### Classical deformations of 2d SCFTs - Part II: Algebra Isomorphisms

#### Posted by Urs Schreiber

It has been shown in Part I (see also hep-th/0401175) that the modes of the $K$-deformed exterior derivative ${d}_{K,\xi}$ on loop space together with their adjoints ${d}_{K,\xi}^{+}$ generate the classical super Virasoro algebra. In the following deformations of ${d}_{K,\xi}$ are studied under which the form of the superconformal algebra is preserved. The new algebra representations obtained this way are identified as corresponding to the massless NS and NS-NS background fields. A further 2-form background is found and T-duality is studied for all these algebras.

*(2.2) Isomorphisms of the superconformal algebra*

From section 2.1.2 of [3] it follows that the general continuous isomorphism of the 0-mode sector ($\xi =1$) of the superconformal algebra is induced by some operator

where $W$ is an operator on loop space of unit reparametrization weight, and looks like

This construction obviously immediately generalizes to the full algebra including all modes

*if*
the crucial relation

remains well defined (i.e. if the modes combine multiplicatively on the left hand side.)

Every operator $W$ which satisfies these conditions therefore induces a classical algebra isomorphism of the superconformal algebra.
However, two different $W$ need
not induce two different isomorphisms.
In particular, *anti-hermitean* ${W}^{\u2020}=-W$ induce
*pure gauge* transformations in the sense that all algebra elements are
transformed by the *same* unitary similarity transformation. Examples for such unitary transformations are given below. They are related to background gauge transformations as well as
to string dualities.
For a detailed discussion of the role of such automorphism in the
general framework
of string duality symmetries see section 7 of
[8].

*2.3 Gravitational background by means of algebra isomorphisms*

First we reconsider the purely gravitational background from the point of view that
its superconformal algebra derives from
the superconformal algebra for *flat* cartesian target space by a deformation
of the above form. For the point particle
limit this was discussed in equations (38)-(42) of [3] and the generalization to loop space is straightforward:

Denote by

the $K$-deformed exterior derivative on *flat* loop space and define the deformation operator by

where $\mathrm{ln}E$ is the logarithm of a vielbein on loop space, regarded as a matrix. This $W$ is constructed so as to satisfy

which yields

It is because of the fact that ${e}^{W}$ interchanges between two different vielbein fields which define two different metric tensors that the index structure becomes a little awkward in the above equations. Since we won’t need these transformations for the further developments we don’t bother to introduce special notation to deal with this issue more cleanly. The point here is just to indicate that a $W$ with the above properties does exist. ${e}^{W}$ transforms all $p$-forms with respect to $E$ to $p$-forms with respect to the flat metric and hence

so that, indeed, this $W$ induces a gravitational field on the target space.

As was discussed on p. 10 of [3] we need to require $\mathrm{det}e=1$, and hence

in order that ${d}_{K,\xi}^{\u2020W}=({d}_{K,{\xi}^{*}}^{W}{)}^{\u2020}$. This is just a condition
on the nature of the coordinate system with respect to which the metric is constructed by the above deformation. Also, recall that, while as an abstract operator ${d}_{K,\xi}$ is of course
*independent* of any metric, its representation in terms of the operators
${X}^{(\mu ,\sigma )},{\partial}_{(\mu ,\sigma )},{\mathcal{E}}^{\u2020\mu},{\mathcal{E}}^{\mu}$ is not, which is what the
above is all about.

Note furthermore, that

According to the discussion in section (2.2)
this implies that the antisymmetric part of $\mathrm{ln}e$ generates a pure gauge transformation
and *only the (traceless) symmetric part* of $\mathrm{ln}e$ is responsible for a perturbation of the gravitational
background. A little reflection shows that the gauge transformation induced by
antisymmetric $\mathrm{ln}e$ is a rotation of the vielbein frame. For further discussion of this
point see pp. 58 of [9].

*(2.4) B-field background*

As in section 2.1.3 of [3] we now consider the Kalb-Ramond $B$-field 2-form

on target space with field strength $H=\mathrm{dB}$. This induces on loop space the 2-form

We will study the deformation operator

on loop space (which is manifestly of reparametrization weight 1) and show that the superconformal algebra that it induces is indeed that found by a canonical treatment of the usual supersymmetric $\sigma $-model with gravitational and Kalb-Ramond background.

