$\beta\gamma$-systems interacting with sigma-models

We find a geometric description of interacting $\beta\gamma$-systems as a null Kac-Moody quotient of a nonlinear sigma-model for systems with varying amounts of supersymmetry.


Introduction
Generalized βγ-systems arise in many contexts -including string theory and conformal fieldtheory; many papers have explored their quantum properties -see, e.g. [1]. In this paper, we explore the geometry of such systems interacting with general nonlinear sigma-models. We restrict our attention to left-moving βγ-systems, but the extension to include right-moving systems is straightforward. Our paper is only indirectly related to the work on chiral bosons -see, e.g. [2]. After completing this work, the relevance of [3] was pointed out to us -it studies quantum and mathematical aspects of certain models related to the ones we describe here; our work focuses on a covariant geometric, albeit classical, description using (supersymmetric) sigma-models.
Consider a free βγ-system, that is a system with bosonic fields with a chiral action 1 S b = d 2 x b∂c , (1.1) which has field equations∂ c = 0 ,∂b = 0 .
( 1.2) We assume that b ≡ b + + has spin one, and c is a scalar. Clearly this system in not a sigma-model, and the target space is not a manifold in the usual sense. We can find a geometric description of this system as follows: we reinterpret b as the gauge connection of a Kac-Moody symmetry on a certain manifold with indefinite signature. We start witĥ S = d 2 x ∂q∂c , (1.3) which is a sigma-model with target space R 1,1 . This has a (right-moving) Kac-Moody symmetry 2 δq = λ , ∂λ = 0 (1.4) (Clearly, it also has a left-moving Kac-Moody symmetry, but we are not interested in it). If we gauge this Kac-Moody symmetry by introducing a connection b ∂q → ∇q := ∂q + b , (1.5) we can choose a gaugeq = 0, and the gauged version of (1.3) reduces to (1.1). We thus have found a geometric interpretation of our βγ-system: it is a chiral or Kac-Moody quotient along a null killing vector of a sigma-model with target space R 1,1 .
In this paper, we generalize this to interacting systems with various amounts of supersymmetry. Throughout this paper, we have assumed that the fields b, c are commuting, as c corresponds to a coordinate on a target space manifold. However, very little changes if we let b, c be anticommuting -we are just studying sigma-models into a target supermanifold.
In Sec. 2, we consider a broad class of generalized bosonic βγ-systems and find their geometric interpretation. In Sec. 3, we repeat the exercise in (1,1) superspace; the couplings to the fermions clearly reflect the underlying geometry in a nontrivial way. In Sec. 4, we increase the supersymmetry to (1,2); in this case the geometric sigma-model is a pseudo SKT geometry (strong Kähler with torsion), and the chiral quotient is different from the usual (1,2) quotient. In Sec. 5, we describe the same system in (2,1) superspace; in this case, the usual quotient gives the βγsystem. One significant difference is that left-moving βγ-systems are necessarily complex in (1,2) superspace but not in (2,1) superspace. In Sec. 6, we consider (2,2) superspace. In this case, these models arise naturally in terms of semichiral superfields, and we find a pseudo generalized Kähler geometry. Finally, in Sec. 7, we discuss our results and further possible developments.

Bosonic models
In this section, we introduce the general bosonic sigma-model interacting with a commuting spin one left-moving βγ-system, and discuss its properties. We then find a geometric sigma-model whose quotient by a null symmetry gives the interacting βγ-system, and discuss its properties. Finally, we discuss various special cases of interest.

Definitions and properties
Let E AB = 1 2 (G AB + B AB ) be the sum of the metric and the B field, and consider where we combine the sigma-model fields φ i with c α and write a generic coordinate As long as it is invertible, we can always choose A α β = δ α β by redefining b, which gives: Then we can absorb E Bα by a shift of b α : Dropping the ′ , we are left with which we call the minimal frame. The action (2.1) then reads The field equations that follow from extremizing (2.6) are 3 where we have used Part of our purpose is to find a geometric interpretation of these equations, which we do below.
We now discuss the formal symmetries of the action (2.6). We expect these to include diffeomorphisms and B field gauge transformations, modified so that they preserve the minimal form of E in (2.5). To this end we note that the action (2.6) is invariant under two symmetries which do not preserve (2.5), and therefore can be used as compensating transformations to restore the minimal frame. The first does not transform the coordinates: The second is any transformation that preserves the sigma-model term in the action and transforms the rest as The B-field transformation 2.12) preserves the action but not the form of E (2.5). To restore the form we add an κ-transformation (2.10) with parameter Thus we find where the operator The reparametrization symmetries 4 preserve the sigma-model part of the action (2.6) but not the form of E (2.5). To restore the form of E, we use a κ (2.10). Since the second term in (2.6) depends on φ A , we also need a µ transformation (2.11) to make the action invariant. The parameters are (2.20) Since E Aα = 0, we need to check that its variation vanishes; using E Aα = 0, we find Thus we find

