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

December 30, 2003

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

Posted by Urs Schreiber

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It has been shown in Part I (see also hep-th/0401175) that the modes of the $K$ -deformed exterior derivative $d_{K, ξ}$ on loop space together with their adjoints $d_{K, ξ}^{+}$ generate the classical super Virasoro algebra. In the following deformations of $d_{K, ξ}$ 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.

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(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 ( $ξ = 1$ ) of the superconformal algebra is induced by some operator

(1)

W = \int d σ W (σ),

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

(2)

d_{K,1} \mapsto d_{K,1}^{W} : = e^{- W} d_{K,1} e^{W}

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

(3)

d_{K, ξ} \mapsto d_{K, ξ}^{W} : = e^{- W} d_{K, ξ} e^{W}

if the crucial relation

(4)

Δ_{K, ξ_{1} ξ_{2}}^{W} = {d_{K, ξ_{1}}^{W}, (d^{W})_{K, ξ_{2}}^{†}}

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 $W^{†} = - W$ 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

(5)

d_{K,1}^{η} : = ℰ^{† (μ, σ)} \partial_{(μ, σ)} + i ℰ_{(μ, σ)} X^{' (μ, σ)}

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

(6)

W = ℰ^{†} \cdot (\ln E) \cdot ℰ .

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

(7)

e^{W} ℰ^{† a} e^{- W} = \sum_{ν} e^{a}_{ν} ℰ^{† (b = ν)},

which yields

(8)

e^{W} ℰ^{† μ} e^{- W} = ℰ^{† (b = μ)} .

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

(9)

d_{K, ξ} = e^{- W} d_{K, ξ}^{η} e^{W},

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 $\det e = 1$ , and hence

(10)

tr \ln e = 0

in order that $d_{K, ξ}^{† W} = (d_{K, ξ^{*}}^{W})^{†}$ . 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 $d_{K, ξ}$ is of course independent of any metric, its representation in terms of the operators $X^{(μ, σ)}, \partial_{(μ, σ)}, ℰ^{† μ}, ℰ^{μ}$ is not, which is what the above is all about.

Note furthermore, that

(11)

W^{†} = \pm W \Leftrightarrow (\ln e)^{T} = \pm \ln e .

According to the discussion in section (2.2) this implies that the antisymmetric part of $\ln e$ generates a pure gauge transformation and only the (traceless) symmetric part of $\ln e$ is responsible for a perturbation of the gravitational background. A little reflection shows that the gauge transformation induced by antisymmetric $\ln e$ 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

(12)

B = \frac{1}{2} B_{μ ν} {dx}^{μ} \land {dx}^{ν}

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

(13)

B_{(μ, σ) (ν, σ^{'})} (X) = B_{μ ν} (X (σ)) δ_{σ, σ^{'}} .

We will study the deformation operator

(14)

W^{(B)} (X) : = \frac{1}{2} B_{(μ, σ) (ν, σ^{'})} (X) ℰ^{† (μ, σ)} ℰ^{† (ν, σ^{'})}

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 $σ$ -model with gravitational and Kalb-Ramond background.

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

(15)

d_{K, ξ}^{(B)} = \int d σ ξ (ℰ^{† μ} {\hat{\nabla}}_{μ} + i T ℰ_{μ} X^{' μ} + \frac{1}{6} H_{α β γ} ℰ^{† α} ℰ^{† β} ℰ^{† γ} - iT ℰ^{† μ} B_{μ ν} X^{' ν}) .

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 $σ$ -model

(16)

S = \frac{T}{2} \int d^{2} ξ d^{2} θ (G_{μ ν} + B_{μ ν}) D_{+} X^{μ} D_{-} X^{ν},

where $X^{μ}$ 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 $W = ℰ^{†} \cdot M \cdot ℰ$ with $M$ a traceless symmetric matrix. If instead we consider a deformation of the same form but for pure trace we get

(17)

W^{(D)} = - \frac{1}{2} \int d σ Φ (X) ℰ^{† μ} ℰ_{μ},

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

The associated superconformal generators are (we suppress the $σ$ dependence and the mode functions $ξ$ from now on)