When calculating the deformation induced by this $W$ one finds

Supercommuting this with its adjoint shows that the consistency condition is satisfied, i.e. the modes of the deformed Laplace-Beltrami operator are well defined.

With hindsight this is no surprise, because the above are precisely the superconformal generators in functional form as found by canonical analysis of the non-linear supersymmetric $\sigma $-model

where ${X}^{\mu}$ are worldsheet superfields. The calculation can be found in section 2 of [10] . (See eqs. (32), (33).)

*(2.5) Dilaton background*

The deformation operator which induces
the gravitational background
was of the form $W={\mathcal{E}}^{\u2020}\cdot M\cdot \mathcal{E}$ with $M$ a traceless
symmetric matrix. If instead we consider a deformation of the same form but for *pure trace* we get

for some real scalar field $\Phi $ on target space. This should therefore induce a dilaton background.

The associated superconformal generators are (we suppress the $\sigma $ dependence and the mode functions $\xi $ from now on)

and their adjoints. It is readily seen that for this deformation the consistency condition is satisfied, so that these operators indeed generate a superconformal algebra.

Comparison of the superpartners of ${\Gamma}_{\pm ,\mu}$

with the bosonic currents obtained from the Born-Infeld action

shows that this has the form expected for the dilaton coupling of a D-string.

* (2.6) Gauge field background *

A gauge field background $A={A}_{\mu}{\mathrm{dx}}^{\mu}$ should express itself via $B\to B+\frac{1}{T}F$, where $F=\mathrm{dA}$, if we assume $A$ to be a $U(1)$ connection for the moment. Since the present discussion so far refers only to closed strings and since closed strings have trivial coupling to $A$ it is to be expected that an $A$-field background manifests itself as a pure gauge transformation in the present context. This motivates to investigate the deformation induced by the anti-hermitean

The associated superconformal generators are found to be

and the respective adjoints. Comparison with the form of the generators found for a B-field background shows that indeed

so that we can identify the background induced by the above $W$ with that of the NS $U(1)$ gauge field. Since ${e}^{W}(X)$ is nothing but the Wilson loop of $A$ around $X$, it is natural to conjecture that for a general (non-abelian) gauge field background $A$ the corresponding deformation is the Wilson loop as well:

where $\mathcal{P}$ indicates path ordering and $\mathrm{Tr}$ the trace in the Lie algebra, as usual.

* (2.8) $C$-field background*

So far we have found deformation operators for all massless NS and NS-NS background fields. One notes a close similarity between the form of these deformation operators and the form of the corresponding vertex operators: The deformation operators for $G$, $B$ and $\Phi $ are bilinear in the form creation/annihilation operators on loop space, with the bilinear form (matrix) seperated into its traceless symmetric, antisymmetric and trace part.

Interestingly, though, there is one more deformation operator obtainable by such a bilinear in the form creation/annihilation operators. It is

It induces the generators

It turns out that these generators, too, respect the consistency condition on the modes of the deformed Laplace-Beltrami operator, so this yields yet another superconformal algebra.

What, though, is the physical interpretation of the field $C$ on spacetime?
It is apparently not the NS 2-form field, because the generators are different and don’t seem to be unitarily equivalent.
A possible guess would therefore be that it is the *RR 2-form* ${C}_{2}$.
The corresponding SCFT should therefore describe a D1-string instead of an
F-string.
This needs to be further examined. A further hint in this direction is that
under a duality transformation which changes the sign of the dilaton the
$C$-field changes roles with the $B$-field. This, and other relations among the classical SCFTs constructed here, is discussed in
Part III.

## Re: Classical deformations of 2d SCFTs - Part II: Algebra Isomorphisms

Hi Urs!

Trying to follow along is hurting my poor brain, but I’m definitely trying! :)

I’m presently attempting to interpret these deformations physically. One thought I’m mulling over is what if instead of deforming the exterior derivative and its adjoint, if you deformed the states they are acting on? Kind of like Schrodinger vs Heisenberg representations.

Then the deformed operators

on these states would seem to be the same as the undeformed operators acting on undeformed states (kind of) :)

It seems like it may be an equivalent way to view the same thing.

Yes, someone is reading your blog entries! :)

Best regards,

Eric