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Superspace Effective Actions for 4D Compactifications of Heterotic and Type II Superstrings

Two-dimensional sigma models are defined for the new manifestly spacetime supersymmetric description of four-dimensional compactified superstrings. The resulting target-superspace effective action is constrained by the way the spacetime dilaton couples to the worldsheet curvature: For the heterotic superstring, the worldsheet curvature couples to the real part of a chiral multiplet, and for Type II it couples to the real part of the sum of a vector multiplet and a tensor hypermultiplet. For the Type II superstring, this contradicts the standard folklore that only a hypermultiplet counts string-loops, explains the peculiar dilaton coupling of Ramond-Ramond fields, and allows the effective action to be easily written in N=2 4D superspace. It also implies that vector multiplet interactions get no quantum corrections, while hypermultiplet interactions can only get corrections if mirror symmetry is non-perturbatively broken.


Introduction
There are two ways to construct low-energy effective actions in string theory.
One can define a two-dimensional sigma model for the string in a curved background, use conformal invariance to determine the equations of motion, and look for an action which provides these equations of motion [1]. Alternatively, one can calculate on-shell S-matrix scattering amplitudes and look for an action which yields these amplitudes [2].
For the superstring in the RNS formalism, both of these methods are made clumsy by the complicated nature of the Ramond fields. For this reason, the part of the effective action coming from the Ramond fields is much less understood than the part coming from the Neveu-Schwarz sector. In light of recent conjectures relating non-perturbative states with the Ramond-Ramond sector of the type II superstring [3], this lack of understanding is especially bothersome. For example, the F 2 term for the graviphoton field strength appears to be independent of the dilaton ϕ, instead of having the expected e −2ϕ dependence of tree-level terms [4].
Recently, a new formalism for the superstring has been discovered with local N=2 worldsheet superconformal invariance [5]. This formalism is related to the RNS formalism by a field-redefinition [6], but has the advantage of being manifestly spacetime supersymmetric. It is especially well-suited for compactifications to four dimensions, where it allows manifestly 4D super-Poincaré invariant quantization [7].
In this paper, we use this new formalism to define a two-dimensional sigma model and construct a superspace effective action for 4D compactifications of heterotic and Type II superstrings. While the heterotic effective action in N=1 4D superspace has already appeared in the literature [8], the Type II effective action in N=2 4D superspace is new. Since spacetime-supersymmetry is manifest, there is no distinction between Ramond and Neveu-Schwarz fields and the previous confusion over the Ramond sector is easily resolved. Furthermore, the requirement that the Fradkin-Tseytlin term for dilaton coupling contains N=2 worldsheet supersymmetry implies certain non-renormalization theorems for the effective action. For those readers who are only interested in the new Type II superspace effective action and the resulting non-renormalization theorems, it may be useful to just read section 3 and section 5, and skip the derivation from the sigma model.
In section 2, we review the new manifestly spacetime supersymmetric formalism for the superstring. This formalism has critical N=2 worldsheet superconformal invariance, and for compactifications to four dimensions, superspace chirality is related to worldsheet chirality [7]. Superstring scattering amplitudes can be computed by calculating correlation functions of BRST-invariant vertex operators on N=2 super-Riemann surfaces [5], and the amplitudes agree with those obtained using the RNS formalism. Since the massless vertex operators are related to the linearized interactions of the sigma model, these vertex operators will be reviewed.
The massless vertex operators are manifestly spacetime supersymmetric and are constructed from prepotentials of the spacetime superfields. For the heterotic superstring, these prepotentials describe N=1 conformal supergravity, a tensor multiplet, super-Yang-Mills multiplets, and chiral scalar multiplets which come from compactification moduli. (We assume throughout this paper that the compactification manifold has no isometries, so all moduli are described by four-dimensional scalars.) For the Type II superstring, the prepotentials describe N=2 conformal supergravity, a tensor hypermultiplet, and chiral or twisted-chiral multiplets from the compactification moduli. The superspace chirality of the compactification moduli superfields comes from the worldsheet chirality or twistedchirality of the relevant N=(2,2) primary fields. As will be discussed, Type II chiral multiplets can be interpreted as vector multiplets [9] while twisted-chiral multiplets can be interpreted as tensor hypermultiplets [10]. Note that tensor hypermultiplets contain the same on-shell component fields as scalar hypermultiplets, which are more commonly used to describe Type II compactification moduli.
Unlike the massless vertex operators, sigma models and effective actions are constructed from superspace gauge fields and field strengths rather than prepotentials. In section 3, we will review the torsion constraints which relate these gauge fields and field strengths to their prepotentials. To analyze these torsion constraints and to facilitate the construction of superspace actions, it will be useful to introduce conformal compensators. (In the bosonic string, the spacetime dilaton which couples to worldsheet curvature plays the role of a conformal compensator [11].) Although there are various types of compensators one can introduce for N=1 and N=2 supergravity, worldsheet supersymmetry of the Fradkin-Tseytlin term in the sigma model uniquely determines the correct type.
As discussed in reference [12], the correct type of compensator can also be determined by requiring factorization of closed superstring states involving worldsheet ghosts.
For the heterotic superstring, the compensator must be a chiral scalar multiplet, which identifies the off-shell theory as matter coupled to old minimal supergravity. For the Type II superstring, two compensators are required, one which is chiral (a vector multiplet) and one which is twisted-chiral (a tensor hypermultiplet). Normally the compensators are gauge-fixed to remove the unphysical spacetime invariances. However, in the superstring sigma model, spacetime conformal invariance is instead removed by gauge-fixing the non-compensator tensor multiplet. (This is similar to the bosonic string sigma model, where the physical scalar is gauge-fixed instead of the dilaton.) For the heterotic superstring, this leaves a spacetime U(1) invariance (which is related to worldsheet U(1) invariance), while for the Type II superstring, this leaves a spacetime U(1)×U (1) invariance (which is related to worldsheet U(1)×U(1) invariance).
In section 4, we use the above spacetime superfields to explicitly construct a sigma model for four-dimensional compactifications of heterotic and Type II superstrings. The sigma model for the heterotic superstring was constructed with the assistance of Jan de Boer, Peter van Nieuwenhuizen, Martin Roček, Ergin Sezgin, Kostas Skenderis, and Kellogg Stelle. Like the standard 4D GS sigma model [13], this sigma model contains a term proportional to 1/α ′ where the superfields couple to their massless vertex operators. In the presence of torsion constraints (which do not put the superfields on-shell), this 1/α ′ term is invariant under classical worldsheet superconformal transformations.
However, unlike the standard GS sigma model, this sigma model also contains a Fradkin-Tseytlin term where the spacetime compensators couple to worldsheet supercurvature. For the heterotic superstring, the spacetime chiral compensator couples to N=(2,0) supercurvature, which is described by a worldsheet chiral superfield [14]. For the Type II superstring, the spacetime chiral and twistedchiral compensators couple to N=(2,2) supercurvature, which is described by a worldsheet chiral and twisted-chiral superfield. Quantum N=2 superconformal invariance of the combined 1/α ′ and Fradkin-Tseytlin terms is expected to imply the equations of motion for the spacetime superfields. This is currently being checked for the heterotic superstring by de Boer and Skenderis.
Finally in section 5, we use some simple properties of the sigma model to construct superspace effective actions. (For those readers only interested in effective actions, they can skip directly to this section, using section 3 as a reference.) Although the heterotic superspace effective action has already appeared in the literature [8], the Type II superspace effective action is new. This N=2 4D superspace action includes a chiral term for vector multiplet interactions and a twisted-chiral term for tensor hypermultiplet interactions. The twisted-chiral term is made SU(2) invariant by introducing harmonic-like variables [15] [16]. Since worldsheet Euler number couples in the Type II sigma model to the sum of the vector and tensor compensators, it is straightforward to determine the string-loop order of each term in the effective action. This type of dilaton coupling contradicts the standard folklore that loops are counted by just a hypermultiplet [17], and explains the dilaton coupling of Ramond-Ramond fields. (By dilaton, we always mean the field which couples to worldsheet curvature, and not the physical scalar which couples like the trace of the metric. The confusion in the literature was caused by the fact that the physical scalar, which does not count string loops, sits in a hypermultiplet.) We then prove various non-renormalization theorems for the Type II effective action, including the theorem that the chiral term for vector multiplet interactions receives no quantum perturbative or non-perturbative corrections. Since mirror symmetry of the sigma model relates chiral and twisted-chiral terms in the effective action, hypermultiplet interactions can only receive quantum corrections if mirror symmetry were broken. This would seem to contradict Type II/heterotic string-duality conjectures, which require that Type II hypermultiplet interactions receive quantum corrections [17]. However, the non-renormalization of hypermultiplet interactions is related to a Pecci-Quinn-like symmetry. If this Pecci-Quinn-like symmetry were non-perturbatively broken by spacetime instantons (which would imply the non-perturbative breaking of mirror symmetry), hypermultiplet interactions could receive quantum corrections.
In section 6, we summarize our results and discuss possible generalizations of this work for the superstring in more than four dimensions.

