A Weyl-invariant action for chiral strings and branes
aa r X i v : . [ h e p - t h ] M a y A Weyl-invariant action for chiral strings and branes
Alex S. Arvanitakis , Department of Applied Mathematics and Theoretical Physics,Centre for Mathematical Sciences, University of Cambridge,Wilberforce Road, Cambridge, CB3 0WA, U.K. Department of Nuclear and Particle Physics,Faculty of Physics, National and Kapodistrian University of Athens,Athens 15784, Greeceemail:
[email protected] , [email protected] Abstract
We introduce a sigma model lagrangian generalising a number of new and oldmodels which can be thought of as chiral, including the Schild string, ambitwistorstrings, and the recently introduced tensionless AdS twistor strings. This “chiral sigmamodel” describes maps from a p -brane worldvolume into a symplectic space and ismanifestly diffeomorphism- and Weyl-invariant despite the absence of a worldvolumemetric. Construction of the Batalin-Vilkovisky master action leads to a BRST operatorunder which the gauge-fixed action is BRST-exact; we discuss whether this implies thatthe chiral brane sigma model defines a topological field theory. ontents p = 1 : Chiral string examples 15 In this paper we will put forward a p -brane sigma model lagrangian and study its basicproperties in a Batalin-Vilkovisky/BRST approach. For lack of a better name we will callthis the “chiral brane model”, even though properly speaking the model is “chiral” only for p = 1. In that case, “chiral” means that the only derivative appearing in the action can bechosen to be a left-handed one, cf. the nomenclature of Siegel et al. in [1, 2].The p = 1 version of the chiral brane model can be gauge-fixed to obtain a numberof previously-known sigma models [3–8], most of which can be considered “chiral” and/ortensionless. We will discuss these models and their precise relation to the chiral branemodel in section 3 later. Let us mention here that a common feature of the actions forthese theories is a lack of manifest worldsheet diffeomorphism invariance, which formed themotivation for constructing the chiral sigma model. We will see shortly that the chiral sigmamodel enjoys manifest diffeomorphism invariance by virtue of a dynamical worldvolumevector field v a , which is a new ingredient compared to the previously-known formulationsof chiral strings, and whose introduction is accompanied by a new local scaling invariance.1his “Weyl invariance” is closely analogous to that of the usual string except in that it actson both worldvolume and target space at the same time.The chiral brane action is: S = S [ z A ; v a ] = 12 Z d p +1 σ (cid:8) Ω AB z A ∂ a z B v a (cid:9) . (1.1)Here σ a , a = 0 , , . . . p are p -brane worldvolume coordinates and the dynamical field z A = z A ( σ ) describes the embedding of the worldvolume Σ into a real symplectic vector spacewith symplectic form Ω AB (assumed to be constant, antisymmetric and nondegenerate).The other dynamical field , v a = v a ( σ ), carries a worldvolume vector index. We will takethe worldvolume Σ to be closed and accordingly drop all total derivatives inside integrals; oneis thus inclined to think of the chiral brane theory as presented in this paper as a Euclideanfield theory, even though there is no worldsheet metric.Later on we will generalise the action (1.1) by adding the following interaction terminvolving a gauge field interacting with a bilinear in z A (where M IAB is constant): S int [ z A ; A I ] = − Z d p +1 σ (cid:8) A I M IAB z A z B (cid:9) . (1.2)Those generalisations will be referred to as “gauged” chiral brane actions. The remarks wemake in this section will also apply to the gauged models after appropriate modifications.One can also consider a complex version of (1.1) which will be relevant for making contactwith the ambitwistor string models in subsection 3.3. In the complex, p = 1 case, (1.1) canbe seen as an action describing a βγ system. A final generalisation, which we will not treatin this paper, is to models with target space supersymmetry, for which we expect similarresults to hold.The action (1.1) is manifestly invariant under worldvolume diffeomorphisms where z A transforms as a worldvolume scalar density of weight +1 / v a transforms as a vectorfield (of weight zero). Perhaps surprisingly, it is also invariant under a local Weyl transfor-mation if z A transforms with weight − v a transforms with weight +2. The infinitesimal “Dynamical” in the sense that it is to be varied; one could call it auxiliary on account of the fact itappears as a lagrange multiplier, but we will refrain from doing so, as “auxiliary” might suggest it can beeliminated. ξ a = ξ a ( σ ) and ω = ω ( σ ) respectively) read δ ξ z A = ξ a ∂ a z A + 12 ( ∂ a ξ a ) z A , δ ξ v a = ξ b ∂ b v a − v b ∂ b ξ a (1.3) δ ω z A = − ωz A , δ ω v a = +2 ωz A . (1.4)Invariance under local Weyl transformations follows from the antisymmetry of Ω AB .It is not entirely clear whether the model can be consistently defined on arbitrary symplec-tic manifold target spaces: while one can always cover any symplectic manifold by Darbouxcharts where the symplectic form is constant, whether such local descriptions can be sewntogether in a consistent way is not obvious. We will anyway not consider this generalisationin this paper.A peculiarity of the gauge transformations just presented is that one can combine themto shift the density weight of the dynamical fields under worldvolume diffeomorphisms. AWeyl transformation with parameter ω = − x∂ a ξ a ; x ∈ R (1.5)can be combined with an infinitesimal diffeomorphism to shift the weight of z A by x andthat of v a by − x . Thus the form of the transformations we exhibited above is not unique.A second peculiarity is that an infinitesimal diffeomorphism with parameter ξ a = v a alwaysvanishes on-shell. We might thus be concerned that the gauge transformations are reducible(i.e. that there are “gauge transformations of gauge transformations”). One could, however,guess that this cannot be the case on account of the fact the space of on-shell vanishinggauge transformations is one dimensional and is spanned by v a .To clarify these issues we will analyse the model in the Batalin-Vilkovisky (BV) formal-ism [9] in section 2, with the result that the gauge transformations close off-shell and areirreducible. The shift in x described in the previous paragraph will be realised by a canonicaltransformation in the BV sense, and thus theories for different values of x are equivalent; assuch we will generally consider the theory with x = 0 only.A fact that falls out of the BV analysis is that S is trivial in the gauge-invariant BRSTcohomology (in the terminology of [13]). After gauge-fixing in the usual way a little morework leads to a gauge-fixed action which is a total BRST variation, i.e. trivial in the BRSTcohomology. One would then conclude that the chiral brane action describes a cohomological3eld theory in the sense of [10]. However we will not be able to prove whether the actionremains trivial in the BRST cohomology after certain ghost variables have been eliminated.Furthermore, the theory described by S clearly possesses local degrees of freedom and itwould thus be highly counterintuitive if it were cohomological at the same time. This point isfurther discussed in subsection 2.1.1. For these reasons the status of the chiral brane theoryas a cohomological field theory is somewhat unclear. The Batalin-Vilkovisky (BV) formalism [9] is an approach for handling gauge theories wherethe algebra of gauge transformations only closes on-shell and/or the gauge transformationsare reducible, and more specifically it is often used to construct BRST operators and BRST-invariant actions for such theories. For the chiral brane sigma model we will also see how itclarifies the structure of the gauge transformations. We will give a very brief outline of theformalism in the next paragraph and refer to the review [11] for details. We also profitedfrom the discussion in [12].The central object in the BV approach is the “master action”. This is constructed fromthe original action S ((1.1) in this case) in a number of steps. One first takes the originalgauge parameters ( ξ a , ω ) and introduces a set of corresponding ghost fields ( c aξ , c ω ), whichare defined to have ghost number gh c aξ = gh c ω = 1. The original fields ( z A , v a ) are definedto have ghost number zero. It is customary to refer to the collection of original fields plusghost fields as just “fields”. For each field φ one then introduces a corresponding antifield ⋆ φ of ghost number gh ⋆ φ = − gh φ − φ in the sense( φ ( σ ) , ⋆ φ ( σ ′ )) = δ p +1 ( σ − σ ′ ) . (2.1)The (anti)fields are bosonic or fermionic depending on their ghost number modulo 2 (assum-ing we started with a purely bosonic theory). The bracket ( − , − ) is known as the antibracketand is graded anticommutative in its two arguments with respect to the grading by ghostnumber. It also satisfies a modified version of the Leibniz rule which can be summarised bysaying the bracket “carries ghost number +1”. The master action S BV = S BV [ φ, ⋆ φ ] is then Such theories are more often called “topological” but we prefer to use this more precise language — afterall the chiral brane action is already “topological” in that does not depend on a worldvolume metric. S BV , S BV ) = 0 (2.2)such that S BV [ φ, ⋆ φ = 0] = S . If the gauge transformations are reducible one needs moreghosts (with corresponding antifields) than described above but we will see that the generalcase is not relevant for this paper.The master action encodes the gauge transformations in its antifield dependence. Specif-ically it is easy to see that the terms linear in antifields of the original fields (i.e. linear in ⋆ z A , ⋆ v a ) must be proportional to the original infinitesimal gauge transformations with gaugeparameter replaced by the corresponding ghost. Higher-order terms in antifields are onlypresent if the gauge algebra only closes on-shell, and more ghost fields than described aboveare needed if the gauge transformations are reducible.For the ungauged chiral brane sigma model action (1.1) a master action solving themaster equation is S BV = S + S (2.3)with S = S [ z A ; v a ] = 12 Z (cid:8) Ω AB z A ∂ a z B v a (cid:9) (2.4)and S = Z (cid:26) ⋆ z A (cid:18) c aξ ∂ a z A + 12 ∂ a c aξ z A − c ω z A (cid:19) + ⋆ v a ( c bξ ∂ b v a − v b ∂ b c aξ + 2 c ω v a ) − ⋆ c ξa c bξ ∂ b c aξ − ⋆ c ω c bξ ∂ b c ω (cid:27) , (2.5)where here and henceforth R = R d p +1 σ unless noted. Notice that both S and S are realbecause we are using the atypical complex conjugation convention (also used in [4, 20]) thatsends ψ ψ → ¯ ψ ¯ ψ if ψ , ψ are fermionic. From the fact the master action is linear inantifields we see that the gauge transformations close off-shell and that they are irreducible.The coefficients of the two terms on the last line are fixed by the master equation.The master equation is equivalent to the invariance of S BV under infinitesimal BRSTtransformations defined as δ BRST Φ ≡ (Φ , S BV Λ) . (2.6)Here Λ is a constant anticommuting parameter of ghost number gh Λ = −
1, where the ghost5umber of Λ was chosen so that δ BRST satisfies the Leibniz rule. We have δ Φ ≡ δ BRST;Λ ( δ BRST;Λ Φ) = 0 (2.7)as a consequence of the master equation and a super Jacobi identity satisfied by the an-tibracket. These BRST transformations are related to but not the same as the ones usuallyemployed in the context of BRST quantisation after the antifields are eliminated.One then defines observables to be functionals F which are BRST-closed ( δ BRST F = 0)modulo BRST-exact ones, i.e. we identify F ∼ F ′ + δ BRST