The Bosonic Geometric Model
To understand the geometry of the model, we use the same strategy as in [4]: We think of b α as a connection and the term as a gauge fixed version of This identifies A α A as the sum of metric and B-field 2.25) in the ungauged sigma-model with additional coordinatesq α . The resulting geometry has a Kac-Moody isometry 5 : ∂E ∂qα = 0. The Lagrangian for this extended (ungauged) model is where we have introducedqα :=q α for convenience. In general EÃB is given bỹ which gives rise to the metricGÃB The nonzero components of the connectionsΓ

The minimal frame
In the particular frame (2.3),(2.5) the matrix (2.28) reduces tõ and we note thatẼα B P B i = 0. The corresponding metric is (2.32) which in general is invertible: In particular, this implies thatGÃB is invertible in the general frame (2.28). We note that vectors of the form (0, v α , 0) and (0, 0,vα) are all null in the metric (2.32). The metric (both in the minimal and the general frame) has signature (n, k, −k) where i = 1 . . . n, and α,α = 1 . . . k, as long as the interaction terms E αj , A α j are not too large. The field equations for the extended sigma-model may be used to write those of the original model as follows where (2.35) is the derivative of (2.7), and we usẽ recall we useqα ≡q α for notational convenience.

Discussion
We have seen that the model with the left-moving fields b α , c α is a chiral quotient (Kac-Moody quotient) of a geometric sigma-model. We have assumed that b α , c α are commuting, but aside from some obvious signs, the discussion would not change if some or all of them were anticommuting -in that case the target space becomes a supermanifold, but the quotient proceeds in the same way.
In the general case (2.1), for E and A to be functions of c, we require c to be a scalar, and hence b is a vector b + + on the world sheet. A particular special case arises when for some functions A α ; then the second term in the action becomes and the functions A α are simply left-moving on-shell. We can change coordinates such that . Then this term looks free, and all the interactions come through the dependence of E on c ′ .
When (2.38) is satisfied, the connections (2.30) take a particularly simple form -the nonvanishing components are:Γ ( 2.40) When inserted into the definition of the curvature ((3.11) below), the curvature has no components with hatted indices.

(1, 1) Supersymmetry
In this section we straightforwardly generalize the bosonic case -both the interacting left-moving βγ-system and the sigma-model whose quotient gives rise to it.

The (1, 1) βγ-system
The Lagrangian (2.1) is immediately generalized to (1, 1) superspace: where the scalars φ and the spinor β are (1, 1) superfields in representations of the supersymmetry algebra given in Appendix A.1. As in the bosonic case, we combine the sigma-model fields φ i with c α and write a generic coordinate Again, we can chose the E and A in the special forms (2.5) and (2.3) using the same arguments to redefine β. Then the action has modified diffeomorphisms (2.22) and B-field symmetries (2.15).
As above, when A α B = A α , B is a gradient, the second term in the action simplifies to and the β field equation implies that the A α are left-moving on shell: To reduce (3.1) to components we shall need the following definitions 6 : The calculation of the component Lagrangian is straight forward albeit not very illuminating.
In its place we follow the strategy of Sec. 2.2 to find the ungauged geometric Lagrangian and reduce that instead.

