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December 23, 2003

Classical deformations of 2d SCFTs - Part I: Loop space

Posted by Urs Schreiber

Motivated by the representation of the classical super Virasoro constraints as generalized Dirac-Kaehler constraints (d±d +)ψ=0 on loop space, examples of the most general continuous deformations de Wde W (with W an even graded reparametrization invariant operator) are considered, which preserve the superconformal algebra. The deformations which induce the three massless NS-NS backgrounds are exhibited. A further 2-form background is found, which is argued to be related to the R-R 2-form. Hints for a manifest realization of S-duality in terms of a algebra isomorphism are discussed. Furthermore, gauge field backgrounds are considered.

The special representation of the deformations used here allows the construction of covariant Hamiltonians generating string evolution in all these backgrounds. As discussed in [3] this can be used to compute curvature corrections to string spectra.

(Here is a preprint of a paper discussing these issues.)

(0) Introduction

The phenomenon that supersymmetric field theories look like Dirac-Kaehler systems when formulated in Schroedinger representation is well studied in the special limits where only a finite number of degrees of freedom are retained, such as the semi-classical quantization of solitons in field theory (see for instance [1] for a brief introduction and further references). That this phenomenon is rooted in the general structure of supersymmetric field theory has been noted long ago in [2]. A way to exploit this fact for the construction of covariant target space Hamiltonians applicable to the computation of curvature corrections of string spectra in nontrivial backgrounds has been proposed in [3].

In the following this technique is looked at more closely. The key idea is that the form of the (classical) superconformal algebra is preserved under the deformation

(1)d Ke Wd Ke W
(2)d K +e W +d K +e W +

when W is an even graded operator that satisfies a certain consistency condition. Here d K is the K-deformed exterior derivative on loop space and K is the Killing vector field on loop space which induces reparametrizations.

The canonical (functional) form of the superconformal generators for all massless NS-NS backgrounds can neatly be expressed this way by deformation operators W that are bilinear in the fermions. It turns out that there is one further bilinear in the fermions which induces a background that is shown to be related to the R-R C 2 field.

It is straightforward to find further deformation operators and hence further backgrounds. Even though at the classical level no equations of motion for the backgrounds are obtained, there is a classical consistency condition which constrains the admissible deformation operators. A few further selected cases are studied here.

This approach for obtaining new superconformal algebras from existing ones by applying deformations is similar in spirit, but rather complementary, to an approach studied by Giannakis and Evans [4]. There, the superconformal generators of one chirality are deformed to lowest order as


Requiring the deformed generators to satisfy the desired algebra to first order shows that δT and δG can be any fields of conformal weight 1 and 1 /2 , respectively. An adaption of this procedure to deformations of the BRST charge itself is discussed in [6].

The advantage of this method over the one discussed in the following is that it operates at the level of quantum SCFTs and has powerful CFT tools at its disposal, such as normal ordering and operator product expansion. The disadvantage is that it only applies perturbatively to the first order in the background fields.

On the other hand, the deformations induced by de Wde We W(G+iG˜)e W preserve the superconformal algebra for arbitrarily large perturbations W. The drawback is that normal ordering is non-trivially affected, too, and without further work the resulting superconformal algebra is only available in classical form, i.e. on the level of Poisson brackets.

(1) Loop space

In this section the technical setup is briefly established, the main purpose being to emphasize the fact that the 0-mode d K of the sum of the left- and the rightmoving supercurrents is indeed the K-deformed exterior derivative on loop space. Weak nilpotency of this K-deformed operator (namely nilpotency up to reparametrizations) is the essential property which implies that the modes of d K and its adjoint generate a superconformal algebra. In this sense the loop space perspective on superstrings highlights a special aspect of the super Virasoro constraint algebra which turns out to be pivotal for the construction of classical deformations of that algebra.

The kinematical configuration space of the closed bosonic string is loop space ℒℳ, the space of parameterized loops in target space . As discussed in section 2.1 of [3] the kinematical configuration space of the closed superstring is therefore the superspace over ℒℳ, which can be identified with the 1-form bundle Ω 1 (ℒℳ). Superstring states in Schroedinger representation are super-functionals on Ω 1 (ℒℳ) and hence section of the form bundle Ω(ℒℳ) over loop space. This is perfectly analogous to the general situation for supersymmetric quantum mechanical systems, i.e. those with a finite-dimensional configuration space. In the following we will make ample use of this formal analogy to finite dimensional systems in order to develop superstring quantum dynamics as a relativistic supersymmetric quantum mechanical system in loop space.