G . (2.8)The cohomology thus defined is called the gauge-invariant BRST cohomology , to contrast itwith the usual BRST cohomology (also known as gauge-fixed BRST cohomology) which wewill focus on later.Usually the classical action S is an observable (it is BRST-closed by virtue of its gauge-invariance). The chiral brane theory however happens to enjoy the peculiar property that S is BRST-exact! This can in fact be verified almost by inspection, as the BRST variationof ⋆ v a v a will include a term proportional to the original lagrangian. In fact after a shortcalculation we find ( − ⋆ v a v a , S BV ) = 12 Ω AB z A ∂ a z B v a + ∂ b ( c bξ ⋆ v a v a ) . (2.9)Thus after integrating over the worldvolume (assumed closed) we find that S is BRST-trivial after dropping the boundary term. More precisely we have just found that S isBRST-trivial in the gauge-invariant BRST cohomology of local functionals. The relationbetween this cohomology and the gauge-fixed BRST cohomology we are actually interestedin can be subtle (see e.g. [13]), so we will revisit this issue after constructing the BRSToperator for the gauge-fixed cohomology.The solution to the master equation is not unique. For the chiral brane sigma model thisambiguity includes the ambiguity in the density weights noted in the Introduction. The BVaction above was written down for a specific choice of weights, so it stands to reason thereshould exist equivalent BV master actions corresponding to each consistent weight choice.We point out that they can be obtained using the canonical transformation defined by the6ermion Ψ x = x Z ⋆ c ω ∂ a c aξ , x ∈ R (2.10)where canonical transformations act in the standard way: e Ψ x Φ = Φ + (Ψ x , Φ) + . . . , (Φ is any (anti)field). (2.11)Ψ x is referred to as a fermion as it must have ghost number -1 if the transformed action isto have ghost number zero. This transformation produces the shifts c ω → c ω − x∂ a c aξ , ⋆ c ξa → ⋆ c ξa − x∂ a ⋆ c ω (2.12)in the BV action. Because this Ψ x transformation is canonical, the BV action thus modifiedsatisfies the master equation for any x .Since these shifts in x are canonical transformations we know that the quantum theorywill be independent of the choice of x (in the absence of BRST anomalies; see e.g. [11]section 8.9). Besides x = 0, another apparently equally natural choice is x = − /
2, forwhich the fields z A transform as worldvolume scalars under diffeomorphisms, while v a turnsinto a vector density of weight 1. However, it seems that x = 0 is singled out if one demandsequivalence with the canonical Hamiltonian formalism (after a fairly innocent-looking choiceof partial gauge fixing), which will be discussed in section 3. The BV action S BV always has a number of local fermionic invariances and thus requiresgauge fixing. These can always fixed by setting the antifields to zero, however doing so in theoriginal BV action simply leads us back to the original action, which has troublesome gaugeinvariances of its own. The way out is to first perform a canonical transformation so thatwhat remains after the antifields vanish has no gauge invariances. In practice, one does notalways eliminate all gauge invariances this way (consider e.g. the usual string in conformalgauge for low-genus worldsheets) but the end action tends to be rather more amenable topath integral methods by virtue of possessing BRST invariance. We will simply view theBV apparatus as a way to obtain the BRST transformations.We gauge fix in the standard way by introducing the “non-minimal sector” fields π a and b a with antifields ⋆ b a and ⋆ π a respectively, with ghost numbers gh π a = 0 and gh b a = −
1. Thefield b a is confusingly known in the literature as the “antighost”, which however should not be7onfused with the c aξ ghost antifield ⋆ c ξa . The solution of the master equation is then modifiedby the addition of the term ⋆ b a π a . Thus the starting point of the gauge fixing procedure isthe “non-minimal action” S BV + Z ⋆ b a π a , (2.13)which clearly satisfies the master equation whenever S BV does. We then consider the gaugefixing fermion Ψ = Z b a ( v a − ˜ v a ) (2.14)where ˜ v a = 0 is nondynamical and its components are assumed to be constant (this gaugecan always be reached locally on the worldvolume as long as v a does not vanish). Withoutloss of generality we can thus use coordinates where ˜ v a = (1 , , . . . T . The canonicaltransformation generated by Ψ amounts to the shifts ⋆ b a → ⋆ b a + ( v a − ˜ v a ) , ⋆ v a → ⋆ v a + b a (2.15)in the non-minimal action. We thus obtain S BV;Ψ = S + Z b a ( c bξ ∂ b v a − v b ∂ b c aξ + 2 c ω v a ) + Z ( v a − ˜ v a ) π a + Z ⋆ b a π a + S (2.16)which is related to the gauge-fixed action S gf by setting all antifields (i.e. all starred fields)to zero: S gf [ φ ] = S BV;Ψ [ φ, ⋆ φ = 0] . (2.17)This has the effect of making the last two terms of S BV;Ψ vanish.The upshot of this so far standard analysis is that the gauge-fixed action inherits theBRST invariance of S BV;Ψ by construction: This is obvious if we rewrite S gf [ φ ] = S + (Ψ , S BV;Ψ ) | ⋆ φ =0 . (2.18)For an irreducible gauge theory such as the chiral brane it is well-known that the followingBRST transformations δ BRST1 φ ≡ ( φ, S BV;Ψ Λ) | ⋆ φ =0 (2.19)square to zero off-shell and thus the above expresion is invariant. We have named thesetransformations “BRST1” for reasons to become apparent.We will now modify these BRST transformations. The modification is by the follow-8ng fermionic “trivial transformation” (in the sense that it vanishes on-shell) with constantanticommuting parameter Λ of ghost number − δ trivial v a = δS BV;Ψ δb a Λ , δ trivial b a = δS BV;Ψ δv a Λ , (2.20)where the functional derivative is defined as δS = R δφ ( δS/δφ ). This is an invariance of S BV;Ψ . The modified BRST transformations acting on S gf are then defined as δ BRST2 φ ≡ (( φ, S BV;Ψ Λ) − δ trivial φ ) | ⋆ φ =0 , ( φ is any field). (2.21)This new BRST variation (which we named “BRST2” to avoid confusion with the one definedin the previous paragraph) automatically satisfies δ = 0 on-shell as a consequence ofthe master equation and the fact δ trivial vanishes on-shell. However for this theory a directcalculation shows that in fact we have δ = 0 off-shell as well, which will be imporantin what follows.We list the BRST transformations in full in the appendix. To verify BRST invariance,we only need calculate δ BRST2 v a = 0 , δ BRST π a = 0 (2.22)and δ BRST2 b a = (cid:18) −
