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The anomalies of 10d (1, 0) supergravity theories can be These anomalies must be cancelled by the anomalies coming cancelled by the Green-Schwarz mechanism [12]. The from the worldsheet degrees of freedom living on the strings. anomaly cancellation allows only 4 choices for gauge groups: A half-BPS string coupled to the 10d supergravity gives rise to an = (0, 8) superconformal field theory (SCFT) at low SO(32) , E E , E U(1)248 ,U(1)496 . (1) N 8 × 8 8 × energy. To find the chirality of the supersymmetry one uses See Appendix A for details. the condition that we start with a chiral theory in 10d, and for The 10d supergravity theories with the former two gauge a BPS string we preserve half the supersymmetries, leading to a definite chirality for the supercurrents on the worldsheet. groups SO(32) and E8 E8 are realized as low energy limits of the type I and the heterotic× string theories. On the other Supersymmetry on the BPS string also shows that the current hand, it was argued in [7] that two other theories with abelian for the group has opposite chirality to that of supersymmetry. gauge factors are not consistent at the quantum level due to We choose conventions so that the supersymmetry current is anomalies in the context of abelian gauge invariance. right-moving and the current for the group is left-moving. We will now propose a novel stringent condition ruling out To cancel the anomaly inflow from the bulk gravity theory, the latter two theories with abelian gauge factors by using 2d the gravitational and the gauge anomalies of the SCFT on a strings coupled to these 10d theories. When 2d strings couple string must be to the 10d supergravity, the worldsheet degrees of freedom in I = Iinflow general develop local gravitational and gauge anomalies. The 4 − 4 worldsheet anomalies can be cancelled by the anomaly inflow 1 1 = Q p (T ) c (SO(8)) + TrF 2 . (6) from the 10d bulk theory toward the 2d strings. In the fol- 2 1 2 − 2 4 i " i # lowing, we will derive the anomaly inflow for 2d strings in X the 10d supergravity by employing the method developed in Here we used the decomposition [13–15]. We will then check if the anomaly inflow can be can- celled by local anomalies in a unitary worldsheet theory, using 1 trR2 = p (T )+ c (SO(8)) , (7) the IR properties of the strings and the resulting effective CFT −2 1 2 2 on them. When this cancellation cannot occur, the 10d su- pergravity becomes an inconsistent theory hosting non-trivial where p1(T2) is the first Pontryagin class of the two-manifold anomalies on the 2d strings. 2 and c2(SO(8)) is the second Chern class of the SO(8) R-symmetryM bundle of the worldsheet theory. Strings are sources for the 2-form tensor field B2, which by assumption of completeness of the spectrum in a gravitational Note that the above result involves the contribution from the theory should exist. Moreover it is easy to show that they are center of mass degrees of freedom. The center of mass modes form a free (0, 8) multiplet (X , λI ) with µ, I = 1, , 8 stable due to the BPS condition. A string with tensor charge µ + ··· Q adds to the 10d action the tensor coupling where Xµ parametrize the motion of strings along 8 trans- I verse directions and λ+ is the right-moving fermion in the 8 SO(8) spinor representation. From this, we read the anomaly str a a S = Q B2 δ(x )dx = Q B. (2) polynomial for the center of mass modes M ∧ M 10 a=1 2 Z Y Z com 1 The 2-form B transforms under the local gauge and the local I = p1(T2) c2(SO(8)) . (8) 4 −6 − Lorentz symmetry [16, 17] (with parameters Λi and Θ respec- tively) as So, the anomaly polynomial of the interacting sector in the 2d worldsheet SCFT is given by I′ = I Icom. 1 4 4 − 4 B B Tr(Λ F ) + tr(ΘR) , (3) Let us now focus on the 2d SCFT on a single string, i.e. 2 → 2 − 4 i i i Q =1. The anomaly polynomial of this CFT is X where Fi denotes the gauge field strengths and R denotes the 2 1 I′ = I Icom = p (T )+ TrF 2 . (9) curvature 2-form of the 10d spacetime. 