Witten Anomaly in 4d Heterotic Compactificaitons with N = 2 Supersymmetry

We showed that there is no SU(2) Witten anomaly in a large class of 4d N = 2 supersymmetric Heterotic string compactifications. The consistency conditions we consider are the modularity of the new supersymmetric index, the integrality of BPS indices, and the discrete Peccei-Quinn shift symmetries. We also found an example where these conditions are not sufficient to show that the theory is anomaly free. This suggests that there are more conditions in the worldsheet SCFT essential for consistent string compactifications.


Introduction
Theoretically consistent string vacua are believed to yield low-energy effective field theories without a theoretical inconsistency. If we find a low-energy inconsistency in a string vacuum, that is taken as a hint of some more theoretical consistency conditions of string theory. Gauge and gravitational anomalies in the effective theories in closed string vacua are known to cancel due to the modular invariance of the CFT [1]. The same class of anomalies in the effective theories in open string vacua cancel locally because of the anomaly inflow mechanism [2,3], and they cancel globally because of the Bianchi identity of Ramond-Ramond fields.
In this article, we address the SU(2) Witten anomalies [4] of the 4d N = 2 supersymmetric effective theories given by Heterotic compactifications. The organization of this article is as follows.
Section 2 is a short review of basic facts about the SU(2) Witten anomaly. Section 3 explains our setup (including the choice of SU(2) gauge symmetry) and some assumptions for the Heterotic compactifications.
In section 4, we show that the Witten anomaly indeed vanishes for the lattice 1 Λ S and the charge vector v 0 , if Λ S has a decomposition U ⊕ W for some even negative definite lattice W and the charge vector v 0 stays within W ∨ . The derivation described in section 4.1 exploits • the modularity of the generating functions 2 Φ and Ψ for BPS indices, defined by the internal worldsheet CFT language, • the integrality of BPS indices, and • the integrality of some coefficients of the leading terms in the prepotential for the effective theory, which follows from the discrete Peccei-Quinn shift symmetries.
Section 4.2 shows that for some choices of the lattice Λ S and the charge vector v 0 , the cancellation of the Witten anomaly follows only by the modularity (the first condition). Section 5 treats an example of Λ S and v 0 for which one cannot show that the Witten anomaly always vanishes only from these conditions listed above. This means that we are missing some other important consistency conditions for the string compactifications.
In the appendix, we review what the discrete Peccei-Quinn shift symmetries imply for the leading coefficients of the prepotential. 1 In our setup, Λ S = U [−1] ⊕ Λ S is the charge lattice of chiral free bosons in the internal worldsheet CFT. The charge vector v 0 ∈ Λ ∨ S specifies the choice of the SU (2) gauge symmetry. See section 3 for more details. 2 See section 3 for the definitions for the modular form Φ. See also 5.1 for the modular form Ψ.

SU (2) Witten Anomaly
Since 4d N = 2 supersymmetric theories are non-chiral, they are free from perturbative triangle anomalies. However there can be nonperturbative anomalies, called Witten anomalies [4]. In N = 2 Sp(k) gauge theories, a half-hypermultiplet massless matter in the representation R with half-integral Dynkin index T (R) produces Witten anomaly. In the case of SU(2) gauge theory, the possibly anomalous representations are those with the spin The theory is free from Witten anomaly if and only if the total number of massless SU(2) half-hypermultiplets in such representations is even, i.e.
where N(R j ) is the number of massless half-hypermultiplets in the spin-j representation R j . This quantity can also be written by N j , the number of massless half-hypermultiplets with the 2j units of the fundamental weight (= SU(2) Cartan charge j): Note that this summation may run over those states in SU(2) full-hypermultiplets because they appear in pairs and have no contributions modulo two. Moreover, if the theory has other U(1) gauge symmetries, only neutral states under those U(1)'s contribute to (3) because U(1)-charged matter has its conjugate and this pair contributes 2.

Heterotic compactification and SU (2) enhancement
Let us consider a Heterotic compactification with 4d N = 2 supersymmetry; we assume that it is without an R 3,1 -filling NS5-brane and other solitons of similar kinds. The internal (c,c) = (22,9) CFT in the NSR formalism with (0, 2) worldsheet supersymmetry for such a compactification is characterized by the conditions in [5]. Let Λ S denote the charge lattice of chiral free bosons that appear in the CFT [6,7], G S := Λ ∨ S / Λ S its discriminant group, and ρ := rank(Λ S ). We assume that the lattice Λ S has a primitive embedding into where U is the even unimodular lattice of signature (1, 1), and a structure 3 Λ S = U[−1] ⊕ Λ S . When a charge vector v 0 ∈ Λ ∨ S satisfies the following conditions, and 4 −2 ≤ v 2 0 < 0, a level-k SU(2) current algebra {J ± , J 3 } appears in the left-mover of the CFT at a complex codimension-1 subspace in the Coulomb branch moduli space; J + has charge v 0 . There is an enhanced SU(2) gauge symmetry in the 4d effective theory (e.g. [8], [9]). The SU(2) Cartan charge is for states whose U(1) charge is v ∈ Λ ∨ S . Since only massless matters contribute to the anomaly, we can focus our attention to the BPS states of the 4d N = 2 supersymmetry algebra. The multiplicities of purely electrically charged BPS states are captured by the new supersymmetric index in the Heterotic worldsheet language: =: where the trace is over the internal (c,c) = (22,9) worldsheet CFT with the right-mover in the Ramond sector, and F R is the zero mode of total U(1) current in the right-mover [5]. The function θ Λ S [−1] is the Siegel theta function for the lattice Λ S [−1]. (See [9] for more details.) The integral coefficients c γ (ν) defined by Φ γ (τ ) η(τ ) 24 =: is relevant to BPS counting. In particular, when v ∈ Λ ∨ S satisfies [v] = 0 ∈ G S and −2 < v 2 < 0, there are n V /H γ BPS vector/half-hyper multiplets of U(1)-charge v, and the following equality holds: The leading Fourier coefficients c γ (ν) for γ ∈ G S , where −1 ≤ ν < 0, are denoted by n γ . 5 The N = 2 supersymmetry in R 3,1 -not more, not less-implies n γ = −2 for γ = 0 ∈ G S . In the following arguments, we implicitly use the integrality of BPS indices c γ (ν).

Λ S = U ⊕ W Cases
We show that the SU(2) Witten anomaly vanishes if Λ S has a decomposition Λ S = U ⊕ W and v 0 ∈ W ∨ ⊂ Λ ∨ S ; here, the lattice W is even and negative definite and is assumed to have a primitive embedding into

Derivation Using also the Discrete Peccei-Quinn Symmetries
In the 4d effective theory, the action of the vector multiplets and the mass of the BPS states of the N = 2 supersymmetry are determined by the prepotential where s is the 4d axion-dilaton complex scalar, and t ∈ Λ S ⊗C collectively denotes the Narain moduli; t is a set of local coordinates of D( Λ S ) := P{℧ ∈ Λ S ⊗ C | ℧ 2 = 0, (℧, ℧) > 0}, and t = e a t a is the component description for an integral basis {e a } of Λ S . Parameters d abc , a ab , b a , and χ reflect more information of the internal space than must the lattice Λ S . Two points in the Coulomb branch moduli space D( Λ S ) should describe the same lattice vertex operator algebra and an identical string vacuum, if they are in a common orbit of As we will review in appendix A, it follows from the unphysical nature of the Γ S action 6 that The vector-valued modular form Φ/η 24 determines a part of d abc [6,10,7,11,12]; 1-loop threshold corrections to a probe gauge group and a gravitational coupling depend on Φ/η 24 , and hence d abc 's on some of the coefficients c γ (ν)'s. It is known that d abc 's would not be integers automatically, if Φ were a generic vector-valued modular form of weight 11 − ρ/2 and in the Weil representation of Mp 2 Z of Λ S with integer-valued c γ (ν) for small ν's. So, the property (12) imposes non-trivial conditions on the BPS state multiplicities.