12 Ω AB z A ∂ a z B + ∂ b ( b a c bξ ) + b b ∂ a c bξ − b a c ω (cid:19) Λ . (2.23)The fact v a is BRST-closed partly motivated our choice of BRST variation). Another moti-vation comes from the fact that with this choice of BRST operator we have S gf Λ = δ BRST2 (cid:18)Z − b a v a (cid:19) + Z ( v a − ˜ v a ) π a Λ . (2.24)Both terms on the right-hand side are individually BRST-invariant.The fields π a and v a can be integrated out together. This imposes the gauge condition v a = ˜ v a = const . in S gf , leaving S gf ′ = Z (cid:26)
12 Ω AB z A ˜ v a ∂ a z B − b b ˜ v a ∂ a c bξ + 2 b a ˜ v a c ω (cid:27) (2.25)after dropping a boundary term. The BRST variation after π a and v a have been eliminatedis still nilpotent off-shell, as was verified by direct calculation. As the BRST variations are9he same, we find: S gf ′ Λ = δ BRST2 (cid:18)Z − b a ˜ v a (cid:19) . (2.26)We have thus recovered the result that the action is BRST-exact, now in the gauge-fixedBRST cohomology defined in terms of the modified BRST variation (2.21) (BRST2) above.The BRST variation BRST2 after π a and v a have been eliminated is still nilpotent off-shell. For this reason, whenever we mention the “gauge-fixed” action and BRST variationsin the rest of the paper (e.g. in the context of the gauged chiral brane models we will discussshortly) we will be referring to S gf ′ (i.e. the gauge-fixed action with π a and v a eliminated)and the variation BRST2 (2.21), except as noted. To summarise: the claim so far is that starting from the action (1.1) and gauge fixing thedynamical variable v a to a nonvanishing constant ˜ v a in a BRST procedure leads to thegauge-fixed, BRST-invariant action S gf ′ [ z A , b a , c aξ , c ω ] = Z ˜ v a (cid:26)
12 Ω AB z A ∂ a z B − b b ∂ a c bξ + 2 b a c ω (cid:27) (2.27)which is BRST-exact in the sense S gf ′ Λ = δ BRST (cid:18)Z − b a ˜ v a (cid:19) (Λ is the constant BRST transformation parameter)(2.28)and where the BRST transformations δ BRST ≡ δ Λ defined in (2.21) (the ones named “BRST2”in the previous section) satisfy δ Λ ( δ Λ z A ) = δ Λ ( δ Λ b a ) = δ Λ ( δ Λ c aξ ) = δ Λ ( δ Λ c ω ) = 0 (2.29)identically, i.e. they square to zero off-shell . The explicit formulas for the BRST variationsare (A.1), (A.2), (A.3), and (A.4) ( excluding the terms with IJ K indices for now since wehave not considered the generalisation to the gauged model yet). Their off-shell nilpotencewas verified by hand and also using the computer algebra programme Cadabra (v. 1.39)[14, 15].A perhaps disturbing feature of that calculation is that while the full gauge-fixed action I am grateful to Paul Townsend for pointing this out to me. S by itself is. One might think thatthis might necessarily be the case: after all, in a standard approach the BRST-invariantgauge-fixed action ( S gf or S gf ′ in the previous section) is obtained from the original action S by the addition of a BRST-exact term. In fact, this is exactly what we found, for the BRSTvariation “BRST1” of the previous section, in formula (2.18) which we reproduce here inmore compact notation: S gf Λ = S Λ + δ BRST1 Ψ . (2.30)where Ψ is the gauge-fixing fermion (2.14), Λ is the constant BRST transformation param-eter, and we are using the gauge-fixed action S gf where the fields π a and v a have not beeneliminated yet. The loophole lies in that when we modified the BRST variations from BRST1to BRST2 by terms proportional to equations of motion, the formula analogous to the abovewas modified to (2.24), reproduced here: S gf Λ = δ BRST2 (cid:18)Z − b a v a (cid:19) + Z ( v a − ˜ v a ) π a Λ . (2.31)We see that under BRST1, S is BRST-closed while the term ( v a − ˜ v a ) π a plus fermions isBRST-exact, while under BRST2 S plus fermions is BRST-exact while ( v a − ˜ v a ) π a by itselfis only BRST-closed. The existence of both off-shell nilpotent BRST variations BRST1 andBRST2 is of course a special feature of the chiral brane theory and appears to be intimatelyrelated to the worldvolume vector field v a : δ BRST1 b a = π a Λ ≈ (cid:18) − z A Ω AB ∂ a z B + . . . (cid:19) Λ = δ BRST2 b a . (2.32)The first equality represents the usual form of the BRST variation δ BRST1 b a of an antighost b a . Using the v a equation of motion, this is equal to δ BRST2 b a .The claim that the action is BRST-exact might come across as counterintuitive, for thefollowing reason: consider the original action (1.1) and partially gauge-fix v a = ( v , v i ) =(1 , − ρ i ) (in coordinates σ a = ( t, σ i ), i = 1 , . . . p ) to get Z dtd p σ (cid:26)
12 Ω AB z A ˙ z B − ρ i (cid:18)
12 Ω AB z A ∂ i z B (cid:19)(cid:27) . (2.33)If the real symplectic vector space with coordinates z A is of dimension 2 d , then after anarbitrary choice of d positions and d momenta we see that the action (2.33) takes the stan-dard form for a constrained Hamiltonian system with p spatial diffeomorphism constraints11nforced by the lagrange multipliers ρ i and therefore describes d − p configuration spacedegrees of freedom (since the constraints displayed are first-class).The conclusion is that the chiral brane theory with the BRST operator BRST2, is ap-parently a cohomological theory with local degrees of freedom! A priori this sounds likea contradiction in terms, especially since the action in fact satisfies the stronger propertyof being a BRST variation and one then expects to be able to set it to zero by deformingthe gauge condition. Indeed, setting ˜ v a = 0 makes the action vanish. However this is nota sensible gauge condition: recall that v a transforms as a worldvolume vector field underdiffeomorphisms and is rescaled by Weyl transformations, so it cannot be set to zero in smallneighbourhoods of any point where it is nonvanishing, and moreover the condition v a = 0does not actually fix the gauge. The best one can do is set v a = ˜ v a = (1 , , , . . . ) T .Another relevant observation is that we have not been able to prove that the gauge-fixedaction remains BRST-trivial after eliminating certain ghost fields by their own equations ofmotion: if we set v a = ˜ v a = (1 , , , . . . ) T in coordinates σ a = ( t, σ i ) , t = σ as before, wecan write the gauge-fixed action as S gf ′ = Z dtd p σ (cid:26)
