Conformal Invariance of (0,2) Sigma Models on Calabi-Yau Manifolds
aa r X i v : . [ h e p - t h ] F e b Conformal Invariance of (0 , Sigma Models onCalabi-Yau Manifolds
Ian T. Jardine † and Callum Quigley † † Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada
Abstract
Long ago, Nemeschansky and Sen demonstrated that the Ricci-flat metric on aCalabi-Yau manifold could be corrected, order by order in perturbation theory, toproduce a conformally invariant (2 ,
2) nonlinear sigma model. Here we extend thisresult to (0 ,
2) sigma models for stable holomorphic vector bundles over Calabi-Yaus. [email protected] [email protected] Introduction
The study of strings propagating on Calabi-Yau backgrounds has a long and rich history.While it was known from early on that Ricci-flat K¨ahler manifolds provided supersymmetricsolutions of the classical supergravity equations, there was a period of confusion regardingtheir fate once α ′ -corrections were taken into account. On the one hand, [1] seemed to provethat the Ricci-flat metric of a Calabi-Yau led to a finite nonlinear sigma model (NLSM), with(2 ,
2) supersymmetry, to all orders in worldsheet perturbation theory. However, explicitcomputation of the (2 ,
2) NLSM beta-function showed that it had a non-zero contributionat four-loop order [2]. The resolution of this seeming paradox was presented in [3], andboiled down to the freedom of adding finite local counterterms at each order in perturbationtheory. In more detail, the authors of [3] showed that one could add corrections to the Ricci-flat metric at each loop order, so that the full, all-order beta function is satisfied. This canbe interpreted as a (non-local) field redefinition of the metric, or as a modification of thesubtraction scheme, so that Ricci-flatness amounts to finiteness to all orders.The discussion of the previous paragraph was entirely in the context of (2 ,
2) NLSMs onCalabi-Yau backgrounds, but it is very natural to ask how it carries over to the heteroticsetting with only (0 ,
2) supersymmetry. In fact, given the intense focus on phenomenolog-ical heterotic models at the time of [3], one easily wonders why such an investigation didnot occur decades ago. One likely explanation is that around the same time it was shownthat generic (0 ,
2) models suffered from non-perturbative instabilities [4], and so investi-gating their perturbative renormalization may have seemed inconsequential. However, wenow know that the superpotential contributions from individual worldsheet instantons canactually sum to zero in many (0 ,
2) models [7–9], thereby eliminating those instabilities.While this provides renewed motivation for studying the issue of finiteness of (0 ,
2) modelson Calabi-Yau backgrounds , during the interim powerful spacetime arguments emergedthat answered the question in the affirmative to all orders in perturbation theory [13]. Inthis note, we revisit this question and confirm the expected result directly using worldsheettechniques. In section 2 we review (0 ,
