The Green-Schwarz mechanism and geometric anomaly relations in 2d (0,2) F-theory vacua

Journal of High Energy Physics, Apr 2018

Timo Weigand, Fengjun Xu

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The Green-Schwarz mechanism and geometric anomaly relations in 2d (0,2) F-theory vacua

HJE The Green-Schwarz mechanism and geometric anomaly relations in 2d (0,2) F-theory vacua Timo Weigand 0 1 2 3 Fengjun Xu 0 1 3 0 CH-1211 Geneva 23 , Switzerland 1 Philosophenweg 19 , 69120, Heidelberg , Germany 2 CERN, Theory Division 3 Institut fu ̈r Theoretische Physik, Ruprecht-Karls-Universita ̈t We study the structure of gauge and gravitational anomalies in 2d N = (0, 2) theories obtained by compactification of F-theory on elliptically fibered Calabi-Yau 5-folds. Abelian gauge anomalies, induced at 1-loop in perturbation theory, are cancelled by a generalized Green-Schwarz mechanism operating at the level of chiral scalar fields in the 2d supergravity theory. We derive closed expressions for the gravitational and the nonabelian and abelian gauge anomalies including the Green-Schwarz counterterms. These expressions involve topological invariants of the underlying elliptic fibration and the gauge background thereon. Cancellation of anomalies in the effective theory predicts intricate topological identities which must hold on every elliptically fibered Calabi-Yau 5-fold. We verify these relations in a non-trivial example, but their proof from a purely mathematical perspective remains as an interesting open problem. Some of the identities we find on elliptic 5-folds are related in an intriguing way to previously studied topological identities governing the structure of anomalies in 6d N = (1, 0) and 4d N = 1 theories obtained from Anomalies in Field and String Theories; F-Theory F-theory. 1 Introduction 2 Anomalies in 2d (0, 2) supergravities 3 F-theory on elliptically fibered Calabi-Yau five-manifolds Gauge symmetries and gauge backgrounds, and 3-branes Matter spectrum from F-theory compactification on CY 5-folds 4 Anomaly equations in F-theory on Calabi-Yau 5-folds Gauge anomalies, Green-Schwarz terms and the 3-7 sector Gravitational anomaly 5 Derivation of the Green-Schwarz terms and 3-7 anomaly 6 Example: SU(5) × U( 1 ) gauge symmetry in F-theory 3.1 3.2 4.1 4.2 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 10d Chern-Simons terms Derivation of the GS term in Type IIB 3-7 anomaly from gauging in Type IIB F-theory lift Relation to 2d effective action Geometric background and 3-7 states Curvature dependent anomaly relations Flux dependent anomaly relations 6.3.1 6.3.2 7 Comparison to 6d and 4d anomaly relations 8 Conclusions and outlook A Conventions A.1 Local anomaly A.2 Type IIB 10D supergravity and brane Chern-Simons actions A.3 Type IIB orientifold compactification with 7-branes B Anomalies and Green-Schwarz term in Type IIB orientifolds C Chirality computation for matter surface flux – i – Introduction Quantum anomalies considerably constrain the structure of chiral gauge theories in even dimensions. Chiral matter is known to induce gauge and gravitational anomalies at the 1-loop level in perturbation theory [1], which jeopardize the consistency of the gauge theory. In the presence of tensor fields the celebrated Green-Schwarz-Sagnotti-West mechanism [2–4] can cancel such 1-loop anomalies provided the anomaly polynomial of the latter factorises suitably. A particularly interesting class of examples of such tensors are the self-dual tensor fields in 4k + 2 dimensions [5]. The ramifications of the anomaly cancellation mechanism have been investigated in great detail, most notably in the context of 6d N = ( 1, 0 ) suSCFT point [11–15], in the context of computing elliptic genera and localisation [16], or with respect to novel types of dualities [17, 18]. Exploring the structure of anomalies of a class of 2d N = (0, 2) supergravities is the goal of this article. If a supergravity theory is engineered by compactifying string theory, the consistency conditions from anomaly cancellation imply a rich set of constraints on the geometry defining the compactification. A prime example of this fruitful interplay between anomalies and geometry is provided by F-theory [19–21]. In this framework, 6d N = ( 1, 0 ) supergravities arise via compactification on elliptically fibered Calabi-Yau 3-folds. Anomaly cancellation then translates into various highly non-trivial relations between topological invariants of the latter [9, 22–25], which would be hard to guess otherwise, and some of which are even harder to prove in full generality. Compactification of F-theory to four dimensions on a Calabi-Yau 4-fold gives rise to an N = 1 supersymmetric theory which is chiral — and hence potentially anomalous — only in the presence of non-trivial gauge backgrounds. This makes it perhaps even more intriguing that the same types of topological relations [26] are responsible for the cancellation of gauge and mixed gauge-gravitational anomalies in six and four-dimensional [27] F-theory compactifications. If one is able to establish the cancellation of anomalies directly from a physical perspective, as has been achieved recently in [28] for four-dimensional F-theory vacua, such reasoning amounts to a physics proof of a number of highly non-trivial topological relations on elliptic fibrations of complex dimension three and four. One of the motivations for this work is to extend this list of topological identities to elliptic fibrations of higher dimension. The 2d (0, 2) supergravity theories considered in this article are obtained by compatifying F-theory on an elliptically fibered Calabi-Yau 5-fold [29, 30]. As we will review in section 3 the theories contain three different coupled sub-sectors: the structure of the gauge theory sector is similar to the 2d (0, 2) GLSMs familiar from the worldsheet formulation of the heterotic string [31, 32]. It includes 2d (0, 2) chiral and Fermi multiplets charged under the in general abelian and non-abelian gauge group factors originating from a topologically twisted theory on 7-branes [29, 30]. D3-branes wrapped around curves on the base of the fibration give rise to additional degrees of freedom. These include a particularly – 1 – fascinating, but largely mysterious sector of Fermi multiplets from the string excitations at the intersection of the D3-branes and the 7-branes [33].1 These two sectors are coupled to a 2d N = (0, 2) supergravity sector [37]. The construction of 2d N = (0, 2) theories has received considerable attention also in other formulations of string theory, most notably via D1 branes probing singularities on Calabi-Yau 4-folds [38–43] and via orientifolds [44, 45]. Various aspects of the non-abelian gauge and the gravitational anomalies in the chiral 2d (0, 2) theory obtained via F-theory have already been addressed in [29, 30, 33, 37, 46]. The non-abelian anomalies induced by the chiral fermions in the 7-brane gauge sector must be cancelled by the anomalies of the 3-7 modes, as indeed verified in globally consistent examples in [29]. The cancellation of all gravitational anomalies for 2d (0,2) supergravities with a trivial gauge theory sector has been proven in [37] with the help of various index theorems. Such theories are obtained by F-theory compactification on smooth, generic Weierstrass models. On the other hand, the structure of gauge anomalies in the presence of abelian gauge theory factors is considerably more involved, and the subject of this article. As in higher dimensions, abelian anomalies induced at 1-loop level need not vanish by themselves provided they are consistently cancelled by a two-dimensional version of the Green-Schwarz mechanism. In general 2d (0, 2) gauge theories, the structure of the Green-Schwarz mechanism has been laid out in [47–49] (see [38, 50] for early work). In the present situation, the Green-Schwarz mechanism operates at the level of real chiral scalar fields which are obtained by Kaluza-Klein reduction of the self-dual 4-form of Type IIB string theory. They enjoy a pseudo-action which is largely analogous to the pseudo-action of the self-dual 2-tensors in 6d N = ( 1, 0 ) supergravities and which we parametrise in general terms in section 2. As one of our main results we carefully derive this pseudo-action in section 5, thereby identifying the structure (and correct normalisation) of the anomalous Green-Schwarz couplings. The latter depend on the non-trivial gauge background and imply a classical gauge variance of the right form to cancel the 1-loop abelian gauge anomalies. A challenge we need to overcome to show anomaly cancellation is that in absence of a perturbative limit the abelian charges of the 3-7 sector modes are notoriously hard to determine in a microscopic approach. Instead of computing the 3-7 anomaly from first principles we extract the anomaly inflow terms onto the worldvolume of the D3-branes in section 5. To this end we start from the Chern-Simons terms of the 10d effective pseudoaction in the presence of brane sources. Uplifting this result to F-theory allows us to quantify the contribution of the 3-7 modes in particular to the gauge anomalies and in turn also to deduce the net charge of the 3-7 modes. One of our main results is to establish a closed expression for the complete gauge and gravitational anomalies of a 2d (0, 2) theory obtained by F-theory compactified on a Calabi-Yau 5-fold. The resulting conditions for anomaly cancellation are summarized in (4.18) and (4.35) of section 4. The structure of anomalies reflected in these equations interpolates between their analogue in 6d and 4d F-theory vacua: in 6d F-theory vacua the anomalies are purely dependent on properties of the elliptic fibration, while in 4d they 1The theory on a D3-brane wrapping a curve [33, 34] or surface [35, 36] in F-theory is interesting by version has recently been constructed in [15]. of the topological identities (4.18) and (4.35) must in fact hold separately, on any elliptically fibered Calabi-Yau 5-fold and for any gauge background satisfying the consistency relations reviewed in section 3. We verify these highly non-trivial anomaly relations in a concrete example fibration for all chirality inducing gauge backgrounds in section 6. It has already been pointed out that, despite their rather different structure at first sight, the gauge anomalies in 6d and 4d boil down to one universal relation in the cohomology ring of an elliptic fibration over a general base, and similarly for the mixed gauge-gravitational anomalies [26].2 This prompts the question if the 2d anomaly relations (4.18) and (4.35) are also equivalent to this universal relation governing the structure of anomalies in four and six dimensions. As we will see in section 7, assuming the 4d/6d relation of [26] implies the flux dependent part of (4.18) and (4.35) for a special class of gauge background. However, it remains for further investigation whether the precise relations extracted in [26] on Calabi-Yau 3-folds and 4-folds follow in turn by anomaly cancellation on Calabi-Yau 5-folds in full generality. 