Appendix B.1.3 of [9] has worked out for a general W which part of the parameters d abc 's are determined by Φ and how. Integrality of some of those d abc 's is translated into [9] (v,a)>0 We can use this condition to show that the Witten anomaly WA ∈ Z 2 vanishes; to do this, let us use a = −v 0 for convenience, and set r 1 = r 2 = r 3 = kv 0 . Then the condition reads Since 2q v is integral and (2q v ) 3 ≡ 2q v mod 2, this condition is nothing but the condition (3) which means the theory is free from Witten anomaly; only neutral states under U(1)'s contribute to (14) as stated earlier, which are states corresponding to v's that are parallel to v 0 . They are indeed massless when the SU(2) is enhanced.

An Approach just Using Modurality of Φ
It is instructive to see how far one can go by using just the property that Φ is an integercoefficient vector-valued modular form, without using the integrality of d abc 's. We consider W and v 0 such that 6 In the language of Heterotic string in the geometric phase, with the structure Λ S = U ⊕ W and W ⊂ E 8 [−1] ⊕2 , the Γ S action on the Coulomb branch moduli (Narain moduli) includes +1 shift of the complex structure of T 2 , +(2π) 2 α ′ shift of T 2 B, and winding gauge transformation of W . In the dual Type IIA language, d abc 's are the intersection numbers of the internal Calabi-Yau threefold. We call these t a → t a + δ a d shifts discrete Peccei-Quinn symmetries in this article.
where e is the generator of the rank-1 lattice W . The charge vector v 0 = 2 2n e satisfies the condition (4) with k = n [9]. In this setup we will see that WA ≡ 0 mod 2 if n / ≡1 mod 4. The integrality of d abc 's turned out to be necessary, however, at least for some cases with n ≡ 1 mod 4.
The coefficients of Φ/η 24 have various linear relations among them. One way to obtain such a relation is to use the fact 7 for a holomorphic vector-valued modular form φ (w) of weight w = 1/2 or 9/2 associated with the dual representation of Mp 2 Z. Here, E 4 and E 6 are Eisenstein series of weight-4 and 6, respectively. We assume that φ (w) has a non-zero q 0 term only in the γ = 0 ∈ G S component, and the coefficient is denoted by [φ (w) ] q 0 . For any given φ (w) , one linear relation on c γ (ν)'s is obtained by comparing coefficients of each term in the Fourier series expansion. Here, where w F is the weight of a modular form F . The linear relations (16) for the q 1 term are combined to yield 8 v=ℓe/(2n) Only BPS states with v v 0 contribute to WA, as stated earlier. So where n ℓ = c [v] (v 2 /2) with v = (ℓ/2n)e ∈ W ∨ . We will use this omitted notation from now on. When n is even [resp. n ≡ 3 mod 4], the relation (18) multiplied by (8n 2 ) [resp. (4n 2 )] and reduced modulo 2 implies that WA = 0 mod 2. When n ≡ 1 mod 4, the relation (18) does not yield useful information on whether WA ≡ 0 mod 2 or not. For example, when n = 1, there is no linear relation among n γ 's, and any n 1 ∈ Z ≥−2 is allowed when c γ (ν)'s with small ν are required to be integers. In the cases with n = 5 and n = 9, on the other hand, there are two and three linear relations among n γ 's, respectively. But they cannot be combined to yield ℓ 2 ≤4n 0<ℓ,odd n ℓ ≡ 0 mod 2.

The Case Λ S = +28
Next we consider the case 9 Λ S = +28 , because this turns out to be minimum degree rank 1 lattice where the theory is possibly anomalous. In this case satisfies and there can be an SU (2) has Cartan charge 1/2. This is the only relevant charged matter. 10 At subspace in the Coulomb branch moduli space where these vectors and matters become massless, the effective field theory is an SU(2) (and U(1) graviphoton) gauge theory with n H 7 half-hyper doublets. Anomaly in this theory is equal to n 7 mod 2.