(18)

d_{K}^{(Φ)} = e^{Φ / 2} ℰ^{† μ} ({\hat{\nabla}}_{μ} - \frac{1}{2} (\partial_{μ} Φ) ℰ^{† ν} ℰ_{ν}) + iT e^{- Φ / 2} X^{' μ} ℰ_{μ}

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 $Γ_{\pm, μ}$

(19)

\mp \frac{1}{2} {d_{K}^{(Φ)} \pm d_{K}^{† (Φ)}, Γ_{\mp, μ}} = e^{Φ / 2} \partial_{μ} \mp iT e^{- Φ / 2} G_{μ ν} X^{' μ} + fermionic terms

with the bosonic currents obtained from the Born-Infeld action

(20)

S = - T \int e^{- Φ} \sqrt{\det G}

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

(2.6) Gauge field background

A gauge field background $A = A_{μ} {dx}^{μ}$ should express itself via $B \to B + \frac{1}{T} F$ , where $F = dA$ , 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

(21)

W = - i A_{(μ, σ)} (X) X^{' (μ, σ)} .

The associated superconformal generators are found to be

(22)

d_{K}^{(A) (B)} = d_{K}^{(B)} - i ℰ^{+ μ} F_{μ ν} X^{' ν}

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

(23)

d_{K}^{(A) (B)} = d_{K}^{(B + \frac{1}{T} F)},

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:

(24)

d_{K}^{(A)} = (Tr 𝒫 e^{i \int A_{μ} X^{' μ}}) d_{K} (Tr 𝒫 e^{- i \int A_{μ} X^{' μ}}),

where $𝒫$ indicates path ordering and $Tr$ 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 $Φ$ 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

(25)

W^{(C)} : = \frac{1}{2} \int d σ C_{μ ν} (X) ℰ^{μ} ℰ^{ν} .

It induces the generators

(26)

d_{K, ξ}^{(C)} = \int d σ ξ (ℰ^{† μ} {\hat{\nabla}}_{μ} + i ℰ_{μ} X^{' μ} - ℰ^{ν} C_{ν}^{μ} {\hat{\nabla}}_{μ} + \frac{1}{2} ℰ^{† α} ℰ^{μ} ℰ^{ν} (\nabla_{α} C_{μ ν}) - \frac{1}{2} C_{ν}^{μ} ℰ^{ν} ℰ^{α} ℰ^{β} (\nabla_{μ} C_{α β}) + \frac{1}{2} C^{α}_{β} ℰ^{β} ℰ^{μ} ℰ^{ν} (\nabla_{α} C_{μ ν})) .

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.

Posted at December 30, 2003 1:17 PM UTC

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

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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.

(1)

∣ ψ (e) 〉 = e^{- e W} ∣ ψ (0) 〉

Then the deformed operators

(2)

d_{e} = e^{- e W} d e^{e W}

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

(3)

d_{e} ∣ ψ (e) 〉 = e^{- eW} d ∣ psi (0) 〉 .

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

Yes, someone is reading your blog entries! :)

Best regards,
Eric

Posted by: Eric on December 31, 2003 1:56 AM | Permalink | Reply to this

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

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Hi Eric!

You wrote:

I`m presently attempting to interpret these deformations physically.

Yes, I think that is a crucial question. As I have discussed, by using appropritate deformation operators $W$ one can obtain functional representations of various superconformal algebras. Except for the obvious cases an important question will always be: What target space backgrounds, what target space physics, do these deformations describe?

Of course this is a familiar question: CFTs don’t necessarily come with a Lagrangian, and given some CFT one may always ask if it comes from some sigma-model.

For instance I should try to figure out if there is a supersymmetric Lagrangian that reproduces the superconformal constraints which I obtained by deforming with

(1)

W = C_{(μ, σ) (ν, σ^{'})} ℰ^{(μ, σ)} ℰ^{(ν, σ^{'})} .

Having this Lagrangian would help clarify the nature of the field that I called $C_{μ ν}$ , I’d hope. (But there should be other ways, too, to clarify this.)

But maybe your question is more general: What do these deformations mean physically in general? The answer is that they induce certain potentials on target space.