Review of the New Superstring Description
By twisting the ghost sector, any critical N=1 string can be "embedded" in a critical N=2 string without changing the physical theory [18]. After performing this embedding for the critical RNS superstring, a field-redefinition allows the resulting N=2 string to be made manifestly spacetime supersymmetric. This N=2 description of the superstring is especially elegant for compactifications to four dimensions, where the critical c = 6 matter sector splits into a c = −3 four-dimensional part and a c = 9 compactification-dependent part [7].
The four-dimensional part of the matter sector contains the spacetime variables, x m (m = 0 to 3), the left-moving fermionic variables, θ α andθ . α (α, . α = 1 to 2), the conjugate left-moving fermionic variables, p α andp . α , and one leftmoving boson ρ (which takes values on a circle of radius 1). For the heterotic GS superstring, one also has the right-moving fermions, ζ q (q = 1 to 32 − 2r), which describe the unbroken gauge degrees of freedom (e.g. the gauge group is E 6 × E 8 when r = 3, and the gauge group is SO(10)×E 8 when r = 4). For the Type II superstring, one needs the right-moving fermionic fields,θ α ,θ . α ,p α ,p . α , and one right-moving bosonρ.
The compactification-dependent part of the matter sector is described as in the RNS formalism by a c = (9, 6+r) N=(2,0) superconformal field theory for the heterotic superstring, and a c = (9, 9) N=(2,2) superconformal field theory for the Type II superstring. Note that because of manifest spacetime supersymmetry, there is no need to perform a GSO projection in either the four-dimensional or compactification-dependent sector.
The strange α ′ dependence of ρ in (2.1)will later be shown to be related to the Fradkin-Tseytlin term.
The left-moving c = 6 stress-energy tensor for this N=2 string is: (d) 2 means 1 2 ǫ αβ d α d β , and [L C , G C ,Ḡ C , J C ] are the left-moving generators of the c = 9 N=2 compactification-dependent stress-energy tensor. As was shown in ref- For the heterotic superstring, the right-moving c = 26 N=0 stress-energy tensor is: whereL C is the right-moving c = 6 + r compactification-dependent stress-energy tensor. For the Type IIB superstring, the right-moving c = 6 N=2 stress-energy tensor is whered α andd . α are obtained from (2.5)by using hatted variables and replacing ∂ − with ∂ + . Since the mirror transformation [21] flips the sign ofĴ C and switches G C withĜ C , the right-moving stress-energy tensor for Type IIA compactifications is The advantage of working with the variables d α and Π m is that they commute with the spacetime supersymmetry generators, When written in terms of the supersymmetric variables, the actions of (2.1)and (2.2)take the more familiar forms α ) and d α differs from (2.5)by terms which vanish on-shell. If the d α and ρ contributions are dropped, the actions in (2.10)and (2.11)are the standard heterotic and Type II Green-Schwarz four-dimensional actions [13].
Although one can formally write an N=2 worldsheet supersymmetric version of these actions as where (e ±± , ξ ± ,ξ ± , a ± ;ξ ± ,ξ ± ,â ± ) are the worldsheet supergravity fields, the quantum behavior of the ρ field makes it difficult to make manifest the worldsheet supersymmetry. Nevertheless, the free-field OPE's of (2.3)make it straightforward to check that [L, G,Ḡ, J] form a c = 6 N=2 superconformal algebra, thereby implying the quantum N=2 superconformal invariance of (2.12).
Scattering amplitudes can be calculated by evaluating correlation functions of physical vertex operators on N=2 super-Riemann surfaces. All vertex operators of zero ghost-number are constructed from U(1)-neutral combinations of the worldsheet matter fields and must carry integer J C charge in order not to have branch cuts with G. The massless vertex operators are simpler than in the RNS formalism, and since they play an essential role in the construction of the two-dimensional sigma model, they will be reviewed in the following sub-section.