4 4 − 4 3 1 2 4 i str i The string action S is not invariant under these local X transformations The left-moving and the right-moving central charges cL,cR and the level ki’s of gauge algebras in the worldsheet SCFT str 1 ′ δΛ,ΘS = Q Tr(ΛiFi) + tr(ΘR) . (4) can be computed from the anomaly polynomial I4. The rel- M −4 2 " i # ative central charge c c is the coefficient of the grav- Z X R L itational anomaly 1 p−(T ) and the right-moving central As a consequence the introduction of 2d strings induces an 24 1 2 charge is c = 3k− where k is the ’t Hooft anomaly co- anomaly inflow along the worldsheet of the strings. The R R R efficient of the superconformal R-symmetry current at the IR anomaly inflow is characterized by the 4-form anomaly poly- fixed point. One finds that ’t Hooft anomalies for the SO(8) nomial which in this case is given by ′ R-symmetry in I4 vanish. The level ki is the coefficient of the gauge anomaly term 1 TrF 2. We then compute 1 4 i Iinflow = Q TrF 2 + trR2 . (5) 4 −4 i i ! cL = 16 , cR =0 , ki =1 . (10) X 3

The central charges are constrained by unitarity conditions G SU(N) SO(N) Sp(N) G2 F4 E6 E7 E8 on 2d CFTs, which can be viewed as IR degrees of freedom λ 1 2 1 2 6 6 12 60 on the strings. The central charge realizing the level-k Kac- Moody algebra of group G is (see, e.g., [18]): TABLE I. Group theory factors

k dimG cG = · , (11) factorizes as k + h∨ − 1 I1 loop = Ω XαXβ , where dimG is the dimension and h∨ is the dual Coxeter num- 8 2 αβ 4 4 ber of group G respectively. The central charge for U(1) cur- α 1 α 2 1 α 2 2 X4 = a trR + bi trFi , (13) rent algebra is cU(1) =1 for any kU(1). For (0, 8) SCFTs, the 2 4 λ i i current algebra for group G is on the left-moving sector. This X tells us that where Ωαβ is a symmetric bilinear form of T +1 tensors with α α R1,T a signature (1,T ), and a and bi are vectors in . ki dimGi The conditions for the factorization can be summarized as ci = · ∨ cL , (12) ki + h ≤ follows: i i i X X H V = 273 29T, a a =9 T, for a unitary CFT on a string. − − · − 0= Bi ni Bi , We find that the 10d supergravity theories with abelian adj − R R R gauge groups contains non-unitary strings violating this in- X equality. The U(1)496 and U(1)248 abelian factors in these λ a b = i Ai ni Ai , theories give rise to too many left-moving modes for the cur- · i 6 adj − R R R ! rent algebras in the worldsheet CFT, and the central charge X 2 of the current algebra exceeds cL = 16, namely i ci > λi i i i bi bi = nRC Cadj , cL. Therefore we conclude that 10d supergravity theories · 3 R − 496 248 P R ! with U(1) and E8 U(1) gauge groups are inconsis- X tent when coupled to 2d× strings, and thus they belong to the b b =2λ λ nij Ai Aj (i = j) , (14) i · j i j R,S R S 6 swampland. On the other hand, the central charges on a sin- R S X, gle string in the 10d supergravities with SO(32) or E8 E8 × where Ωαβ is used for the inner product of two vectors, like gauge group saturate the bound (12) as i ci = cL = 16, so α β v w = Ωαβv w . Here V and H are the number of vec- the string can consistently couple to these 10d theories. · i P tor and hyper multiplets, and nR denotes the number of hy- permultiplets in the representation R for gauge group Gi and i i i AR,BR, CR are group-theory factors for each representation III. STRINGS IN 6D N = (1, 0) SUPERGRAVITY defined as follows: 2 2 4 4 2 2 trRF = ARtrF , trRF = BRtrF + CR(trF ) . We now turn to six-dimensional supergravity theories pre- (15) serving 8 supersymmetries. There are four kinds of massless supermultiplets appearing in such theories: a gravity multi- When these conditions are satisfied, the perturbative anomaly plet, tensor multiplets, vector multiplets, and