In its most elementary form this is exhibited by the deformation of the $D = 1$ , $N = 2$ theory (simple supersymmetric quantum mechanics of a point particle) originally introduced by Witten in his work on Morse theory:

(2)

d \to d^{W} : = e^{- W} d e^{W} .

When you compute the deformed Laplace-Beltrami operator $H = {d^{W}, (d^{W})^{+}}$ you see that it looks like

(3)

H = {d, d^{+}} + (grad W)^{2} + fermionic terms .

The first term is just the kinetic term. The second term appears due to the deformation and it is just an ordinary scalar potential. Depending on the state that this Hamiltonian is acted on there are further contributions to the potential from the last term.

So a scalar deformation yields a scalar potential. You could now replace the scalar function $W$ with something else, for instance a 2-form $B$ . Calculating the deformation of $d$ , $d^{+}$ and ${d, d^{+}}$ as above shows that the resulting theory is that describing a point particle in a background with torsion. In fact this is the point particle limit of a string in a Kalb-Ramond field background.

So in general the deformations should create fields in spacetime that interact with the point/string which is described by the operators that are being deformed.

It is well known how such background field interaction is described in terms of vertex operators in the path integral formulation. There should therefore be a close relation between the deformation operators $W$ that I have listed and the corresponding vertex operators. As I have mentioned, one indeed sees many similarities. In particular the scalar coefficients that enter the $W$ s for gravitational, Kalb-Ramond, dilaton, and gauge field backgrounds are precisely those that enter the corresponding vertex operators. But the remianing operator structure is a little different, though not totally unrelated. Closed string backgrounds are given by deformation operators that are bilinears in the loop space form creators/annihilators, while the open string gauge field background is given by something poportional to $X^{'}$ (the tangent to the loop).

It should also be possibe to relate this to the technique of marginal deformations of CFTs, somehow.

You furthermore wrote:

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.

(4) $∣ ψ_{e} 〉 = \exp (- eW) ∣ ψ_{0} 〉$

Then the deformed operators

(5) $d_{e} = \exp (- eW) d \exp (eW)$

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

(6) $d_{e} ∣ ψ_{e} 〉 = \exp (- eW) d ∣ {psi}_{0} 〉 .$

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

Yes, definitely. Note however, that what you write in general only holds for $d_{e}$ , not for $d_{e}^{+}$ , or vice versa. A very crucial point of this whole deformation business is that unless $W$ is anti-hermitean the operators $d$ and $d^{+}$ are not subject to the same similarity transformation. That is because

(7)

d_{e} = e^{- e W} d e^{eW}

but

(8)

d_{e}^{+} = e^{e W^{+}} d^{+} e^{- e W^{+}},

which again is due to the requirement

(9)

(d_{e})^{+} = d_{e}^{+},

that is necessary to ensure that the Hamiltonian is self-adjoint. (Here $(\cdot)^{+}$ is of course supposed to be the adjoint of $(\cdot)$ . I do not know how to produce the TeX \dag with itex.)

For non anti-hermitean $W$ you can choose states in a way that cancels the deformation on $d_{e}$ , as you noted:

(10)

d_{e} ∣ ψ_{e} 〉 = \exp (- eW) d ∣ {psi}_{0} 〉 .

But for these same states there will be no such cancellation for $d_{e}^{+}$ :

(11)

d_{e}^{+} ∣ ψ_{e} 〉 = \exp ({eW}^{+}) d^{+} e^{- e W^{+}} e^{- eW} ∣ {psi}_{0} 〉

unless it happens that $W$ is anti-hermitean $W^{+} = - W$ .

Therefore the deformations where $W$ is anti-hermitean are special. They correspond to unitary transformations which affect all operators of the constraint algebra in the same way. Indeed the fact that a unitary transformation of the operators can be completely canceled by a unitary transformation of the states, the way that you have indicated, shows that these deformation don’t produce new phyics, but lead to pure gauge transformations in a generalized sense.

In my discussion I had given two mildly interesting examples of such gauge tranformations. One is the deformation

(12)

W = ℰ^{+} \cdot M \cdot ℰ

for $M$ a real antisymmetric matrix $M^{T} = - M$ . This deformation induces Lorentz rotations on the vielbein field which of course leave the metric unaffacted and hence have no effect on physics. This is the gauge freedom of local Lorentz rotations.