Massless vertex operators
For the heterotic superstring, all massless vertex operators which are independent of the compactification are constructed from the real spacetime superfields V I (x, θ,θ) and V m (x, θ,θ), where V I is the prepotential for super Yang-Mills (I=1 to d labels the group index) and V m is the prepotential for N=1 supergravity plus a tensor multiplet (m = 0 to 3 is a spacetime vector index).
In integrated form, these vertex operators are given by [7] dz + dz − {Ḡ, [G, V P ]}O P (2.13) where O I = j I for the super Yang-Mills vertex operator (j I is the right-moving current constructed from the ζ q 's) and O m = Π m Gauge transformations which leave this vertex operator invariant are where A Im are the gluons, ψ Iα are the gluinos, and D I is the real auxiliary field. In Wess-Zumino gauge, the remaining 12+12 component fields of V m are described by For scalars, Ω (i) has dimension ( 1 2 , 1) and the vertex operator is where c.c. means complex conjugate and we are ignoring double poles in the OPE with G (these double poles vanish on-shell). For vectors, Ω (i) has dimension ( 1 2 , 1 2 ) and the vertex operator is where q = 1 to 16 − 2r. For spinors, Ω (i) has dimension ( 1 2 , r 8 ) and the vertex operator is As was shown in reference [22], these SO(16 − 2r) representations combine into representations of the maximal unbroken subgroup of E 8 (e.g. for r = 3, they combine into representations of E 6 ).
The compactification-dependent vertex operators are on-shell when M (i) x + = x + iθθ. (We will supress SO(16 − 2r) indices from now on.) For the Type II superstring, all compactification-independent massless vertex operators are constructed from a single real superfield, U (x, θ,θ,θ,θ) [23]. In integrated form, these vertex operators are given by [24] dz where all of the ∇'s act on U .
Gauge transformations which leave this vertex operator invariant are which can gauge-fix 96+96 component fields. In Wess-Zumino gauge, the remaining 32+32 component fields are described by From the superspin 1 piece of U representing the conformal supergravity multiplet, h mn is the traceless graviton, χ mα andχ mβ are the gravitinos, A U(1) m is the U(1) gauge field, A jk m are the three SU(2) gauge fields, T mn is the auxiliary tensor, ψ α andψ β are the auxiliary fermions, and D is the auxiliary scalar. From the superspin 0 piece of U representing the tensor multiplet, l jk is the SU(2) triplet, b mn is the anti-symmetric tensor, ξ α andξ β are the dilatinos, and y is a complex auxiliary scalar. This vertex operator is physical when U satisfies the N=2 primary conditions where ∇'s act only on M (i) 's and G C 's act only on Ω (i) 's. For Type IIA compact- tc andĜ C is switched withĜ C in the above vertex operators.
c , which implies the 8+8 component expansion of a vector multiplet field strength: tc , which implies the 8 + 8 component expansion of a tensor hypermultiplet field strength: tc , imply the usual polarization conditions and equations of motion for these component fields.

Superspace Description of the Massless Spectrum
From the previous section, we have seen that massless vertex operators which are independent of the compactification are constructed from superspace prepotentials. For the heterotic superstring, the N=1 supergravity and tensor multiplets are described by the superspin 3 2 and 1 2 parts of a real prepotential V m , and the super-Yang-Mills multiplet is described by a real prepotential V I . For the Type II superstring, the N=2 supergravity and tensor multiplets are described by the superspin 1 and 0 parts of a real prepotential U .

