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1 c 2006 International Press Adv. Theor. Math. Phys. 10 (2006) Two-dimensional twisted sigma models and the theory of chiral differential operators Meng-Chwan Tan Department of Physics, National University of Singapore, S(119260), Singapore Abstract In this paper, we study the perturbative aspects of a twisted version of the two-dimensional (0, 2) heterotic sigma model on a holomorphic gauge bundle E over a complex, hermitian manifold X. We show that the model can be naturally described in terms of the mathematical theory of Chiral Differential Operators. In particular, the physical anomalies of the sigma model can be reinterpreted in terms of an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on X. One can also obtain a novel understanding of the sigma model 1-loop beta-function solely in terms of holomorphic data. At the (2, 2) locus, where the obstruction vanishes for any smooth manifold X, we obtain a purely mathematical description of the half-twisted variant of the topological A-model and (if c 1 (X) = 0) its elliptic genus. By studying e-print archive:

2 760 MENG-CHWAN TAN the half-twisted (2, 2) model on X = CP 1, one can show that a subset of the infinite-dimensional space of physical operators generates an underlying superaffine Lie algebra. Furthermore, on a non-kähler, parallelized, group manifold with torsion, we uncover a direct relationship between the modulus of the corresponding sheaves of chiral de Rham complex and the level of the underlying WZW theory. 1 Introduction The mathematical theory of Chiral Differential Operators (CDOs) is a fairly well-developed subject that aims to provide a rigorous mathematical construction of conformal fields theories, possibly associated with sigma models in two-dimensions, without resorting to mathematically non-rigorous methods such as the path integral. It was first introduced and studied in a series of seminal papers by Malikov and coworkers [1 5] and in [6] by Beilinson and Drinfeld, whereby a more algebraic approach to this construction was taken in the latter. These developments have found interesting applications in various fields of geometry and representation theory such as mirror symmetry [7] and the study of elliptic genera [8 10], just to name a few. However, the explicit interpretation of the theory of CDOs, in terms of the physical models it is supposed to describe, has been somewhat unclear, that is until recently. In the pioneering papers of Kapustin [11] and Witten [12], initial steps were taken to provide a physical interpretation of some of the mathematical results in the general theory of CDOs. In [11], it was argued that on a Calabi Yau manifold X, the mathematical theory of a CDO, known as the chiral de Rham (CDR) complex can be identified with the infinite-volume limit of a half-twisted variant of the topological A-model. And in [12], the perturbative limit of a half-twisted (0, 2) sigma model with right-moving fermions was studied, where its interpretation in terms of the theory of a CDO that is a purely bosonic version of the CDR was elucidated. And even more recently, an explicit computation (on P 1 ) was carried out by Frenkel et al. in [13] to verify mathematically, the identification of the CDR as the half-twisted sigma model in perturbation theory. In this paper, we will consider a generalization of the model considered in [12] to include left-moving worldsheet fermions valued in a holomorphic gauge bundle over the target space. To this end, we will be studying the perturbative aspects of a twisted version of the two-dimensional (0, 2) heterotic sigma model. Our main goal is to seek a physical interpretation of the mathematical theory of a general class of CDOs, constructed from generic

3 TWISTED SIGMA MODELS 761 vertex superalgebras, by Malikov and coworkers in [3, 5]. In turn, we hope to obtain some novel insights into the physics via a reinterpretation of some established mathematical results. Additional motivation for this work also come from the fact that this generalization is important in the heterotic string theory. In fact, other various aspects of similar models have been extensively studied in the physics literature. Of particular physical importance would be the results obtained by Katz and Sharpe in [15], which suggest that under certain conditions, the correlation functions of physical operators in the model considered can be related to the Yukawa couplings in heterotic string compactifications. Various twisted heterotic sigma models were also used in [14] to ascertain the criteria for conformal invariance in (0, 2) models. Last but not the least, the existence of a topological heterotic ring of ground operators (which reduces to the (a, c) ring of an untwisted (2, 2) model at the (2, 2) locus) in the conformal and massive limits of an isomorphic model was also investigated in [16]. This presents new possibilities for the application of physical insights in mathematics and vice-versa. To this end, we shall generalize Witten s approach in [12]. 1.1 A brief summary and plan of the paper A brief summary and plan of the paper is as follows. First, in Section 2, we will review the two-dimensional heterotic sigma model with (0, 2) supersymmetry on a rank-r holomorphic gauge bundle E over a Kähler manifold X. We will then introduce a twisted variant of the model, obtained via a redefinition of the spins of the relevant worldsheet fields. Next, in Section 3, we will focus on the space of physical operators of this twisted sigma model. In particular, we will study the properties of the chiral algebra furnished by these operators. In addition, we will show how the moduli of the chiral algebra arise when we include a non-kähler deformation of X. In Section 4, we will discuss, from a purely physical perspective, the anomalies of this specific model. The main aim in doing so is to prepare for the observations and results that we will make and find in the next section. In Section 5, we will introduce the notion of a sheaf of perturbative observables. An alternative description of the chiral algebra of physical operators in terms of the elements of a Cech cohomology group will also be presented. Thereafter, we will show that the twisted model on a local patch of the target space can be described in terms of a free bc-βγ system, where in order to give a complete description of the model on the entire target space, it will first