hypermultiplets. factorizes and it can be cancelled by adding to the action the 6d supergravity theories may have anomalies, which are char- Green-Schwarz term acterized by an 8-form anomaly polynomial I8, from the chi- S = Ω Bα Xβ . (16) ral fields in these multiplets. GS αβ 2 ∧ 4 Let us consider a gravity theory coupled to T tensor mul- Z This term induces tree-level anomalies of the form IGS = tiplets and vector multiplets of the gauge group G = Gi, 8 i 1 α β as well as hypermultiplets transforming in representation R 2 ΩαβX4 X4 that exactly cancels the factorized anomaly − − of the gauge group. The chiral fields such as the self-dualQ I1 loop. So, 6d supergravity theories satisfying the condi- ± 8 and anti-self dual two-forms Bµν , a gravitino, and other chi- tions in equation (14) have no apparent quantum anomalies ral fermions in this theory contribute to the anomalies for and seem to be consistent. Extensive lists of would-be con- the gauge and Lorentz transformations. Such anomalies can sistent 6d supergravity theories are given in various literature exactly be computed by evaluating 1-loop box diagrams for [8, 20–27] (see [4] for a review). the chiral fields with four external gravitational and gauge sources. Consistent quantum supergravity theories must be free of such anomalies. Thus non-vanishing 1-loop anomalies A. Central charges of 2d (0, 4) SCFTs on strings must be cancelled for the 6d theories that are consistent at the quantum level, which leads to quite stringent constraints. Let us now consider 2d strings in 6d supergravity theory The 1-loop anomalies can be cancelled by the Green- without manifest anomalies. We will discuss additional con- Schwarz-Sagnotti mechanism [19] if the anomaly polynomial ditions from the 6d/2d coupled system. Strings are sources 4

α for the two-form fields B2 and thus should exist by assump- the SU(2) R-symmetry in the (0, 4) superconformal algebra. tion of completeness of the spectrum in a gravitational theory. This SU(2)I is descended from the SU(2) R-symmetry of the We shall consider BPS strings preserving half supersymme- local 6d SCFTs or LSTs, but it is broken in the full supergrav- tries. The worldsheet theory on those strings gives rise to 2d ity theory. The free theory with the center of mass degrees (0, 4) SCFT at low energy. As discussed in the 10d cases, the of freedom we discussed above also has the same accidental degrees of freedom living on the string worldsheet can have SU(2)I symmetry. non-zero anomalies and these anomalies must be cancelled It is therefore crucial to identify the right R-symmetry in through the anomaly inflow mechanism. The anomaly inflow the IR SCFTs. Only after this we can extract the correct cen- in 6d SCFTs was studied in [28, 29] (See also [30] forgeneral- tral charges in the IR SCFTs. From now on we will focus ization to 6d supergravities from F-theory compactification). on the strings in the 6d supergravity theory that give rise to See Appendix B for a brief review on the anomaly inflow to a single interacting SCFT at low energy without the acciden- 2d strings in 6d SCFTs and 6d supergravity theories. tal SU(2)I symmetry. The IR SCFTs on such supergravity The 2d SCFT on strings with charge Qα in the 6d super- strings (not strings in local 6d SCFTs or LSTs) have the (0, 4) gravity theory has the anomaly polynomial of this form superconformal algebra with an SU(2)R R-symmetry. The conditions for this type of strings will be given below. The α 1 α 2 1 α 2 1 β right-moving central charge cR of these SCFTs can then be I4 =ΩαβQ a trR + bi TrFi + Q χ4(N4) 2 4 2 read off from the anomaly coefficient of the SU(2)R symme- i ! X try. For a non-degenerate 2d SCFT on a supergravity string, Q a 1 = · p (T )+ Q b TrF 2 the central charges cL,cR are given by − 4 1 2 4 · i i i X cL =3Q Q 9Q a +2 , cR =3Q Q 3Q a . (20) Q Q Q a Q Q + Q