2 Anomalies in 2d (0, 2) supergravities Consider an N = (0, 2) supersymmetric theory in two dimensions with gauge group and matter fields in representations Here rI denotes an irreducible representation of the simple gauge group factor GI and q = (q1, . . . , qnU( 1 ) ) are the charges under the Abelian gauge group factors. We are interested in the structure of the gauge and gravitational anomalies in such a theory. These are induced by chiral matter at the 1-loop level. In a general D-dimensional quantum field theory, the gauge and gravitational anomalies can be described by a gauge invariant anomaly polynomial of degree D/2 + 1 in the gauge field strength F and the curvature two-form R, where the sum is over all matter fields with spin s which have zero-modes in representation R with multiplicity ns(R). In particular, a chiral fermion, corresponding to s = 1/2, contributes with 2By contrast, the purely gravitational anomaly in 6d has no direct counterpart in 4d. See, however, [51]. ID+2 = X ns(R)Is(R)|D+2 , R,s I1/2(R) = −trR e−F Aˆ(T) , – 3 – where Aˆ(T) is the A-roof genus and F denotes the hermitian gauge field strength. An anti-chiral fermion contributes with the opposite sign. For more details on our conventions we refer to appendix A. In D = 2 dimensions, the 1-loop anomaly polynomial from the charged matter sector is hence a 4-form. Correspondingly, the anomaly contribution from chiral and anti-chiral fermions in the theory sums up to I4 = X(n+(R) − n−(R)) R 1 − 2 trR(F )2 + 1 24 p1(T) dim(R) , where the first Pontryagin class of the tangent bundle is defined as p1(T) = − 21 trR2. For future purposes we express the anomaly polynomial for the non-abelian, the abelian and the gravitational anomaly as I4|GI = −AI trfundFI2 = − 2 I4|AB = −AAB F AF B = − 2 I4|grav = 1 24 Agrav p1(T ) = 1 24 rI R R 1 X c(r2I) χ(rI ) trfundFI2 1 X qA(R) qB(R) dim(R) χ(R) F AF B X χ(R) dim(R) p1(T) , (2.5) (2.6) (2.7) (2.8) (2.9) with χ(R) denoting the chiral index of zero-modes in representation R. In the first line we have related the trace in a representation rI of the simple gauge group factor GI to the trace in the fundamental representation via trrI F 2 = c(r2I) trfundF 2 . In general, the 1-loop induced quantum anomaly need not be vanishing in a consistent theory provided the tree-level action contains gauge variant terms, the Green-Schwarz counter-terms, which cancel the anomaly encoded by ID+2. For this cancellation to be possible, the 1-loop anomaly polynomial ID+2 of the matter sector must factorize suitably. In two dimensions, the Green-Schwarz counterterms derive from gauge variant interactions of scalar fields. The structure of the possible Green-Schwarz terms in a general 2d N = (0, 2) supersymmetric field theory has been analyzed in [47–49] (see [38, 50] for early work). In this paper, however, we are interested in the specific 2d N = (0, 2) effective theory obtained by compactification of F-theory on an elliptically fibered Calabi-Yau 5-fold [29, 30]. In these theories a gauge theory with gauge group (2.1) is coupled to a 2d N = (0, 2) supergravity sector.3 The latter contains a set of real axionic scalar fields cα arising from the Kaluza-Klein (KK) reduction of the F-theory/Type IIB Ramond-Ramond forms C4 [37].4 3The gauge theory in question arises from spacetime-filling 7-branes. In addition, the compactification contains spacetime-filling D3-branes, but the associated gauge fields are projected out due to SL(2, Z) monodromies along the D3-brane worldvolume [29, 33]. 4As discussed in [37], these scalars split into n+ chiral and n− anti-chiral real scalars. Out of these n+ pairs of real chiral and anti-chiral scalars form non-chiral real scalars, which constitute the imaginary part of the bosonic component of a corresponding number of 2d (0, 2) chiral multiplets. The remaining τ = n− −n+ anti-chiral real scalars form 2d (0, 2) tensor multiplets and contribute, together with the gravitino, to the gravitational anomaly at 1-loop level according to the general formulae reviewed in appendix A. This contribution to the 1-loop anomaly is in addition to the classical gauge variance of the Green-Schwarz action discussed in this section. – 4 – As we will derive in detail in section 5, their pseudo-action can be parametrized as Schwarz action [3, 4] of self-dual tensor fields in D = 6 (see e.g. [52]) and, in fact, D = 10 dimensions, with the role of the gauge invariant self-dual field strengths being played here by the 1-forms Hα = Dcα. These are subject to the self-duality condition gαβ ∗ Hα = ΩαβHβ . dHα = Xα , Xβ = ΘβA F A Hα = Dcα = dcα + ΘαAAA . AA c α → AA + dλA → cα − ΘαAλA The second term in the action constitutes the Green-Schwarz coupling, which is responsible for the non-standard Bianchi identity where we used that Ωαβ is a constant matrix. The Green-Schwarz couplings will be found to take the form depending on the background flux. This identifies Hα as β with F A the field strength associated with the gauge group factor U( 1 )A and with ΘA The axionic shift symmetry of the chiral scalars is gauged by the abelian vector AA according to the transformation rule (2.11) such that the covariant derivative Dcα is gauge invariant. As a result, the pseudo-action picks up a gauge variation of the form with I2( 1 ),GS a gauge invariant 2-form. By the standard descent procedure, it defines an anomaly-polynomial I4GS encoding the contribution to the total anomaly from the GreenSchwarz sector. Concretely, the descent equations IGS = dI3GS, 4 δλI3GS = dI2( 1 ),GS(λ) imply 1 2 2πI4GS = ΩαβXαXβ = ΩαβΘαAΘβB F AF B . 1 2 – 5 – Consistency of the theory then requires that I4 + I4GS = 0 . This is possible only if the non-abelian and gravitational anomalies vanish by themselves and the abelian anomalies factorise suitably. The resulting constraints on the spectrum take the following form: Non-abelian : Abelian : Gravitational : 2 R 1 X dim(R) χ(R) qA(R) qB(R) = 1 4π ΩαβΘαAΘβB RI 2 1 X χ(rI ) c(r2I) = 0 X dim(R) χ(R) = 0 . R (2.19) (2.20a) (2.20b) (2.20c) Note that, unlike in higher dimensions, the 2d GS mechanism operates entirely at the level of the abelian gauge group factors: in (4k + 2) dimensions the analogue of (2.14) is the gauge invariant field strength associated with the self-dual rank (2k + 1)-tensor fields, and the correction term in the covariant action involves the Chern-Simons (2k + 2)-forms associated with the gauge and diffeomorphism group. In 2d the Chern-Simons form is proportional to the trace over the gauge connection and must hence be abelian. Therefore the 2d non-abelian and gravitational anomalies from the chiral sector at 1-loop must vanish by themselves; likewise there can be no mixed gravitational-gauge anomalies induced at 1-loop. Furthermore, let us point out that in the 2d (0, 2) theories of the type considered here the gauging (2.15) of the scalars is directly related to the anomalous Green-Schwarz coupling (2.13). This is a notable difference to the implementation of the Green-Schwarz mechanism in the more general 2d (0, 2) gauge theories of [47], where these two are in principle independent. Before we proceed, we would like to comment on the scalar fields cα. In principle, all of the axionic scalar fields cα obtained from the Type IIB RR fields Cp can contribute to the Green-Schwarz mechansim. However, as in 6d and 4d F-theory compactifications, the gauging of the scalar fields from C2 is encoded via a geometric Stu¨ckelberg mechanism in terms of non-harmonic forms, at least in the description via the dual M-theory [53]. In this work we will we will only focus on the Green-Schwarz mechanism associated with the scalar fields arising from the RR potential C4, which will be seen to depend on the background flux. 3 F-theory on elliptically fibered Calabi-Yau five-manifolds In this section we provide some background material on N = (0, 2) supersymmetric compactifications of F-theory to two dimensions. The reader familiar with this type of constructions from [29, 30] can safely skip this summary. – 6 – We consider a 2d (0, 2) supersymmetric theory describing a vacuum of F-theory compactified on an elliptically fibered Calabi-Yau 5-fold X5 [29, 30] with projection π : X5 → B4 . The base B4 is a smooth complex 4-dimensional K¨ahler manifold, which is to be identified with the physical compactification space of F-theory. Via F/M-theory duality, F-theory on B4 is related to the supersymmetric quantum mechanics [54] obtained by compactification For simplicity we assume that X5 has a global section [z = 0] so that it can be described HJEP04(218)7 of M-theory on X5. by a Weierstrass equation y2 = x3 + f x z4 + g z6 . Δ = 4f 3 + 27g2 = 0 Here the projective coordinates [x : y : z] parametrise the fiber ambient space P2,3,1 and f, g are sections of the fourth and sixth power of the anti-canonical bundle K¯ of the base. The discriminant locus C3 = AiI ∧ [EiI ] + . . . . 5To avoid clutter we will mostly avoid the hat above π in the sequel. – 7 – (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) specifies the location of the 7-branes. The non-abelian gauge group factors GI in (2.1) are associated with 7-branes wrapping divisors WI , which are complex 3-dimensional components of the discriminant locus Δ = 0 in the base. We assume that the Kodaira singularities in the fibre above WI admit a crepant resolution5 πˆ : Xˆ5 → B4 . The resolution replaces the singularity over WI by a chain of rational curves. After taking into account monodromy effects, which appear for non-simply laced groups, this allows one to identify a collection Pi1I , iI = 1, . . . , rk(gI ) of independent rational curves in the resolved fiber which can be associated with the simple roots αiI of the Lie algebra gI underlying GI in the following sense: the fibration of Pi1I over WI — more precisely of the image of Pi1I under monodromies in the non-simply laced case — defines a resolution divisor EiI with the property that [EiI ] · [Pj1J ] = −δIJ CiI jJ . Here [EiI ] denotes the homology class of the divisor EiI and unless noted otherwise, all intersection products are taken on Xˆ5. The matrix CiI jI is the Cartan matrix of gI (in conventions where the entries on its diagonal are +2). Via duality with M-theory, M2branes wrapping the fibral curves Pj1J give rise to states associated with the simple roots −αiI , and the Cartan U( 1 )iI gauge field arises by KK reduction of the M-theory 3-form as λ 1 2 2 1 In this sense the resolution divisors [EiI ] can be identified with the generators TiI of the Cartan subgroup of GI in the so-called co-root