Gauge fields and field strengths
Although prepotentials are the most compact superspace representations for these multiplets, they are inconvenient for constructing super-reparameterization invariant quantities. In the sigma model and effective action, it will be more con- where j = ± are SU(2)-indices which can be raised and lowered using the ǫ jk tensor. Comparing with the notation from the previous section, The field strengths of these superspace gauge fields are obtained from the graded commutators is the covariant derivative, w AB C is the spin connection, Γ J A are the U(N) connections (N=1 for heterotic and N=2 for Type II), M D C are the Lorentz generators, T I are the Yang-Mills group generators, Y J are the U(N) group generators, T AB C is the torsion, R ABC D is the supercurvature, F I AB is the super Yang-Mills field strength, f J AB is the U(N) field strength, and H M NP is the tensor field strength. As in ordinary gravity, the connections w AB C and Γ J A are not independent superfields and are related to E A M by torsion constraints.
However, since E A M , B M N , and A M contain more component fields than the prepotentials, one needs to impose further torsion constraints to remove the additional degrees of freedom. These constraints will also be needed for worldsheet supersymmetry of the sigma model and are given explicitly as constraints 1) and 2) in equations (4.3) and (4.4). After imposing them, the above field strengths can be expressed in terms of the reduced field strengths of the following tables: Heterotic multiplet tensor vector supergravity where the N=1 reduced field strengths are defined by Type II multiplet tensor vector supergravity where the N=2 reduced field strengths are defined by Note that we have not discussed the chiral prepotential Ξ for N=1 and N=2 tensor multiplets. (V m or U can only be used as the tensor prepotential in certain gauges.) The N=2 vector multiplet will appear when we discuss compensators and compactification-dependent states.
Bianchi identities imply various conditions on the above reduced field strengths. For example, for the heterotic superstring, L is a real N=1 linear superfield and W I α is an N=1 chiral superfield satisfying For the Type II superstring, Bianchi identities imply that L jk is a real N=2 linear superfield and W is an N=2 chiral superfield satisfying Implications of Bianchi identities for the reduced N=1 and N=2 supergravity field strengths can be found in references [25] and [26]. Although it is not widely appreciated, the bosonic string also makes use of a conformal compensator field, ϕ, which couples to worldsheet curvature and will be called the dilaton [11]. Recall that the physical massless vertex operator of the bosonic string is dz + dz − (g mn + b mn )∂ + x m ∂ − x n , and g mn splits at linearized level into a traceless piece h mn = g mn − 1 D η mn η pq g pq representing conformal supergravity, and a physical scalar l = η mn g mn which will not be called the dilaton. The vertex operator for the dilaton, on the other hand, is constructed from worldsheet ghosts which, when integrated out, give coupling to worldsheet curvature.

Compensators
The usual version of the low-energy effective action for the bosonic string is The relative coefficients of these terms can be determined by T-duality which transforms g mn into b mn . In the supersymmetric cases, they are already fixed by supersymmetry [12].) Here Φ is related to the more common form of the dilaton field used in string theory by (The fact that this is the T-duality invariant combination follows from the fact that Φ must soak up the √ −g measure, since √ −g is not T-duality invariant.) At this point we have not yet seperated out the trace of g mn ; this we now perform with full nonlinearity by the Weyl rescaling g mn → l 2 g mn (3.10) (We do not scale Φ, and leave Φ-dependence out of the metric rescaling, so that Φ stays out of T-duality transformations [12].) The result is In general relativity one normally breaks the scale invariance introduced by this rescaling by gauge-fixing Φ = (−g) 1/4 (ϕ = 0), or in this case the slightly modified gauge Φ = (−g) 1/4 l −1 , which produces an action with the standard form for the Einstein term, and the right sign for the scalar term: However, one can also break scale invariance by choosing the "string gauge" l = 1, which returns us to the form of the string effective action before Weyl rescaling.
Although Φ naively appears to be a physical scalar in this string gauge, it is easily identified as a compensating scalar by its "wrong-sign" kinetic term. On the other hand, the physical scalar with a "right-sign" kinetic term has turned in string gauge into the determinant of the metric.
For N=1 and N=2 supergravity, there are various possible types of conformal compensators [25] [27]. However, as will be shown in the following section, worldsheet supersymmetry of the Fradkin-Tseytlin term uniquely determines the correct type. (As discussed in [12], the correct compensator can also be deter- The other Type II compensator is described by superspace twisted-chiral and twisted-anti-chiral superfields, Φ tc andΦ tc , satisfying∇ . jk must carry conformal weight +2. Note that W (0) carries U(1) weight +1 and is an SU(2) singlet, while L (0) jk carries U(1) weight zero and is an SU(2) triplet.

Compactification-dependent states
Finally, we shall give the superfield description of the compactification dependent massless states, whose vertex operators are explicitly described in section 2. Since these moduli superfields describe deformations of the compactification manifold, we shall assume throughout this paper that the values of these moduli are small enough so that the massless spectrum is not modified.

Sigma Models
Because we know the action in a flat background and the massless vertex operators from section 2, it is straightforward to construct a two-dimensional sigma model for the superstring in a curved background. This sigma model should be spacetime super-reparameterization and gauge invariant, and at linearized level, its interactions should reproduce the massless vertex operators.
When the spacetime superfields satisfy their equations of motion, the sigma model action is expected to be N=2 superconformally invariant at the quantum level, i.e., the components of the N=2 stress-energy tensor form a c = 6 N=2 superconformal algebra. These superspace equations of motion are currently being computed for the heterotic superstring by de Boer and Skenderis, which will be the first covariant β-function computation in a fermionic background.

The classical term
The two-dimensional sigma model is constructed from the spacetime superfields of section 3 and splits into a classically worldsheet N=2 superconformally invariant term and a Fradkin-Tseytlin term. Using the action in a flat background and the vertex operators of section 2, it is easy to guess the classically worldsheet superconformally invariant term:  in the supergravity vertex operator of (2.21), one sees that at linearized level, Note that the Type IIA sigma model action is obtained from the Type IIB action by switching Ω Up to field redefinitions (e.g.  3) T α+ c