4 762 MENG-CHWAN TAN be necessary to study its local symmetries. Using these local symmetries, one can then glue together the free conformal field theories (each defined on a local patch of the target space by the free bc-βγ system) to obtain a globally defined sheaf of CDOs or vertex superalgebras which span a subset of the chiral algebra of the model. It is at this juncture that one observes the mathematical obstruction to a global definition of the sheaf (and hence the existence of the underlying theory) to be the physical anomaly of the model itself. Via an example, we will be able to obtain a novel understanding of the non-zero 1-loop beta-function of the twisted heterotic sigma model solely in terms of holomorphic data. In Section 6, we will study the twisted model at the (2, 2) locus where E = TX, for which the obstruction to a global definition of the sheaf of vertex superalgebras vanishes for any smooth manifold X. In doing so, we obtain a purely mathematical description of the half-twisted variant of Witten s topological A-model [17] in terms of a theory of a class of conformal vertex superalgebras called the CDR, which for a target space with vanishing first Chern class such as a Calabi Yau manifold, acquires the structure of a topological vertex algebra. Our results therefore serve as an alternative verification and generalization of Kapustin s findings in [11] and Frenkel and Losev s computation in [13]. Using the CFT state-operator correspondence in the Calabi Yau case, one can make contact with the mathematical definition of the elliptic genus introduced in [8 10] solely via physical considerations. In Section 7, we will analyse, as examples, sheaves of CDR that describe the physics of the half-twisted (2, 2) model on two different smooth manifolds. The main aim is to illustrate the rather abstract discussion in the preceding sections. By studying the sheaves of CDR on CP 1, we find that a subset of the infinite-dimensional space of physical operators furnishes an underlying superaffine Lie algebra. As in Section 5, we will be able to obtain a novel understanding of the non-zero 1-loop beta-function of the half-twisted sigma model solely in terms of holomorphic data. Furthermore, for the half-twisted (2, 2) model on a non-kähler, parallelized, smooth manifold with torsion such as S 3 S 1, a study of the corresponding sheaf of CDR reveals a direct relationship between the modulus of sheaves and the level of the underlying SU(2) WZW theory. 1.2 Beyond perturbation theory As pointed out in [12], instanton effects can change the picture radically, triggering a spontaneous breaking of supersymmetry, hence making the chiral

5 TWISTED SIGMA MODELS 763 algebra trivial as the elliptic genus vanishes. Thus, out of the perturbation theory, the sigma model may no longer be described by the theory of CDOs. This non-perturbative consideration is beyond the scope of the present paper. However, we do hope to address it in a future publication. 2 A twisted heterotic sigma model 2.1 The heterotic sigma model with (0, 2) supersymmetry To begin, let us first recall the two-dimensional heterotic non-linear sigma model with (0, 2) supersymmetry on a rank-r holomorphic gauge bundle E over a Kähler manifold X. It governs maps Φ : Σ X, with Σ being the worldsheet Riemann surface. By picking local coordinates z, z on Σ, and φ i, φī on X, the map Φ can then be described locally via the functions φ i (z, z) and φī(z, z). Let K and K be the canonical and anti-canonical bundles of Σ (the bundles of one-forms of types (1, 0) and (0, 1), respectively), whereby the spinor bundles of Σ with opposite chiralities are given by K 1/2 and K 1/2. Let TX and TX be the holomorphic and anti-holomorphic tangent bundle of X. The left-moving fermi fields of the model consist of λ a and λ a, which are smooth sections of the bundles K 1/2 Φ E and K 1/2 Φ E, respectively. On the other hand, the right-moving fermi fields consist of ψ i and ψī, which are smooth sections of the bundles K 1/2 Φ TX and K 1/2 Φ TX, respectively. Here, ψ i and ψī are superpartners of the scalar fields φ i and φī, while λ a and λ a are superpartners to a set of auxiliary scalar fields l a and l a, which are in turn smooth sections of the bundles K 1/2 K 1/2 Φ E and K 1/2 K 1/2 Φ E, respectively. Let g be the hermitian metric on X. The action is then given by S = d 2 z Σ ( 1 2 g i j( z φ i z φ j + z φ i z φ j )+g i jψ i D z ψ j + λ a D z λ a +F a bi j(φ)λ a λ b ψ i ψ j l a l a), (2.1) whereby i, ī =1,...,n= dim C X, a =1,...,r, 1 d 2 z = idz d z, and F a bi j (φ) =A a bi, j(φ) is the curvature two-form of the holomorphic gauge bundle E with connection A. In addition, D z is the operator on K 1/2 φ TX 1 As we will be studying the sigma model in the peturbative limit, worldsheet instantons are absent, and one considers only (degree zero) constant maps Φ, such that Σ Φ c 1(E) = 0. Since the selection rule from the requirement of anomaly cancellation states that the number of λ a s must be given by Σ Φ c 1(E)+r(1 g), where g is the genus of Σ, we find that at string tree level, the number of λ a s must be given by r.

6 764 MENG-CHWAN TAN using the pull-back of the Levi Civita connection on TX, while D z is the operator on K 1/2 Φ E using the pull-back of the connection A on E. In formulas (using a local trivialization of K 1/2 and K 1/2, respectively), we have 2 D z ψ j = z ψ j +Γ j l k zφ lψ k, (2.2) and D z λ a = z λ a + A a bi(φ) z φ i λ b. (2.3) Here, is the affine connection of X, while A Γ j l k a bi(φ) is the connection on E in component form. The infinitesimal transformation of the fields generated by the supercharge Q + under the first right-moving supersymmetry is given by δφ i =0, δφī = ɛ ψī, δψī =0, δψ i = ɛ z φ i, δλ a =0, δλ a = ɛ l a, (2.4) δl a =0, δl a = ɛ (D z λ a + F a bi j(φ)λ b ψ i ψ j ), while the infinitesimal transformation of the fields generated by the supercharge Q + under the second right-moving supersymmetry is given by δφ i = ɛ ψ i, δλ a = ɛ (l a + A a bi(φ)λ b ψ i), δψ i =0, δl a = ɛ A a bi(φ)l b ψ i, δφī =0, δψī = ɛ z φī, (2.5) δλ a =0, δl a = ɛ z λ a, where ɛ and ɛ are anti-holomorphic sections of K 1/2. Since we are considering a holomorphic vector bundle E, the supersymmetry algebra is trivially satisfied Twisting the model Classically, action (2.1), and therefore the model that it describes, possesses a left-moving flavour symmetry and a right-moving R-symmetry, giving rise 2 Note that we have used a flat metric and hence vanishing spin connection on the Riemann surface Σ in writing these formulas. 3 The supersymmetry algebra is satisfied, provided the (2, 0) part of the curvature vanishes, i.e., A a b[i,j] A a c[ia c bj] = 0. For a real gauge field A such that A i = Aī, this just means that E must be a holomorphic vector bundle [18].