The main technical consequence of the infinite dimensionality are the well known divergencies of certain objects, such as the Ricci-Tensor and the Laplace-Beltrami operator, which inhibit the naive implementation of quantum mechanics on ℒℳ. But of course these are just the well known infinities that arise, when working in the Heisenberg (CFT) instead of in the Schroedinger picture, from operator ordering effects, and which are removed by imposing normal ordering. Since the choice of Schroedinger or Heisenberg picture is just one of language, the same normal ordering (now expressed in terms of functional operators instead of Fock space operators) takes care of infinities in loop space. We will therefore not have much more to say about this issue here. The main result of the following sections are various (deformed) representations of the super-Virasoro algebra on loop space (corresponding to different spacetime backgrounds), and will be derived in their classical (Poisson-bracket) form without considering normal ordering effects. The quantization of these classical algebras, which will not be discussed here, would yield anomalies and equations of motion for the background fields in order to cancel these.

A mathematical discussion of aspects of loop space can for instance be found in [6] [7]. A rigorous treatment of some of the objects discussed below is also given in [8].

(1.1) Definitions

Let (,g) be a pseudo-Riemannian manifold, the target space, with metric g, and let ℒℳ be its loop space consisting of smooth embeddings of the circle into :

(5)ℒℳ:=C (S 1 ,ℒℳ).

The tangent space T Xℒℳ of ℒℳ at a loop X:S 1 is the space of vector fields along that loop. The metric on induces a metric on T Xℒℳ: Let g(p)=g μν(p)dx μdx ν be the metric tensor on . Then we choose for the metric on ℒℳ at a point X the mapping

(8)=dσg μν(X(σ))U μ(σ)V ν(σ).

(For the intended applications Tℒℳ is actually too small, since there will be need to deal with distributional vector fields on loop space. Therefore one really considers T¯ℒℳ, the completion of Tℒℳ at each point X with respect to the norm induced by the inner product.)

For brevity, whenever we refer to “loop space” in the following, we mean ℒℳ equipped with the induced metric described above.

To abbreviate the notation, formal multi-indices (μ,σ) may be introduced and we write equivalently

(9)U μ(σ):=U (μ,σ)

for a vector UT Xℒℳ, and similarly for higher-rank tensors on loop space.

Extending physicist’s Ricci-calculus to the infinite-dimensional setting in the obvious way, we also write:

(10)dσU μ(σ)V μ(σ):=U (μ,σ)V (μ,σ).

For this to make sense we need to know how to “shift” the continuous index σ. Because of

(11)dσg μν(X(σ))U μ(σ)V ν(σ)=dσdσ δ(σ,σ )g μν(X(σ))U μ(σ)V ν(σ )

it makes sense to write the metric tensor on loop space as

(12)G (μ,σ)(ν,σ )(X):=g μν(X(σ))δ(σ,σ ).


(13)U,V=U (μ,σ)G (μ,σ)(ν,σ )V (ν,σ )


(14)V (μ,σ)=G (μ,σ)(ν,σ )V (ν,σ )=V μ(σ).

Consequently, it is natural to write

(15)δ(σ,σ ):=δ σ σ =δ σ σ=δ σ,σ =δ σ,σ .

A (holonomic) basis for T Xℒℳ may now be written as

(16) (μ,σ):=δδX μ(σ),

where the expression on the right denotes the functional derivative, so that

(17) (μ,σ)X (ν,σ )=δ (μ,σ) (ν,σ )=δ μ νδ(σ,σ ).

By analogy, many concepts known from finite dimensional geometry carry over to the infinite dimensional case of loop spaces. Problems arise when traces over the continuous “index” σ are taken, like for contractions of the Riemann tensor, which leads to undefined diverging expressions. These are taken care of by the usual normal-ordering of quantum field theory.

(1.2) Differential geometry on loop space

With the metric on loop space in hand the usual objects of differential geometry can be derived for loop space. Simple calculations yield the Levi-Civita connection as well as the Riemann curvature, which will be frequently needed later on. The exterior algebra over loop space is introduced and the exterior derivative and its adjoint, which play the central role in the construction of the superconformal algebra in section 2, are constructed in terms of operators on the exterior bundle Ω(ℒℳ). Furthermore isometries on loop space are considered, both the one coming from reparametrization of loops as well as those induced from target space. The former leads to the reparametrization constraint on strings, while the latter is crucial for the the Hamiltonian evolution on loop space along the lines of [3].

(1.2.1) Basic geometric data The basic geometric data on loop space is obtained straightforwardly by the usual computations. For instance the inverse metric is obviously

(18)G (μ,σ)(ν,σ )(X)=g μν(X(σ))δ(σ,σ ).

A vielbein field e a=e a μdx μ on gives rise to a vielbein field E (a,σ) on loop space:

(19)E (a,σ) (μ,σ )(X):=e a μ(X(σ)δ σ σ

which satisfies

(20)E (a,σ) (μ,σ )E (b,σ)(μ,σ )=η abδ σ,σ :=η (a,σ)(b,σ ).