12 Ω AB z A ˙ z B − b ˙ c ξ − b i ˙ c iξ + 2 b c ω (cid:27) (2.34)where we have split b a = ( b , b i ) and c aξ = ( c ξ , c iξ ) T . Then b and c ω can be jointly eliminatedto set b = 0 and c ω = ˙ c ξ / S gf ′′ = Z dtd p σ (cid:26)
12 Ω AB z A ˙ z B − b i ˙ c iξ (cid:27) . (2.35)Since b = b a ˜ v a has been set to zero there appears to be no candidate expression which wecould vary to obtain S gf ′′ .The status of the chiral brane theory as a cohomological field theory thus appears todepend on the choice of ghost fields to be integrated over in the path integral. This isultimately a choice of path integral measure. It is, in particular, possible that consistencyrequires a path integral measure which involves integrations over extra fermion variables.This was actually found to be the case in [16] for the p = 0 case of the chiral brane, which is— in the ungauged case — simply a particle on phase space with vanishing Hamiltonian. Itis therefore conceivable that the full set of ghosts ( c ω , b a , c aξ ) are required by consistency, asopposed to the minimal set ( b i , c i ); in that case whether c ω and b a ˜ v a can be integrated outshould be investigated carefully. Such quantum considerations would, however, have to be12he subject of another paper.Before ending this subsection we note that similar non-conclusions follow for the gaugedchiral brane model, which we will consider next. The gauged chiral brane model has bosonic action S ′ [ z A ; v a , A I ] = S + S int = 12 Z d p +1 σ (cid:8) Ω AB z A ∂ a z B v a − A I M IAB z A z B (cid:9) . (2.36)We will assume the matrices M IAB are constant and symmetric in AB . They are to bethought of as determining a Lie algebra of gauge transformations acting on z A , for which A I is the gauge field and lagrange multiplier. The most straightforward way to see this isto make an arbitrary definition of worldvolume time t and gauge fix v a ∂ a = ∂ t . When thisis the case we have an ordinary phase space action in Hamiltonian form for the phase spacespanned by z A with canonical Poisson brackets { z A , z B } = Ω AB . Then T I ≡ M IAB z A z B (2.37)define a number of constraints on the phase space spanned by z A , and those constraints arefirst-class in the sense of Dirac (and thus correspond to gauge transformations) if the Poissonbracket algebra closes, i.e. (the factor of 2 is conventional) { T I , T J } = 2 f IJ K T K ⇐⇒ M KAB f IJ K = M IAC Ω CD M JBD . (2.38)The quantities f IJ K are thus constant when the algebra closes and are interpreted as thestructure constants of the Lie algebra of the T I constraints. The Jacobi identity for the f IJ K is implied by that of the Poisson bracket.At this point one could construct the BV master action for this gauged model and pro-ceed as before to obtain the gauge-fixed BRST-invariant action, analogous to S gf ′ above. Inthat derivation however, the fact δ = 0 off-shell was only shown by direct calculation.We therefore found it more economical to simply deform the BRST transformations BRST2(2.21) of the ungauged model by adding terms involving the new fields and ghosts (corre-sponding to the new gauge invariances in the gauged theory), and then constrain the relativecoefficients by demanding that the new transformation be nilpotent, rather than go through13he convoluted procedure of the previous subsection.Let us sketch how this works. In the gauge A I = 0, which is an admissible gaugechoice for the above action since the gauge transformation of A I , in contrast to that of v a , is inhomogeneous, the number of possible deformation terms is rather limited: the newterms can only depend on c I (the ghost for the gauge transformations generated by T I ), thecorresponding antighost b I (of ghost number − M IAB , f IJ K ,and Ω AB as well as its inverse Ω AB . At the same time we constrain the BRST variationsto be quadratic in the fields because we expect the putative BV action to be purely cubic,like in the ungauged theory. This fact along with some ghost number counting implies thatthe candidate deformation terms in the BRST variation of the original antighost, b a , are( b I ∂ a c I )Λ and ( ∂ a b I c I )Λ. Assuming that such an off-shell nilpotent BRST variation existswe can therefore immediately write down a gauge-fixed BRST-exact action: S = Z (cid:26)
12 Ω AB z A ˜ v a ∂ a z B − b b ˜ v a ∂ a c bξ + 2 b a ˜ v a c ω − b I ˜ v a ∂ a c I (cid:27) , (2.39) δ BRST (cid:18) − Z b a ˜ v a (cid:19) = S Λ . (2.40)The coefficient on the last term of (2.39) is fixed by the detailed calculation of δ BRST outlinedin the appendix.The only conditions which must hold in order that δ BRST squares to zero off-shell werefound to beΩ AC Ω BC = δ AB , M KAB f IJ K = M IAC Ω CD M JBD , f [ IJ L f K ] LM = 0 . (2.41)The first two are the definitions of Ω AB and f IJ K respectively, while the last is the Jacobiidentity for f IJ K . As the f IJ K are the same structure constants (as in (2.38)) appearingin the canonical Hamiltonian analysis we outlined above , we find that any gauged chiralbrane sigma model has a BRST-exact action (and thus describes a cohomological field theory,subject to the caveats of subsection (2.1.1)) whenever the constraints T I close into a first-class constraint algebra . One can usually determine this closure at a glance. It is then easyto find examples of gauged chiral p -brane sigma models in the literature, at least for the case p = 1. 