basis, whose trace over the fundamental representation of GI is normalised such that trfundTiI TjJ = δIJ λI CiI jI with CiI jI = CiI jI . (3.7) The quantity λI denotes the Dynkin index in the fundamental representation and is tabulated in table 1. Note that for simply-laced groups CiI jI = CiI jI . The geometric manifestation of this identification is the important relation π∗([EiI ] · [EjJ ]) = −δIJ CiI jI [WI ] = −Tr TiI TjJ [WI ] , where Tr is related to the trace in the fundamental representation via 1 λI Tr = trfund . The push-forward π∗([EiI ] · [EjJ ]) to the base of the fibration is defined by requiring that [EiI ] ·Xˆ5 [EjJ ] ·Xˆ5 [Dα] ·Xˆ5 [Dβ] ·Xˆ5 [Dγ ] = π∗([EiI ] ·Xˆ5 [EjJ ]) ·B4 [Dαb] ·B4 [Dβb] ·B4 [Dγb] for any basis of vertical divisors [Dα] = π∗[Dαb], where Dαb is a divisor on B4. Each non-Cartan Abelian gauge group factor U( 1 )A is associated with a global rational section SA of Xˆ5 in addition to the zero-section S0. To each SA one can assign an element [UA] ∈ CH1(Xˆ5) through the Shioda map UA = SA − S0 − DA + X kiI EiI . iI The vertical divisor DA and the in general fractional coefficients kiI are chosen such that UA satisfies the transversality conditions [UA] ·Xˆ5 [Dα] ·Xˆ5 [Dβ] ·Xˆ5 [Dγ ] ·Xˆ5 [Dδ] = 0 [UA] ·Xˆ5 [S0] ·Xˆ5 [Dα] ·Xˆ5 [Dβ] ·Xˆ5 [Dγ ] = 0 [UA] ·Xˆ5 [EiI ] ·Xˆ5 [Dα] ·Xˆ5 [Dβ] ·Xˆ5 [Dγ ] = 0 , which must hold for every vertical divisor [Dα] = π∗Dαb. In analogy with the relation (3.8), one can define the so-called height pairing [25, 55] π∗([UA] ·Xˆ5 [UB]) = −Tr TATB [DAB] . The objects TA, TB are the generators of U( 1 )A and U( 1 )B and DAB is a divisor on the base of the fibration. Unlike the divisor WI , even for A = B this divisor is not one of – 8 – (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) By F/M-duality, the G4 fluxes are subject to the transversality constraints Z ˆ X5 ∗ G4 ∧S0 ∧π ω4 = 0 and G4 ∧π∗ω6 = 0 , ∀ ω4 ∈ H4(B4), ω6 ∈ H6(B4) . (3.16) Z Z ˆ X5 ˆ X5 G4 ∧ EiI ∧ π ω4 = 0 ∗ the irreducible components of the discriminant Δ (in the sense that Δ would factorise into the union of various irreducible such DAA). Nonetheless, we will see that it plays a very analogous role for the structure of anomalies also for F-theory compactifications to 2d. A crucial ingredient in F/M-theory compactifications on Calabi-Yau five-folds is the gauge background for the field strength G4 = dC3 of the M-theory 3-form potential field. As in compactifications to four dimensions, the full gauge background is an element of the Deligne cohomology group HD4(Xˆ5, Z(2)) and can be parametrized by equivalence classes of rational complex codimension-2-cycles [56, 57], which form the second Chow group CH2(Xˆ5). The field strength of G4 as such takes values in H4(Xˆ5). It is subject to the Freed-Witten quantization condition [58] HJEP04(218)7 If this flux satisfies in addition the constraint In order to preserve two supercharges in the M/F-theory compactification on Xˆ5, the ( 3, 1 ) and ( 1, 3 ) Hodge components of H4(Xˆ5) must vanish [54] and hence 1 G4 + 2 c2(Y5) ∈ H4(Xˆ5, Z) . 1 G4 + 2 c2(Y5) ∈ H4(Xˆ5, Z) ∩ H2,2(Xˆ5) . it leaves the gauge group factor GI unbroken. Higher curvature corrections in the M-theory effective action induce a curvature dependent tadpole for the M-theory 3-form C3. In the dual F-theory these curvature corrections subsume the curvature contributions to the Chern-Simons action of the 7-branes (including, in the perturbative limit, the orientifold planes). In a consistent M-theory vacuum this tadpole must be cancelled by the inclusion of background flux G4 and/or by M2-branes wrapping a curve class on Xˆ5 determined by the tadpole equation [54]. The projection of this curve class to the base B4 describes,6 in the dual F-theory, the class wrapped by background D3-branes filling in addition the extended directions along R1,1. The projected class is given by [29, 54] 6The M2-brane states along the fibral component of this class are related to momentum modes along the circle S1 arising in F/M-theory duality [37]. – 9 – (3.14) (3.15) (3.17) (3.18) The charged chiral matter fields whose contributions to the 1-loop anomalies we will be studying arise from three sources [29, 30]: 7-brane bulk matter propagating along the nonabelian divisors WI , 7-brane codimension-two matter localised along the intersections of various discriminant components or self-intersections of the discriminant, and finally Fermi multiplets at the pointlike intersection of D3-branes with the 7-branes. Due to the chiral nature of the 2d (0, 2) theory, all three types of matter are chiral even for vanishing gauge backgrounds. The bulk matter fields transform, in the absence of gauge flux, in the adjoint representation of GI . In the dual M-theory quantum mechanics, this matter arises from M2-branes wrapping suitable combinations of resolution Pi1I in the fiber over WI . For non-vanishing gauge backgrounds, which can be described by a non-trivial principal gauge bundle L, the original gauge group GI can be broken into a product of some sub-groups. The spectrum decomposea into irreducible representations R of the unbroken gauge factors GI → HI Adj(GI ) → Adj(HI ) ⊕ M R R Note that if R 6= R¯ , each representation is accompanied by its complex conjugate. The matter fields organise into 2d (0, 2) chiral multiplets, which contain one complex boson and a complex chiral Weyl fermion, as well as Fermi multiplets, which contain one complex anti-chiral Weyl fermion. Each of these matter fields is counted by a certain cohomology group on WI involving the vector bundle LR. The chiral index of massless matter in a given complex representation, defined as the difference of chiral and anti-chiral fermions in complex representation R, is then given by [29, 30] χ(R) = − Z WI 1 12 c1(WI ) rk(LR) c2(WI ) + ch2(LR) . (3.21) For real representations, this expression is to be multiplied with a factor of 21 . In particular, the chiral index of the adjoint representation depends purely on the geometry and takes the form χ(Adj(HI )) = − 214 R WI c1(WI )c2(WI ). Extra matter states in representation R of Gtot localizes on complex 2-dimensional surfaces CR on B4. This occurs whenever some of the rational curves Pi1I in the fiber split over CR. Group theoretically, this signifies the splitting of the associated simple roots into weights of representation R. The associated charged matter fields arise from M2-branes wrapped on suitable linear combinations of fibral curves over CR, which in fact span the weight lattice of the gauge theory. Hence to each state in representation R we can associate a matter 3-cycle SRa which is given by a linear combination of fibral curves over CR and carries a weight vector βiaI , a = 1, . . . , dim(R), such that π∗([EiI ] · [SRa]) = βiaI [CR] . These matter states also organize both into chiral and Fermi multiplets and are counted by cohomology groups of a vector bundle LR which derives from the gauge background. If the surface CR on B4 is smooth, the chiral index of this type of matter follows from an index theorem as [29, 30] χ(R) = Otherwise one has to perform a suitable normalisation in order to be able to apply the index theorem, and this will lead to correction terms as exemplified in [29]. HJEP04(218)7 The third type of massless matter arises from 3-7 string states at the intersection of the 7-branes with the spacetime-filling D3-branes wrapping the curve class [C] in (3.18). Matter in the 3-7 sector comes in 2d (0,2) Fermi multiplets [29, 30]. In purely perturbative setups, each intersection point of [C] with one of the D7-branes carries a single Fermi multiplet in the fundamental representation of the D7-brane gauge group. However, monodromy effects along the 3-brane worldvolume considerably obscure such a simple interpretation of the 3-7 modes in non-perturbative setups [29, 33]. As one of our results, we will see how the structure of 2d anomalies sheds new light on the structure of 3-7 modes, including, in particular, their charges under the non-Cartan abelian gauge factors. 4 Anomaly equations in F-theory on Calabi-Yau 5-folds In this section we present closed expressions for the anomaly cancellation conditions in 2d (0, 2) F-theory vacua. We begin in section 4.1 by deriving a formula for the chiral index of charged matter states in the presence of 4-form flux G4 in the dual M-theory, which is uniformly valid for the bulk and the localised 7-7 modes. We also shed some more light on the counting of 3-7 modes. Together with the Green-Schwarz counterterms this leads to formula (4.18) for the cancellation of all gauge anomalies. In section 4.2 we extend the gravitational anomaly cancellation conditions of [37] to situations with non-trivial 7-branes and fluxes, leading us to condition (4.34). 4.1 Gauge anomalies, Green-Schwarz terms and the 3-7 sector Recall from the previous section that in this paper we assume the existence of a smooth crepant resolution Xˆ5, which describes the dual M-theory on its Coulomb branch. This forces us, as usual in this context, to restrict ourselves to Abelian gauge backgrounds G4. In particular, the vector bundles appearing in the expressions (3.21) and (3.23) are complex line bundles. For simplicity of presentation we first assume that the gauge flux G4 does not break any of the non-abelian gauge group factors. The chiral index (3.23) of the localised matter can be split into a purely geometric and a flux dependent contribution χ(R) = χgeom(R) + χflux(R) χgeom(R) = − 12 CR 1 Z χflux(R) = Z CR 21 c12(LR) . ch2(CR) = 1 Z We stress that this expression is correct provided the matter surfaces CR on B4 are smooth. The line bundle LR on CR to which a state with weight vector βa(R) couples is obtained from G4 by first integrating G4 over the fiber of the matter 3-cycle SRa and then projecting onto the surface CR. This gives rise to a divisor class on CR which is to be identified, similarly to the procedure in F-theory on Calabi-Yau 4-folds [56, 57], with Note that for gauge invariant flux, the result is the same for each of the matter 3-cycles SRa and hence correctly defines the line bundle associated with representation R. This allows us to rewrite χflux(R) explicitly in terms of G4 as χflux(R) = 1 2 π∗(G4 · SRa) ·CR π∗(G4 · SRa) , where ·CR denotes the intersection product on CR. Next, consider the bulk modes. For gauge invariant flux, this sector contributes only states in the adjoint representation of GI (which due to the quadratic nature of the anomalies nonetheless contribute to the anomaly), and according to (3.21) their chiral index is given by χbulk(R = adjI ) = − 24 1 Z WI c1(WI )c2(WI ) . It is useful to note that χbulk(R) is formally identical to the flux-independent part of the chirality of a localised state whose matter locus is given by the canonical divisor on WI , i.e. the complex 2-cycle on WI in the class Indeed, by adjunction, using the short exact sequence [Ccan] = −c1(WI ) = +c1(KWI ) . 