a · − · · − · · − · c2(R)+ · · c2(l) . (17) − 2 2 The central charges ki and kl for the bulk gauge symmetries Gi and SU(2)l can also be extracted from the anomaly poly- 2 In this computation, we used the decomposition trR = nomial. We find 1 p (T )+ c (l)+ c (R). − 2 1 2 2 2 1 This result involves the contribution from the center of mass ki = Q bi , kl = (Q Q + Q a + 2) . (21) degrees of freedom which decouples in the IR SCFT. The · 2 · · center of mass modes consist of 4 bosons common to left- A large class of 6d (1, 0) supergravity theories can be en- and right-movers and 4 right-moving fermions and they form gineered in F-theory on elliptic Calabi-Yau 3-folds. In the a free hypermultiplet (Xaa˙ , λa) where a, a˙ are indices for context of F-theory, the 2d SCFT with string charge Q arises SU(2)l SU(2)R. They contribute to the anomaly as as a low energy theory on a D3-brane wrapping genus g curve × C = Q in the base B of the 3-fold. We can comparethe above com 1 results against the central charges of the strings coming from I4 = p1(T2) c2(l) . (18) −12 − D3-branesin F-theory. The 2d SCFT for a D3-brane wrapping Therefore the anomaly polynomial of the 2d worldsheet the- a genus g curve C inside B has the central charges [31] (See ory after removing the center of mass contributions becomes also [30]) ′ ′ ′ I = I Icom cL =3C C 9K C +6 , cR =3C C 3K C +6 , 4 4 − 4 · − · · − · (22) 3Q a 1 1 2 = · − p1(T2)+ Q biTrFi − 12 4 · where K is the canonical class of B,and it has a SU(2)l cur- i ′ X rent algebra at level k = g 1. Here the genus g of the curve Q Q Q a Q Q+Q a+2 l − · − · c (R)+ · · c (l) . (19) C can be computed by the Riemann-Roch theorem − 2 2 2 2 C C + K C =2g 2 . (23) The central charges of the 2d SCFT can be extracted from · · − the anomaly polynomial as discussed in the previous section. These results again include the contribution from the center The relative central charges cR cL is again the coefficient of mass modes; 4 left- and 4 right-movingbosons and 4 right- of the gravitational anomaly. The− right-moving central charge moving fermions. The central charges of the center of mass com com cR is associated to the anomaly coefficient of the R-symmetry modes are cL =4,cR =6 and, as discussed in [31], they com current. Here, we should be careful about the R-symmetry at contribute to the SU(2)l current algebra by kl = 1. ′ ′ − ′ the IR fixed point. It is possible that an accidental symmetry One can easily see that the central charges cL,cR, kl in F- emerges at low energy and it takes over the role of the R- theory models after removing the center of mass contributions symmetry in the IR (0, 4) superconformal algebra. It is also are in perfect agreement with the central charges of 2d SCFTs possible that a 2d worldsheet theory degenerates to a product from the anomaly inflow given in (20) and (21). To see this of distinct SCFTs carrying different IR R-symmetries. agreement, one needs to identify the inner product Ω among Indeed, this happensfor the strings in local 6d SCFTs or lit- tensors with the intersection form in H2(B, Z), and map the tle string theories (LSTs) embedded in the supergravity theo- vector a to the canonical class K in the base of the elliptic ries. The 2d SCFTs on such stringshave an accidental SU(2)I CY3. This comparison confirms our anomaly inflow compu- symmetry in the decouping limit and this symmetry becomes tation for 2d strings in 6d supergravity theories. 5