The (1, 1) Geometric model
The Lagrangian for this higher-dimensional sigma-model is where the geometry is as in Sec. 2.2 with all fields now superfields. In particular, we have We define components as We collect terms and integrate by parts to get: CDÃB is the Riemann curvature of Γ (+) : Separating out the i, α andα components is not particularly rewarding. However, we observe that it follows from the relations (2.30) and the fact that ∂ ∂qα is an isometry, that theÃ andB indices of R (+) ABCD can never beα orβ. Since the metricGÃB is invertible, we can eliminate the auxiliary fields FÃ: The details are given in the minimal frame in Appendix B.
The∇-covariant derivatives in (3.10) arẽ whileB =β yieldsGÃβ∇ Similarily we have for the∇ terms in (3.10): and forB =βGÃβ∇ Using these formulae we rewrite the action (3.9) as To make contact with (3.1) we first gauge the Kac-Moody isometry ∂ ∂qα by replacing (recall (3.7) tells us φα ≡qα) D +qα → ∇ +qα := D +qα + β α+ , (3.20) in analogy to (2.24), and choose a gauge where Comparing the components ofqα from (3.8) to those of β α+ in (3.5) we see from (3.21) that in our gauge. With this identification it is clear that the auxiliary fields agree In addition we find from (3.23) that if we substitute β α+ = D +qα , we get In the action,qα and ψα − only appear in these combinations. We thus find the components of (3.1) with all F auxiliary fields eliminated: We note that η is a fermionic auxiliary field whose equation 3). Thus we have found a geometric form of the component action corresponding to (3.1), including complicated interaction terms of the fermions. We also observe that when A α B = A α , B holds, the b + + , β + terms collapse to the component expansion of the semifree action (3.3): (3.28)

(1, 2) Supersymmetry
For the bosonic and the (1, 1) models, the relation between the sigma-model and its gauge-fixed reduction is straightforward. When we go to (1,2) supersymmetry, the natural extensions do not have the same clear relation.

The (1, 2) βγ-system
Our starting point is the (1, 2) action for a βγ-system coupled to a sigma-model: The supersymmetry algebra is given in Appendix A.2, and J is a diagonal matrix such that J 2 = −1: it is +i on holomorphic vectors and −i on antiholomorphic vectors.
Reducing (4.1) to (1, 1) components, as described in Appendix A.2, we find (3.1) with nonzero components: where we have chosen a particular gauge for the B-field in E [6]. More covariantly, we can write: There are two ways we can satisfy A α B = A α , B (cf. (2.38)) : when A a , B = 0, then A a is antichiral: D − A a = 0. Then we can make a change of coordinates to replacecā by A a . The β equations of motionD − A a = 0 imply that A a is left-moving as in (3.4); the complex conjugate works in the same way.
An alternative is to use a real 7 A α ; since the β field equation impliesD − A a = 0 and theβ field equation implies D − Aā = 0, then A α is left-moving.
In contrast to the previous cases in Secs. 2 and 3, here we can only shift β by chiral functions due to (4.2), which means we cannot choose the minimal form (2.5) in (1,2) superspace.

The (1, 2) Geometric model
Alternatively, we start from a general (1,2) sigma-model with isometries generated by ∂ ∂qâ : 5) where now Because of (4.3), the isometries (and their complex conjugates) imply thatẼ has the form (2.28) We could try to gauge the imaginary part of the isometries in chiral representation as described in [7]; in contrast to the case of (1, 1) superspace above, this does not give the correct quotient model, and so we need another procedure.
The key observation is that the action (4.5) actually has a Kac-Moody symmetry: we can shiftq a ≡qâ by right moving chiral parameters λâ obeying This can be promoted to a local symmetry with a (1, 2) chiral gauge parameter Λâ by introducing a novel chiral connection βâ + ≡ β a+ obeyingD − β a+ = 0, which gives D +qa → ∇ +qa := D +qa + β a+ , (4.10) where δq a = Λ a , δβ 11) and similarly for the complex conjugate. When we choose the gaugeq =q = 0, we recover (4.1) with A a ≡ kâ. This is the correct complexified version of the (1, 1) story.

(2, 1) Supersymmetry
It is interesting to describe the same geometry in (2,1) superspace. Here the description of the βγ-system is quite different; in particular, as the complex structure appears in the opposite sector, there is no need to complexify the βγ-system. The quotient needed to descend from the geometric model to the βγ-system is the usual quotient [7], as in the bosonic and (1, 1) cases.

The (2, 1) βγ-system
Our starting point is the (2, 1) action for a βγ-system coupled to a sigma-model; in this case, the form of the action appears geometric, but the ghost fields c α are described by unconstrained scalar fields X α .
whereas X α are unconstrained, and J is a complex structure as in the previous section. The supersymmetry algebra is given in Appendix A.3.