0 → TCcan → TWI → NCcan/WI → 0 and the resulting relation one computes c(TCcan ) = c(TWI )/c(NCcan/WI ) = (1 + c1(WI ) + c2(WI ))/(1 − c1(WI )), c1(Ccan) = 2c1(WI ) c2(Ccan) = c2(WI ) + 2c12(WI ) . (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) This implies that Z 1 The additional factor of 12 in (4.4) is due to the fact that the adjoint is a real representation. More generally, and in complete analogy to the description of bulk modes in compactifications on Calabi-Yau 4-folds [57], we can associate to a bulk matter state associated with the root ρI the 3-cycle (4.11) HJEP04(218)7 SρI = X aˆiI EiI |KWI . iI The parameters aˆiI are related to the coefficients in the expansion of the root ρI in terms of the simple roots αiI .7 Geometrically, the fiber of SρI is given by the corresponding linear combination of fibral rational curves Pi1I . An M2-brane wrapped along this linear combination of fibral curves gives rise to a state whose Cartan charges are given precisely by the root ρI . For gauge invariant flux satisfying (3.17), the line bundle π∗(SρI · G4) vanishes by construction. Hence the expression for the bulk and the localised chirality are completely analogous and both types of matter will from now on be treated on the same footing. This conclusion persists if the gauge background breaks some or all of the simple gauge group factors GI . In this case, the adjoint representation for the bulk matter or the representations associated with the localised matter decompose into irreducible representations of the unbroken subgroup. The operation (4.2) now leads to a well-defined line bundle for each of these individual representations, for bulk and localised matter alike. Next, we consider the contribution from the 3-7 modes. As it turns out, to each representation R one can associate a divisor D37(R) on B4 such that the chiral index of 3-7 states in representation R is given by χ3−7(R) = − 1 1 24 π∗(c4(Xˆ5)) − 2 π∗(G4 · G4) ·B4 D37(R) . (4.12) The expression in brackets is the curve class [C], defined in (3.18), wrapped by the spacetime-filling D3-branes. For instance, for a perturbative gauge group GI = SU(N ), each intersection point of [C] with the 7-brane divisor WI hosts a (negative chirality) Fermi multiplet in representation R = (N) [29, 30] and therefore D37(R = (N)) = WI . For non-perturbative gauge groups and for Abelian non-Cartan groups U( 1 )A determining the representation and charge of the 3-7 strings from first principles is more obscure due to subtle SL(2, Z) monodromy effects on the worldvolume of the D3-brane along C [33]. However, in the next section we will derive that in the presence of extra U( 1 )A gauge group factors the net contribution to the U( 1 )A − U( 1 )B anomaly (2.7) from the 3-7 sector takes 7For simply laced Lie algebras, ρI = PiI aˆiI αiI . For non-simply laced Lie algebras, fractional corrections must be included to take into account monodromy effects, as explained e.g. in appendix A of [25]. the form AAB|3−7 = X qA(R) qB(R) dim(R) χ3−7(R) 1 2 1 R,3−7 1 1 = 2 24 π∗(c4(Xˆ5)) − 2 π∗(G4 · G4) ·B4 π∗(UA · UB) . Here we recall that UA and UB generate the respective U( 1 ) factors via the Shioda map (3.11) and that the height-pairing π∗(UA · UB) had been introduced in (3.13). More generally, our results imply that the right-hand side correctly captures the contribution to the anomaly also of the Cartan U( 1 ) group for non-perturbative gauge groups. Let us to collectively denote set of divisors generating any of the Cartan U( 1 )iI or non-Cartan U( 1 )A gauge symmetries. Then our claim is that the contribution to the gauge anomaly due to 3-7 modes can be summarized as AΛΣ|3−7 = 2 R,a 1 X βΛa(R) βΣa(R) χ3−7(R) = 1 1 1 2 24 π∗(c4(Xˆ5))− 2 π∗(G4 ·G4) ·B4 π∗(FΛ ·FΣ) . If the index Λ = iI refers to a Cartan U( 1 )iI , the object βiaI (R) denotes the weights associated with representation R with respect this U( 1 )iI , and for Λ = A we define βAa(R) = qA(R). We will come back to the interpretation of this formula at the end of this section. As the final ingredient we will derive, in section 5, the Green-Schwarz counterterms appearing on the righthand side of (2.20b). These are found to be purely flux dependent and of the form 1 4π ΩαβΘαΣΘβΛ = 1 2 π∗(G4 · FΣ) ·B4 π∗(G4 · FΛ) . For instance, if we let FΛ = UA, FΣ = UB refer to non-Cartan Abelian groups, then this describes the Green-Schwarz counterterms for the U( 1 )A − U( 1 )B anomalies. For FΛ = EiI , FΣ = EjI , the right-hand side is non-vanishing only if the gauge background G4 breaks the simple gauge group factors GI and GJ , in which case it computes the counterterms for the U( 1 )iI − U( 1 )jJ anomaly. For gauge invariant flux, on the other hand, no such Green-Schwarz terms are induced, in agreement with expectations. With this preparation we can now rewrite the gauge anomaly equations (2.20a), (2.20b) in a rather suggestive form. Since the anomaly equations must hold for arbitrary gauge background G4 and since the flux independent terms only give a constant off-set, the flux dependent and the flux independent contributions to the anomalies must vanish separately. The requirement (2.20a), (2.20b) of cancellation of all gauge anomalies therefore results in (4.13) (4.14) (4.15) (4.16) (4.17) two independent identities: R,a The two terms in (4.18a) respectively represent the flux independent anomaly contribution from the 7-7 sector, (4.1), and from the 3-7 sector, (4.16). In (4.18b) we have collected the flux dependent 3-7 and the Green-Schwarz contribution to the anomaly in the brackets in the second and third line to illustrate the striking formal similarity between them. We will understand this similarity in the next section. Let us now come back to the interpretation of (4.16). For FΛ = EiI , FΣ = EjI this equation allows us to deduce the net contribution to the anomalies due to 3-7 strings charged under the non-abelian gauge group factors, which, as noted already, can be rather obscure due to monodromy effects. To interpret this expression, recall the crucial identity (3.8). If we assume that each geometric intersection point [C] ·B4 WI hosts an (anti-chiral) Fermi multiplet in representation R, then for consistency this representation must satisfy This is to be contrasted with the fact that for any representation R of a simple group GI X βiaI (R)βjaI (R) =! CiI jI . a X βiaI (R)βjaI (R) = trRTiI TjI = λI c(R2) CiI jI a (4.19) (4.20) R with TiI denoting the Cartan generators in the coroot basis. The Dynkin index λI for the fundamental representation of GI is collected, for all simple groups, in table 1, and c(2) normalizes the trace with respect to the fundamental representation as in (2.9). By definition, the smallest value of c(R2) occurs for the fundamental representation cfund = 1. (2) Hence unless λI = 1 or λI = 2, the interpretation in terms of 3-7 modes necessarily involves ‘fractional’ Fermi multiplets.8 This is in agreement with the observation of [29] that e.g. for GI = E6, the net contribution to the anomaly from the 3-7 sectors corresponds to that of a 16 -fractional Fermi multiplet per intersection point. 4.2 Gravitational anomaly The gravitational anomaly for F-theory compactified on a smooth Weierstrass model X5 without any 7-brane gauge group and background flux has already been discussed in [37]. 8The case λI = 2 requires, for consistency, that the fundamental representation be real and hence contributes with a factor of 21 to compensate for λI . Table 1 confirms that this is indeed the case for all simple algebras with λI = 2. The anomaly polynomial receives contributions from the moduli sector, from the 2d (0, 2) supergravity multiplet as well as from the 3-7 sector, 1 Note that Agrav|mod includes what would be called in Type IIB language the contributions from the closed string moduli sector, from the moduli associated with the 7-branes (which however by assumption carry no gauge group), and from τ (B4) many 2d (0, 2) tensor χ1(X5) = − 24 X5 1 Z c5(X5) = Z B4 90c14 + 3c12c2 − 2 c1c3 1 with ci = c1(B4). Furthermore the signature τ (B4) counts the difference of self-dual and anti-self-dual 4-forms on B4 and is related to the Hodge numbers of B4 as τ (B4) = b4+(B4) − b4−(B4) = 48 + 2h1,1(B4) + 2h3,1(B4) − 2h2,1(B4) . The D3-brane class appearing fixed by the tadpole on a smooth Weierstrass model without flux is [C] = 214 π∗(c4(X5)). As shown in [37] with the help of various index theorems, the total anomaly can be evaluated as 1 I4,grav = 24 p1(T ) (−24χ0(B4) + 24) ≡ 0 , where the last equality holds because h0,i(B4) for i 6= 0 if B4 is to admit a smooth Calabi Yau Weierstrass fibration over it. Suppose now that the fibration contains in addition a non-trivial 7-brane gauge group and charged 7-7 matter, and let us also switch a non-trivial flux background G4. For simplicity assume first that the supersymmetry condition that G4 be of pure (2, 2) Hodge type [54] does not constrain the moduli of the compactification. In analogy with G4 flux on Calabi-Yau 4-folds, this is guaranteed whenever G4 ∈ Hv2e,2rt(Xˆ5), the primary vertical subspace of H2,2(Xˆ5) generated by products of ( 1, 1 ) forms.9 In this situation the gravitational anomaly generalizes as follows: first, we must now work on the resolution Xˆ5 of the 9The space of (2, 2) forms on Calabi-Yau 5-folds deserves further study beyond the scope of this article. In particular it remains to investigate in more detail whether a similar split into horizontal and vertical subspaces exists as on Calabi-Yau 4-folds. In any event if G4 is a sum of (2, 2) forms obtained as the product of two ( 1, 1 ) forms, the Hodge type does not vary. singular Weierstrass model describing the more general 7-brane configuration. In particular the D3-brane curve class changes to [C] = 214 π∗(c4(Xˆ5)) − 12 π∗(G4 · G4) with c4(Xˆ5) evaluated on the resolved space Xˆ5. Second, we must add the anomaly contribution from the non-trivial 7-7 sector. This sector includes the localised matter in some representation R of the total gauge group as well as the bulk matter in the adjoint representation (or its decomposition if the flux background breaks the non-abelian gauge symmetry). Each massless multiplet in the bulk sector contributes dim(adj) many states to the anomaly. Of these, rk(G) many states are associated with the Cartan subgroup of the gauge group and are in fact encoded already in the contribution from the ‘moduli sector’. More precisely, if we replace in (4.22) the contribution χ1(X5) by χ1(Xˆ5), the resulting expression Agrav|mod now includes the anomaly from the rk(G) = h1,1(Xˆ5) − (h1,1(B4) − 1) many vector multiplets associated with the Cartan subgroup as well as the ‘open string moduli’ in the Cartan, which enter the values of h1,p(Xˆ5). As a result, the