper involves the notion of superconformal function on a super Riemann surface M, which is there defined to be a cross-section of a certain (1,1)-dimensional vector bundle E_m on M which is holomorphic and whose local representatives (in the coordinate system labelled α) have components $(g_{\alpha:}\gamma_{\alpha})$ where

Equation

The main theorem is:

Theorem 1. The space $\cal{SC}(M)$ of superconformal functions on the super Riemann surface M has the structure

Equation

(Here B is the Grassmann algebra on which the super Riemann surface is modelled.)

For theorem 1 of to be valid, it is necessary to put an extra condition on a superconformal function: there must exist at all points on the super Riemann surface coordinate systems in which the components $(g_{\alpha:}\gamma_{\alpha})$ take the simpler form

Equation

This extra condition precludes the existence of superconformal functions of the form Equation , which spoil the vector space structure of the space of superconformal functions. (They also are contrary to the whole motivation for the notion of superconformal function, which is to validate in a direct manner the holomorphic quantization of the spinning string.)

The strategy of the proof of the theorem is to explicitly construct an isomorphism Φ between the desired super vector space and the space of superconformal functions. The map Φ in the proof of theorem 1 given in our paper is in fact not well-defined, because the quantity $r_{\alpha[ij]}'$ is not uniquely determined by equation (26), but only up to an element $s_\alpha$ of $H^0(M,K)$ ; additionally, it is $H^2(M_{[\emptyset]:\cal{O})$ (rather than $H^2(M_{[\emptyset]:\Bbb{C})$ ) which is trivial. Thus $K_{\alpha\beta ijk}$ is in $C^1(M,\cal{O})$$ and so its derivatives may not be zero; however, since $H^2(M_{[\emptyset]:\Bbb{C})$ is one-dimensional, the non-constant part of $K_{\alpha\beta ijk}$ can be absorbed by the freedom in r'. To see this, note that the change $\delta A_{\alpha\beta ijk}$ in $A_{\alpha\beta ijk}$ corresponding to a change $s_\alpha$ in $r_\alpha$ satisfies the cocycle condition

Equation

so that the integral of $X_{\alpha\beta\gamma}$$ is in $H^2(M_{[\emptyset]:\Bbb{C})$ . Since this space is one-dimensional, $s_\alpha$ can be chosen so that the corresponding $K_{\alpha\beta}$ are constant. This condition uniquely determines s up to a cross-section of E_M of the form $(0,\zeta_\alpha t'(z_\alpha))$ , which is precluded by the extra condition included in the definition of superconformal bundle, so that with this extra condition theorem 1 is valid.