Another example is the deformation which creates an NS gauge field on spacetime

(13)

W = - i A_{(μ, σ)} X^{' (μ, σ)} .

Obviously this $W$ is anti-hermitean. The reason is that the closed string (and all what I said applies to the closed string only so far) does not couple to the $A_{μ}$ field. So turning on this field on spacetime must not change the physics of closed strings propagating in this background - and indeed it does not. When moving this deformation from the operators to the states one sees that the deformation induces a pure phase shift on the states, which is not observable.

Posted by: Urs Schreiber on December 31, 2003 3:21 PM | Permalink | Reply to this

dagger

I do not know how to produce the TeX \dag with itex

Hmmm. That could be a good symbol to add to itex.

If you can’t find the symbol you want in the WebTeX manual, you can always use the corresponding XHTML or numeric entity (in this case, &dagger; or †).

Posted by: Jacques Distler on December 31, 2003 3:55 PM | Permalink | Reply to this

MathML entities

Oh yeah. Here’s the complete list of MathML Named Entities. Theoretically, each of these should have an itex equivalent. Alas, only a fraction of them are currently implemented. And some of them are probably implemented incorrectly (i.e., the TeX symbol gets mapped to the wrong, or even to a nonexistent, MathML entity).

There is much debugging to be done in itex, so please let me know when you find missing, or incorrect symbols. I’ll try to correct them, as time permits.

Posted by: Jacques Distler on December 31, 2003 4:07 PM | Permalink | Reply to this

Re: MathML entities

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Ok, I have replaced the $(\cdot)^{+}$ with $(\cdot)^{†}$ using &dagger; in the blog entry.

The only other thing that I have noted so far is that \; is not accepted, only \, is.

I have to run now, the party is about to start!! :-)

Have a happy new year!

Posted by: Urs Schreiber on December 31, 2003 4:45 PM | Permalink | Reply to this

Re: MathML entities

Ok, I have replaced the (•)⁺ with (•)^† using &dagger; in the blog entry.

I’ve added \dagger → &dagger; and \ddagger → &ddagger; to the itex2MML conversions.

The only other thing that I have noted so far is that \; is not accepted, only \, is.

The WebTeX manual on spaces lists what’s available in that regard. Personally, I feel we’re so far from pixel-perfect rendering of MathML, that I’m not about to spend alot of time tweaking the spacing in my formulas (or the itex2MML support for others to tweak their formulas).

I have to run now, the party is about to start!! :-)

Happy New Year!

Posted by: Jacques Distler on December 31, 2003 8:42 PM | Permalink | Reply to this

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

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Hi Urs!

Thanks for your reply. I had a feeling I would get in trouble with the adjoint :)

I’m still searching for an alternative way to view this because it seems troubling to me to deform the exterior derivative. The reason being that I associate the exterior derivate with Stokes theorem, i.e. $d$ is that thing which makes Stokes’ theorem valid. If you deform $d$ , then it seems like you are also deforming the boundary map $\partial$ .

Seeing how this deformation can take you from Minkowski space to a more general semi-Riemannian manifold makes me wonder if ALL deformations can be cast into the form of a modification of the Hodge star. Something like

(1)

⋆_{e} = ⋆ \exp (eW)

maybe. Then with the undeformed $d$ and the deformed adjoint

(2)

d_{e}^{†} = \exp (- eW) d^{†} \exp (eW)

it would seem to me that you’d still have a completely valid supersymmetry [note: what little I know about supersymmetry is limited to a few pages of Witten’s Morse theory paper :)]. Would we be losing anything else? If you could somehow reproduce your results with a mere deformation of the Hodge star, then that would seem more natural to me and would leave Stokes’ theorem (the most beautiful theorem of all mathematics) untouched :)

Cheers!

Eric

Posted by: Eric on January 2, 2004 2:38 PM | Permalink | Reply to this

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

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Ok. I was thinking about this some more.