total gravitational anomaly polynomial is now 1 24 p1(T ) (Agrav|7−7 + Agrav|mod + Agrav|uni + Agrav|3−7) with the individual contributions R Agrav|7−7 = X dim(R)χ(R) − rk(G)χ(adj) Agrav|mod = −τ (B4) + χ1(Xˆ5) − 2χ1(B4), Agrav|uni = 24 Agrav|3−7 = −6c1(B4) · 1 1 24 π∗(c4(Xˆ5)) − 2 π∗(G4 · G4) . (4.29) (4.30) (4.31) (4.32) (4.33) Note that the topological invariants χ1(Xˆ5) and c4(Xˆ5) contain correction terms in addition to the base classes appearing for the case of a smooth Weierstrass model which depend on the resolution divisors and extra sections (if present). The vanishing of the total gravitational anomaly implies that these individual contributions must cancel each other, Agrav|7−7 + Agrav|mod + Agrav|uni + Agrav|3−7 = 0 . (4.34) This leads to a set of topological identities which must hold for every resolution Xˆ5 of an elliptically fibered Calabi-Yau 5-fold, and for every consistent configuration of background fluxes thereon, as specified above. Note that the flux background enters not only through the 3-brane class in A3−7, but also because the chiral indices in the 7-brane sector split as χ(R) = χ(R)|geom + χ(R)|flux as in (4.1). In principle, if the Hodge type of G4 were to vary over the moduli space, the supersymmetry condition G4 ∈ H2,2(Xˆ5) would induce a potential for some of the moduli [54] and hence modify the number of uncharged massless fields. According to our assumptions, this does not occur for the choice of flux considered here and the uncharged sector contributes to the anomaly as above. Then the anomaly equations split into the independent sets of equations X dim(R)χ(R)|geom − rk(G)χ(adj)|geom − τ (B4) + χ1(Xˆ5) − 2χ1(B4) + 24 R 1 − 4 c1(B4) · π∗c4(Xˆ5) = 0 − 6c1 · π∗(G4 · G4) = X π∗(G4 · SRa) ·CR π∗(G4 · SRa) R,a (4.35a) (4.35b) (4.36) (4.37) (4.38a) (4.38b) In the second equation, which accounts for the flux dependent anomaly contribution, we do not need to treat the 7-brane states in the Cartan separately as their chirality is not affected by the flux background. The flux independent contribution can be analysed further if the fibration Xˆ5 is smoothly connected to a smooth Weierstrass model X5. In the terminology of [59], this means that the F-theory model does not contain any non-Higgsable clusters and hence after the blowdown of the resolution divisors the gauge symmetry can be completely Higgsed. In that case we know already from (4.28) that the anomalies on the resulting smooth Weierstrass model X5 cancel for G4 = 0. Let us therefore define 1 24 π∗c5(Xˆ5)−π∗c5(X5) = − 24 π∗c5(Xˆ5)− 90c14 +3c12c2 − 2 c1c3 . 1 The anomaly equations can then be rewritten as −6c1 · Δ[C] + Δχ1 = − 12 R X dim(R) ch2(CR) Z CR Z C(adj) + rk(G) ch2(C(adj)) −6c1 · π∗(G4 · G4) = X π∗(G4 · SRa) ·CR π∗(G4 · SRa) 1 1 1 12 It is interesting to speculate about the effect of G4 fluxes which are not automatically of (2, 2) Hodge type. The supersymmetry condition (3.15) is reflected in a dynamical potential which is expected to render some of the supergravity moduli massive [54]. The resulting change in the gravitational anomaly compared to the fluxless geometry must be compensated by a suitable modification of the remaining uncharged spectrum. Indeed, the flux contributes at the same time to the D3-brane tadpole and hence changes the D3-brane curve class [C] compared to the fluxless compactification. This changes the number of massless Fermi multiplets in the 3-7 sector. The net number of moduli stabilized in the presence of flux must equal the change in the number of 3-7 modes. This interesting effect has no analogue in 6d or 4d F-theory vacua: in 6d there is no background flux, and in 4d there is no purely gravitational anomaly. The various 7-brane codimension-two matter loci CR have been listed in (6.6), and in the present example they are all smooth [29] such that the index theorem can be applied as in (4.1). Noticing the matter surfaces (6.6) CR can always be written as intersections of two divisors A, B of the base B4, With the adjunction formula we can obtain χ(CR) = 1 24 A · B · (2c2 − c12 + A2 + B2) Applying to (6.6), we find the following flux independent part of the chiral indices for the matter surfaces, χ(101)|geom = χ(53)|geom = χ(5−2)|geom = χ(15)|geom = 1 1 24 1 12 1 24 24 c1W 2c2 + W 2 W (3c1 − 2W ) −12c1W + 8c12 + 2c2 + 5W 2 W (5c1 − 3W ) −15c1W + 12c12 + c2 + 5W 2 (4c1 − 3W ) (3c1 − 2W ) 24c12 + 2c2 − 36c1W + 13W 2 . (6.13) (6.14) HJEP04(218)7 The first equation in (2.20), i.e. the purely non-abelian SU(5) gauge anomaly, has been verified in [29]. For this analysis to be self-contained, let us briefly recap the computation as a warmup. With the appropriate anomaly coefficients (2.9), c(120) = 3, c(52) = 1, the matter from the 7-brane codimension-two loci contributes to the non-abelian anomaly (2.6) 3 2 1 2 ASU(5)|surface,geom = χ(101)|geom + χ(53)|geom + χ(5−2)|geom . (6.15) The chiral matter from the 7-brane bulk transforms in the adjoint with c(224) = 10 and contributes ASU(5)|bulk,geom = 5χ(240)|geom = − 24 W (c1 − W ) (W (W − c1) + c2) , (6.16) where we have used (4.4). In addition, there is another contribution from anti-chiral fermions generated in the 3-7 sector. These modes transform in representation 5q1 and their chiral index is given by minus the point-wise intersection number −[W ] · [C] with [C] = 214 π∗(c4(Xˆ5)) in the absence of flux. With the help of (6.3), their SU(5) anomaly contribution follows as ASU(5)|3−7,geom = 2 χ3−7(5q1)|geom = − 2 W · 24 π∗(c4(Xˆ5)) 1 1 = − 48 W · (144c13 − 264c12W + 12c1c2 + 162c1W 2 − 30W 3) . (6.17) Then the pure non-abelian SU(5) anomalies, in the absence of G4 fluxes, indeed cancel, ASU(5)|3−7,geom + ASU(5)|bulk,geom + ASU(5)|surface,geom = 0 . (6.18) Now we switch gear to check the cancellation of the U( 1 )A gauge anomaly. As we have discussed above, there are two types of charged matter states in the 3-7 sector with 5 1 1 2 1 as it must since the Green-Schwarz counterterms vanish in absence of flux. Finally, let us compute the gravitational anomalies. In absence of flux, gravitational anomaly cancellation is equivalent to (4.38a) over a generic base B4. This equation involves the Chern class c5(Xˆ5) and c4(Xˆ5) of the resolved Calabi-Yau five-fold Xˆ5. With the help of (6.2), we find (6.19) (6.20) (6.21) (6.23) (6.24) Δχ1(Xˆ5) = −66c14 −61c13W −c12c2 + −6c1Δ[C] = 54c14 +66c13W − 81c21W 2 2 235c21W 2 4 2 15c1W 3 23c1c2W 12 − 76c1W 3 3 + 3c2W 2 4 + 17W 4 4 Summing both terms up perfectly matches the r.h.s. of (4.38a), = − 12c14 −5c13W +c12c2 − 73c21W 2 4 23c1c2W 12 (6.22) In summary, we have checked that in this example with the absence of G4 fluxes, all types of anomalies are cancelled by themselves and in agreement with (2.20). 6.3 Flux dependent anomaly relations In the SU(5) × U( 1 )A model defined by (6.1), there only exist two types of gauge invariant 4-form fluxes G4 ∈ Hv2e,2rt(Xˆ5) compatible with the SU(5) × U( 1 )A gauge group [66]. We choose a basis of fluxes as different U( 1 )A charges. With the help of (6.9), their combined contribution to the abelian anomalies is AU( 1 )|3−7,geom = 25 q12 χ3−7(5q1)|geom + 12 q22χ3−7(1q2)|geom = − 48 π∗(c4(Xˆ5))·(−30W +50c1) . 1 This perfectly cancels the anomalies from the 7-7 sector, AU( 1 )|geom = 2 R 1 X qA2(R)dim(R)χ(R)|geom 1 2 = = 0 10χ(101)+20χ(5−2)+45χ(53)+25χ(15)+(5q12 χ3−7(5q1)+q22χ3−7(1q2)) |geom G4A = π∗(F ) · [UA] G4λ = −λ E2 · E4 + (2E1 − E2 + E3 − 2E4) · c1 . Here [UA] is the 2-form class dual to the non-Cartan U( 1 )A divisor UA defined in (6.7), λ is a constant and F ∈ H1,1(B4) is an arbitrary class parametrizing the flux. Both λ and F are to be chosen such that G4 + 21 c2(Xˆ5) ∈ H2(Xˆ5, Z). We now analyze the anomaly relations, including the Green-Schwarz terms, for both of these flux backgrounds in turn. 6.3.1 We begin with the flux background (6.23). The cancellation of non-abelian SU(5) gauge anomalies in the presence of G4A has already been verified in [29] so that we can focus on evaluating (4.18b), or equivalently (2.20b), for the U( 1 )A anomaly. To compute the flux dependent chiral index of the 7-brane various matter states, we need to extract the line bundle LR defined in (4.2) on the 7-brane codimension-two matter loci. Since G4A is simply the gauge flux associated with the non-Cartan factor U( 1 )A, we know that π∗(G4A · SRa) = qA(R)F |CR. It follows that c1(L101) = F |C101 , c1(L53) = 3 F |C53 , c1(L5−2) = −2F |C5−2 , c1(L15) = 5 F |C15 HJEP04(218)7 and therefore The anomaly contribution (4.16) from the 3-7-brane sector is χ(101)|flux = χ(53)|flux = F 2, (3F )2, χ(5−2)|flux = χ(15)|flux = 1 Z 1 Z AU( 1 )|3−7,flux = − 14 F 2 ·B4 π∗([UA] · [UA]) ·B4 π∗([UA] · [UA]) (6.25) (6.26) (6.27) (6.28) (6.30) (6.31) (6.32) (6.33) with π∗([UA] · [UA]) = −DA as in (6.8). Altogether this gives for the l.h.s. of (2.20b) AU( 1 )|flux = AU( 1 )|7−7,flux +AU( 1 )|3−7,flux = (10χ(101)|flux +20χ(5−2)|flux +45χ(53)|flux +25χ(15)|flux)− 41 F 2DA2 (6.29) 1 2 2 = 1 F 2(50c1 −30W )2 . This is to be compared to the r.h.s. of (2.20b) given by the Green-Schwarz counterterms (4.17) 1 1 2 = 1 F 2(50c1 − 30W )2 . 4π ΩαβΘαAΘβB = 2 π∗(G4 · G4) ·B4 π∗([UA] · [UA]) = 2 F ·B4 F ·B4 π∗([UA] · [UA])2 1 Hence (2.20b) and therefore (4.18b) hold. above expressions, the l.h.s. of (4.38b) yields Finally, let us switch to cancellation of the purely gravitational anomaly. Given the − 6c1 · π∗(G4A · G4A) = −6c1 · F · F · (−DA) = −6c1F 2(−50c1 + 30W ) which perfectly matches the r.h.s. of (4.38b) given by 2 (10χ(101)|flux +5χ(53)|flux +5χ(5−2)|flux +χ(15)|flux) = 6c1F 2(50c1 −30W ) . 