14 p = 1 : Chiral string examples Here we will show out how the bosonic gauged chiral string action (2.36) is related to anumber of different first order string actions which have previously appeared in the literaturein various contexts. In all cases this will be done by an appropriate choice of the targetsymplectic space alongside a choice of gauge-fixing for the components of v a , and possiblyredefinitions of z A . A lagrangian for the tensionless string in Minkowski space was first proposed by Schildapproximately 40 years ago [3]. We will look at a version given later by Lindstrom, Sundborgand Theodoridis [17] in its phase space form: S [ X µ , P µ ; λ, ρ ] = Z dtdσ (cid:8) ∂ t X µ P µ − λ (cid:0) P (cid:1) − ρ ( ∂ σ X µ P µ ) (cid:9) , (3.1)where λ and ρ are lagrange multipliers. This is related to the action for the usual (tensile)string through the replacement λ ( P ) → λ ( P + ( T ∂ σ X ) ), where T is the tension.This tensionless string action is obtained from (2.36) if we set z A = X µ φ − P µ ! , Ω AB = −
11 0 ! , v a = v t v σ ! = φ − ρ ! , A I = 2 λφ ,M IAB z A z B = φ − P , (3.2)and integrate by parts, where φ (which will drop out) is constrained by the requirement ∂ a v a = 0 . (3.3)We can view this requirement as a gauge-fixing condition for the local Weyl transformationsif v a is assumed to transform as a worldsheet vector density of weight +1 so that ∂ a v a = 0 is adiffeomorphism invariant condition. With that assumption z A and thus X µ must transformas a scalar; then the remaining gauge transformations match fully if we assume P µ transformsas a scalar density of weight +1.We note that v a can be identified up to a Weyl transformation with the vector density V a (of weight +1 /
2) appearing in the tensionless string action after the momenta P µ have15een eliminated: S [ X µ ; V a ] = 12 Z d σ (cid:8) V a V b ∂ a X µ ∂ b X µ (cid:9) , V a = 12 √ λ − ρ ! . (3.4)In [17], after comparison with the tensile string, V a was interpreted as the null eigenvectorthe worldsheet metric acquires in the tensionless ( T →
0) limit. As V a and v a are alwaysproportional the same interpretation holds for v a .It is not entirely clear whether the gauge-fixed BRST-invariant action (2.39) constructedfor the gauged chiral string (2.36) of this paper is automatically equivalent to what one wouldobtain from the first-order form of the Schild action (3.1) directly, as we have not shownwhether all of the necessary redefinitions can be realised as canonical transformations in theBV formalism. However, work by Sundborg [18] suggests that the theory of the Schild stringshould be thought of as “topological”, and its realisation as a cohomological field theorythrough the BRST operator defined in this paper would accord quite naturally with thissuggestion. A twistor action for strings (as well as particles or p -branes, but we will focus on strings)moving through D -dimensional Anti-de Sitter spacetime (AdS D ) was recently put forwardin [4]. The twistor reformulation applies only to tensionless strings, this time in the sense T R → T is the string tension and R is the AdS D radius. Therefore, in contrast to the Schildstring, we are now setting to zero the dimensionless quantity ( T R ), as opposed to T . Inthe context of AdS/CFT, this limit corresponds to vanishing ’t Hooft coupling on the CFTside (see e.g. the discussion by Tseytlin [19]). This AdS D twistor string action is S [ Z ; ρ, A ] = Z dtdσ (cid:26) tr R (cid:20) Z † Ω ∂ t Z − A (cid:18) Z † Ω Z (cid:19)(cid:21) − ρ tr R (cid:20) Z † Ω ∂ σ Z (cid:21)(cid:27) . (3.6)The action depends on the twistor variable Z , which is a 4 × K = R , C , H (where H represents the quaternions). The choice of K determines thedimensionality of the AdS D spacetime through D = dim K + 3. The model has an O (2; K ) = O (2) , U (2) , Spin(5) gauge invariance for K = R , C , H respectively, which is enforced in the16ction by a lagrange multiplier A which is a 2 × K -antihermitian matrix, on top of amore standard worldvolume spatial diffeomorphism invariance enforced by the real lagrangemultiplier ρ . The matrix Ω is the standard 4 × K -hermitian conjugation. We refer to [4,20] for more details on this division-algebra notation.The AdS D twistor string action (3.6) is a special case of the gauged chiral string action(2.36) where v a has been fixed to v a = v t v σ ! = − ρ ! (3.7)and z A is identified with the real components of Z . Since the bilinear form tr R [ Z † Ω Z ]is antisymmetric and nondegenerate in Z and Z , and since the O (2; K ) constraints havealready been verified to close among themselves in [4], the considerations of the previoussection imply that the gauge-fixed BRST action for the AdS D twistor string can be chosento take the form (2.39).The field Z of the AdS D twistor string action (3.6) naturally transforms as a worldsheetscalar density of weight +1 / z A of the gauged chiral stringfor the choice x = 0. This follows from a straightforward calculation of the gauge transfor-mations generated from the Poisson brackets of the constraint tr R [ Z † Ω ∂ σ Z ] enforced by ρ .In this sense, x = 0 is singled out for this model, and no canonical transformation (in theBV sense) is necessary to make the gauge transformations of the gauged chiral string matchthose of the AdS D twistor string. There is thus no reason to suspect the two actions describedifferent theories. We note that this reformulation of the AdS D twistor string clarifies theworldsheet diffeomorphism invariance of the model, which is somewhat opaque in the originalaction (3.6). The name “ambitwistor string” refers to a class of models introduced by Mason and Skinnerin [5]. The simplest version of the original model is S [ X µ , P µ ; e ] = Z Σ (cid:8) P µ ¯ ∂X µ − eP (cid:9) . (3.8)This looks similar to the Lindstrom-Sundborg-Theodoridis action for the tensionless Minkowskistring (3.1) which we treated above, as has already been discussed in the literature [21].17owever there are some important differences which are relevant if we are to understand theprecise relation to the gauged chiral string action (2.36).The worldsheet Σ of the ambitwistor string is assumed to be a Riemann surface withholomorphic coordinate σ ∈ C (with conjugate ¯ σ ). ¯ ∂ then denotes ∂ ¯ σ . The dynamical fields( X µ , P µ , e ) appearing in the ambitwistor string action (3.8) are all complex-valued, and thusthe action itself is complex, in contrast to the Lindstrom-Sundborg-Theodoridis action (3.1),the gauged chiral string action (2.36), and all other action functionals which have appearedin this paper so far . The ¯ ∂ operator is also complex but that is less of an issue since ¯ ∂ couldarise from Wick rotating a derivative along a worldsheet lightcone direction.To relate the ambitwistor string to our gauged chiral string one must thus considera complexified version of (2.36), where z A , v a and A I are all complex. Fortunately thecalculations of section 2 are unchanged for the complexified model. We can then consider(2.36) in holomorphic ( σ, ¯ σ ) coordinates and set z A = X µ P µ ! , Ω AB = −
11 0 ! , v a = v σ v ¯ σ ! = ! , A I = 2 e , M IAB z A z B = P (3.9)to obtain the bosonic ambitwistor string action (3.8). While the actions do match after gaugefixing, the variables z A and X µ , P µ transform differently under worldsheet diffeomorphismsand, much like we saw for the Schild string in subsection 3.1, it is not clear that the BRST-invariant action (2.39) derived for the (complexified) gauged chiral string is also appropriatefor this ambitwistor string.There is another class of ambitwistor string models however, whose interpretation asgauged chiral strings does not suffer from this ambiguity. These are the “four-dimensionalambitwistor strings” of [6]. The bosonic action of that model reads S [ Z A ′ , W A ′ ; a ] = Z Σ n W A ′ ¯ ∂Z A ′ − Z A ′ ¯ ∂W A ′ + a ( Z A ′ W A ′ ) o (3.10)where Z A ′ ∈ T ∼ = C and W A ′ ∈ T ∗ are worldsheet spinors, i.e. transform with densityweight +1 /
2, as do the z A of the gauged chiral string (with the choice x = 0). One can thus The AdS D twistor string action is also real. This can be seen immediately from the presence of thetr R [ − ] (real trace) operation in (3.6). z A = Z A ′ W A ′ ! , Ω AB = −
11 0 ! , v a = v σ v ¯ σ ! = ! , A I = − a , M IAB z A z B = Z A ′ W A ′ . (3.11)An interesting point arises if we consider the Weyl transformations of the complexifiedchiral string model (before the gauge fixing we described just now), which send ( Z A ′ , W A ′ ) → ( ωZ A ′ , ωW A ′ ) where the local scaling ω = ω ( σ ) is now complex-valued. At the same time,the constraint Z A ′ W A ′ = 0 will also generate a local scaling, where now Z and W transformoppositely, i.e. ( Z A ′ , W A ′ ) → ( ω ′ Z A ′ , ( ω ′ ) − W A ′ ). After we quotient by both transformations,and ignoring the fact that the ω transformation also acts on v a , we find that the target spaceof the model is the locus Z A ′ W A ′ = 0 in PT × PT ∗ , i.e. projective ambitwistor space. By now it should be clear how string actions with ultralocal, quadratic constraints can be seenas special cases of the (gauged) chiral string action (2.36). For two further examples, we pointout the SU (2 , tensionless twistor string (i.e. the one with K = C ) described above by removingthe U (2) constraints), and the Hohm-Siegel-Zwiebach string [8], at least in its “halved” form(formula (2.16) of that paper). We have introduced a sigma model action (1.1) describing maps from a p -brane worldvolumeinto a symplectic target space. Besides its relevance for a number of previously consideredtheories, this action is interesting for some of its technical features: its Weyl invarianceacting on both target space as well as the worldvolume vector field v a , its diffeomorphisminvariance in spite of the absence of a worldvolume metric and the closely related fact that itcan apparently be written as a total BRST variation (after the ghost sector is introduced).The last claim is subject to technical caveats detailed in subsection 2.1.1. To sum-marise that discussion: it is true that there is a choice of off-shell nilpotent BRST operator(“BRST2”, defined in (2.21) for the ungauged model and explicitly by formulas (A.1), (A.2),(A.3), (A.4), (A.5) and (A.6) in the general case) such that the action is BRST-trivial, but19e have not been able to verify that this remains true after some ghost variables are elimi-nated. The choice of ghost fields to be integrated over in the path integral is closely relatedto the choice of measure and as such, this issue would be clarified when the quantum theoryof the chiral brane model is formulated, perhaps along the lines of [16].Assuming that the BRST operator we define is in fact appropriate, the chiral branetheory is an example of a cohomological field theory (also known as “topological”). Suchtheories enjoy a number of special properties. To see some of them, let us briefly reviewan argument due to Witten [22] (reviewed in [10] and [23]) that shows the semi-classicalapproximation to the path integral for such theories is exact. Let us assume the absence ofBRST anomalies. In the path integral formulation this is equivalent to assuming the measureis BRST invariant [24, 25]. If we also assume that the gauge-fixed action S entering the pathintegral is BRST invariant, invariance of the measure can be equivalently expressed as ∀O : h δ BRST