this is the most general sigma model action which is invariant under spacetime super-reparameterizations, under the gauge transformations (δA
The first type of constraints are the usual representation preserving constraints which allow a consistent definition of chiral (and twisted-chiral) superfields. The second type are conventional constraints which define the vector components of the super-vierbein and super Yang-Mills gauge field, E a M and A I a , in terms of the spinor components. Note that conventional constraints for the connections w AB C , Γ A , and Γ jk A can be defined arbitrarily since these superfields never appear in the sigma model action. Therefore conformal and SU (2)/U (1) invariance is broken in the Type II sigma model by gauge-fixing L jk = δ jk (which as will be shown in the following paragraphs, implies N αβ = G a jk δ jk = S ++ = S −− = 0). The conformal-breaking constraint also implies that P αβ = W αβ + ǫ αβ S +− and that Q α In the tensor calculus approach to constructing actions for N=2 supergravity coupled to matter [27], a "minimal multiplet" is employed as a starting point, consisting of conformal supergravity coupled to a vector multiplet, in the scale (+U(1)) gauge W (0) = 1. Since for string theory it is more useful to apply the string gauge L jk = δ jk , it is instructive to compare the field content of the two gauges. As for N=1 [25], both gauges imply the vanishing of some of the torsions arising from the superconformal covariant derivatives. From (3.5), the dimension 1 torsions are originally W αβ , S (jk) , N αβ , and G j k a (hermitian).
The basic procedure is: (1) Fix one of the two superfields W (0) or L jk .
(Although one could also fix the tensor compensator L (0) jk , this would give the same field content as fixing the non-compensator L jk .) That breaks scale invariance (and more), but it doesn't introduce a dimension 0 scalar or dimension 1 2 spinor (which come from fixing the second compensator), so you get no new tensors. there is the auxiliary field W αβ , so that torsion will always survive. There are also the U(2) gauge fields G j k a , which will become gauge invariant only if they find scalars to eat. (2) A vector multiplet has a gauge vector, with the dimension 1 field strength F αβ , as well as the hermitian auxiliary fields D (jk) . It also has a complex physical scalar of dimension 0, whose real part is eaten by the conformal graviton, and whose imaginary part is eaten by the U(1) gauge vector.
(3) A tensor multiplet has a dimension 1 field strength H a , and a complex scalar auxiliary field. It also has the hermitian physical fields L (jk) of dimension 0, of which one is eaten by the graviton, the remaining two by the SU (2) where Ω Y M is the super Yang-Mills Chern-Simons form satisfying ((∇) 2 + R)Ω Y M = W I α W Iα , Ω Lorentz is the super Yang-Mills Chern-Simons form satisfying ((∇) 2 + R)Ω Lorentz = W αβγ W αβγ , c 1 and c 2 are compactification-dependent constants, and W αβγ is the chiral field strength of conformal supergravity.
Since L = 1 breaks Yang-Mills and Lorentz invariance, it is more convenient to choose the gaugeL = 1, which implies R = c 1 W I α W Iα + c 2 W αβγ W αβγ from the linear condition on L. Therefore, in order to preserve manifest Yang-Mills and Lorentz invariance, the third type of constraints in (4.3)should be modified for the heterotic superstring to includes the contribution of the Chern-Simons forms (A I M is the Yang-Mills gauge field and w M B C is the Lorentz spin connection).
Because these modifications to the heterotic torsion constraints are higherorder in α ′ , they can not be checked using classical worldsheet superconformal invariance. To justify them, one should check that with these modified constraints,

The Fradkin-Tseytlin term
So the classical term in the sigma model action is (4.1)or (4.2)with the constraints of (4. f is the field strength of the worldsheet U(1) gauge field, and χ,χ are the field strengths of the worldsheet gravitini [28].
If and only if Φ = e −φ is superspace chiral (and therefore {G, φ} = 0), one can construct the worldsheet supersymmetric Fradkin-Tseytlin term where ǫ is the super-worldsheet chiral density and e is the ordinary worldsheet density, we ignore possible double poles in the OPE of G with V (as in the vertex operators of section 2.2), and φ κ is a worldsheet and spacetime superfield defined such that Q κ φ k = [G, φ κ ] andQκφ κ = [Ḡ, φ κ ] where Q κ andQκ are the worldsheet supersymmetry generators. (On a flat worldsheet, Note that the Fradkin-Tseytlin term can be written independently of Γ A since When the heterotic Fradkin-Tseytlin term is in N=(2,0) superconformal gauge, φ −φ couples to a − in the same way as the ρ field in (2.1), so one can combine these couplings into dz + dz − ∂ + (ρ + i(φ −φ))a − . This term is invariant under the spacetime U(1) transformation if one also transforms the worldsheet variables Although it may look unusual for worldsheet variables to transform under a spacetime gauge transformation, it is similar to the transformation of heterotic chiral fermions, ζ q , under a Yang-Mills gauge transformation.
However, the kinetic term for the ρ variable, dz + dz − ∂ − ρ∂ + ρ, is not invariant under the transformation of (4.11). We must therefore modify the kinetic (Although it may seem strange to have a term in the sigma model action which is quadratic in φ, this also occurs in the bosonic string if one couples the dilaton ϕ to worldsheet ghosts, and then integrates out the ghosts to obtain dz + dz − (∂ + ϕ∂ − ϕ + ϕr) [29].) Note that the equation of motion for a − now implies that the right-moving part of ρ satisfies For the Type II superstring, the logarithms of the conformal compensators couple to N=(2,2) worldsheet supercurvature, which is described by a worldsheet chiral superfield Σ c and its complex conjugateΣ c , and by a worldsheet twistedchiral superfield Σ tc and its complex conjugateΣ tc . ( In components, Σ c = b + κ(χ + ψ) +κ(χ +ψ) + κκ(r + i(f +f ) + d) and where r is the two-dimensional curvature, f andf are the right and left-moving worldsheet U(1) field strengths, χ andχ are the complex gravitino field-strenghs, b and c are complex weight 1 auxiliary fields, ψ andψ are complex weight 3 2 auxiliary fields, and d is a real weight 2 auxiliary field. These additional auxiliary fields are present since we are using the U(1)×U(1) formulation of N=(2,2) supergravity.
If, and only if, Φ c = e −φ c is superspace chiral and Φ tc = e −φ tc is superspace twisted-chiral, one can construct the following Type II worldsheet Fradkin-Tseytlin term: where ǫ c and ǫ tc super-worldsheet chiral and twisted-chiral densities [19], φ c κκ and φ tc κκ are defined as in the heterotic case, and S aux describes the coupling of φ to the worldsheet auxiliary fields.
Although classical worldsheet supersymmetry requires that Φ c = e −φ c and Φ tc = e −φ tc are chiral and twisted-chiral, it does not require that they are restricted superfields, i.e. that they satisfy the reality constraints (∇) 2 Φ c = (∇) 2Φ c and (∇) 2 Φ tc = (∇) 2Φ tc . As in the case of the superfields for the compactification moduli, quantum N = (2, 2) superconformal invariance is expected to imply these reality conditions (or a suitable modification), as well as the equations of motion.
Note that S aux vanishes when φ c +φ c = φ tc +φ tc (i.e. when Φ cΦc = Φ tcΦtc ) since d couples to (φ c +φ c −φ tc −φ tc ), ψ couples to ∇ α (φ c −φ tc ) = ∇ α (φ c +φ c −φ tc −φ tc ), and b couples to ∇ α∇α φ c = ∇ α∇α (φ c +φ c − φ tc −φ tc ). Because S aux violates N=(2,2) superconformal invariance in a manner which does not appear to be cancelled by anomalies in the classical term of the sigma model, Φ cΦc = Φ tcΦtc is expected to be the on-shell superspace equation of motion (at least to lowest order in α ′ ). Note that when L jk = δ jk and all CY fields are set to zero, Φ cΦc = Φ tcΦtc is indeed the equation of motion implied by varying the weight 2 conformal supergravity scalar in the low-energy Type II effective action at the end of section 5.
From the form of the Type II Fradkin-Tseytlin term, it is reasonable to call the θ =θ =θ =θ = 0 component of φ c +φ c + φ tc +φ tc the dilaton, and the θ =θ =θ =θ = 0 component of φ c −φ c ± (φ tc −φ tc ) the axions. Just as the dilaton zero mode couples to the worldsheet Euler number, the axion zero modes couple to the worldsheet right and left-moving U(1) instanton number.
As in the heterotic case, one can combine the Type II Fradkin-Tseytlin term with the kinetic terms for ρ andρ to obtain (4.14) which is invariant under the spacetime U(1)×U (1) transformations Note that S aux is separately invariant under these transformations since it only depends on the U(1)×U(1) invariant combination φ c +φ c − φ tc −φ tc .