6.3.2 4λ flux Verifying the anomalies in the presence of flux of the form G4λ is slightly more involved. In the sequel we heavily build on the analysis of [57], where this gauge background is described, in a compactification to four dimensions, as a ‘matter surface flux’. Since the fiber structure is the same, we can extend these results to F-theory compactification on an elliptic 5fold. Since we are now working over a base of complex dimension four, extra technical complications arise in the computation of the chiral index for the 7-brane ammeter, which we will solve in appendix C. π∗(G4λ · SRa), given this time by Key to computing the 7-brane matter chiralities is again the induced line bundle LR = A derivation can be found in section 5 of [57]. By Poincar´e duality, the objects [Yi] describe curve classes on the respective matter codimension-two loci on the base, defined as the intersection loci C53 ∩ C101 = Y2 , C5−2 ∩ C101 = Y1 + Y2 , C5−2 ∩ C53 = Y2 + Y3 . The first Chern classes of the line bundles L101 and L53 can be expressed as the pullback of divisor classes from W to the respective matter loci, c1(L101 ) = c1(L53 ) = λ λ 5 (−3([Y2] + [Y1]) + 7[Y2]) |C101 = 5 (−2[Y2]) |C53 = 5 (−2c1) |C53 . λ λ 5 (6c1 − 5W ) |C101 Hence we can straightforwardly compute the associated chiralities as integrals on B4 χ(C101 )|flux = χ(C53 )|flux = 1 Z λ 2 50 λ 2 50 c21(L101 ) = W · c1 · (6c1 − 5W )2 , c21(L53 ) = W · (3c1 − 2W ) · 4c12 . (6.34) (6.35) (6.36) (6.37) (6.38) (6.39) (6.40) By contrast, c1(L5−2 ) cannot be interpreted as the class of a complete intersection of a base divisor with C5−2 [57]. Each of the classes Yi defines a divisor class on C5−2 , dual to a curve. The technical difficulty is that Y1 and Y2 separately cannot be written as the pullback of a divisor class from the 7-brane divisor W to C5−2 . Rather, on W , the curves Yi are given by intersections Y1 = a1 ∩ a2,1|W , Y2 = a1 ∩ a3,2|W , Y3 = a4,3 ∩ a3,2|W , (6.41) induced by G4λ take the form where the flux dependent piece of the 3-brane class reads χ3−7(5q1 ) = −[C]|flux · W where the class of these Tate coefficients have been listed in (6.4). In appendix C we will discuss how to evaluate the chirality of 5−2 despite this complication, our final result being χ(C5−2 )|flux = c21(L5−2 ) = − 25 c1 · W · 60c12 − 79c1W + 25W 2 . In light of the discussion of section 6.1, the chiral indices for the 3-7 matter states as (6.42) (6.43) (6.44) (6.45) (6.47) (6.48) (6.49) With this the Green-Schwarz terms are 1 4π ΩαβΘαAΘβB = 1 2 π∗(G4λ · [UA]) · π∗(G4λ · [UA]) = 2 1 λ2 c21 · W 2 . W · c1 · (6c1 − 5W ) . result for π∗(G4λ · G4λ) = λ π∗(G4λ · S1a01 can then be read off from (6.37). To derive this latter result, recall from section 4.3 of [57] that up to irrelevant correction terms G4λ for λ = 1 is the class associated with one of the matter fibrations S1a01 . The We are finally in a position to check the cancellation of anomalies in the presence of G4λ, beginning with the pure non-abelian gauge anomaly. Note the G4λ background does not induce any chirality for the 7-brane bulk matter. Together with the above explicit expressions for chiral indices in the 7-brane and the 3-7 sector, one can easily confirm that ASU(5)|flux = χ(C101 )|flux + χ(C53 )|flux + χ(C5−2 )|flux + 1 2 χ3−7(5q1 )|flux = 0 . (6.46) Next we turn to the G4λ dependent part of the abelian gauge anomalies. The combined 1-loop anomaly from the 7-7 and the 3-7 matter evaluates to AU( 1 )A |flux = 1 X dim(R) qA2(R) χ(R)|flux 10χ(101)+20χ(5−2)+45χ(53)+25χ(15)+5q12 χ3−7(5q1 )+q22χ3−7(1q2 ) |flux For the 3-7 contribution we can either use (6.43) with the charge assignments (6.11), or directly evaluate the G4λ dependent component of (4.14). The combined 1-loop anomaly forms the l.h.s. of (2.20b) and must be cancelled by the Green-Schwarz terms (4.17) appearing on the r.h.s. . To compute the latter, we make again use of the interpretation of G4λ as one of the matter fibrations Sa(101). Intersection this with the U( 1 )A generator UA in the fiber reproduces the U( 1 )A charge of 101 and therefore π∗(G4 · [UA]) = λ C101 · W = λ c1 · W . This perfectly cancels the 1-loop anomalies (6.47) and hence verifies the G4λ dependent part of (2.20b) or equivalently (4.18b). As for the cancellation of the gravitational anomalies, with the help of (6.45), the l.h.s. of (4.38b) becomes − 6c1 · π∗(G4λ · G4λ) = − 65 λ2 c21 · W · (6c1 − 5W ), which is again exactly equal to the r.h.s. of (4.38b) 2 (10χ(101) + 5χ(53) + 5χ(5−2) + χ(15))|flux = − 56 λ2 c21 · W · (6c1 − 5W ) . (6.50) (6.51) (7.1) (7.2) 7 Comparison to 6d and 4d anomaly relations In this final section we compare the 2d anomaly relations (4.18) and (4.35) to their analogue in a 6d or 4d F-theory compactification on an elliptic fibration Xˆ3 or Xˆ4, respectively. The cancellation of all gauge and mixed gauge-gravitational anomalies in both these classes of theories is captured by two relations, each valid in H4(Xˆ3) or H4(Xˆ4), of the form X βΓa(R) βΛa(R) βΣa(R)SRa − 3 F(Γ · π∗π∗(FΛ · FΣ)) = 0 X βΛa(R) SRa + 6 FΛ · c1 = 0 . These two homological relations have been shown in [26] to be equivalent to the intersection theoretic identities derived from the requirement of gauge and mixed gauge-gravitational anomaly cancellation in 6d [25] and 4d [27] F-theory vacua. In addition the cancellation of purely gravitational anomalies in 6d F-theory vacua poses an extra constraint on the geometry of Xˆ3, which has no direct counterpart in 4d.18 Interestingly enough, however, apart from this latter point anomaly cancellation in 6d and 4d F-theory vacua is based on the same type of homological relations. While a general proof of these relations from first principles, and without relying on anomaly cancellation, is not yet available in the literature, these relations can be verified in explicit examples.19 The details of such a verification appear to be completely independent of the choice of base of the elliptic fibration, including its dimension [26]. This raises the question if the same type of relations also holds on elliptically fibred Calabi-Yau 5-folds and if they play any role in anomaly cancellation in the associated 2d (0,2) theories. The situation in compactifications to two dimensions looks rather more involved at first sight: as we have shown in section 4, there are two types of independent anomaly relations, (4.18), associated with the cancellation of the gauge anomaly, and another two, (4.35), for the pure gravitational anomaly. We will now see that the flux dependent part of these anomaly relations, (4.18b) and (4.35b), is in fact closely related in form to (7.1) and (7.2). 18This relation is given, for example, as equation (3.8) in [25], and proven generally in [23]. 19On the other hand, [28] proves anomaly cancellation in 4d F-theory vacua by comparison with the dual M-theory. Combined with the above statement this is a physics proof of (7.1) and (7.2) on elliptic Calabi-Yau 4-folds. Consider first relation (4.18b) for the cancellation of the flux dependent part of the 2d gauge anomalies, X βΛa(R)βΣa(R)π∗(G4 · SRa) ·CR π∗(G4 · SRa) = π∗(G4 · G4) ·B4 π∗(FΛ · FΣ) + π∗(G4 · FΣ) ·B4 π∗(G4 · FΛ)+π∗(G4 · FΛ) ·B4 π∗(FΣ · G4) . A priori (7.3) holds for every transversal flux G4, i.e. for every element G4 ∈ H2,2(Xˆ5) satisfying (3.16), including potentially non gauge invariant fluxes. Our first observation is that this relation can be generalized to X βΛa(R)βΣa(R)π∗(G(41) ·SRa)·CR π∗(G(42) ·SRa) valid for all transversal fluxes G( 1 ) and G(42): to see this, insert the ansatz G4 = G(41) + G(2) 4 4 into (7.3). This gives three types of contributions, one depending quadratically on G( 1 ) and on G(42), respectively, and a cross-term involving G( 1 ) and G(42). Since the quadratic 4 4 terms vanish by themselves thanks to (7.3), this is enough to establish the more general relation (7.4). Let us now specialise one of the fluxes appearing in (7.4) to (7.3) (7.4) (7.5) (7.6) (7.7) (7.8) (7.9) (7.10) (7.11) (7.12) and analyze the resulting identity further by repeatedly using the projection formulae G( 1 ) = π∗D · FΓ 4 with D ∈ H1,1(B4) π∗(π∗A ·Xˆ5 B) = A ·B4 π∗(B) π∗(E) ·B4 F = E ·Xˆ5 π∗(F ) for suitable cohomology classes on B4 and Xˆ5. In the sequel, unless specified explicitly, the symbol · denotes the intersection product on Xˆ5. Then with (7.5) the first term on the r.h.s. takes the form π∗(G(41) · G(42)) ·B4 π∗(FΛ · FΣ) = D ·B4 π∗(FΓ · G(42)) ·B4 π∗(FΛ · FΣ) = G(42) · FΓ · π∗(D ·B4 π∗(FΛ · FΣ)) = π∗D · G(42) · FΓ · π∗π∗(FΛ · FΣ) . Similar manipulations for the remaining two other terms on the r.h.s. of (7.3) yield As for the l.h.s., observe that r.h.s. of (7.3) = 3 π∗D · G(42) · F(Γ · π∗π∗(FΛ · FΣ)) . π∗(G(41) · SRa) = π∗(π∗D · FΓ · SRa) = βΓa(R) (D ·B4 CR) . Here we are using that in expressions of this form, the intersection of the divisor FΓ with the matter 3-cycle SRa in the fibre reproduces the charge βΓa of the associated state with respect to U( 1 )Γ. As explained around (4.2), the expression on the right of (7.12) is the first Chern class of the line bundle induced by the specific flux G( 1 ) to which the matter states on CR couple. For the special choice (7.5) this line bundle is the pullback of a line bundle from B4. With this understanding, the intersection product appearing on the l.h.s. 4 can be further simplified as π∗(G(41) · SRa) ·CR π∗(G(42) · SRa) = βΓa(R) π∗D · G(42) · SRa . Altogether we have thus evaluated (7.4), for the special choice (7.5), to  R,a π∗D · G(42) ·  X βΓa(R) βΛa(R) βΣa(R) SRa − 3 F(Γ · π∗π∗(FΛ · FΣ)) = 0 .  (7.13) (7.14) (7.15) HJEP04(218)7 Repeating the same steps for the flux dependent gravitational anomaly relation (4.35b) leads to π∗D · G(42) ·    R,a X βΛa(R) SRa + 6 FΛ · c1 = 0 . The terms in brackets are identical in form with the linear combinations of 4-form classes which are guaranteed to vanish on an elliptically fibered Calabi-Yau 3-fold and 4-fold by anomaly cancellation according to (7.1) and (7.2). We conclude that if the relations (7.1) and (7.2) hold also within H4(Xˆ5), as suggested by the results of [26], this implies cancellation of the flux dependent part of the anomalies in 2d F-theory vacua for the special choice of flux (7.5). For more general fluxes, however, the constraints imposed on anomaly cancellation on a Calabi-Yau 5-fold seem to be stronger. In particular, a direct comparison with (7.1) and (7.2) is made difficult by the fact that (4.18b) and (4.35b) are quadratic in fluxes and a priori involve the intersection product on the matter loci CR, not on B4. For general G4 backgrounds, this makes a difference, as we have seen in section 6.3. Furthermore, anomaly cancellation in 2d predicts the flux independent relations (4.18a) and (4.35a). Condition (4.35a) can be viewed as analogous, though very different in form, to the geometric condition on cancellation of the purely gravitational anomalies in 6d referred to in footnote 18. It would be very interesting to investigate if a deconstruction of the topological invariants appearing in (4.18a) and (4.35a), similar to the procedure applied for the Euler characteristic on Calabi-Yau 3-folds in [23, 24], can lead to a geometric proof of these identities. 