2) models, their beta-functions, and setup our con-ventions. Section 3 is the main body of this work where we prove our claim. We summarizeour results and outline some open questions in section 4. We thank Jacques Distler for explaining some of this historical context and providing his insights. Investigations early on, however, did not reveal any obstructions at the first few orders [5, 6]. See, however, [10] for some important limitations on the extent of these claims. There has also been renewed interest in heterotic geometry from the target space perspective, and thecorresponding moduli problem, see e.g. [11, 12]. Review of the (0 , NLSM and beta functionals (0 , superspace and superfields We will work in Euclidean (0 ,
2) superspace with coordinates ( z, ¯ z, θ, ¯ θ ). We assign θ a U (1) R charge q R = +1 and ¯ θ a charge q R = −
1, and take the fermionic integration measuresuch that R d θ θ ¯ θ = 1. The covariant derivatives and supercharges are given by: D = ∂ θ + ¯ θ ¯ ∂, ¯ D = ∂ ¯ θ + θ ¯ ∂, (2.1) Q = − ∂ θ + ¯ θ ¯ ∂, ¯ Q = − ∂ ¯ θ + θ ¯ ∂, (2.2)with non-trivial anti-commutators { D, ¯ D } = 2 ¯ ∂, { Q, ¯ Q } = − ∂. (2.3)Chiral fields are annihilated by ¯ D , and the scalar and Fermi chiral fields have the componentexpansions Φ = φ + √ θψ + θ ¯ θ ¯ ∂φ, Γ = γ + √ θF + θ ¯ θ ¯ ∂γ. (2.4)Anti-chiral fields on the other hand are annihilated by D , and they have correspondingcomponent expansions¯Φ = ¯ φ − √ θ ¯ ψ − θ ¯ θ ¯ ∂φ, ¯Γ = ¯ γ + √ θ ¯ F − θ ¯ θ ¯ ∂ ¯ γ. (2.5)We assign q R = 0 to all the (lowest components of the) chiral fields. To simplify the details,we will demand the existence of a non-anomalous global U (1) L symmetry, under whichΦ , ¯Φ , Γ , ¯Γ have charges q L = 0 , , +1 , −
1, respectively. It is straightforward to generalize tocases without the U (1) L symmetry. (0 , NLSM
The most general, renormalizable, U (1) R × U (1) L invariant, (0 ,
2) supersymmetric actionis given by S NLSM = − πα ′ Z d zd θ h (cid:0) K i (Φ , ¯Φ) ∂ Φ i − K ¯ ı (Φ , ¯Φ) ∂ ¯Φ ¯ ı (cid:1) − h a ¯ b (Φ , ¯Φ)¯Γ ¯ b Γ a i . (2.6)The (1 , K = K i d φ i (with complex conjugate K ∗ = K ¯ ı d φ ¯ ı ) is the (0 ,
2) analog ofthe (2 ,
2) K¨ahler potential, and is only defined up to shifts by holomorphic (1 , K (Φ , ¯Φ) → K (Φ , ¯Φ) + K ′ (Φ). The action is also invariant under K → K + i ∂f , for any2eal-valued function f . Holomorphic redefinitions of the Fermi fields, Γ a → M ab (Φ)Γ b ,imply that the Hermitian metric h a ¯ b is only defined up to transformations of the form h → M † hM . In particular, rescaling h a ¯ b by a constant factor leads to an equivalenttheory; this will be relevant at a later stage.Expanding the supersymmetric sigma model action (2 .
6) in components, we obtain S NLSM = 12 πα ′ Z d z (cid:2) g i ¯ (cid:0) ∂φ i ¯ ∂ ¯ φ ¯ + ¯ ∂φ i ∂ ¯ φ ¯ (cid:1) + B i ¯ (cid:0) ∂φ i ¯ ∂ ¯ φ ¯ − ¯ ∂φ i ∂ ¯ φ ¯ (cid:1) + g i ¯ ¯ ψ ¯ (cid:0) ∂ψ i + (cid:0) Γ − ijk ∂φ j + Γ − i ¯ k ∂φ ¯ (cid:1) ψ k (cid:1) + h a ¯ b ¯ γ ¯ b (cid:0) ¯ ∂γ a + A ai b ¯ ∂φ i γ b (cid:1) + F i ¯ a ¯ b ψ i ¯ ψ ¯ ¯ γ ¯ b γ a i , (2.7)where the couplings in the above action are determined by K i and h a ¯ b : g i ¯ = ∂ ( i K ¯ ) , B i ¯ = ∂ [¯ K i ] , A ai b = h a ¯ a ∂ i h b ¯ a , F i ¯ a ¯ b = h a ¯ a ∂ i A ¯ a ¯ ¯ b , and the connection Γ − is defined by Γ ± = Γ ± H, where Γ is the usual Christoffel connectionfor the metric g and H = dB is the tree-level torsion. In particular, we haveΓ − ijk = Γ ijk − H ijk = g i ¯ ∂ j g k ¯ , Γ − i ¯ k = Γ i ¯ k − H i ¯ k = − H i ¯ k , (2.8)along with their complex conjugates, and the rest of the components vanish.Geometrically, the sigma model action describes maps of a two-dimensional worldsheetΣ into a target manifold M , equipped with a metric, g , and B -field. (0 ,