The Dirac operator is basically the square root of the Laplacian, but you don’t call

(1)

D = d + d^{†}

a modified $d$ , right? Similarly, you can think of

(2)

d_{K} = d + i_{K}

as a square root of the Lie derivative along K. It is not really a modified exterior derivative either, just as the Dirac operator is not a modified exterior derivative. Is there a name for the square root of a Lie derivative? :)

Eric

Posted by: Eric on January 2, 2004 9:16 PM | Permalink | Reply to this

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

Is there a name for the square root of a Lie derivative? :)

Yes! It is called a supersymmetry generator! :-)

At least that’s true when the Lie derivative is along a Killing vector field.

Posted by: Urs Schreiber on January 2, 2004 9:20 PM | Permalink | Reply to this

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

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I am tired, having spent the whole day with preparing, solving and typing exercise sheets for the course that I am tutoring. There is some backlog from christmas and I have to work ahead because I’ll be in Barcelona at the RTN Winter School on Strings, Supergravity and Gauge Fields next week. (If I recall correctly Robert Helling is involved in the organization, so maybe I’ll meet him there. (?)) But before quitting I need to do at least one expedient thing today, so let me try to say something appropriate in response to your comments:

I’m still searching for an alternative way to view this because it seems troubling to me to deform the exterior derivative. The reason being that I associate the exterior derivate with Stokes theorem, i.e. $d$ is that thing which makes Stokes’ theorem valid. If you deform d , then it seems like you are also deforming the boundary map $\partial$ .

Yes, we have talked about this in private email. If we let $〈 ω, S 〉$ be the pairing of the differential form $ω$ with the chain $S$ and let $(\cdot)^{T}$ denote the dual of an operator with respect to this pairing, so that

(1)

d^{T} = \partial

then the deformation

(2)

d_{e} = e^{- eW} d e^{eW}

of the exterior derivative induces a similar deformation on the boundary operator $\partial$ which sends chains to their boundary:

(3)

\partial_{e} : = (d_{e})^{T} = e^{{eW}^{T}} \partial e^{- {eW}^{T}} .

The deformation of the meaning of boundary and coboundary is of course completely equivalent to that of $d$ and $d^{†}$ .

Seeing how this deformation can take you from Minkowski space to a more general semi-Riemannian manifold makes me wonder if ALL deformations can be cast into the form of a modification of the Hodge star.

I need to think more about what this does really mean. Right now I can come up only with these two observations, which are maybe related to what you have in mind:

If you have a physical system governed by constraints of the form

(4)

e^{- W} d e^{W} ∣ ψ 〉 = 0

(5)

e^{W^{†}} d^{†} e^{- W^{†}} ∣ ψ 〉 = 0

and you cannot stand the sight of the first equation then you can of course always perform an isomorphism

(6)

A \to e^{W} A e^{- W}

on the entire algebra and send states to

(7)

∣ ψ 〉 \to ∣ \tilde{ψ} 〉 : = e^{W} ∣ ψ 〉 .

The above constraints are equivalent to

(8)

d ∣ \tilde{ψ} 〉 = 0

(9)

d^{†} e^{- W^{†}} e^{- W} ∣ \tilde{ψ} 〉 = 0 .

This even preserves the supersymmetry algebra, up to the isomorphism.

If you want to preserve the form of the relation

(10)

d^{†} = \pm ⋆^{- 1} d ⋆

while deforming, then you could define

(11)

⋆_{e} : = e^{- W} ⋆ e^{- W^{†}} .

This gives

(12)

d_{e}^{†} = \pm ⋆_{e}^{- 1} d_{e} ⋆_{e} .

But maybe you are looking for something deeper than the above equivalences.

Posted by: Urs Schreiber on January 2, 2004 9:46 PM | Permalink | Reply to this

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

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Hello!

I hope my elementary questions are not too boring for everyone.

I have no idea what “D = 1, N = 2” means, but in your paper hep-th/0311064, you say that the “D = 1, N = 2 supersymmetry algebra may be represented by operators $d$ and $d^{†}$ , which satisfy

(1)

{d, d} = {d^{†}, d^{†}} = 0, {d, d^{†}} = Δ,

as well as $(d)^{†} = d^{†}$ .”

When I then think about your example deformation that produces a semi-Riemannian manifold from flat Minkowski space, the thought that comes to my mind is that this seems like it may be achieved simply by a deformation of the Hodge star, which incorporates information about the metric. On the condition that this statement makes any sense at all (which I’m not sure it does), then I am wondering if the other example deformations that you demonstrate can also be expressed as simple deformations of the Hodge star.