The (2, 1) Geometric model
In (2, 1) superspace, the geometric sigma-model is straightforward to find. Just as in (2.24), we identify X as a connection gauging a symmetry of a general (2, 1) sigma-model by letting where c is a chiral superfield:D + c = 0 , D +c = 0 .
Thus the ungauged geometric sigma-model is found by letting X α → c α +c α (5.9) and gives an action To compare to the (1, 1) geometric model, we need to interpret c +c as the real ghost field c and i(c − c) asq in (3.7): (5.12) In this basisẼ has the form (2.28)ẼÃB with the components of E and A given (5.3).
The sigma-model that we get after (5.9) has the obvious null isometry: This is actually a Kac-Moody symmetry, because c α +c α is invariant under We can gauge the symmetry following [7] -we start by introducing an unconstrained real scalar superfield V , which we identify with X and let c α +c α → X α + c α +c α . (5.16) This combination is now gauge invariant under the complexified gauge transformations: Because only this combination enters in the gauged action, the gauge connection Γ − does not appear in the action. Hence when we choose the gauge c =c = 0, we recover (5.1).

Models with only right semichirals
As pointed out in [8] a model with only right semichiral fields describes a multiplet of free left moving bosons and left moving fermions. Here we briefly recapitulate this. We use a notation consistent with the previous sections of this paper, albeit differing from the literature on semichiral multiplets [9] and label the right semichiral fields by indices {α} ≡ {a,ā}: The (2,2) action is The (2,2) field equations that follow from this arē (6.3) and the complex conjugate 8 . In the last equality we assume that K ab is invertible. Using the results of Appendix A, we find that (6.3) corresponds to the (1, 1) equations: (6.4) where Ψ α + := −J α β Q + X β .
Observe that when there is an isometry, e.g., when K(ϕ,φ, X +X), A a = Aā as discussed below (4.2); then (2.38) is satisfied, and A α is left-moving. This can be seen directly in (2,2) superspace, as the X,X field equations imply D − K X =D − K X = 0 (cf. Sec. 6.1).
We now substitute (6.11) into (4.4); we must remember to identify J B A from Sec. 4 with J (−) . We then find the geometric quantities E and A which are used to write the (1, 1) superspace action: (6.12)

Reduction to (2, 1) superspace
The reduction of the model to (2,1) superspace is simpler. We use (A.25) and (A.27) to find (6.13) Here φ i are (anti)chiral (2, 1) superfields, J (−) is as discussed in Sec. 6.2.1, and X are complex unconstrained (2,1) superfields. To compare to Sec. 5, we could decompose them into their real and imaginary parts, but it is more convenient to keep the complex coordinates. We need to recall the J i j in Sec. 5 is now J (+) . Then we find Computing the (1, 1) quantities by substituting these into (5.3) gives exactly the same answer as above, namely (6.12).

The (2, 2) Geometric model
To relate the βγ-system to a (2, 2) sigma-model, we mimic the ALP construction of [4]. This is based on the interpretation of semichiral superfields as gauge fields for certain symmetries in a sigma-model with chiral and twisted chiral superfields. We thus consider the action where X a := Φ a +χ a ,Xā :=Φā + χā (6.16) with Φ and χ chiral and twisted chiral fields, respectively. The target space geometry is thus a torsionful geometry with a left and a right complex structure covariantly constant with respect to two torsionful connections 10 .
The action is invariant under a complex Kac-Moody symmetry that preserves X α : where The quotient described below is analogous to what we found in Sec. 4.2, namely a novel gauging for Kac-Moody symmetries.
To reduce to to (1,2), we use We find (4.5) with 6.20) whereĴ is J (+) when written in a coordinates ϕ, X, Y . Writing out the various indicies we have: and similarly for the complex conjugates. The ± is + for chiral superfields and − for twisted antichiral superfields, which are both chiral with respect to J (−) ; see (6.8). Identifying Yα :=qα, we recover a special case of (4.5).
Just as in the (1,2) case, the standard gauging [12] does not reduce the model to (6.5); instead, we gauge the Kac-Moody symmetry (6.17) as in [4]. We introduce a right semichiral field X α K(ϕ i , X α ) → K(ϕ i , X α + X α ) . (6.22) This potential is now invariant under where Λ a is chiral andΛ a is twisted antichiral. Clearly we can then choose a gauge where we gauge away φ a ,χ a ; then and we recover the form (6.5), now with knowledge about the underlying sigma-model geometry.