8 Conclusions and outlook In this work we have provided closed expressions for the gravitational and gauge anomalies in 2d N = (0, 2) compactifications of F-theory on elliptically fibered Calabi-Yau 5-folds. In particular, we have derived the Green-Schwarz counterterms for the cancellation of abelian gauge anomalies. The Green-Schwarz mechanism operates in a manner very similar to its 6d N = ( 1, 0 ) cousin: dimensional reduction of the self-dual Type IIB 4-form results in real chiral scalar fields whose axionic shift symmetry is gauged and whose Chern-Simons type couplings hence become anomalous. We have uplifted our results for the gauging and the couplings to an expression valid in the most general context of F-theory on elliptically fibered Calabi-Yau 5-folds. Anomaly cancellation in the 2d (0, 2) supergravity is then equivalent to (4.18) for the gauge and (4.35) for the gravitational part. Each equation splits into a purely geometric and a flux dependent identity. These must hold separately on every elliptic Calabi-Yau 5-fold and for every consistent background of G4 fluxes. We have verified this explicitly in a family of fibrations and for all vertical gauge fluxes thereon. It is instructive to compare these 2d anomaly cancellation conditions to their analogue in 6d and 4d F-theory vacua in the form put forward in [25] and [27], respectively. The structure of anomalies as such becomes more and more constraining in higher-dimensional field theories. At the same time the engineering of the quantum field theory in terms of the internal geometry becomes more intricate as the dimension of the compactification space increases, and hence the number of large spacetime dimensions decreases. Correspondingly, the topological identities governing anomaly cancellation on elliptic 5-folds contain considerably more structure compared to their analogues in 4d and 6d F-theory compactifications. For once, the anomaly relations in 6d N = ( 1, 0 ) F-theory vacua are only sensitive to the topology of the elliptic fibration, while in 4d N = 1 theories they are linearly dependent on a gauge flux. In 2d N = (0, 2) F-theories, both a purely topological and a flux dependent contribution arises. The latter is, in fact, quadratic in the gauge background. Despite differences in structure, the 6d and 4d gauge anomaly relations of [25] and [27] can be reduced to one single identity [26], valid in the cohomology ring H2,2(Xˆn) of an elliptically fibered Calabi-Yau n-fold, with n = 3 and 4, respectively. The same is true for their mixed gauge-gravitational counterparts. One motivation for the present work was to investigate these universal identities, (7.1) and (7.2), with respect to anomaly cancellation in 2d F-theories. The flux-dependent parts of (4.18) and (4.35) exhibit striking similarities to (7.1) and (7.2). We have shown that if the 6d and 4d universal relations hold also in the cohomology ring of an elliptic 5-fold, as suggested by the examples studied in [26], they imply the flux dependent anomaly relations at least for the subset of gauge backgrounds associated with massless U( 1 ) gauge groups. It would be very interesting to study further if also the converse is true, i.e. if the 2d relations allow us to establish a relation in the cohomlogy ring of elliptic 5-folds governing the 4d and 6d anomalies as well. The flux-independent anomaly relations, on the other hand, seem not to be related in a straightforward manner to the structure of anomalies in higher dimensions. In fact, already in 6d N = ( 1, 0 ) Ftheory vacua, cancellation of the purely gravitational anomalies implies another topological identity with no counterpart in 4d. This relation has been proven for generic Weierstrass model in [23] using a deconstruction of the Euler characteristic of elliptic 3-folds. It would be worthwhile exploring if a similar proof is possible on Calabi-Yau 5-folds. The structure of anomalies in 6d and 4d F-theory vacua is closely related to the ChernSimons terms in the dual M-theory in five [52, 69, 70] and three dimensions [27, 71, 72]. In [28] this reasoning has lead to a proof of anomaly cancellation in 4d N = 1 vacua obtained as F-theory on an elliptic Calabi-Yau 4-fold. It would be very interesting to extend such reasoning also to the 2d case. The Chern-Simons terms in the dual 1d N = 2 SuperQuantum-Mechanics have been analyzed in [29] and expressed geometrically in terms of data of the Calabi-Yau 5-fold. As expected, the similarities between the resulting identities such as (10.8) in [29] and the 2d anomaly conditions are striking. At a more technical level, the expressions for the anomalies presented in this work are valid under the assumption that the loci on the base hosting massless matter are smooth. Quite frequently, this assumption is violated, and an application of the usual index theorems requires a normalization of the singular loci [29]. We leave it for future investigations to establish the anomaly relations in such more general situations. Likewise, in the presence of Q-factorial terminal singularities in the fiber the precise counting of uncharged massless states in terms of topological invariants will change. In 6d F-theory vacua, this leads to a modification of the condition for cancellation of the gravitational anomaly [73, 74], and similar effects are expected to play a role in 2d models. Our focus in this work has been on the implications of anomaly cancellation rather than on the structure of the effective 2d N = (0, 2) supergravity per se. The axionic gaugings induced by the flux background, as derived in this context, give rise to a K¨ahler moduli dependent D-term, as noted already in [29]. What remains to be clarified is a careful definition of the chiral variables in the supergravity sector and a comparison of the Green-Schwarz action to the superspace formulation put forward in 2d (0,2) gauge theories in [47–49]. This will also determine the correct normalization of the D-term. At the level of the supersymmetry conditions induced by the flux, we have made, in passing, an interesting observation: extrapolating from the situation on Calabi-Yau 4-folds we expect the existence of G4 backgrounds which are not automatically of (2, 2) Hodge type and would hence break supersymmetry [54]. More precisely, whenever H2,2(Xˆ5) contains (2, 2) forms which are not products of ( 1, 1 ) forms, it is expected that the Hodge type of a 4-form varies over the complex structure moduli space. This would constrain some of the complex structure of the 5-fold [54]. This makes it tempting to speculate that the contribution of the supergravity sector to the purely gravitational anomaly should change compared to a background without flux. At the same time, the flux dependent contribution to the D3-brane tadpole modifies the class of the D3-branes in the background and therefore also the anomaly contribution from the sector of 3-7 string modes. For consistency, both effects have to cancel each other, which is in principle possible due to the opposite chirality of the fields involved. In this sense the net effect of complex moduli stabilization would be topological, in stark contrast to the situation in 4d N = 1 compactifications. More work on elliptically fibered 5-folds is needed to flesh out the details behind this phenomenon. Acknowledgments We thank Seung-Joo Lee and Diego Regalado for important discussions, and Martin Bies, Antonella Grassi, Craig Lawrie, Christoph Mayrhofer and Sakura Sch¨afer-Nameki in addition for collaboration on related topics. The work of T.W. and F.X. is partially supported by DFG Transregio TR33 ‘The Dark Universe’ and by DFG under GK ‘Particle Physics Beyond the Standard Model’. – 39 – Conventions In this appendix we collect our conventions for the technical computations in this paper. A.1 Local anomaly Our conventions for the anomaly polynomial mostly follow [75]. Consider a quantum field theory in D = 2r-dimensional Minkowski space M2r with quantum effective action S[A], where Aα is the connection associated with a local symmetry of S with gauge parameter ǫα. The anomaly Aα is defined as the gauge variation HJEP04(218)7 where the 2r-form I2(1r)(ǫ) is related to (2r + 2)-form I2r+2 via the Stora-Zumino descent relations I2r+2 = dI2r+1, δǫI2r+1 = dI2(1r)(ǫ) . In our sign conventions, the anomaly polynomial I2r+2 of a complex chiral Weyl fermion in representation R takes the form Is=1/2(R)|2r+2 = −trRe−F Aˆ(T)|2r+2 . Here F is the hermitian field strength associated with the gauge potential A and T denotes the tangent bundle to spacetime. Its curvature 2-form R is the curvature associated with the spin connection. Furthermore, in 2r = 4k + 2 dimensions, a self-dual r-tensor contributes to the gravitational anomalies with 1 Is.d.|2r+2 = − 8 L(T)|2r+2 . The A-roof genus and the Hirzebruch L-genus above can be expressed as 1 1 Aˆ(T) = 1 − 24 p1(T) + . . . = 1 − 24 (c12(T) − 2c2(T)) + . . . L(T) = 1 + 3 p1(T) + . . . = 1 + 3 (c12(T) − 2c2(T)) + . . . . 1 1 We will oftentimes write the first Pontrjagin class of the tangent bundle as Note that we have included an overall minus sign in (A.4) and (A.5) compared to the conventions used in [75]. The reason is that in the quantum field theory we are analyzing the chiral fermion fields arise as the zero-modes of strings on the worldvolume of 7- and 3-branes. The anomalies induced by these modes on the brane worldvolume must be 1 p1(T) = − 2 trR ∧ R . (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) SIIB = 2π  Z d10x e−2φ(√−gR + 4∂M φ ∂M φ) − 2 1 Z e−2φH3 ∧ ∗H3 − 4 p=0 1 X4 Z F2p+1 ∧ ∗F2p+1 − 2 C4 ∧ H3 ∧ F3 . Here we are working in conventions where the string length ℓs = 2π√α′ ≡ 1 and the field strengths are defined as 1 H3 = dB2, F1 = dC0, F3 = dC2 − C0 dB2, F5 = dC4 − 2 C2 ∧ dB2 + 2 B2 ∧ dC2, together with the duality relations F9 = ∗F1, F7 = − ∗ F3, F5 = ∗F5, which hold at the level of equations of motion. The Chern-Simons action for the D7-branes and the O7-plane takes the form 1 2p (A.8) (A.9) (A.10) cancelled via an anomaly inflow mechanism by the anomalous Chern-Simons action of the branes. This relates the sign of the 1-loop anomalies to the sign conventions used for the Chern-Simons brane actions. As we will discuss below, the sign of the 7-brane ChernSimons action is fixed as in (A.10) by the convention that the 7-brane couples magnetically to the axio-dilaton, which is usually defined in F-theory as τ = C0 + ie−φ (rather than −C0 + ieφ). The sign chosen in (A.10) conforms with this convention. In order for the anomalies of chiral fermions in the worldvolume of a D7-brane to be cancelled by anomaly inflow, we must then adopt the convention (A.4). A.2 Type IIB 10D supergravity and brane Chern-Simons actions The bosonic part of the 10d Type IIB supergravity pseudo-action in its democratic form is given by SD7 = − 2 2π Z D7 Tr eiF X C2p SO7 = 16π Z 2 X C2p O7 2p s L( 14 TO7) . L( 14 NO7) s Aˆ(TD7) Aˆ(ND7) Since we are working in the democratic formulation, where each RR gauge potential is accompanied by its magnetic dual, the Chern-Simons action has to include a factor of 21 [61], which we are making manifest in (A.10). This factor is crucial in order to obtain the correctly normalized anomaly inflow terms, and, as we find in the main text, also to reproduce the correctly normalised Green-Schwarz counterterms. As stressed already, the minus sign in front of the Chern-Simons action of the D7-branes ensures that in the above conventions for the supergravity fields the D7-brane couples magnetically to the axio-dilaton τ = C0 + ie−φ. Note furthermore that we are writing the brane action in terms of Tr = λ1 trfund, where the Dynkin index λ is given in table 1. Finally, TD7 and ND7 denote the tangent and normal space to the 7-brane along D7, and similarly for the O7-plane. The