Effective Actions
The most straightforward method for constructing the effective action is to use β-function methods to compute the low-energy equations of motion, and then look for an action which yields these equations. Although this β-function method is necessary for computing the explicit form of the action, we can learn a lot just by requiring that the action has the symmetries predicted by the sigma model.
(By "effective action", we always mean the true effective action, rather than the possibly anomalous Wilson effective action.) Since the superfields in the sigma model are conformally gauge-fixed byL = 1 or L jk = δ jk , it will be convenient to first construct superspace actions in this conformal gauge, and then remove the gauge-fixing condition to obtain conformally-invariant superspace actions. In terms of the conformally gauge-fixed superfields which appear in the heterotic sigma model, the most general N=1 superspace effective action which is U(1) invariant and satisfies the above properties is:

Heterotic superspace effective actions
where K p g,n I = (K p g,−n I ) * is a general superfield with U(1) weight n I and conformal weight p, and F g is a chiral superfield with U(1) weight g and conformal weight 3g (the conformal weight of an N=1 chiral superfield must be three times its U(1) weight). We have explicitly separated out the Φ andΦ zero modes in (5.1), so K p g,n I and F g are defined to be independent of these zero modes. Note that while the real d 2 θd 2θ integral is written with a factor of E −1 = sdet(E A M ) −1 to make it a density, this real factor can not appear in the chiral d 2 θ integrand. However, in a chiral basis where∇ .
α , Φ acts as the corresponding density. (We will always be using a chiral basis when defining chiral integrals.) In fact, all truly chiral superfields are such densities, with density weight corresponding directly to conformal weight. This simplifying feature of chiral integrals is one reason why their component evaluation is simpler than integrals over full superspace. A similar procedure can also be applied to non-supersymmetric theories, with gravity written as conformal gravity plus a compensating scalar: By writing all fields as densities, all factors of √ −g can be removed from the action.
This procedure is useful for string theory, because it is the density form of the dilaton field, with the same weight as (−g) 1 4 , that is invariant under T-duality. For example, the low-energy action of the bosonic closed string can be written with this dilaton field as the only density.
(Note that K p g,n I may involve scale-invariant functions ofL such as ∂ m (logL).) Since the U(1) weight g of F g must be absorbed by Φ 2−2g , F g can only occur with coupling λ 2g−2 . This means that F g cannot receive perturbative or non-perturbative quantum corrections. In other words, after expanding Φ and L around their vacuum expectation values (which can be set to one by rescaling λ and α ′ ), F g appears only once in the perturbative expansion. Of course, this does not prevent quantum corrections if a chiral F -term can also be written as a non-chiral D-term, which could then receive quantum corrections through U(1)neutral factors of ΦΦ.
Note that any chiral F -term can be written as a non-chiral D-term if one allows the non-local operator (∇) 2 / . However, non-local 1/ terms in the effective action come from anomalies, which are not expected to receive quantum corrections [30]. Furthermore, some chiral F -terms can be written as local nonchiral D-terms by pulling a factor of (∇) 2 off a field-strength. Normally, this would not lead to quantum corrections, since adding ΦΦ to the D-term would break gauge invariance. However, in the heterotic effective action where the tensor multiplet transforms anomalously under Lorentz and Yang-Mills gauge transformations, this type of D-term can sometimes lead to quantum corrections.
For example, the F -term can also be written as the non-chiral D-term where (∇) 2 Ω Y M = W I α W αI and τ g (M (i) ) is a chiral function of the compactification moduli. This non-chiral D-term can get corrections, for example, from which would give quantum corrections to τ g of λ 2nτ g (M (i) ,M (i) ). So the F -term is not protected against quantum corrections. Note, however, that because (5.5)is not gauge-invariant, it must occur in the combination This means that quantum corrections to the F -term of (5.3)are related by the proportionality constant c 1 /c 2 to quantum corrections of the F -term At low energies, only K 0 0,0 contributes to the non-chiral D-term in the treelevel effective action. (n I = 0 since U(1)-charged superfields have non-zero conformal weight.) Since only the compactification moduli have conformal weight zero, K 0 0,0 must be some function K 0 (M (i) ,M (i) ). The tree-level low-energy effective action is therefore The supergravity, tensor, and super-Yang-Mills actions come from the D-term, while the cosmological constant and Yukawa couplings come from the F -term.
Note that the kinetic term for Yang-Mills fields gets tree-level contributions from the D-term λ −2 d 4 x d 2 θd 2θ c 1 Ω Y M K(M (i) ,M (i) ) and gets one-loop contributions from the F -term d 4 x d 2 θ W I α W αI F 1 (M (i) ). Although many aspects of the heterotic superspace effective action have already been discussed in reference [8], the Type II superspace effective action has not yet appeared in the literature.