The obvious choice for a deformed Hodge star would seem to be

(2)

⋆_{e} = ⋆ \exp (eW),

which would give rise to a modified inner product

(3)

[A, B]_{e} : = \int_{M} A \land ⋆_{e} B .

At the risk of rambling nonsensically, I’d then speculate that the adjoint exterior derivative of $d$ with respect to this modified inner product (which may not even be an inner product any more), is given by

(4)

d_{e}^{†} = \exp (- eW) d^{†} \exp (eW) .

Unless I am missing something obvious (which is likely), then the operators $d$ and $d_{e}^{†}$ satisfy the conditions above for a D = 1, N = 2 supersymmetry algeba (whatever that is) :)

If this makes any sense so far, then I’m wondering if any of the deformations you consider might also be obtained simply from $d$ and $d_{e}^{†}$ ?

Phew! :)

Thanks again,

Eric

Posted by: Eric on January 3, 2004 4:38 AM | Permalink | Reply to this

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

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Hello again,

Sorry for responding to myself, but a thought just occured to me soon after I hit the “submit” button.

I was thinking about in what situations my deformed inner product

(1)

[A, B]_{e} = \int_{M} A \land ⋆_{e} B

would actually be an inner product. The answer is obvious to me now. Since

(2)

[A, B]_{e} = [A, \exp (eW) B],

it follows that a necessary condition for $[A, B]_{e}$ to be an inner product is that $\exp (eW)$ must be Hermitian with respect to the undeformed inner product.

This seems slightly interesting because I think you mentioned that if it were anti-Hermitian, it could be gauged away so only the Hermitian part leads to new physics.

Best regards,

Eric

Posted by: Eric on January 3, 2004 5:08 AM | Permalink | Reply to this

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

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Hi Eric,

you write:

I hope my elementary questions are not too boring for everyone.

Please continue asking elementary questions if you feel like it! :-) Even though I am eager to sort out the more string specific implications of what I have written in that draft (and hoping for comments by the string theorists at the Coffe Table) there are of course also more general questions raised by the insight that the original deformation by Witten which may have seemed like a mere weird mathematical trick without further meaning, turns out to be the general recipe for producing superconformal algebras for all kinds of string background fields. This fact certainly seems to make a further analysis of the ‘moral meaning’ of these deformations worthwhile. And with your insistence on clarifying the precise implication of these deformations on inner products and Hodge duality you put your finger precisely at the heart of the matter, I believe.

I have no idea what “D = 1, N = 2” means,

This designates the supersymmetry algebra of $D = 1$ dimensions with $N = 2$ supercharges. This is the rather trivial algebra

(1)

{D^{A}, D^{B}} = 2 δ^{AB} Δ

where $A, B \in {1, \dots, N}$ are indices, $D^{A}$ are the $N = 2$ supersymmetry generators and $Δ$ is the single (because of $D = 1$ ) bosonic generator.

When I then think about your example deformation that produces a semi-Riemannian manifold from flat Minkowski space, the thought that comes to my mind is that this seems like it may be achieved simply by a deformation of the Hodge star, which incorporates information about the metric.

I see how it may seem to be this way, but I am not sure if this can be given a precise meaning. This deformation which gives curved space from flat space is both simple and a little subtle. The subtle point is that an operator $d$ which is quite independent of the metric still receives a deformation when the metric is modified. This is due to the fact that it is represented in terms of operators which are not independent of the metric, and the deformation has to take account of that. I feel that a clean mathematical formulation of this is missing and desireable, but so far the best I have come up with is what I have written in that paper. In the context of what I have written there the Hodge star operator receives no deformation under the action of $W^{G}$ .

To see what I mean by this, use the formulation of Hodge star deformations that I have mentioned in another recent comment in this thread here: For arbitrary deformations the relation

(2)

d^{†} = - \bar{⋆} d {\bar{⋆}}^{- 1}

is preserved if we introduce the deformed Hodge star ${\bar{⋆}}_{e}$ defined by

(3)

{\bar{⋆}}_{e} : = e^{W^{†}} \bar{⋆} e^{W} .