For the Levi-Civita connection one finds:

(21)Γ (μσ)(ασ )(βσ )(X)=1 2 (δδX μ(σ)G (α,σ )(β,σ )(X)+δδX β(σ )G (μ,σ)(α,σ )(X)δδX α(σ )G (β,σ )(μ,σ)(X))
(22)=Γ μαβ(X(σ))δ(σ,σ )δ(σ ,σ ),

Similarly one finds the Riemann tensor on loop space as

(23)R (μ,σ 1 )(ν,σ 2 ) (α,σ 3 ) (β,σ 4 )(X)=R μν α β(X(σ 1 ))δ(σ 1 ,σ 2 )δ(σ 2 ,σ 3 )δ(σ 3 ,σ 4 ).

(1.2.2)Exterior and Clifford algebra over loop space

The anticommuting fields +(μ,σ), (μ,σ), satisfying the CAR

(24){ +(μ,σ), +(ν,σ )}=0
(25){ (μ,σ), (ν,σ )}=0
(26){ (μ,σ), +(ν,σ )}=δ (ν,σ ) (μ,σ),

are assumed to exist over loop space, in analogy with the form creators and form annihilators c +μ, c μ on the exterior bundle in finite dimensions (in the notation of appendix A of [3]). (For a mathematically rigorous treatment of the continuous CAR compare [7] and references given there.) From them the Clifford fields

(27)Γ ± (μ,σ):= +(μ,σ)± (μ,σ)

are obtained, which satisfy

(28){Γ ± (μ,σ),Γ ± (ν,σ )}=±2 G (μ,σ)(ν,σ )
(29){Γ ± (μ,σ),Γ (ν,σ )}=0 .

The above operators will frequently be needed with respect to some orthonormal frame E (a,σ):

(30)Γ ± (a,σ):=E (a,σ) (μ,σ )Γ ± (μ,σ ).

Just like in the finite dimensional case, the following derivative operators can now be defined:

The covariant derivative operator (cf. A.2 in [3]) on the exterior bundle over loop space is

(31)̂ (μ,σ)= (μ,σ) cΓ (μ,σ) (α,σ ) (β,σ ) +(β,σ ) (α,σ )

or alternatively

(32)̂ (μ,σ)= (μ,σ)ω μ a b(X(σ)) +b(σ) a(σ).

One should note well the difference between the functional derivative (μ,σ) c which commutes with the coordinate frame forms ([ (μ,σ) c, +ν]=0 ) and the functional derivative (μ,σ) which instead commutes with the ONB frame forms ([ (μ,σ), +a]=0 ). See (A.29) of [3] for more details.

In terms of these operators the exterior derivative and coderivative on loop space read, respectively (A.39)

(33)d= +(μ,σ) (μ,σ) c= +(μ,σ)̂ (μ,σ)
(34)d += (μ,σ)̂ (μ,σ).

These operators are the basis for the construction of the superconformal algebras in section 2.

(1.2.3) Isometries

Regardless of the symmetries of the metric g on , loop space (ℒℳ,G) has an isometry generated by the reparametrization flow vector field K, which is defined by:

(35)K (μ,σ)(X)=TX μ(σ).

Here T is just a constant, the string tension, which we include for later convenience.

The flow generated by this vector field obviously rotates the loops around. Since the loop space metric is diagonal in the parameter σ, this leaves the geometry of loop space invariant, and the vector field K satisfies Killing’s equation

(36)G (ν,σ )(λ,σ ) (μ,σ)K (λ,σ )+G (μ,σ)(λ,σ ) (ν,σ )K (λ,σ )=0 ,

as is readily checked.

The operator inducing the Lie-derivative along K on differential forms is (see section A.4 of [3])

(37) K={ +(μ,σ) (μ,σ) c, (ν,σ )X (ν,σ )}=X (μ,σ) (μ,σ) c+ +(μ,σ) (μ,σ).

This operator will be seen to be an essential ingredient of the super-Virasoro algebra in section 2.

Apart from this generic isometry, every symmetry of the target space manifold gives rise to a family of symmetries on ℒℳ: Let v be any Killing vector on target space,

(38) (μv ν)=0 ,

then every vector V on loop space of the form

(39)V ξ(X)=V ξ (μ,σ)(X) (μ,σ):=v μ(X(σ))ξ σ (μ,σ),

(where ξ σ=ξ(σ) is some differentiable function S 1 C), is a Killing vector on loop space. For the commutators of these Killing fields one finds

(40)[V ξ 1 ,V ξ 2 ]=0
(41)[V ξ,K]=V ξ .