2) supersymmetryguarantees that M is a complex manifold [14, 15], with fundamental form ω = ig i ¯ dφ i d ¯ φ ¯ , related to H by H = dB = i ( ¯ ∂ − ∂ ) ω. (2.9)The right-moving fermions, ψ , transform as sections of (the pullback of) T M , coupledto the H -twisted connection Γ − . The left-moving fermions, γ , transform as sections of a(stable) holomorphic bundle E → M , where E is equipped with a Hermitian metric h α ¯ β ,and holomorphic connection A i with curvature F i ¯ . To simplify our analysis we will assumethat E is stable, as opposed to just poly-stable or semi-stable. The fact that U (1) L isanomaly-free translates into the statement that c ( E ) = 0. Similarly, if U (1) R is anomaly-free then c ( M ) = 0 and the theory flows to a non-trivial (0 ,
2) SCFT in the infrared. Non-Hermitian metrics, with components h ab and h ¯ a ¯ b , are forbidden by the U (1) L symmetry. The auxiliary fields appear in the action L aux = ( F a + A ai b ψ i γ a ) h a ¯ b ( ¯ F ¯ b − A ¯ b ¯ ¯ e ¯ ψ ¯ γ ¯ e ) , which vanisheson shell. .3 Beta functionals of the (0,2) NLSM The one-loop beta functionals for (0 ,
2) NLSMs were derived in [16] by demanding (0 , δ Ψ M = ∇ ( − ) M ǫ = 0 , (2.10) δλ = (cid:0) ∂ (cid:30) ϕ − H (cid:30) (cid:1) ǫ = 0 , (2.11) δχ = F (cid:30) ǫ = 0 . (2.12)Given the relation between (0 ,
2) worldsheet and N = 1 spacetime supersymmetries, itis rather natural that manifestly (0 ,
2) supersymmetric beta functions are related to thespacetime BPS equations.We will restrict ourselves to flat worldsheets, Σ = C , and therefore will not be sensitiveto the coupling of the dilaton or its corresponding beta functional. In other words, we willonly study the conditions for finiteness of the SCFTs that the NLSMs flow to, rather thantackle the more involved question of (super-)Weyl invariance of the full heterotic stringworldsheet. Therefore, the (0 ,
2) NLSMs we consider are determined by two only couplingfunctionals: K i and h a ¯ b , each with their corresponding beta functional β Ki and β ha ¯ b . Let uswrite β (1) for the one-loop contribution to the beta functionals and ∆ β for the sum of allhigher loop contributions: β = β (1) + ∆ β . We will compare our approach to [16] furtherin what follows. For our starting point, we will take the one-loop NLSM beta functionalsto be given by β K (1) i = c g j ¯ k ( ∂ j g i ¯ k − H ij ¯ k ) (2.13) β h (1) a ¯ b = c ′ g i ¯ F i ¯ a ¯ b , (2.14)where c and c ′ are some known constants whose precise values we do not require. Our β (1) h coincides with that of [16], and its vanishing clearly implies the spacetime gaugino equa-tion (2 . β (1) K is a little more subtle, because it corresponds to a linear combinationof the two remaining beta functionals of [16]. One member of that pair reads ∂ i ϕ = H ij ¯ k g j ¯ k , (2.15) While the ∆ β are scheme dependent, once we fix a renormalization scheme their expressions are unique. It bears pointing out that iβ K (1) i = i (Γ − jij − Γ − ¯ i ¯ ) := Γ − i is the induced connection on the canonicalbundle [17]. When Γ − i is flat, then ∇ ( − ) has SU ( n ) holonomy as required by the gravitino equation (2 . ϕ is the dilaton field, and this is equivalent to the dilatino equation (2 . ,
2) SCFTs defined on the plane, it should come as no surprise that weare not sensitive to (2 . .