7 TWISTED SIGMA MODELS 765 to a U(1) L U(1) R global symmetry group. Denoting (q L,q R ) to be the leftand right-moving charges of the fields under this symmetry group, we find that λ a and λ a have charges (±1, 0), ψī and ψ i have charges (0, ±1), and l a and l a have charges (±1, ±1), respectively. Quantum mechanically however, these symmetries are anomalous because of non-perturbative worldsheet instantons; the charge violations for the left- and right-moving global symmetries are given by Δq L = Σ Φ c 1 (E) and Δq R = Σ Φ c 1 (TX), respectively. In order to define a twisted variant of the model, the spins of the various fields need to be shifted by a linear combination of their corresponding leftand right-moving charges (q L,q R ) under the global U(1) L U(1) R symmetry group; by considering a shift in the spin S via S S [(1 2s)q L +(2 s 1)q R ] (where s and s are real numbers), the various fields of the twisted model will transform as smooth sections of the following bundles: ( λ a Γ K (1 s) Φ E ), λ a Γ(K s Φ E), ( ) ( ) ψ i Γ K (1 s) Φ TX, ψī Γ K s Φ TX, (2.6) ( l a Γ K (1 s) K s Φ E ) ( ), l a Γ K s K (1 s) Φ E. Notice that for s = s = 1 2, the fields transform as smooth sections of the same tensored bundles defining the original heterotic sigma model, i.e., we get back the untwisted model. In order for a twisted model to be physically consistent, one must ensure that the new Lorentz symmetry (which has been modified from the original due to the twist) continues to be non-anomalous quantum mechanically. Note that similar to the untwisted case, the U(1) L and U(1) R symmetries are anomalous in the quantum theory. The charge violations on a genus-g Riemann surface Σ are given by Δq L = r(1 2s)(1 g)+ Φ c 1 (E), (2.7) Σ Δq R = n(2 s 1)(g 1) + Φ c 1 (TX). (2.8) As we will show in Section 5.3, physically consistent models must obey the condition c 1 (E) =c 1 (TX). Hence, we see from (2.7) and (2.8) that an example of a non-anomalous combination of global currents that one can use to twist the model with, is 1 2 (J L J R ), where s = s = 0. If one has the additional condition that c 1 (E) =c 1 (TX) = 0, i.e., X is a Calabi Yau, one can also consider the non-anomalous current combination 1 2 (J L + J R ), where s = 0 and s =1. Σ

8 766 MENG-CHWAN TAN Note at this point that we would like to study a twisted model which can be related to the half-twisted variant of the topological A-model at the (2, 2) locus where E = TX. To this end, we shall study the twisted variant of the heterotic sigma model defined by s = s = 0, i.e., we consider the twisted model associated with the current combination 1 2 (J L J R ). Hence, as required, the various fields in this twisted model of interest will transform as smooth sections of the following bundles: λ a Γ(Φ E), λ za Γ(K Φ E ), ψz ī Γ ( K Φ TX ), ψī Γ ( Φ TX ), (2.9) lz ā Γ ( K Φ E ), l za Γ(K Φ E ). Notice that we have included additional indices in the above fields so as to reflect their new geometrical characteristics on Σ; fields without a z or z index transform as worldsheet scalars, while fields with a z or z index transform as (1, 0) or (0, 1) forms on the worldsheet. In addition, as reflected by the a, i, and ī indices, all fields continue to be valued in the pull-back of the corresponding bundles on X. Thus, the action of the twisted variant of the two-dimensional heterotic sigma model is given by ( 1 S twist = d 2 z Σ 2 g i j( z φ i z φ j + z φ i φ j z )+g i jψ zd ī z ψ j ) + λ za D z λ a + F a bi j(φ)λ za λ b ψ zψ j ī l za lz ā. (2.10) A twisted theory is the same as an untwisted one when defined on a Σ which is flat. Hence, locally (where one has the liberty to select a flat metric), the twisting does nothing at all. However, what happens non-locally may be non-trivial. In particular, note that globally, the supersymmetry parameters ɛ and ɛ must now be interpreted as sections of different line bundles; in the twisted model, the transformation laws given by (2.4) and (2.5) are still valid, and because of the shift in the spins of the various fields, we find that for the laws to remain physically consistent, ɛ must now be a function on Σ while ɛ must be a section of the non-trivial bundle K 1. One can therefore canonically pick ɛ to be a constant and ɛ to vanish, i.e., the twisted variant of the two-dimensional heterotic sigma model has just one canonical global fermionic symmetry generated by the supercharge Q +. Hence, the infinitesimal transformation of the (twisted) fields under this single canonical symmetry must read (after setting ɛ to 1) δφ i =0, δφī = ψī, δψī =0, δψ ī z = z φ i,

9 TWISTED SIGMA MODELS 767 δλ a =0, δλ za = l za, (2.11) ( ) δl za =0, δlz ā = D z λ a + F a bi j(φ)λ b ψzψ j ī. From the (0, 2) supersymmetry algebra, we have Q 2 + = 0. In addition, (after twisting) Q + transforms as a scalar. Consequently, we find that the symmetry is nilpotent i.e., δ 2 = 0 (off-shell) and behaves as a BRST-like symmetry. Note at this point that the transformation laws of (2.11) can be expressed in terms of the BRST operator Q +, whereby δw = {Q +,W} for any field W. One can then show that action (2.10) can be written as S twist = d 2 z {Q +,V} + S top (2.12) where while S top = 1 2 Σ V = g i jψ ī z z φ j λ za l ā z, (2.13) g i j Σ ( z φ i z φ j z φ i z φ j ) (2.14) is Σ Φ (K), the integral of the pull-back to Σ of the (1, 1) Kähler form K = i 2 g i jdφ i dφ j. Notice that since Q 2 + = 0, the first term on the RHS of (2.12) is invariant under the transformation generated by Q +. In addition, because dk =0on akähler manifold, Σ Φ (K) depends only on the cohomology class of K and the homotopy class of Φ (Σ), i.e., the class of maps Φ. Consequently, S top is a topological term, invariant under local field deformations and the transformation δ. Thus, the action given in (2.12) is invariant under the BRST symmetry as required. Moreover, for the transformation laws of (2.11) to be physically consistent, Q + must have charge (0, +1) under the global U(1) L U(1) R gauge group. Since V has a corresponding charge of (0, 1), while K has a zero charge, S twist in (2.12) continues to be invariant under the U(1) L U(1) R symmetry group at the classical level. As mentioned in the introduction, we will be studying the twisted model in perturbation theory, where one does an expansion in the inverse of the large-radius limit. Hence, only the degree-zero maps of the term Σ Φ (K) contribute to the path integral factor e S twist. Therefore, in the perturbative limit, one can set Σ Φ (K) = 0 since dk = 0, and the model will be independent of the Kähler structure of X. This also means that one is free to study an equivalent action obtained by setting S top in (2.12) to zero. After eliminating the l za l ā z term via its own equation of motion l ā z = 0, the