It is not known if there exists a Heterotic compactification with Λ S = +28 ; no consistency condition is known to rule out such a Heterotic compactification, however. So we assume that there is one, and end up with a conclusion that modularity of some invariants of this compactification and the conditions (12, 40, 42) combined fail to predict that the 9 For any n ∈ N >0 , a primitive embedding Λ S ֒→ U ⊕3 ⊕ E 8 [−1] ⊕2 exists. 10 Charged matter v ∈ Λ S ∨ that contributes to Witten anomaly satisfies v v 0 , −2 < (v, v) < 0, q v ∈ 1/2 + Z. v 1 and −v 1 are the only elements satisfying these conditions. SU(2) Witten anomaly vanishes automatically. This is a clear indication that there are more theoretical consistency conditions in the worldsheet CFT than those properties we use in this analysis.

Modular Forms Φ and Ψ
Suppose that there is a Heterotic compactification with Λ S = +28 . Then two vector-valued modular forms Φ and Ψ extract invariants of this string vacuum; Φ has appeared already in this article; the other one, Ψ, is of weight 25/2, and also in the Weil representation of Mp 2 Z associated with the lattice Λ S = +28 . The modular form Ψ as an invariant of a Heterotic-Type IIA dual vacuum appears manifestly already in [7]; see [9] for references and treatment of Ψ for a general Λ S . The parameters d abc are determined by the Fourier coefficients of both Φ and Ψ, and hence we need to deal with both 11 to exploit the condition (12).
(25) 11 In the case of Λ S = U ⊕ W , some of d abc 's depend only on the Fourier coefficients of Φ, not both. It just happens that the condition (12) for such d abc 's is enough to guarantee that WA ≡ 0 mod 2. 12 To work out those Fourier coefficients with ν ≤ 1 in terms of the 14 independent n γ 's, and also to derive (23, 25), we used holomorphic Jacobi forms of index-14 and weight 4, 6, · · · , 12, and 16. See sections 2.4 and 3.1.2 of [9]. This allowed range is non-empty. For example, n 0 = −2, n 12 = 3, n 7 = n 10 = n * with 0 ≤ n * ≤ 47, and n γ = 0 for all other γ's.
Similarly, the vector space of vector-valued modular forms of weight 25/2 and Weil representation for +28 is of 17 dimensions, which we compute by using the Riemann-Roch theorem. It turns out that the Fourier coefficients c Ψ γ (ν)'s in are parametrized by the 15 c Ψ γ (ν) = c Ψ −γ (ν)'s with −1 ≤ ν < 0, and two more, c γ=0 (0) and c γ=1 (1/56); we have seen that 13 all of c Ψ γ (ν) with ν ≤ 1 are integers as long as all the 17 independent parameters are integers. Those 17 extra parameters of the Heterotic vacuum in question cannot be arbitrary integers, however. They are subject to the conditions [9] where d γ (ν)'s are the Fourier coefficients of (ΦE 2 −Ψ)/η 24 , and the leading coefficients (d γ (ν)'s with −1 ≤ ν < 0) are denoted by m γ . d 0 (0) and d 1 (1/56) are simply denoted by d 0 and d 1 in the following.
The property (12) implies 15 that d 0 ∈ 24Z, not just d 0 ∈ 12Z as in (27). This implies that the probe gauge group is free from the SU(2) Witten anomaly (see discussions in [9]), but n 7 can be an arbitrary integer, so the SU(2) gauge group for the v 0 in (20) may have non-vanishing Witten anomaly.