Oi ≡ Z D φ ( δ BRST O ) exp( iS/ ~ ) = 0 . (4.1)Therefore, if V denotes any product of BRST-invariant operators (i.e. δ BRST V = 0), while O is still an arbitrary operator, we must have ∀O : hV δ BRST Oi = 0 . (4.2)Now consider varying any correlation function hVi with respect to ~ . If S = δ BRST B as isthe case for the chiral brane then ∂∂ ~ hVi = − i ~ − hV δ BRST B i = 0 . (4.3)Therefore, assuming these formal manipulations are not obstructed (by e.g. the nonexistenceof an appropriate path integral measure), the theory is independent of the value of ~ andwe can calculate in the limit ~ →
0, where the path integral localises on the solutions of theequations of motion. A similar argument implies the theory cannot (continuously) dependon the gauge choice ˜ v a . Since a worldvolume metric only enters the theory through thatgauge choice it should follow that the theory also does not depend on that. Some of theseproperties were anticipated in [21] (for the ambitwistor string).Given that the Schild string (3.1) and the AdS D twistor string (3.6) actions were bothderived from tensionless limits of the usual tensile string, the above properties of the BRSToperator suggested in this paper appear sensible: after all the string tension, or more ac-20urately its inverse α ′ ∝ T − , plays the role of ~ , but as we have already taken the limit α ′ → ∞ in deriving those actions, it would be strange if another loop-counting parametersomehow appeared in these theories. It is thus fortunate that these theories turn out tobe independent of the value of the prefactor in the action. Note that we are referring toworldsheet rather than target space loops here: it should be possible to obtain target spaceloop corrections by considering worldsheets of arbitrary genus. In effect what the previousargument shows is that there is no α ′ expansion, but there might still be a g S expansion. Ofcourse these conclusions will only follow if the quantum theories are critical (i.e. no BRSTanomalies), which will only be true in certain dimensions and/or for some particular numberof supersymmetries. For this reason it would be interesting to consider the supersymmetricgeneralisation of the chiral brane action.On the other hand, it is not clear whether the choice of BRST operator proposed in thispaper is appropriate for ambitwistor strings for a number of reasons. For one, there appearsto be no consensus in the literature on whether the diffeomorphisms of the ambitwistor modelare gauged [26], whereas the diffeomorphisms of the chiral string introduced in this paper are,of course, gauged. For another, the argument above is only valid in the absence of BRSTanomalies, i.e. in the critical dimension, while ambitwistor strings are often consideredoutside of their critical dimension (e.g. in [6]). Ambitwistor string amplitudes do seemto enjoy certain localisation properties however and it would be natural to assume thislocalisation is realised through the mechanism described in this paper. Acknowledgements
This work was prompted by an interesting discussion involving Alec Barns–Graham, DavidSkinner, and Jack Williams.I would like to thank Kai Roehrig for a clarification regarding ambitwistor strings, Ko-standinos Sfetsos and the Faculty of Physics at the University of Athens for their kindhospitality, and Paul Townsend for discussions. I would also like to thank Kenny Wong forbringing the review [23] to my attention. 21
BRST transformations
The BRST transformations under which the gauge-fixed action (2.39) for the gauged chiralstring is invariant are δ BRST z A = (cid:18) c bξ ∂ b z A + 12 ∂ b c bξ z A − c ω z A + c I M IBC Ω AB z C (cid:19) Λ (A.1) δ BRST c a = ( − c bξ ∂ b c aξ )Λ (A.2) δ BRST c ω = ( − c bξ ∂ b c ω )Λ (A.3) δ BRST b a = (cid:18) −
12 Ω AB z A ∂ a z B + ∂ b ( b a c bξ ) + b b ∂ a c bξ − b a c ω + b I ∂ a c I (cid:19) Λ (A.4) δ BRST c I = (cid:18) − f JK I c J c K − c aξ ∂ a c I (cid:19) Λ (A.5) δ BRST b I = (cid:18) M IAB z A z B − f IJ K c J b K − c aξ ∂ a b I − ∂ a c aξ b I + 2 c ω b I (cid:19) Λ (A.6) δ BRST ˜ v a = 0 . (A.7)Λ is the constant, anticommuting BRST transformation parameter of ghost number gh Λ = − IJ K indices are absent. The BRST transformationsfor the gauged model were constructed by perturbing those of the ungauged one by quadraticterms involving
IJ K indices, and then demanding that the resulting BRST transformationsquares to zero off-shell , i.e. δ ≡ δ BRST;Λ δ BRST;Λ φ = 0 (A.8)for any Λ , Λ and where φ is any of the above fields. The vanishing of the left-hand side fixesthe coefficients in the BRST variation. The values for those coefficients were obtained withthe help of the computer algebra programme Cadabra (v. 1.39) [14, 15], and the off-shellnilpotence of the BRST variation derived thereby was subsequently verified by hand.22ore precisely: we considered the ansatz δ BRST b a = (cid:18) −
12 Ω AB z A ∂ a z B + ∂ b ( b a c bξ ) + ∂ b ∂ a c bξ − b a c ω + θ b I ∂ a c I (cid:19) Λ (A.9) δ BRST c I = (cid:18) κ f JKI c J c K + α c aξ ∂ a c I + β c ω c I + γ ∂ a c aξ c I (cid:19) Λ (A.10) δ BRST b I = (cid:18) µ M IAB z A z B + λ f IJ K c J b K + ζ c aξ ∂ a b I + ǫ ∂ a c aξ b I + η c ω b I (cid:19) (A.11)where the variations for the other fields are as above and the real parameters { α, β, γ, ǫ, ζ, η, θ, κ, λ, µ } were determined along with the relations between the constant tensors M IAB , Ω AB , Ω AB and f IJ K (assumed to have the (anti)symmetry properties described in the text). We then found δ z A = 0 ⇐⇒ α = − , β = 0 , γ = 0 , (cid:18) κ M KAB f IJ K + M IAC Ω CD M JBD (cid:19) Ω EA = 0 , (A.12) δ c I = 0 = ⇒ f [ IJ K f L ] K M = 0 , (A.13) δ b I = 0 = ⇒ ǫ = − , ζ = − , η = − , λ = κ , (A.14) δ b a = 0 = ⇒ Ω AC Ω BC = + δ AB , µθ = 1 , (A.15)where in deriving each implication we have already substituted in the values of the parametersfixed above. The parameters κ and either µ or θ are arbitrary, and were fixed to κ = − , θ = µ = 1 to produce the transformations above. The upshot is that the BRST transformationsare nilpotent if conditions (2.41) on M IAB , Ω AB , Ω AB and f IJ K are satisfied.
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