10 768 MENG-CHWAN TAN equivalent action in perturbation theory reads ( ) S pert = d 2 z g i j φ j z z φ i + g i jψ zd ī ψ j z + λ za D z λ a + F a bi jλ za λ b ψzψ j ī, Σ (2.15) where it can also written as S pert = d 2 z {Q +,V}. (2.16) Σ Note that the original symmetries of the theory persist despite limiting ourselves to the perturbation theory; even though S top = 0, from (2.16), one finds that S pert is invariant under the nilpotent BRST symmetry generated by Q +. It is also invariant under the U(1) L U(1) R global symmetry. S pert shall henceforth be the action of interest in all our subsequent discussions. 3 Chiral algebras from the twisted heterotic sigma model 3.1 The chiral algebra Classically, the model is conformally invariant. The trace of the stress tensor from S pert vanishes, i.e., T z z = 0. The other non-zero components of the stress tensor, at the classical level, are given by T zz = g i j z φ i φ j z + λ za D z λ a, (3.1) and ( ) T z z = g i j z φ i z φ j + g i jψ z ī z ψ j +Γ j l k z φ lψ k. (3.2) Furthermore, one can go on to show that T z z = {Q +, g i jψ z z ī φ j }, (3.3) and ( ) [Q +,T zz ]=l za D z λ a + g i jd ψ j z + F a bi j(φ)λ za λ b ψ j z φ i = 0 (on shell). (3.4) From (3.4) and (3.3), we see that all components of the stress tensor are Q + - invariant; T zz is an operator in the Q + -cohomology while T z z is Q + -exact and thus trivial in Q + -cohomology. The fact that T zz is not Q + -exact even at the classical level implies that the twisted model is not a 2D topological field theory; rather, it is a 2D conformal field theory. This is because the original model has (0, 2) and not (2, 2) supersymmetry. On the other hand, the fact that T z z is Q + -exact has some non-trivial consequences on the nature of the local operators in the Q + -cohomology. Let us discuss this further.

11 TWISTED SIGMA MODELS 769 We say that a local operator O inserted at the origin has dimension (n, m) if under a rescaling z λz, z λz (which is a conformal symmetry of the classical theory), it transforms as n+m / z n z m, that is, as λ n λ m. Classical local operators have dimensions (n, m), where n and m are nonnegative integers. 4 However, only local operators with m = 0 survive in Q + -cohomology. The reason for the last statement is that the rescaling of z is generated by L 0 = d z zt z z. As we noted in the previous paragraph, T z z is of the form {Q +,...}, so L 0 = {Q +,V 0 } for some V 0. If O is to be admissible as a local physical operator, it must at least be true that {Q +, O} = 0. Consequently, [ L 0, O] ={Q +, [V 0, O]}. Since the eigenvalue of L 0 on O is m, we have [ L 0, O] =mo. Therefore, if m 0, it follows that O is Q + -exact and thus trivial in Q + -cohomology. By a similar argument, we can show that O, as an element of the Q + - cohomology, varies holomorphically with z. Indeed, since the momentum operator (which acts on O as z ) is given by L 1, the term z O will be given by the commutator [ L 1, O]. Since L 1 = d zt z z, we will have L 1 = {Q +,V 1 } for some V 1. Hence, because O is physical such that {Q +, O} = 0, it will be true that z O = {Q +, [V 1, O]} and thus vanishes in Q + -cohomology. The observations that we have so far are based solely on classical grounds. The question that one might then ask is whether these observations will continue to hold when we eventually consider the quantum theory. The key point to note is that if it is true classically that a cohomology vanishes, it should continue to do so in perturbation theory, when quantum effects are small enough. Since the above observations were made based on the classical fact that T z z vanishes in Q + -cohomology, they will continue to hold at the quantum level. Let us look at the quantum theory more closely The quantum theory Quantum mechanically, the conformal structure of the theory is violated by a non-zero 1-loop β-function; renormalization adds to the classical action S pert a term of the form: Δ 1-loop = c 1 R i j φ j z ψz ī + c 2 g i j F a bi jλ za l b z (3.5) for some divergent constants c 1,2, where R i j is the Ricci tensor of X. In the Calabi Yau case, one can choose a Ricci-flat metric and a solution to the Uhlenbeck Yau equation, g i j F a bi j = 0, such that Δ 1-loop vanishes and 4 Anomalous dimensions under RG flow may shift the values of n and m quantum mechanically, but the spin given by (n m), being an intrinsic property, remains unchanged.