Chern-Simons action for a D3-brane carries a relative sign compared to the 7-brane action, SD3 = 2π Z 2 D3 2p s Aˆ(TD3) Aˆ(ND3) . The gauge invariant field strength F above is defined as F = i(F + 2πφ∗B2I) . Compared to expressions oftentimes used in the literature we have absorbed a factor of 21π in the definition of F . The NS-NS two-form field B2 is pulled back to the brane via φ . We will always set B2 = 0 in this article, but one should bear in mind that it appears in ∗ various consistency conditions as detailed e.g. in [76]. We will sometimes decompose where the minus sign is due to the worldsheet parity action. Fi → Fi′ = −σ∗Fi , (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) (A.17) so that F denotes the gauge invariant field strength of the gauge field in non-compact flat space while F¯ stands for the internal flux background. Note that it is the hermitian field strength F which appears in the anomaly polynomial (A.4). Finally, the curvature terms in the above Chern-Simons actions enjoy the expansion s Aˆ(TD7) = 1 + Here we have used the definitions (A.6) together with the fact that c1(TD) = −c1(ND) by adjunction on the Calabi-Yau space on which we compactify the Type IIB theory A.3 Type IIB orientifold compactification with 7-branes In a Type IIB orientifold compactification on a Calabi-Yau 4-fold X4, the orientifold projection Ω(−1)FLσ acts as in the more familiar case of compactification on a 3-fold, as summarized e.g. in [77]. In particular, the p-form fields transform under the combined action of worldsheet parity Ω and left-moving femrion number (−1)FL as Ω(−1)FL : (C0, B2, C2, C4, C6) → (C0, −B2, −C2, C4, −C6) . The holomorphic involution σ acts only on the internal space X4 such that the K¨ahler form J and the holomorphic top-form Ω4,0 transform as σ : J → J , Ω4,0 → −Ω4,0 . The cohomology groups H(p,q)(X4) split into two eigenspaces H(p,q)(X4) = H+(p,q)(X4) L H−(p,q)(X4) under the action of σ. In performing the dimensional reduction, the orientifold even and odd form fields are expanded along a basis of the invariant and anti-invariant cohomology groups. on the orientifold image brane Under the orientifold action the field strength on each brane is mapped to its cousin Anomalies and Green-Schwarz term in Type IIB orientifolds In this appendix we verify our intermediate results (5.20) for the Green-Schwarz terms in Type IIB orientifolds. Together with our confirmation of the final F-theoretic expressions in the explicit example of section 6, this also supports our rules explained in 5.4 for the correct uplift to F-theory. The setup we analyze is identical to the one in appendix C.2 of [29], which we now briefly summarize. Consider a Type IIB orientifold on a general Calabi-Yau 4-fold X4 with gauge group (SU(n) × U( 1 )a) × U( 1 )b. The brane configuration consists of n 7-branes wrapping a divisor W and one extra D7-brane along the divisor V , each accompanied by their orientifold images wrapped along W ′ and V ′, respectively. We assume that all brane divisors are smooth. In order to cancel the D7-tadpole, it is required that The D3-tadpole cancellation condition fully determines the spacetime-filling D3-brane system wrapped along a total curve class [C] plus orientifold image brane [C′] as None of these assumptions is essential, but dropping them would require some modifications of the anomaly computation. We are now in a position to determine the contribution to the U( 1 )a − U( 1 )a and the U( 1 )b − U( 1 )b anomaly due to the chiral matter states. Since our primary interest here is 20Otherwise, a D5-bane tadpole cancellation must be imposed on the gauge background. [V ] = [V ′], to prevent the gauge potentials associated with U( 1 )a and U( 1 )b from acquiring a mass, in absence of flux, via the geometric Stu¨ckelberg mechanism.20 We simplify the calculation of the U( 1 )a anomaly contribution further by assuming This implies that there exists no intersection locus of W and W plane, which would carry matter in the symmetric representation of SU(n). This would lead to extra complications in the computation of the chiral spectrum, which we avoid by requiring (B.5). For the same reason we make the simplifying assumption that ′ away from the O7[V ] · [V ′] = [V ] · [O7] . n 24 n 24 [W ] · c2(W ) + 1 1 with U( 1 )a and U( 1 )b, respectively. For simplicity, we require Here LW and LV denote line bundles on W and V whose structure groups are identified [O7] · c2(O7) + n ch2(LW ) · [W ] + ch2(LV ) · [V ] [O7] · c2(O7) + n ch2(L′W ) · [W ′] + ch2(L′V ) · [V ′] . (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) U( 1 )2a anomaly (c1(LV ) = 0) = − 12 R ch2(L)qa2dim(R) − 21 [W ] · [V ] · 12 c21(LW ) n − 21 [W ] · [V ′] · 12 c21(LW ) n − 12 [W ′] · [V ′] · 21 c12(LW ) n − 21 [W ′] · [V ] · 12 c21(LW ) n = − 12 R ch2(L)qb2dim(R) − 21 [W ] · [V ] · 21 c12(LV ) n − 21 [W ] · [V ′] · 21 c12(LV ) n − 21 [W ′] · [V ′] · 12 c21(LV ) n − 12 [W ′] · [V ] · 21 c12(LW ) n 0 12 n(n − 1)(2,0) −2 × 21 [W ] · [W ′] · 12 c21(L2w) × 22 × 21 n(n − 1) to check the Green-Schwarz counterterm (5.20) and its normalization relative to the 1-loop anomalies, it suffices to focus on the flux-dependent contribution of these states. The chiral spectrum from the D7-D7 brane sector and the flux dependent part of its contribution to the anomalies are listed in table 2, and similarly for the 3-7 sector in table 3. Note that we have omitted matter in the adjoint representation, which is not charged under U( 1 )a and U( 1 )b. We adapt the convention (A.4) for the anomaly polynomial so that there is overall factor of −1 in front of every term in table 2, while in table 3 we have taken into account the antichiral nature of the 3-7 matter, which hence contributes with a +1. Merely to save some writing, we have assumed, in the column containing the U( 1 )2a anomalies, that LV = 0, and similarly in the column containing the U( 1 )b2 anomalies that LW = 0. Furthermore, with our assumption (B.5) all matter on W ∩W ′ transforms in the anti-symmetric representation of U(n), while due to (B.6) the states on V ∩ V ′ are all projected out (as there exists no anti-symmetric representation of U( 1 )b). The total anomaly from the 7-7 sector is then obtained by summing over all states in table 2 and dividing the final result by two. The division by two is due to the orientifold quotient. Table 2 contains sectors in this upstairs picture which are pairwise identified under the involution. To offset for this overall factor of 21 in the invariant sector W ∩ W ′ we are including a factor of 2 for these states in table 2. From (B.2) we read off the flux-dependent term part of the 3-brane class [C], 1 [C]|flux = The 7-7 and 3-7 sector contribution to the U( 1 )a − U( 1 )a anomaly is hence, for with Aa = − 2 1 1 1 4 × 4 1 (B.7) (B.8) (B.9) c1(LV ) = 0 for simplicity, 4 I1−loop U( 1 )2a = F 2d a ∧ Fa2d Aa c21(LW ) n [W ] · [V ] + 4 × 4 2 n(n − 1) c12(L2W ) [W ] · 4[O7] 1 − 4 × 2 n[W ] · 2 nc12(LW ) · [W ] = − 2 [W ] · c12(LW ) · n[V ] + 4n2 [O7] − 4n [O7] − n2 [W ] = −n2 [W ]2 · c12(LW ) . V ′ T C V ′ T C n¯( −1,1 ) n¯(−1,−1) 1(0,−1) 1(0,−1) 1( 0,1 ) 1( 0,1 ) = + 12 W · Cq2dim(R) + 12 [W ] · [C] × 12 × n + 12 [W ] · [C] × 12 × n + 12 [W ] · [C] × 12 × n + 12 [W ] · [C] × 12 × n 0 0 0 0 = + 12 V · Cq2dim(R) 0 0 0 0 + 12 [V ] · [C] × 12 × 1 + 12 [V ] · [C′] × 12 × 1 + 12 [V ′] · [C] × 12 × 1 + 12 [V ′] · [C′] × 12 × 1 In the last line we have used that n[W ] + [V ] = 4[O7], (B.10) This 1-loop anomaly is precisely cancelled by the Green-Schwarz term contribution (5.20) because the trace over the diagonal U( 1 )a ⊂ U(n) evaluates to TrF¯a = trIn F¯a and hence IGS 4 U( 1 )2a = 1 4 TraTraFa2d ∧ Fa2d 4 F¯a · [W ] = F 2d a ∧ Fa2d n2c1(LW ) · [W ] . (B.11) Similarly, the 1-loop U( 1 )b2 anomaly induced by the chiral matter, for c1(LW ) = 0, Ab = − 2 1 1 4 × 4 c21(LV ) n [W ] · [V ] − 4 × 2 1 = −[V ]2 · c12(LV ) is correctly cancelled by the GS term contribution (5.20). C Chirality computation for matter surface flux In this appendix we compute flux dependent part of the chiral index (6.42) induced for states in representation 5−2 by the gauge background G4λ in the SU(5) × U( 1 )A model of section 6.3. The matter surface C5−2 ⊂ W ⊂ B4 is cut out by the locus P ∩ W on B4 with P := {a1a4,3 − a2,1a3,2 = 0} . The classes in which the Tate polynomials ai,j take their value are listed in (6.4). As discussed in section 6.3, our task amounts to computing c21(L5−2 ) = (−2[Y2] + 3[Y1])2 , (B.12) (C.1) (C.2) λ2 Z 50 C5−2 where [Y1] and [Y2] denote the classes of eponymous curves on the surface C5−2 ⊂ W ⊂ B4. These curves cannot be expressed as the complete intersection of the surface C5−2 with a divisor from B4, but are defined by the complete intersection of 7-brane divisor W with two divisors on B4. Concretely, from (6.41) we read off Yi = Ai ∩ Bi with Z Y1 [Y1] = 2c1 ·W ·(2c1 −W )·(W −c1), [Y2] = c1 ·W ·(3c1 −2W )·(W −c1) , (C.13) in terms of the intersection product on B4, where we are using (C.4) and (6.4). of Yi on C5−2 are computed via We hence need to evaluate the intersections RC5−2 [Yi]·[Yj ] for i = 1, 2. The self-intersections Z C5−2 Z Yi Z Yi where the first Chern class of the normal bundle NYi⊂C5−2 is computed via the normal bundle short exact sequence 0 → NYi⊂C5−2 → NYi⊂W → NC5−2 ⊂W → 0 . The normal bundles are given as NYi⊂W = O(Ai) ⊕ O(Bi) NC5−2 ⊂W = O(P ) , where O(Ai) defines a line bundle of first Chern class [Ai]|C5−2 on C5−2 and O(P ) is a line bundle on W of first Chern class [P ]|W . This gives Collecting the terms of first order yields c(NYi⊂C5−2 ) = C5−2 c(NYi⊂W ) c(NC5−2 ⊂W ) C5−2 The integral (C.5) can now be expressed as an integral directly on W , Z Yi c1(NYi⊂C5−2 ) = ([Ai] ·W [Bi]) ·W (−[P ]|W + [Ai] + [Bi]) . Since all involved classes are defined on or can be extended to B4, this evaluates to Z Y2 (C.3) (C.5) (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) In particular, with [a1] = c1, Z C5−2 Z C5−2 Z C5−2 Z C5−2 ([Y1] + [Y2])2 = [a1] · [a1] = c12 · W · (5c1 − 3W ) , where the last intersection is taken on B4. The idea is then to express the cross-term as ([Y1] + [Y2])2 − [Y1]2 − [Y2]2 = c1 · W · (6c12 − 7c1W + 2W 2) . The remaining task is to compute the cross-term RC5−2 [Y1] · [Y2]. We note that even though the curves Yi cannot individually be written as the complete intersection of a divisor with the divisor P defining C5−2 , the combination Y1 + Y2 is of this simpler form: indeed Since C101 = {a1 = 0} we can then write on C5−2 for Y1 + Y2 Y1 + Y2 = C5−2 ∩ C101 . } ⊂ C5−2 . Plugging everything into (C.2) leads to the final result (6.42). Open Access. 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Timo Weigand, Fengjun Xu. The Green-Schwarz mechanism and geometric anomaly relations in 2d (0,2) F-theory vacua, Journal of High Energy Physics, 2018, 107, DOI: 10.1007/JHEP04(2018)107