15) as a constraint that defines ϕ forour models (at least at leading order in α ′ ). We remark that (2 .
15) can always be solvedlocally, and so it is indeed a valid definition of ϕ in a CFT. If we wish to promote such asolution to string theory, we must further ensure that ϕ is globally defined, and this mayobstruct such a lift. Fortunately, when including worldsheet supergravity the field ϕ isintrinsically defined by its coupling to the worldsheet curvature, and this issue is avoided.Using the constraint (2 . β (1) K repro-duces the correct beta functionals for the physical couplings g and B : β g (1) i ¯ = R i ¯ + 2 ∇ i ∇ ¯ ϕ − H iMN H ¯ MN = − ∂ ( i β K (1)¯ ) , (2.16) β B (1) i ¯ = ∇ M (cid:0) e − ϕ H M i ¯ (cid:1) = ∂ [ i β g (1)¯ ] . (2.17)We take this result as support of our starting point (2 . . ,
2) generalization of the key lemma from [3].
Lemma 2.18.
Let E → M be a stable holomorphic vector bundle with data (˜ g, ˜ B, ˜ A ) ,derived from ˜ K and ˜ h , with vanishing one-loop beta functionals. Then, there exist couplings K i = ˜ K i − δK i , h a ¯ b = ˜ h a ¯ b − δh a ¯ b , with corresponding data ( g, B, A ) such that β K ( g, B, A ) = β K (1) ( g, B ) + ∆ β K ( g, B, A ) = β K (1) (˜ g, ˜ B ) , (2.19) β h ( g, B, A ) = β h (1) ( g, A ) + ∆ β h ( g, B, A ) = β h (1) (˜ g, ˜ A ) , (2.20) In particular, the full beta-functionals for ( g, B, A ) can be made to vanish. H The validity of a perturbative loop/ α ′ expansion for a general (0 ,
2) NLSM is doubtful atbest. It has been shown by many authors that the existence of non-vanishing H -flux inthe tree-level action leads to string scale cycles in the geometry, and therefore a breakdown For example, this occurs for torsional NLSMs on S × S . See Appendix C of [18] for more details.
5f a large-volume expansion [19–23]. Therefore, we will content ourselves with the moreconservative goal of proving the above lemma in cases where H vanishes classically. Moreprecisely, we will assume the validity of α ′ perturbation theory, where the α ′ → and that H → M isCalabi-Yau to leading order in α ′ . However, except in the case of the standard embeddingwith A = Γ, H -flux will be generated by loop effects such as the Bianchi identity, dH = α ′ R + ∧ R + − tr F ∧ F ) , (3.1)and (2 .
9) will ensure that the full metric, g , is no longer K¨ahler. In the simplified setting where H = 0 classically, it is straightforward to prove the lemma.The general approach will be within the framework of α ′ perturbation theory. We will firstshow that the lemma holds to lowest order in α ′ , and then extend this to higher ordersinductively by using the results of the earlier steps. In this way, we will prove the lemmato all order in α ′ .We will first tackle β K . First, note that (2 .
9) allows us to write β K (1) i = c (cid:16) ∂ i log det g − H ij ¯ k g j ¯ k (cid:17) = c (cid:16) ∂ i log det g − ∂ [ i g j ]¯ k g j ¯ k (cid:17) . (3.2)Next, we can rewrite (2 .