12 770 MENG-CHWAN TAN the original action is restored. In this case, the classical observations made above continue to hold true. On the other hand, in the massive models where c 1 (X) 0, there is no way to set Δ 1-loop to zero. Conformal invariance is necessarily lost, and there is non-trivial RG running. However, one can continue to express T z z as {Q +,...}, i.e., it remains Q + -exact and thus continues to vanish in Q + -cohomology. Hence, the above observations about the holomorphic nature of the local operators having dimension (n, 0) continue to hold in the quantum theory. We would also like to bring to the reader s attention another important feature of the Q + -cohomology at the quantum level. Recall that classically, we had [Q +,T zz ] = 0 via the classical equations of motion. Notice that the classical expression for T zz is not modified at the quantum level (at least up to 1-loop), since even in the non-calabi Yau case, the additional term of Δ 1-loop in the quantum action does not contribute to T zz. However, due to 1-loop corrections to the action of Q +, we have, at the quantum level [Q +,T zz ]= z (R i j z φ i ψ j )+ (3.6) (where is also a partial derivative of some terms with respect to z). Note that the term on the RHS of (3.6) cannot be eliminated through the equations of motion in the quantum theory. Neither can we modify T zz (by subtracting a total derivative term) such that it continues to be Q + -invariant. This implies that in a massive model, operators do not remain in the Q + -cohomology after general holomorphic coordinate transformations on the worldsheet, i.e., the model is not conformal at the level of the Q + -cohomology. 5 However, T zz continues to be holomorphic in z up to Q + -trivial terms; from the conservation of the stress tensor, we have z T zz = z T z z, and T z z, while no longer zero, is now given by T z z = {Q +,G z z } for some G z z, i.e., z T z z continues to be Q + -exact, and z T zz 0inQ + -cohomology. The holomorphy of T zz, together with relation (3.6), has further implications for the Q + -cohomology of local operators; by a Laurent expansion of T zz, 6 one can use (3.6) to show that [Q +,L 1 ]=0. This means that operators remain in the Q + -cohomology after global translations on the worldsheet. In addition, recall that Q + is a scalar with spin zero in the twisted model. As shown few paragraphs before, we have the condition L 0 = 0. Let the spin be S, where S = L 0 L 0. Therefore, 5 In Section 5.7, we will examine more closely, from a different point of view, the 1-loop correction to the action of Q + associated with the beta-function, where (3.6) will appear in a different guise. 6 Since we are working modulo Q + -trivial operators, it suffices for T zz to be holomorphic up to Q + -trivial terms before an expansion in terms of Laurent coefficients is permitted.

13 TWISTED SIGMA MODELS 771 [Q +,S] = 0 implies that [Q +,L 0 ] = 0. In other words, operators remain in the Q + -cohomology after global dilatations of the worldsheet coordinates. One can also make the following observations about the correlation functions of these local operators. First, note that {Q +,W} = 0 for any W and recall that for any local physical operator O α, we have {Q +, O α } = 0. Since the z operator on Σ is given by L 1 = d z T z z, where T z z = {Q +, }, we find that z O 1 (z 1 )O 2 (z 2 ) O s (z s ) is given by d z {Q +, } O 1 (z 1 ) O 2 (z 2 ) O s (z s ) = d z {Q +, i O i(z i )} = 0. Thus, the correlation functions are always holomorphic in z. Secondly, T z z = {Q +,G z z } for some G z z in the massive models. Hence, the variation of the correlation functions due to a change in the scale of Σ will be given by O 1 (z 1 )O 2 (z 2 ) O s (z s ){Q +,G z z } = {Q +, i O i(z i ) G z z } = 0. In other words, the correlation functions of local physical operators will continue to be invariant under arbitrary scalings of Σ. Thus, the correlation functions are always independent of the Kähler structure on Σ and depend only on its complex structure A holomorphic chiral algebra A Let O(z) and Õ(z )betwoq + -closed operators such that their product is Q + -closed as well. Now, consider their operator product expansion (OPE) O(z)Õ(z ) k f k (z z )O k (z ), (3.7) in which the explicit form of the coefficients f k must be such that the scaling dimensions and U(1) L U(1) R charges of the operators agree on both sides of the OPE. In general, f k is not holomorphic in z. However, if we work modulo Q + -exact operators in passing to the Q + -cohomology, the f k s which are non-holomorphic and are thus not annihilated by / z drop out from the OPE because they multiply operators O k which are Q + -exact. This is true because / z acts on the LHS of (3.7) to give terms which are cohomologically trivial. 7 In other words, we can take the f k s to be holomorphic coefficients in studying the Q + -cohomology. Thus, the OPE of (3.7) has a holomorphic structure. In summary, we have established that the Q + -cohomology of holomorphic local operators has a natural structure of a holomorphic chiral algebra (as defined in the mathematical literature), which we shall henceforth call A; it is always preserved under global translations and dilatations, though (unlike 7 Since {Q +, O} = 0, we have zo = {Q +,V(z)} for some V (z), as argued before. Hence zo(z) Õ(z )={Q +,V(z)Õ(z )}.

14 772 MENG-CHWAN TAN the usual physical notion of a chiral algebra) it may not be preserved under general holomorphic coordinate transformations on the Riemann surface Σ. Likewise, the OPEs of the chiral algebra of local operators obey the usual relations of holomorphy, associativity, and invariance under translations and scalings of z, but not necessarily invariance under arbitrary holomorphic reparameterizations of z. The local operators are of dimension (n, 0) for n 0, and the chiral algebra of such operators requires a flat metric up to scaling on Σ to be defined. 8 Therefore, the chiral algebra that we have obtained can only be globally defined on a Riemann surface of genus 1, or be locally defined on an arbitrary but curved Σ. To define the chiral algebra globally on a surface of higher genus requires more in-depth analysis and is potentially obstructed by an anomaly involving c 1 (Σ) and (c 1 (E) c 1 (X)), which we will discuss in Sections 4 and 5.6. Last but not least, as is familiar for chiral algebras, the correlation functions of these operators depend on Σ only via its complex structure. The correlation functions are holomorphic in the parameters of the theory and are therefore protected from perturbative corrections. 3.2 The moduli of the chiral algebra Here, we shall consider the moduli of the chiral algebra A. To this end, let us first make some additional observations about A. First, notice that the metric g i j of the target space X appears in the classical action S pert inside a term of the form {Q +, }. Similarly, the fibre metric h a b of the holomorphic vector bundle E, which appears implicitly in the expression λ za lz ā of V in (2.13), also sits inside a term of the form {Q +, }. Hence, in passing to the Q + -cohomology, we find that the chiral algebra is independent of the metrics on X and the fibre space of E. Secondly, note that the chiral algebra does depend on the complex structure of X and the holomorphic structure of E because they enter in the definition of the fields and the fermionic symmetry transformation laws of (2.11). As we are not going to study how the chiral algebra behaves under a continuous deformation of the bundle E, its dependence on the holomorphic structure of E shall be irrelevant to us, at least in this paper. Note also that the chiral algebra varies holomorphically with the complex structure of X; one can show, using the form of S pert in (2.16), that if J denotes the complex structure of X, an anti-holomorphic derivative / J changes S pert by a term of the form {Q +,...}. 8 Notice that we have implicitly assumed the flat metric on Σ in all of our analysis thus far.