Type II superspace effective action
From the form of the Type II sigma model, the dimensionless coupling constant λ (which couples to string loops) can be absorbed in the effective action by rescaling the chiral and twisted-chiral compensators Φ c → λΦ c and Φ tc → λΦ tc .
As explained in section 3, Φ c can be identified with a vector field strength W (0) and Φ tc can be identified with a tensor field strength L tc .) Since the graviphoton field strength is one of the components of W (0) , its kinetic term (which is quadratic in W (0) ) appears at order λ −2 , as expected for a tree-level contribution.
jk , α ′ can be absorbed in the conformally-invariant action by rescaling For the Type II superstring, mirror symmetry further restricts the superspace effective action. Mirror transformations relate Type IIA compactifications to Type IIB compactifications by switchingĜ C withĜ C (the right-moving fermionic generators for the compactification N=2) and switching Ω (i) c with Ω (i) tc (the worldsheet chiral and twisted-chiral primary fields). From the form of the Type II sigma model of (4.2)and (4.14), this switch can be undone by switching tc (one must also switchρ with −ρ,d α withd . α , and Π 0 + with −Π 0 + , but this is just a redefinition of worldsheet variables). So after gauge-fixing L jk = δ jk and imposing the standard reality conditions on Φ c and Φ tc , mirror symmetry relates Type IIA effective actions to where c (0) and c (i) are independent constants. This implies that before gaugefixing L jk , the Type II effective action must be invariant under which is the unique conformally and SU(2)-invariant generalization of (5.9).
In terms of the conformally and SU(2)/U(1) gauge-fixed superfields appearing in the Type II sigma model, the most general N=2 superspace action which is U(1)×U(1) invariant and satisfies the above properties is: where K p g,n I ,n I ,s = (K p g,−n I ,−n I ,s ) * is a general superfield with U(1)×U(1) weight [n I +n I , n I −n I ] and conformal weight p (s is unrestricted), F c g is a chiral superfield with U(1)×U(1) weight [2g, 0] and conformal weight 2g (the conformal weight of an N=2 chiral superfield must be equal its U(1) weight), and F tc g is a twistedchiral superfield with U(1)×U(1) weight [0, 2g] and conformal weight 2g (although the conformal weight of an N=2 twisted-chiral superfield is unrestricted when L jk is gauge-fixed, mirror symmetry forces F tc g to have the same conformal weight as F c g ). We have explicitly separated out the Φ zero modes in (5.11), so K p g,n I ,n I ,s , F c g , and F tc g are defined to be independent of these zero modes (however, they can depend on the moduli M As in the heterotic superstring, F c g and F tc g do not receive perturbative quantum corrections since they must appear in the effective action only at order λ 2g−2 so that their U(1)×U(1) charge is compensated by the U(1)×U (1)  We are assuming that the chiral and twisted-chiral F -terms in (5.11)can not be written as non-chiral D-terms, which could then receive quantum corrections from U(1)-neutral combinations of Φ c and Φ tc . (See the previous subsection for why such D-terms are unlikely.) Note that if a twisted-chiral F -term can be written as such a D-term, then mirror symmetry implies that the mirror chiral F -term can also be written as such a D-term. So in the unlikely event that F tc g receives perturbative corrections, so does F c g . However, as will be discussed at the end of this section, if mirror symmetry were non-perturbatively broken, F tc g might receive non-perturbative corrections without F c g receiving corrections. One type of g-loop chiral and twisted-chiral term is where P αβ and Q α . β are the chiral and twisted-chiral supergravity field strengths which appear in the sigma model of (4.2). These terms describe the scattering of two gravitons with either 2g − 2 graviphotons or 2g − 2 hypermultiplets, and occur only at g string-loops in the perturbative S-matrix [31] [24]. τ c g and τ tc g are functions which depend only on topological properties of the compactification manifold, and since mirror symmetry exchanges P αβ P αβ with Q α . β Q α . β , τ c g for Type IIA compactifications is equal to τ tc g for Type IIB compactifications, and τ tc g for Type IIA compactifications is equal to τ c g for Type IIB compactifications. By removing the gauge-fixing condition on L jk , the non-chiral and chiral terms in (5.11)can easily be written in conformally and SU(2)-invariant form as where y jȳk is defined as ǫ jk + L jk √ L lm L lm in the non-chiral integrand, and K p g,n I ,n I ,s (y,ȳ) is defined by contracting all SU(2) + indices with y's and all SU(2) − indices withȳ's. Note that there are an equal number of y's andȳ's in the integrand because of U(1) invariance.
However, in order to write the twisted-chiral term of (5.11)in conformally and SU(2)-invariant form, one needs to introduce two independent complex variables, u + and u − [15]. Using u ± and any linear superfield h j 1 ...j n satisfying one can define a "harmonic" superfield of degree n,h(u + , u − ) = h j 1 ...j n u j 1 ...u j n , which satisfies Note that the product of two harmonic superfields of degree n 1 and n 2 is a harmonic superfield of degree n 1 + n 2 .
Using a harmonic superfieldh of degree 2 (which has conformal weight 2 by N=2 superspace rules), one can then define the conformally and SU(2)-invariant action [16] d 4 x u j du j d 4 θ ♮h = d 4 x u j du j (v j dθ j ) 2 (v k dθ k ) 2 (v l u l ) 4h (5.16) where v j is arbitrary (becauseh is harmonic, the integral is independent of v j ) and du is defined as a contour integral in CP 1 around a pole ofh. (Note that the action is invariant under the complex projective transformation u j → qu j .) In order to reproduce the twisted-chiral term of (5.11),h should be defined ash = λ 2g−2 (α ′ ) g−1 (L (0) ) 2−2gF g L (5.17) jk ,L = u j u k L jk , andF tc is a harmonic superfield of degree 4g which satisfiesF g (u − = 1, u + = 0) = F tc g when L jk is gauge-fixed to δ jk . (Note that ∇ −Fg =∇ −Fg = 0 when u − = 1 and u + = 0, so F tc g is a twisted-chiral superfield of U(1)×U (1)  where the contour integral is performed around the pole ofL = u j u k L jk .
Note that (5.18) tc =L (i) /L (0) . Although the chiral part of this low-energy action agrees with the component action of [27], the "harmonic" part of (5.21)allows more general tensor hypermultiplet couplings than the "improved tensor" action of [27]. Note that unlike the improved tensor action of [27], (5.21)contains the Pecci-Quinn-like symmetry of (5.10).
Although N=2 supersymmetry also allows the cosmological term where V jk(0) is the prepotential of the vector compensator, this term breaks the Pecci-Quinn-like symmetry and is therefore perturbatively forbidden.