If this Heterotic compactification has a Type IIA dual, and the dual is in a geometric phase, then it has a property that (see discussion around (42)) This only leads to The SU(2) gauge group associated with v 0 in (20) may have only an even number of halfhypermultiplets in the adjoint representation. Consequently there is no restriction on n 7 other than (24). The modular nature of the elliptic genus was enough in proving that the low-energy gauge theory of a Heterotic compactification is free from perturbative anomalies [1]. By examining the SU(2) Witten anomaly in the low-energy effective theory, however, we found that there is more consistency conditions in the internal space worldsheet SCFT other than the modularity and integrality 16 of the invariants Φ and Ψ and the unphysical discrete Peccei-Quinn shift symmetries t a → t a + δ a d .
14 By following the arguments in [9], one can see that the extra parameters d γ (ν)'s of Ψ determine (c 2 ) a and d abc only through at most ρ independent linear combinations. In the case of a lattice Λ S with a given rank ρ and larger G S , there tends to be more independent d γ (ν)'s than ρ, as in (29). We have multiple (mutually non-exclusive) interpretations for what is happening. In the language of dual Type IIA string theory, (a) there may be multiple Calabi-Yau three-folds in a common diffeomorphism class that are distinct in their symplectic/complex structure, (b) there may be multiple different hypermultiplet-moduli tunings of a Calabi-Yau three-fold X so that a singularity develops along the same curve class, and (c) there are more theoretical constraints on d γ (ν)'s than those discussed in [9]. 15 The parameters (c 2 ) 1 and d 111 have also been worked out in terms of the coefficients of Φ and Ψ in the case of Λ S = +2 [9] and Λ S = U . The property (12) implies d 0 /24 ∈ Z, and hence the absence of the SU (2) Witten anomaly in the probe gauge group also in the case of Λ S = +2 and U . 16 To be rigorous, we have only exploited the integrality of c γ (ν)'s and c Ψ γ (ν)'s with small ν's in this article. Note also, the modular invariance of a Heterotic string compactification may be more than the modularity of that of Φ and Ψ.

A Consequences of the Discrete Peccei-Quinn Shift Symmetries
Here, we think of a Heterotic compactification described in section 3. The lattice Λ S is not necessarily of the form U ⊕ W . We neither assume that the (c,c) = (22,9) CFT has a Lagrangian-based description in the Heterotic language, nor assume that the Type IIA dual is in a geometric phase. By exploiting the fact that the Γ S action on D( Λ S ) should be unphysical, we will derive (12) (cf. [14,15,16,17]). Consider changing the Narain moduli t a continuously to (t ′ ) a = t a + δ a d for some d = 1, · · · , ρ. There is an isometry it is straightforward to verify that this g (d) is in the subgroup Γ S ⊂ Isom( Λ S ). Thus, we should have arrived at the original vacuum at the end. The BPS mass spectrum should be the same before and after this deformation.
States from F1 Heterotic string have purely electric charges under the (2 + ρ) U(1) vector fields in the 4d effective theory; the electric charges (simply referred to as charges outside of this appendix) v fill the free abelian group Λ el = Λ ∨ S . States with magnetic charges under the (2 + ρ) U(1) vector fields arise from NS5-branes and objects of similar kinds; the magnetic charges m fill the free abelian group Λ mg = Λ S . The Dirac-Zwanziger quantization condition is satisfied, with a canonical symplectic form introduced on the free abelian group Λ el ⊕ Λ mg = Λ ∨ S ⊕ Λ S [6]. So, the transformation g (d) on Λ ∨ S = Λ el should be lifted to a consistent actiong (d) on Λ el ⊕ Λ mg , and the spectrum of the central charges remain the same before and after the deformations.
The central charge 17 of the 4d N = 2 supersymmetry algebra is given by (e.g., [15,16]) where I, J range from 0, 1, · · · , ρ, ρ + 1; a, b = 1, · · · , ρ. We use a prepotential of the form A rationale for assuming this form of F even when the Type IIA dual is not necessarily associated with a geometric phase is found toward the end of this appendix. The liftg (d) acting on (X A , F A ) T should bẽ with For (g −1 (d) ) T to be interpreted as relabeling of electric and magnetic charges (v A , m A ) ∈ Λ el ⊕ Λ mg , and the parameters a ab and b a should be chosen so that 18 For the fractional part of a ab to be determined consistently, parameters of the prepotential should also satisfy d abb + d aab ≡ 0 mod 2, ∀ a, b ∈ 1, · · · , ρ.