15 TWISTED SIGMA MODELS 773 We shall now consider adding to S pert a term which will represent the moduli of the chiral algebra A. As we will show shortly in Section 3.3, this term results in a non-kähler deformation of the target space X. Thus, X will be a complex, hermitian manifold in all our following discussions. To proceed, let T = 1 2 T ijdφ i dφ j be any two-form on X that is of type (2, 0). 9 The term that deforms S pert will then be given by S T = d 2 z {Q +,T ij ψz ī z φ j }. (3.8) Σ By construction, S T is Q + -invariant. Moreover, since it has vanishing (q L,q R ) charges, it is also invariant under the global U(1) L U(1) R symmetry group. Hence, as required, the addition of S T preserves the classical symmetries of the theory. Explicitly, we then have ( S T = d 2 z T ij, kψ kψ z ī z φ j T ij z φ i z φ j), (3.9) Σ where T ij, k = T ij / φ k. Note that since d 2 z = idz d z, we can write the second term on the RHS of (3.9) as S (2) T = i T ij dφ i dφ j = i Φ (T ). (3.10) 2 Σ Σ Recall that in perturbation theory, we are considering degree-zero maps Φ with no multiplicity. Hence, for S (2) T to be non-vanishing (in contrast to the closed Kähler form K that we encountered in Section 2.2), T must not be closed, i.e., dt 0. In other words, one must have a non-zero flux H = dt. As T is of type (2, 0), H will be a three-form of type (3, 0) (2, 1). Notice here that the first term on the RHS of (3.9) is expressed in terms of H, since T ij, k is simply the (2, 1) part of H. In fact, S (2) T can also be written in terms of H as follows. Suppose that C is a three-manifold whose boundary is Σ and over which the map Φ : Σ X extends. Then, if T is globally defined as a (2, 0)-form, the relation H = dt implies, via Stoke s theorem, that S (2) T = i Φ (H). (3.11) C Hence, we see that S T can be expressed solely in terms of the three-form flux H (modulo terms that do not affect the perturbation theory). Note at this point that we do not actually want to limit ourselves to the case that T is globally defined; as is clear from (3.8), if T were to be 9 As we will see shortly, the restriction of T to be a gauge field of type (2, 0), will enable us to associate the moduli of the chiral algebra with the moduli of sheaves of vertex superalgebras studied in the mathematical literature [3, 5] as desired.

16 774 MENG-CHWAN TAN globally defined, S T and therefore the moduli of the chiral algebra would vanish in Q + -cohomology. Fortunately, the RHS of (3.11) makes sense as long as H is globally defined, with the extra condition that H be closed, since C cannot be the boundary of a four-manifold. 10 Therefore, (as will be shown shortly via Poincaré s lemma), it suffices for T to be locally defined such that H = dt is true only locally. Hence, T must be interpreted an a two-form gauge field in the string theory (or a non-trivial connection on gerbes in mathematical theories). In the quantum theory, a shift in the (Euclidean) action S E by an integral multiple of 2πi is irrelevant as the path integral factor is e S E. Hence, the effective range of the continuous moduli of H is such that 0 S (2) T < 2πi. Also, the continuous U(1) L and U(1) R symmetries of the classical theory reduce to discrete symmetries in the quantum theory due to worldsheet instantons. In order for the discrete symmetries to remain anomaly-free, H 2π must be an integral cohomology class, i.e., 1 2π C Φ (H) Z. Hence, the continuous moduli of H present in perturbation theory may be absent in the non-perturbative theory. Since we are only considering the physics in the perturbative regime, we will not see this effect. In writing S (2) T in terms of H, we have made the assumption that Φ extends over some three-manifold C with boundary Σ. Since in perturbation theory, one considers only topologically trivial maps Φ which can be extended over any chosen C, the assumption is valid. Non-perturbatively however, one must also consider the contributions coming from topologically non-trivial maps as well. Thus, an extension of the map over C may not exist. Therefore, the current definition of S (2) T will not suffice. Notice also, that T cannot be completely determined as a two-form gauge field by its curvature H = dt, as one may add a flat two-form gauge field to T where H does not change at all. This indeterminacy of T is inconsequential in perturbation theory as S (2) T can be made to depend solely on H via (3.11). Non-perturbatively on the other hand, because C may not exist, S (2) T can only be expressed in terms of T and not H, as in (6.9). The explicit details of T will then be important. Since the sheaf of CDOs or vertex superalgebras, as defined in the mathematical literature, only depends on H, the theory of CDOs can only be used to describe the physics of the twisted model in perturbation theory. This dependency of the sheaf of vertex superalgebras on H is also what motivates us to express S T entirely in terms of H. 10 From homology theory, the boundary of a boundary is empty. Hence, since Σ exists as the boundary of C, the three-manifold C itself cannot be a boundary of a higher dimensional four-manifold.

17 TWISTED SIGMA MODELS Moduli As mentioned earlier, T must be locally defined only such that the expression H = dt is valid only locally. At the same time, H must be globally defined and closed so that (3.11) can be consistent. Fortunately, one can show that a globally defined, closed three-form H of type (3, 0) (2, 1) can be expressed locally as H = dt, where T is a locally defined two-form of type (2, 0). To demonstrate this, let us first select any local two-form Y such that H = dy. Poincaré s lemma asserts that Y will exist because H is closed and globally defined. In general, Y is given by a sum of terms Y (2,0) + Y (1,1) + Y (0,2) of the stated types. Since in our application, H has no component of type (0, 3), it will mean that Y (0,2) = 0. By the version of Poincaré s lemma, we then have Y (0,2) = η, where η is a one-form of type (0, 1). Let Ỹ = Y dη, so that we have H = dy = dỹ, where Ỹ = Ỹ (2,0) + Ỹ (1,1). Since H has no component of type (1, 2) either, it will mean that Ỹ (1,1) = 0. By the version of Poincaré s lemma again, we will have Ỹ (1,1) = ζ, where ζ is a one-form of type (1, 0). By defining T = Ỹ dζ, we have H = dỹ = dt, where T is a two-form of type (2, 0) as promised. Recall that if T and therefore H = dt is globally defined, S T will vanish in Q + -cohomology. Hence, in perturbation theory, the moduli of the chiral algebra derived from the twisted heterotic sigma model on a holomorphic vector bundle E over a complex hermitian manifold X- are parameterized by a closed three-form H of type (3, 0) (2, 1) modulo forms that can be written globally as H = dt, where T is a form of type (2, 0). In other words, the moduli are represented by some cohomology class that H represents. Nonperturbatively, however, the picture can be very different; for the topologically non-trivial maps of higher degree, S (2) T can only be expressed in terms of T via (6.9), where even flat T fields will be important. In addition, H must be an integral class, which means that the moduli in the non-perturbative theory must be discrete and not continuous. The analysis is beyond the scope of the present paper, and we shall not expound on it further Interpretation via H 1 (X,Ω 2,cl X ) Now, we would like to determine the type of cohomology class that H represents. To this end, let U a, a =1,...,sbe a collection of small open sets providing a good cover of X such that their mutual intersections are open sets as well. Suppose that we have a globally defined closed three-form H of type (3, 0) (2, 1) that can be expressed as H = dt locally, where T is a twoform of type (2, 0) which is locally-defined. This means that on each U a,