19) as c − ∆ β Ki = ∂ i log det(˜ g/g ) + 4 g j ¯ k ∂ [ i g j ]¯ k = − ∂ i Tr log (cid:0) − ˜ g − δg (cid:1) − g j ¯ k ∂ [ i δg j ]¯ k (3.3)= X n> n ∂ i Tr (cid:0) ˜ g − δg (cid:1) n − X n ≥ ˜ g j ¯ ℓ (cid:2)(cid:0) ˜ g − δg (cid:1) n (cid:3) ¯ k ¯ ℓ ∂ [ i δg j ]¯ k , where we have expanded g = ˜ g − δg and used the fact that ˜ g is K¨ahler. As in [3], thisequation may be solved iteratively for δK , order by order in α ′ , in terms of the input data c − ∆ β K . To lowest order, this equation is simply c − ∆ β Ki = ∂ i (˜ g j ¯ k δg j ¯ k ) − g j ¯ k ∂ [ i δg j ]¯ k = ∂ i (cid:16) ˜ g j ¯ k ∂ ( j δK ¯ k ) (cid:17) + 2˜ g j ¯ k ∂ ¯ k δB ij , (3.4) This rules out the only known class of truly torsional backgrounds, based on the total space T → K For instance, the second-order correction will depend on the tree-level data and quadratically on theone-loop correction. δB ij = ∂ [ j δK i ] . We can see that the righthand side is the sum of ∂ -exactand co-exact pieces: c − ∆ β K = ∂ (cid:16) ˜ ∇ · δK (cid:17) + 2 ∂ † B, (3.5)where ∂ † = ∗ ¯ ∂ ∗ is the adjoint of ∂ . Because the target space M is simply connected, theHodge decomposition theorem tells us that every 1-form has such a decomposition. Inparticular, the left hand side can be expressed as c − ∆ β K = ∂f + 2 ∂ † χ (3.6)for some scalar function f and a (2 , χ . The factor of 2 is merely used for convenience.Note that f is only defined up to the addition of a constant, while χ is only well-definedmodulo a co-closed 2-form, γ . Applying the Hodge decomposition once again to γ , we seethat γ must be co-exact, and so χ ∼ χ + ∂ † ρ for ρ a (3 , .
5) and (3 .
6) are equivalent to the pair ˜ ∇ · δK = f, ∂ ( δK ) = χ, (3.7)where again f ∼ f + const. and χ ∼ χ + ∂ † ρ . On a simply connected manifold, any 1-form iscompletely determined by its divergence and exterior derivative, therefore δK is in principlefixed to lowest order. To see this in detail, first we can decompose δK itself as δK = ∂k + ∂ † κ. (3.8)We can assume k is real scalar function, since any imaginary component only adds a totalderivative to the action, as noted below (2 . ˜ ∇ · δK = (cid:0) ∂ † δK + ¯ ∂ † δ ¯ K (cid:1) = (cid:0) ∂ † ∂ + ¯ ∂ † ¯ ∂ (cid:1) k = (∆ ∂ + ∆ ¯ ∂ ) k = ∆ ∂ k, (3.9)because ˜ g is K¨ahler and on a K¨ahler manifold ∆ ∂ = ∆ ¯ ∂ = ∆ d . Here we have introduced theLaplace-Beltrami operator ∆ ∂ = ∂∂ † + ∂ † ∂ , with similar expression for the other Laplacians.Next, we can use the ambiguity κ ∼ κ + ∂ † α to remove the co-exact piece of κ from itsHodge decomposition, ensuring that κ is ∂ -exact and therefore closed. In other words, ∂ ( δK ) = ∂∂ † κ = ∆ ∂ κ , and so (3 .
7) can be written∆ ∂ k = f, ∆ ∂ κ = χ. (3.10)These equations can always be solved locally by inverting the Laplacian, provided f and χ contain no zero-modes of ∆ ∂ . For χ this is trivial, as there are no harmonic (2 ,
0) forms on7 , while for the function f we can use the ambiguity f ∼ f + const to ensure that f hasno constant term. We will discuss the existence of global solutions in the following section.Thus we obtain the lowest order to solution for δK : δK = ∂ (∆ − ∂ f ) + ∂ † (∆ − ∂ χ ) , (3.11)where f and χ are determined from the input data c − ∆ β K via (3 . . δK order by order in the expansion.Turning now to β h , in a similar fashion we rewrite (2 .