Discussion
We have found a geometric way of understanding βγ-systems coupled to sigma-models with varying amounts of supersymmetry: as quotients along null Kac-Moody isometries of conventional sigma-models.
We have studied the case with only left-moving β and γ, and have only concerned ourselves with the classical geometric aspects -in particular, we have not concerned ourselves with quantization and sigma-model anomalies, as discussed, e.g., in [1], [2], [3]. We expect the inclusion of right-moving βγ-systems to be straightforward; by describing left-moving βγ-systems in both (1,2) and (2,1) superspace, the methods to treat the right-moving systems are apparent.
For (2,2) supersymmetric models, we have only considered sigma-models described by chiral and twisted chiral superfields; we expect the extension to the general case, including further left and right semichiral superfields, to be straightforward. Other superfield representations, namely complex linear and twisted complex linear superfields are equivalent to models with chiral and twisted chiral superfields.
It would be interesting to see if these considerations can be extended in any way to "higher dimensional βγ-systems" [13]. and to Lars Hierta's foundation for partial support. MR thanks NSF-PHY-1915093 for partial support.

A Superspaces
In these appendices, we discuss the superspace for various superalgebras. Sigma-models have target space geometries that depend on the amount of supersymmetry. For (1,1), the geometry is (pseudo)Riemannian with a natural connection with torsion; for (1,2) or (2,1), the geometry is (pseudo) strong Kähler with torsion; and for (2,2), the geometry is (pseudo) generalized Kähler.

A.1 (1, 1) superspace
The (1, 1) superalgebra is generated by spinor derivatives D ± that obey The (1,1) superfields are unconstrained, and gauging is done with a spinor connection D ± → ∇ ± = D ± + β ± . The superspace action is written using the measure D + D − as follows: The (1,2) superalgebra is generated by the real spinor derivative D + and the complex spinor derivatives D − ,D − .
Right-(anti)chiral superfields obeyD − φ = 0, D −φ = 0, resp. Usual gauging involves a left-spinor connection β + and a real potential V -see [7] for the details of the analogous (2, 1) case. As shown in Sec. 4.2, we need a different kind of gauging that is suitable for Kac-Moody symmetries.
We reduce to (1,1) using 4) from which it follows the superspace measure becomes When we push in Q − to find the (1, 1) action for chiral superfields, we use, e.g., 6) which can be written covariantly for {φ i } = {φ i ,φī} as A.3 (2, 1) superspace The (2, 1) superalgebra is generated by the real spinor derivative D − and the complex spinor derivatives D + ,D + .
Left-(anti)chiral superfields obeyD + φ = 0, D +φ = 0, resp. Usual gauging involves a left-spinor connection β − and a real potential V -see [7] for the details. As shown in Sec. 5.2, we need a different kind of gauging that is suitable for left Kac-Moody symmetries generated by parameters obeying∂λ = 0.
We reduce to (1, 1) using from which it follows the superspace measure becomes When we push in Q + to find the (1, 1) action for chiral superfields, we use, e.g., which can be written covariantly for {φ i } = {φ i ,φī} as On the other hand, for an unconstrained superfield X, Q + X is independent as a (1, 1) superfield: .13) A.4 (2, 2) superspace The (2, 2) algebra of covariant derivatives is (A.14) and the complex conjugate relations.
Chiral superfields Φ a satisfy:D ± Φ a = D ±Φā = 0 , (A.15) but in d = 2 we may also introduce twisted chiral fields χ that satisfȳ as well as left and right semichiral superfields; in this paper we only use 11 right semichiral superfields which obeyD To display the physical content we may rewrite an action in (1,2) superspace. By analogy to (A.4), we descend to (1,2) superspace by defining the left-handed real spinor derivative (A.18) and the generator of second supersymmetry They satisfy The (2, 2) measure reduces to In (1,2) superspace, all superfields are either unconstrained or chiral; we now explain how (2,2) superfields decompose into their (1,2) components. From (A.15), we find where J is the canonical complex structure (diagonal +i, −i). Similarly, from (A.16), we find However, Φ,χ are the (1, 2) chiral superfields, which we collectively denote as φ. To distinguish (2,2) and (1,2) chirality properties, we use the notation J (+) and J (−) as explained in Sec. 6.2.

B Minimal frame components
Here we work out the detailed form of various quantities in the minimal frame of Sec. 2.3 (in particular, see (2.5), (2.3)). For the bosonic auxiliary field equations, when the indicesB = B in (3.12), the equations read Choosing B = β and B = j in turn in (B.1) yields ForB =β (3.12) readsG The∇-covariant derivatives in (3.10) arẽ (B.5) whileB =β yieldsGÃβ∇ Similarily we have for the∇ terms in (3.10): (B.8) and forB =β We next work out the details of the component action in the minimal frame.