18 776 MENG-CHWAN TAN we will have a (2, 0)-form T a, such that H a = dt a. On each open double intersection U a U b, let us define T ab = T a T b, where T ab = T ba (3.12) for each a,b, and T ab + T bc + T ca = 0 (3.13) for each a, b and c. Since H is globally defined, H a = H b on the intersection U a U b, so that dt ab = 0. This implies that T ab = T ab = 0. Notice that since on each U a, we have H a = dt a, the shift given by T a T a + S a, where ds a = 0 (and therefore S a = S a = 0), leaves each H a invariant. In other words, in describing H, we have an equivalence relation T ab T ab = T ab + S a S b. (3.14) Let us proceed to describe T ab more precisely. In order to do so, let us first denote Ω 2 X as the sheaf of (2, 0)-forms on X and Ω2,cl X as the sheaf of such forms that are annihilated by. (The label cl is short for closed and refers to forms that are closed in the sense of being annihilated by. We will occasionally write this as Ω 2,cl when there is no ambiguity.) A holomorphic section of Ω 2,cl X in a given set U X isa(2, 0)-form on U that is annihilated by both and. Likewise, Ω n,cl X is the sheaf whose sections are (n, 0) forms that are -closed, such that its holomorphic sections are also annihilated by. Since it was shown in the last paragraph that each T a Ω 2 X and that T ab = T ab = 0 in each double intersection U a U b, we find that T ab must be a holomorphic section of Ω 2,cl X. Next, notice from the equivalence relation (3.14) that T ab 0ifwecan express T ab = S b S a in U a U b, where S a and S b are holomorphic in each U a and U b respectively (since S a = S b = 0). Hence, the non-vanishing T ab s are those which obey the identities (3.12) and (3.13), modulo those that can be expressed as T ab = S b S a. In other words, T ab is an element of the Cech cohomology group H 1 (X, Ω 2,cl X ). Now, if H is globally given by the exact form H = dt, it would mean that T is globally defined and as such, T a = T b = T in each U a U b, whereupon all T ab s must vanish. Thus we have obtained a map between the space of closed three-forms H of type (3, 0) (2, 1), modulo forms that can be written globally as dt for T of type (2, 0), to the Cech cohomology group H 1 (X, Ω 2,cl X ). One could have also run everything backwards, starting with an element of H 1 (X, Ω 2,cl X ) and using the partition of unity subordinate to the cover U a of X to construct the inverse of this map [12]. However, since this argument is also standard in the mathematical literature in relating a and Cech cohomology, we shall skip it for brevity. Therefore, we can conclude

19 TWISTED SIGMA MODELS 777 that H represents an element of H 1 (X, Ω 2,cl X ). Hence, in perturbation theory, the moduli of the chiral algebra derived from the twisted heterotic sigma model on a holomorphic vector bundle E over a complex hermitian manifold X, like the moduli of sheaves of vertex superalgebras studied in the mathematical literature [3, 5], are associated with H 1 (X, Ω 2,cl X ). 3.3 The moduli as a non-kähler deformation of X As shown above, in order to incorporate the moduli so that we can obtain a family of chiral algebras, we need to turn on the three-form H-flux. As we will show in this section, this will in turn result in a non-kähler deformation of the target space X. The motivation for our present discussion rests on the fact that this observation will be important when we discuss the physical application of our results at the (2, 2) locus in Section 7.2. The moduli s connection with a non-kähler deformation of X can be made manifest through the twisted heterotic sigma model s relation to a unitary model with (0, 2) supersymmetry. Thus, let us first review some known results [19, 20] about (0, 2) supersymmetry. A unitary model with (0, 2) supersymmetry can be constructed by enlarging the worldsheet Σ to a supermanifold Σ with bosonic coordinates z, z and fermionic coordinates θ +, θ +. (The + superscript in θ +, θ + just indicates that they transform as sections of the positive chirality spin bundle of Σ.) The two supersymmetry generators that act geometrically in Σ are given by Q + = θ + iθ+ z, Q + = θ + i θ + z, (3.15) where Q 2 + = Q 2 + = 0 and {Q +, Q + } = 2i / z. To construct supersymmetric Lagrangians that are invariant under Q + and Q +, we note the fact that these operators anti-commute with the supersymmetric derivatives D + = θ + + iθ+ z, D + = θ + + i θ + z, (3.16)