20) as c ′− ∆ β ha ¯ b = ˜ g i ¯ ˜ F i ¯ a ¯ b − g i ¯ F i ¯ a ¯ b (3.12)= ˜ g i ¯ ˜ F i ¯ a ¯ b − X m,n ≥ ˜ g i ¯ k (cid:2)(cid:0) ˜ g − δg (cid:1) m (cid:3) ¯ ¯ k (˜ h a ¯ a − δh a ¯ a ) ∂ i (cid:18) ˜ h ¯ ac h(cid:16) ˜ h − δh (cid:17) n i bc ∂ ¯ (˜ h b ¯ b − δh b ¯ b ) (cid:19) . To lowest orders in δh and δg , this can be written as D A δh a ¯ b = c ′− ∆ β ha ¯ b + ˜ F i ¯ a ¯ b δg i ¯ , (3.13)where D A is gauge-covariant Laplacian, and we have used β h (1) a ¯ b (˜ g, ˜ A ) = ˜ g i ¯ ˜ F i ¯ a ¯ b = 0. Sincewe have assumed the bundle E is stable, we have dim H ( End E ) = 1 (see, for example,Cor. 1.2.8 of [25]), so D A has a unique zero-mode corresponding to the uniform rescalingof h a ¯ b . However, as mentioned in section 2.2, such scaling of the bundle metric can beabsorbed into the Fermi fields. So up to this unphysical ambiguity, we can solve (3 . δh uniquely in terms of the lowest order solution for δg from the previousparagraphs and the input ∆ β h . As before, the lowest order solution can be plugged backinto (3 .
12) to get an iterative solution for δh .In summary, we have demonstrated how to construct local, perturbative solutions for δK and δh , order by order in the α ′ /loop expansion, in terms of ∆ β K and ∆ β h . In thefollowing section we will show that these local solutions actually patch together into well-defined global ones. This will complete the proof that the corrected sigma model data, K = ˜ K − δK and h = ˜ h − δh , have vanishing beta functionals. With the existence of a local solution to the set of equations defining K i and h a ¯ b shown,we will now argue that this solution is globally defined. The key point to notice is thatthe solutions for δK i and δh a ¯ b are determined entirely in terms of the higher loop betafunctionals ∆ β K and ∆ β h . We will now show that the ∆ β are globally defined, so that δK δh are as well. The arguments presented in this section closely parallel original onesof [3] for the simpler (2 ,
2) case.We will begin with ∆ β K . First, recall that β gi ¯ = − ∂ (¯ β Ki ) (3.14)must have a covariant form, since it provides the metric equations of motion, and so definesa global tensor on M . On the other hand, β K itself need not be globally defined, as a shiftby a holomorphic one-form is allowed. So when we go from one patch to another, ∆ β K must satisfy: ∆ β Ki ′ − ∂z j ∂z ′ i ∆ β Kj = f i ( z ) , (3.15)∆ β K ¯ ′ − ∂z ¯ ı ∂z ′ ¯ ∆ β K ¯ i = g ¯ (¯ z ) , (3.16)for set of (anti-)holomorphic functions f i and g ¯ . Furthermore, we must consider the possi-bility for gauge transformation between the patches. Since β gi ¯ is a globally defined singlet,then ∂ (¯ ∆ β i ) must also be a globally defined singlet. So we get the relationship between thetwo patches from a gauge transformation as∆ β Ki ′ − ∆ β Ki = ˜ f i ( z ) , (3.17)∆ β K ¯ ′ − ∆ β K ¯ j = ˜ g ¯ (¯ z ) , (3.18)for some other (anti-)holomorphic functions ˜ f and ˜ g .Let us focus on ∆ β Ki , though similar arguments apply for ∆ β K ¯ . On general grounds,we expect ∆ β K to be expressible as a product of derivatives of K i , K ¯ and h a ¯ b , with indicescontracted by the metrics g i ¯ and h a ¯ b . There are four classes of factors that could make