Conclusions
In this paper, we have constructed manifestly spacetime supersymmetric sigma models and effective actions for 4D compactifications of heterotic and Type II superstrings. For the heterotic superstring, the sigma model can be found in equations (4.1)and (4.12), and the effective action in (5.2)and (5.8). For the Type II superstring, the sigma model can be found in (4.2)and (4.14), and the effective action in (5.13), (5.18), and (5.21).
We have also proven various non-renormalization theorems for the superspace effective action, including the theorem that chiral F -terms receive no perturbative or non-perturbative corrections. For the Type II superstring, mirror symmetry implies that twisted-chiral F -terms (which describe hypermultiplet interactions) are also unrenormalized. However, it is plausible that mirror symmetry is nonperturbatively broken by spacetime instantons, which would allow hypermultiplet interactions to receive non-perturbative corrections.
Our results in this paper were based on the observation that string loops in the 4D heterotic superstring are counted by an N=1 chiral compensator, and string loops in the 4D Type II superstring are counted by an N=2 vector compensator and an N=2 tensor compensator. This explains the coupling of Ramond-Ramond fields, and contradicts the standard folklore that Type II string loops are counted by just a hypermultiplet [17]. The mistaken folklore was caused by confusing the physical scalar (which sits in a hypermultiplet and couples like the determinant of the metric) with the dilaton compensator (which couples to the worldsheet curvature).
Based on a similar incorrect reasoning, the standard folklore also claims that for the 4D heterotic superstring with N = 2 spacetime supersymmetry, string loops are counted by a vector multiplet [17]. Although we are presently unsure of the precise supersymmetry multiplet for the dilaton compensator in this N=2 heterotic case, we believe that it is not just a vector multiplet, and therefore the standard folklore is again incorrect.
An obvious question is if our techniques can be generalized to the tendimensional uncompactified superstring. Although there does exist an N=2 worldsheet-supersymmetric description of the uncompactified superstring, this description is not manifestly SO(9,1) invariant [5]. This makes it difficult to extend the worldsheet action in a flat superspace background to a sigma model action in a curved superspace background. Nevertheless, we conjecture that if such an extension were performed (probably using harmonic variables to make Lorentz invariance manifest), the 10D supergravity theory would contain a spacetime conformal compensator which couples to worldsheet supercurvature in the sigma model. Hopefully, a proper understanding of this compensator will lead to an off-shell superspace description of 10D supergravity.
A less ambitious question is if our techniques can be generalized to sixdimensional compactifications of the superstring (which, after toroidal compactification, would be useful for understanding the N=2 4D heterotic superstring).
Since there already exists an N=2 worldsheet-supersymmetric description of the superstring with manifest SO(5,1) invariance [24], the answer is probably yes. As was shown with Cumrun Vafa, this six-dimensional superstring actually contains N=4 worldsheet supersymmetry, which can be made manifest by introducing SU(2)/U(1) harmonic variables. It is likely that, as in references [32] and [15], these harmonic variables will be useful for constructing manifestly spacetime supersymmetric sigma models and effective actions. A 6D superspace effective action would be very useful for studying the recent string-duality conjectures which relate the 6D heterotic and Type II superstrings [3][4] [17].