20 778 MENG-CHWAN TAN and commute with z and z. Moreover, the measure d 2 z dθ d θ is also supersymmetric, i.e., it is invariant under the action of Q + and Q +. Consequently, any action constructed using only the superfields and their supersymmetric and/or partial derivatives, together with the measure, will be supersymmetric. To construct such an action, we can describe the theory using chiral superfields Φ in the supermanifold Σ, which obey D + Φ i = D + Φī = 0. They can be expanded as Φ i = φ i + 2θ + ρ i + i θ + θ + z φ i, Φī = φī 2 θ + ρī+ + i θ + θ + z φī. (3.17) Here, φ i and φī are scalar fields on Σ which define a map φ :Σ X; they serve as the (local) holomorphic and anti-holomorphic complex coordinates on X, respectively. ρ i + and ρī+ are the fermionic superpartners of the φ fields on Σ with positive chirality, and they transform as sections of the pull-backs φ (TX) and φ (TX), respectively. To ascertain how the various component fields of Φ transform under the two supersymmetries generated by Q + and Q +, we must compute how the Φ superfields transform under the action of the Q + and Q + operators defined in (3.15) and compare its corresponding components with that in the original superfields. In particular, Q + generates the field transformations δφ i =0, δφī = 2 ρī+, δρ i + = i 2 z φ i, (3.18) δρī+ =0. Notice at this point that the non-zero transformations in equation (2.11) of Section 2.2 are given by δφī = ψī and δψ ī z = z φ i on-shell. These coincide with the non-zero transformations in (3.18) if we set ψī = 2ρī+ and ψ ī z = iρ i +/ 2. Hence, we see that the structure in a unitary (0, 2) model is a specialization of the structure in the twisted heterotic sigma model studied in Section 2.2, with Q + corresponding to Q +. This should not be surprising since we started off with a heterotic sigma model with (0, 2) supersymmetry anyway. As our following arguments do not require us to refer to the field transformations generated by the other supercharge Q +, we shall omit them for brevity.

21 TWISTED SIGMA MODELS 779 Now let K = K i (φ j,φ j )dφ i bea(1, 0)-form on X, with complex conjugate K = K ī (φ j,φ j )dφī. A supersymmetric action can then be written as [20] ( S = d 2 z d θ + dθ + i 2 K i(φ, Φ) z Φ i + i ) 2 Kī (Φ, Φ) z Φī. (3.19) Note that action density of (3.19) must be a local expression, so that the action will be invariant under additional transformations of K, whence one can obtain the required invariant geometrical objects that can then be globally defined. In particular, since the action density of (3.19) is a local expression, we are free to discard exact forms after integrating parts. This can be shown via a superspace extension of Poincaré s lemma. 11 Consequently, the transformations of K and K in the target space X, which correspond to transformations in superspace that leave the action invariant, will be given by K K + Λ, K K Λ, (3.20) where Λ = Λ(φ i,φī) is some imaginary zero-form. In fact, under (3.20), the corresponding superfield transformations will be given by K i (Φ, Φ) K i (Φ, Φ) + Φ iλ(φ, Φ), K ī (Φ, Φ) K ī (Φ, Φ) Φ īλ(φ, Φ), (3.21) and one can show that the action density changes by the total derivative i z Λ(Φ, Φ), which can be integrated to zero. By integrating the θ + and θ + variables in (3.19), we get an action given by an integral over the z and z variables of a Lagrangian written in terms of fields on Σ. Of particular interest would be the hermitian metric of X found in the Lagrangian and defined by ds 2 = g i jdφ i dφ j. It is given by g i j = i K j + jk i. (3.22) Notice that it is invariant under (3.20); hence, it can be globally defined. Associate to this metric is a globally defined (1, 1)-form on X that is invariant under (3.20) and given by ω T = i ( ) K K. (3.23) 2 Note that ω T is the analogue of a Kähler (1, 1)-form ω onakähler manifold. However, from (3.23), we find that in contrast to ω, which obeys ω = ω = 11 Via a superspace extension of Poincare s lemma, we learn that locally exact forms Ŷ on Σ are globally closed. Let Σ be the boundary of a higher dimensional supermanifold Ĉ. Hence, via Stoke s theorem in superspace, we find that Σ Ŷ = dŷ = 0, where d is Ĉ the exterior derivative operator in superspace.

22 780 MENG-CHWAN TAN 0, ω T obeys the weaker condition ω T = 0 (3.24) instead. This just reflects the well-known fact that the target space of a model with (0, 2) supersymmetry is in general hermitian and non-kähler. In order to relate the twisted heterotic sigma model to the unitary model reviewed here, we will need to express (3.19) in the form d 2 z {Q +,V} that S pert takes in Section 2.2. To do so, we must first convert (3.19) to an integral of an ordinary Lagrangian over z and z. The standard way to do this is to perform the integral over θ and θ. However, a convenient shortcut is to note that for any superfield W, we can make the replacement d 2 z d θ + dθ + W = d 2 2 z θ + θ + W θ+ =θ + =0 = d 2 z D + D + W θ + = θ + =0. (3.25) The rationale for the first step is that for a fermionic variable θ, dθ W = ( W/ θ) θ=0. The rationale for the second step is that the D s differ from the / θ s by z terms, which vanish upon integration by parts. Now, since Q + differs from D + and / θ + by a total derivative, we can rewrite (3.25) and therefore action (3.19) as S = d 2 z {Q +, [D +,W]} θ + = θ + =0, (3.26) where W = i 2 K i(φ, Φ) z Φ i + i 2 Kī (Φ, Φ) z Φī. Since we are identifying Q + with Q + of S pert = d 2 z {Q +,V}, we see that V = D + W = id + (K i (Φ, Φ) z Φ i K ī (Φ, Φ) z Φī)/2. To compute V, note that since D + Φ=0, we have D + (K i z Φ i K ī z Φī) =K i,j D + Φ j z Φ i + K i z D + Φ i K ī,i D + Φ i z Φī. By subtracting the total derivative z (K i D + Φ i ) which will not contribute to the action, we get V = i 2 (K i, j + K j,i) z Φ j D+ Φ i i 2 (K i,j K j,i ) D + Φ j z Φ i. To set θ + = θ + = 0 in (3.26), we just need to set Φ i = φ i, Φī = φī, and D + Φ i = 2ρ i + =2iψz, ī and let Q + act ) as in (3.18). Hence, V = ( ( jk i + i K j)ψ z ī φ j z +( i K j j K i )ψz ī z φ j. By setting the λ za lz ā term in equation (2.13) to zero via the equation of motion lz ā = 0, we can read off the hermitian metric g i j used in Section 2.2 to construct the basic Lagrangian in S pert, as well as the field called T in Section 3.2. We have g i j = i K j + jk i, as claimed in (3.22) above, and T ij =( i K j j K i ). From the last statement, and the definition T = 1 2 T ijdφ i dφ j in Section 3.2, we see that T = K asa(2, 0)-form. Thus, it follows from (3.23) that the curvature of the two-form field T is H = dt = K =2i ω T. By virtue