upour beta function, ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m K i , (3.19) ∂ i ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m K k n +1 , (3.20) ∂ i ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m K ¯ l m +1 , (3.21) ∂ i ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m h a ¯ b . (3.22)To simplify our analysis, we will make one further reasonable assumption: the B -field, B i ¯ = ∂ [¯ K i ] , can only appear in the beta functions through the invariant combination H = dB + α ′ ( CS ( ω + ) − CS ( A )) + O ( α ′ ). So that even if individual factors in the list above9ave anomalous transformation properties, because of the Green-Schwarz mechanism, theseeffects will cancel out of the full beta functions. Thus we will not need to examine anomaloustransformations of the above factors, and we can restrict our attention to conventionaldiffeomorphisms and gauge transformations only.When we consider a spacetime transformation between patches of one of the factorslisted above, two types of terms can arise. We will follow [3] and refer to these types ashomogeneous and inhomogeneous. Homogeneous terms are the standard tensor transfor-mations, with each spacetime index contracted with a Jacobian matrix, while the inhomo-geneous terms involve higher derivatives between the coordinates on the patches. Sincewe contract all the indices except i using g j ¯ k , the homogeneous terms will all cancel, savefor a single Jacobian factor associated with the uncontracted i index. In other words, thehomogenous terms satisfy the transformation law (3 .
15) with f i = 0.What about the inhomogeneous terms, can they generate a non-zero f i ? Note that aninhomogeneous term with a factor of the form ∂ r z ′ k ∂z j ...∂z j r (3.23)will have more indices on the bottom then on the top. Since we must contract all indicesexcept i , this implies that the inhomogeneous terms must have factors involving the inversemetric. This is a function of both z and ¯ z , which cannot appear on the righthand sideof (3 . ∂ r z ′ k ∂z i ∂z j ...∂z j r − . (3.24)If r >
1, then the same argument from above applies. However, we could have a term thatinvolves ∂ z ′ k ∂z i ∂z j (3.25)Since this has a balanced set of indices, we can imagine contracting this with a functionwith only holomorphic dependence. The only term available would be another Jacobianfactor, giving us two possibilities. The first would involve a term like ∂ z ′ k ∂z i ∂z j ∂z j ∂z ′ k = ∂ z ′ k ∂z i ∂z ′ k = 0 . (3.26)The other would be ∂ z ′ k ∂z i ∂z j ∂z ′ j ∂z k . (3.27)However, this type of term would not show up from a coordinate transformation, as wemust have z ′ contracting with z ′ and z contracting with z . So, the inhomogeneous terms10nvolving (3 .
24) do not show up. Since the homogeneous term implies f i ( z ) = 0 andthe inhomogeneous terms cannot contribute a non-zero f i , then we conclude that ∆ β K isglobally defined under diffeomorphisms.We now turn to gauge transformations. Note that terms involving (3 . .
21) do notinvolve gauge indices and so will not transform under gauge transformations. This leavesonly terms involving (3 . β K does not have any gauge indices so we mustcontract up terms with (3 .