It is easy to see that this operator makes the follwoing relation hold true:

(4)

d_{e}^{†} = - {\bar{⋆}}_{e} d_{e} {\bar{⋆}}_{e}^{- 1} .

As I said, this holds for arbitrary deformations. So let’s see what happens in the case of gravitational deformations with $W = W^{(G)}$ : Due to the special form of this deformation operator together with the tracelessness condition (p. 13 of my draft or p. 10 of hep-th/0311064) one sees that $W^{(B)}$ is anti Hodge self dual:

(5)

\bar{⋆} W^{(G)} \bar{⋆} = - W^{(G)} .

But this implies that

(6)

{\bar{⋆}}^{(G)} = e^{W^{†}} \bar{⋆} e^{W} = e^{W} e^{- W} \bar{⋆} = \bar{⋆}

and hence the Hodge star remains invariant under the gravitational deformation. This should not be too surprising if you recall how the Hodge star operator is constructed from the ONB form creators and annihilators (as in equation (A.14) of hep-th/0311064) and that what was an ONB form creator/annihilator before the deformation is one also after the deformation. The deformation actually only affects how the coordinate basis creators/annihilators are defined in terms of the ONB creators/annihilators.

Posted by: Urs Schreiber on January 3, 2004 4:24 PM | Permalink | Reply to this

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

I am not involved with the RTN school. At some point I thought about attending but I will stay here in Cambridge although the program looks very interesting.

Robert

Posted by: Robert on January 6, 2004 12:41 PM | Permalink | Reply to this

Read the post itex2MML Plugin Update
Weblog: Musings
Excerpt: A minor update to the itex2MML executable, used in my plugin. I added a few more MathML entities.
Tracked: January 1, 2004 7:52 PM

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

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Hi Urs and everyone,

I’m writing a new comment because for some reason I am getting some kind of parsing error when I try to respond to your Jan 3, 4:24 PM UTC post.

Regarding my elementary questions…

Well, if my elementary questions are detracting from some possibly more interesting string discussions then it really would not hurt my feelings at all if you (or anyone else) asked me to take it to private email (like we usually do anyway) :) I just thought that this might be a place to discuss things in the open.

Back to semi technical things…

I know I am horrible at explaining myself clearly and I don’t know if I am getting my point across very well here either. My point is that I see the “geometrical language” as providing a nice way to separate topological concepts from metric dependent concepts. In the case of (sourcefree) Maxwell’s equations we have

(1)

dF = 0,

which is purely topological and

(2)

δ F = 0,

which is metric dependent (I am switching to $δ = d^{†}$ for convenience). I don’t know if I can prove this, but I am pretty sure that any deformations of the geometry (which would seem to give rise to potentials) should be incorporated into the Hodge star via the transformation

(3)

⋆_{e} = ⋆ \exp (eW) .

In your responses, you have given a different deformation of the Hodge star, which I can’t seem to understand. In my opinion, a modified Hodge star should give rise to a modified inner product and a modified $δ$ . I guess I am trying to push the idea that perhaps all of the deformations you present can be cast into a mere deformation of the Hodge star (and corresponding deformed inner product). It seems to me that keeping $d$ undeformed while deforming the Hodge star (and consequently the inner product and adjoint exterior derivative) would not affect the supersymmetry algebra.

I guess I should buckle down and see if this is a case for Witten’s simple deformation :)

On a slightly different note…

I was reading your hep-th/0311064 again and another thought came to me. Essentially, it seems like you are trying to use some Killing vector to define a flow of time and then you separate the equations of motion into a Schrodinger-like term plus a spatial constraint. My experience has shown me that artificially separating a “spacetime” phenomena into “space” + “time” typically ends up making things excessively complicated.

I’m sure that the following computation is not original, but I haven’t seen it before and thought it was kind of cute.

If my hunch is correct and we should not be deforming $d$ , then we can try to do some perturbation theory based on the field $F$ and the adjoint $δ$ , i.e.