The reparametrization Killing vector K will be used to deform the exterior derivative on loop space as discussed in section 2.1.1 of [3], and a target space induced Killing vector V ξ will serve as a generator of parameter evolution on loop space along the lines of section 2.2 of [3]. There it was found in equation (88) that the condition

(42)[K,V ξ]=0

needs to be satisfied for this to work. Due to the above commutators this means that one needs to choose ξ=const, i.e. use the integral lines of V ξ=1 as the “time”-parameter on loop space. This is only natural: It means that every point on the loop is evolved equally along the Killing vector field v on target space.

(2) Classical superconformal generators for various backgrounds

(2.1) Purely gravitational background

In this section it is described how to obtain representations of the classical superconformal algebra on loop space. For a trivial background the construction itself is relatively trivial and, possibly in different notation, well known. The point that shall be emphasized here is that the identification of super-Virasoro generators with modes of the deformed exterior(co-)derivative on loop space allows a convenient treatment of curved backgrounds as well as more general non-trivial background fields.

As was reviewed in section 2.1.1 of [3], (which is based on [2]) one obtains from the exterior derivative and its adjoint on a manifold the generators of a global D=2 , N=1 superalgebra by deforming with a Killing vector. The generic Killing vector field on loop space is the reparametrization vector K. Using this to deform the exterior derivative and its adjoint as in equation (19) of [3] yields the operators

(43)d K:=d+i (μ,σ)X (μ,σ)= +(μ,σ)̂ (μ,σ)+i (μ,σ)X (μ,σ)
(44)d K +:=d +i (μ,σ) +X (μ,σ)= (μ,σ)̂ (μ,σ)i (μ,σ) +X (μ,σ).

The corresponding modes are

(45)d K,ξ=dσξ( +μ̂ μ+i μX μ)
(46)d K,ξ +=dσξ( μ̂ μ+i μ +X μ).

Their auto-anticommutator gives

(47){d K,ξ 1 ,d K,ξ 2 }= K,ξ 1 ξ 2 ,

where is the operator inducing the Lie derivative on differential forms over loop space.

Any field A(σ) is said to have reparametrization weight w(A) if

(48)[ K,ξ,A(σ)]=(ξA +wξ A)(σ).

In particular

(49)w(X μ)=0

(50)w(X μ)=1
(51)w( μ)=1
(52)w(Γ ± a)=1 /2 .

The weight is additive under operator multiplication, so that w(d K)=3 /2 and hence

(53)[ K,ξ 1 ,d K,ξ 2 ]=d K,(1 2 ξ 1 ξ 2 ξ 1 ξ 2 )


(54)[ K,ξ 1 , K,ξ 2 ]= K,(ξ 1 ξ 2 ξ 1 ξ 2 ).

Similar equations hold for d K +.

This yields part of the sought-after algebra. In order to find the rest, a very simple and apparently unproblematic, but rather crucial step is to now define the modes of the deformed Laplace-Beltrami operator as the right hand side of

(55){d K,ξ 1 ,d K,ξ 2 +}=Δ K,ξ 1 ξ 2 .

For this definition to make sense one needs to check that the consistency condition

(56){d K,ξ 1 ξ 3 ,d K,ξ 2 +}={d K,ξ 1 ,d K,ξ 2 ξ 3 +}

is satisfied. This is trivial in the present case, but becomes a crucial condition for more general backgrounds.

The remaining supercommutators all follow from the Jacobi identity, and one can check that the resulting algebra is indeed the superconformal one.

Posted at December 23, 2003 7:37 PM UTC

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Re: Classical deformations of 2d SCFTs - Part I: Loop space

Hi everybody!

This is an experiment. I felt like discussing the deformation issue which I had raised recently in more detail. What you see here is the first part of a paper-like discussion of what I have in mind. I have called it ‘Part I’ because it is barely more than an introduction so far. The main reason why the text stops after section (2.1) is that I haven’t found the time to compile more material, but also because the ‘preview’ function of the blog didn’t work for a slightly longer text. Some experimenting showed that apparently there was a certain number of characters or of lines beyond which hitting ‘preview’ didn’t produce any answer from the server. (?)

The really interesting (in my opinion) stuff of this deformation business will have to wait for Part II, which I hope to complete before next year :-) (unless of course the members of the Coffe Table don’t approve of such extensive blogging). I hope I don’t have to emphasize that I would very much appreciate any comments on what I have written. Even though this is tentatively in the form of a paper I am not at that stage yet. On the other hand, I can fully understand if you would rather think and talk about topological strings and twistors. But, since this is a coffe table, I thought I’d give it a try.

I whish you all a merry christmas and nice holidays!

All the best,

Posted by: Urs Schreiber on December 23, 2003 7:52 PM | Permalink | Reply to this
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