22) with factors of h a ¯ b . Under a gauge transformation, we recallthat h ′ a ¯ b = U ca h c ¯ d U ¯ d ¯ b . (3.28)So now we consider the full term ∂ i ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m ( U ca h c ¯ d U ¯ d ¯ b ) . (3.29)We note that undifferentiated factors of U will cancel against the transformation of h a ¯ b ,and the difference between patches of such terms will vanish. So we only have to worryabout terms that involve derivatives acting on U . Note, however, that the all derivativesexcept ∂ i must be contracted with g j ¯ k , which we have noted before is not holomorphic andwould violate (3 . ∂ i h a ¯ b , however, are not ruled out. Therefore, theonly terms that could possibly lead to non-zero holomorphic differences are contractionsof h a ¯ b , h a ¯ b , and A aib . The only terms one can construct from these basic building blockswith the correct index structure are, schematically, ∂ i tr ( hh...h ) = 0 and tr ( A i h...h ) = 0,assuming that gauge group is semi-simple. So we have shown that ∆ β K is a globally definedgauge singlet, as well.We now go through the same arguments for ∆ β h . Analogous to the previous case, wemay insist that β Aia ¯ b = ∂ i β ha ¯ b and β A ¯ a ¯ b = ∂ ¯ β ha ¯ b (3.30)are globally defined. Satisfying both of these constraints restricts the patching conditionfor ∆ β h to be: ∆ β ha ¯ b ′ − ∆ β a ¯ b = C a ¯ b (3.31)where C is a constant matrix. Now we have three classes of factors that could appear inour beta function: ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m K k n +1 , (3.32) ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m K ¯ l m +1 , (3.33) ∂ k ...∂ k n ∂ ¯ l ...∂ ¯ l m h a ¯ b . (3.34)11he same argument as before applies for diffeomorphisms. In fact it is even simpler, asall of the spacetime indices must be contracted, and the argument is very similar to theoriginal one found in [3]. Gauge transformations will be a bit different. Since the fac-tors (3 .
32) and (3 .
33) do not have any gauge dependence, they are invariant under gaugetransformations. So we will only need to worry about (3 . C a ¯ b that transform under the correctrepresentation, except the trivial case C a ¯ b = 0.In summary, we have shown that ∆ β K , and ∆ β h are globally defined tensors, underboth spacetime diffeomorphisms and gauge transformations. Our proof required assumingthat the full beta functions β K and β h only depend on the B -field though the invariantthree-form H . We do not expect this assumption to affect our result. Since the ∆ β are thedata that determine δK and δh , we conclude that these corrections are also globally defined.Thus, the full beta functions for the corrected couplings K = ˜ K − δK and h = ˜ h − δh canbe made to vanish. In this short note, we have shown that the (0 ,
2) NLSMs on Calabi-Yau backgrounds,equipped with a stable holomorphic bundle, can be made to be finite to all orders inperturbation theory. We have closely paralleled the approach of [3] in the (2 ,
2) setting,by demonstrating the existence of a set of counterterms to the sigma model couplings thatmake the beta functions vanish to all orders.There are a few unresolved issues, that we leave as open problems for future study. Thefirst question is how our results carry over when the (0 ,
2) NLSMs are coupled to worldsheetsupergravity. This will require studying the full Weyl symmetry, along with couplings tothe dilaton field and its associated beta function. This would appear to be a prerequisite forgoing beyond the restriction of demanding H = 0 at the classical level, and only allowingfor H to be generated by loop effects. While tree-level H -flux is problematic for a controlled α ′ -expansion, it may be possible to avoid these complications since we are only interestedin the general structure of the theory, and not necessarily perturbing about some particulartorsional background. Finally, we have assumed that anomalous transformations will notappear in the beta functions. While we believe this to be true, a cautious reader mightbe wary. Since the anomaly only shows up at one-loop, they could potentially affect thetwo-loop beta functions and this should be checked directly.12 cknowledgements This project originated from a discussion between Jacques Distler, Ilarion Melnikov andCQ at the workshop
Heterotic Strings and (0,2) QFT , hosted by the Mitchell Instituteat TAMU in April, 2014. CQ would like the thank the organizers and participants forproviding a stimulating and exciting environment. We especially thank Ilarion Melnikovfor helpful suggestions at various stages of this paper. We also thank Marc-Antoine Fiset,Stefan Groot Nibbelink, and Eirik Svanes for helpful discussions and feedback on earlierdrafts. IJ is supported by a NSERC Discovery grant. CQ was supported by a NSERCfellowship.
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