(40) 17 The mass of BPS states of charge Q = (v I , m I ) is given by m = |Z|/ √ G N , where G N is the 4d Newton constant. 18 Difference ∆a ab ∈ Z and ∆b a ∈ Z is absorbed by a relabeling of the charges of the form (v ′ I , m ′ I ) = (v I + w IJ m I , m I ) with w IJ ∈ Z and w IJ = w JI . Therefore, both (12) and (40) must be satisfied in a Heterotic compactification under consideration.
In the matching calculation using the Heterotic 1-loop threshold corrections to the gauge coupling of a probe level-1 gauge group, 19 the parameters d abc and the 1-loop corrected gauge kinetic function f (R) = (s + d ′ a t a ) of the probe gauge group are determined only modulo shift d abc → d abc + [(δn a )C bc + cyclic] and f (R) → f (R) + (δn a )t a (cf [18]). 20 Because the parameters d ′ a are regarded as part of the cubic term coefficients of the prepotential in the Coulomb branch of the probe gauge group, it follows that d ′ a ∈ Z. This indicates-based on the presentation in §3.1.3 of [9]-that the property (12) is equivalent to the integrality of d A rationale for the form of prepotential (10) is the following. Think of the dual Type IIA compactification, and then the third derivative of the prepotential F should be equal to the three-point function of the corresponding A-model, regardless of whether the Type IIA compactification is associated with a geometric phase. Due to the unphysical nature of Peccei-Quinn shift t i for some rank-(ρ + 1) free abelian group L; κ ijk and K β,ijk are constant parameters. Thus, F is a polynomial at most cubic in the special coordinates t i X , besides the exponential terms as in (10). Presumably it is possible to translate this argument into the language of Heterotic string.
The prepotential (10) is not in the most general form one expects from the argument above. The structure of the s-linear term in the cubic part (κ ijk /6)t i X t j X t k X in F is due to that of the tree-level gauge kinetic function of the (ρ + 2) 4d vector fields. The absence of s 2 terms and s 3 terms in the cubic part, as well as the s-independence of the quadratic part −(a ij /2)t i X t j X and the linear part −b i t i X in (10) is explained by the fact that the combination v I X I in an appropriate frame should be proportional to the right-moving momenta in the Heterotic string. When a Type IIA dual exists, and is in a geometric phase, all of those structures are translated into the following properties: (i) the fiber K3 class divisor D s of a Calabi-Yau three-fold with a regular K3-fibration satisfies D 2 s = 0, and (ii) D s · D a · D b = C ab [19]; (iii) we know in a geometric phase [14,17,20] that b (ρ+1) +Z = (24) −1 X c 2 (T X)D s +Z = Z and a a(ρ+1) + Z = 2 −1 D a · D s · D s + Z = Z [19].
If a Type IIA dual exists, and is in a geometric phase, the property (12) is a trivial statement that all the triple divisor intersection numbers are integers. The property (40) is also known as one of the necessary conditions for a real 6-manifold [21]. In a geometric phase, it is known [14,22] that the parameters b a + Z are equal to (c 2 ) a /24 + Z appearing in the gravitational coupling (28). So the second property in (39) is read as which is also one of the necessary conditions for a real 6-manifold [21]. The authors have not been able to build an argument relating the fractional part of the parameter b a in F and (c 2 ) a /24 for the coefficient (c 2 ) a in (28) without assuming that a Type IIA compactification is in a geometric phase. 21,22 Despite this caveat, it is tempting to include (42) as one of the properties of a Heterotic compactification introduced in section 3.