(4)

F = \sum_{m} F^{(m)}

and

(5)

δ = \sum_{m} δ^{(m)},

where the superscript $(m)$ denotes terms of order $m$ in some small perturbation parameter. The topological equation remains unchanged and we have

(6)

{dF}^{(m)} = 0

for all $m$ . However, the metric dependent equation becomes

(7)

δ F = δ^{(0)} F^{(0)} + [δ^{(0)} F^{(1)} + δ^{(1)} F^{(0)}] + O (e^{2}) = 0

so that

(8)

δ^{(0)} F^{(0)} = 0,

which is not all that surprising, but the inetresting thing is we have

(9)

δ^{(0)} F^{(1)} = - δ^{(1)} F^{(0)} .

The physical interpretation of this is kind of neat. The first order perturbation of the electromagnetic field $F^{(1)}$ is the field due to the presence of a source derived from the deformed adjoint $δ^{(1)}$ applied to the undeformed field $F^{(0)}$ . In other words, the deformation of $δ$ produces a current

(10)

j^{(1)} = - δ^{(1)} F^{(0)}

so that at first order in the perturbation we have

(11)

{dF}^{(1)} = 0

and

(12)

δ^{(0)} F^{(1)} = j^{(1)} .

Of course the pattern is now obvious. The field $F^{(1)}$ is going to give rise to a new current $j^{(2)}$ and so on so that we have

(13)

{dF}^{(m)} = 0

and

(14)

δ^{(0)} F^{(m)} = j^{(m)} .

This is pretty neat.

Sorry if I bored you with my reinventing the wheel, but I can’t help but think this is somehow important for what you are doing :)

Best regards,

Eric

Posted by: Eric on January 3, 2004 6:01 PM | Permalink | Reply to this

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

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Hi Eric,

you wrote:

I’m writing a new comment because for some reason I am getting some kind of parsing error when I try to respond to your Jan 3, 4:24 PM UTC post.

Maybe there is a limit to the depth of nested comments?

I just thought that this might be a place to discuss things in the open.

Yes, certainly, I believe this is what this blog is intended for. No, you are not distracting me, for sure!

In your responses, you have given a different deformation of the Hodge star, which I can’t seem to understand. In my opinion, a modified Hodge star should give rise to a modified inner product and a modified $δ$ . I guess I am trying to push the idea that perhaps all of the deformations you present can be cast into a mere deformation of the Hodge star (and corresponding deformed inner product). It seems to me that keeping $d$ undeformed while deforming the Hodge star (and consequently the inner product and adjoint exterior derivative) would not affect the supersymmetry algebra.

Ok. In a previous comment I had offered two interpretations of your desire to keep $d$ undeformed. It seems like you are complaining about the second one. ;-) What you write above sounds to me like it harmonizes with the first one, however.

So let’s start with the ordinary version of the deformation

(1)

d_{e} = e^{- W} d e^{W}

(2)

δ_{e} = e^{W^{†}} δ e^{- W^{†}}

and then apply the isomorphism

(3)

A \mapsto A_{i} : = e^{W} A e^{- W}

to every operator in sight. This manifestly preserves all the algebraic relations between all operators and yields

(4)

d_{e} \mapsto d_{e, i} = d

(5)

δ_{e} \mapsto δ_{e, i} = e^{W} e^{W^{†}} δ e^{- W^{†}} e^{- W} .

This way $d$ remains $d$ , as desired. The resulting relation between $d_{e, i}$ and $δ_{e, i}$ can now indeed be expressed by means of a modified inner product. As you have already indicated, let

(6)

〈 \cdot ∣ \cdot 〉_{i} : = 〈 \cdot ∣ e^{- W^{†}} e^{- W} \cdot 〉,

where on the right we have the ordinary Hodge inner product, and write $A^{†_{i}}$ for the adjoint of an operator $A$ with respect to $〈 \cdot ∣ \cdot 〉_{i}$ .

This is nice, because $〈 \cdot ∣ \cdot 〉_{i}$ is just the inner product on the deformed states

(7)

ψ \mapsto ψ_{i} = e^{W} ψ

that reproduces the ordinary inner product on the ordinary states.

Now, indeed, we get

(8)

δ_{e, i} = d^{†_{i}} .

As you have already pointed out, since

(9)

〈 α ∣ β 〉 = \int α \land ⋆ β

we have

(10)

〈 α ∣ β 〉_{i} = \int α \land ⋆ e^{-}

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December 30, 2003