B. Consistency conditions in (20). However, this is not the case. Note that the central charge cR above is obtained by assuming the R-symmetry of We shall now show that the consistency of 2d worldsheet the low energy (0, 4) SCFT is the SU(2)R. As discussed, the theories encoded in the central charges imposes additional strings in local 6d SCFTs or LSTs have an accidental SU(2)I conditions on 6d supergravity theories. symmetry and this becomes the R-symmetry of the low energy Let us consider the moduli space of a 6d supergravity the- SCFT. Therefore cR in such strings is different from what we ory that is parametrizedby scalar fields in the tensor multiplets computed above. The central charges of various worldsheet as well as the scalar field in the hypermultiplet controlling the theories in 6d SCFTs are computed in the literature [28, 29], overall volume of the tensor moduli space. From supergrav- and one can check that those theories have positive central ity considerations, for this moduli space being well-defined, charges cR,cL with respect to the SU(2)I R-symmetry. we should be able to find a linear combination of these scalar We are interested in the configurations of a single string fields, which we call J, satisfying in the 6d supergravity that have SU(2)R as the R-symmetry in the superconformal algebra and that do not degenerate to J J > 0 , J bi > 0 , J a> 0 . (24) a product of disconnected 2d SCFTs at low energy. A sin- · · − · gle string state has no bosonic zero mode along the transverse This J plays the role of the central charge in the supersym- R4 directions except the center of mass degrees of freedom. metry algebra for the -fields. The first condition stands for B This implies that, after removing the center of mass modes, the metric positivity of the tensor branch along . The second J the worldsheet theory on a string contains the SU(2) current one is the condition for the gauge kinetic term along J to have l algebra realized on the left-movers. So the SU(2) central proper sign on the tensor moduli [19]. Otherwise, the gauge l charges should be non-negative, i.e. kl 0. In F-theory com- kinetic term has a wrong sign and it leads to an instability. The pactification, this condition becomes a trivial≥ condition saying last condition ensures, through supersymmetry, the positivity that g 0 for a string wrapped on a genus g curve Q. The of the Gauss-Bonnet term in gravity [32]. While there have central≥ charge conditions c 0 and k 0 on these SCFTs been attempts to prove the positivity of the curvature-squared R l can be summarized as ≥ ≥ corrections in D> 4 using e.g. unitarity [33], the singular UV behavior due to graviton exchange prevents one from making Q Q 1 ,Q Q + Q a 2 . (25) such spectral decomposition argument [34]. Here, we note · ≥− · · ≥− that even if we impose this last condition, there seem to be There are more conditions associated to the flavor central charges k = Q b . The flavor central charge measures the infinitely many anomaly-free 6d supergravity theories (see [4] i · i for a review). We thus assume its validity, leaving a derivation index of the bulk fields charged under the gauge group Gi on for future work. the string background with charge Q. So it counts the number In F-theory realization [35], this combination J corre- of zero modes at the intersection between the tensor carrying sponds to a K¨ahler form J H1,1(B) of the base B. The the gauge group Gi and the tensor labelled by the string charge above conditions on J define∈ a positive-definite K¨ahler cone Q. Unless the string degenerates to an instanton string of the group G , namely unless Q b , the flavor central charge can on B. We will call J a K¨ahler form for all 6d theories regard- i ∼ i less of whether it has an F-theory realization. receive contributions only from fermionic zero modes which The tensions of 2d BPS strings are determined with respect are in the left-movingsector. This means that the flavor central to the K¨ahler form J. This imposes a condition Q J 0 charges of the 2d SCFTs on non-degenerate strings (not in on the string charge Q. A worldsheet theory has non-negative· ≥ local 6d SCFTs or LSTs) in 6d supergravities should be non- tension only if Q J 0. negative. In other words, · ≥ The strings with Q J 0 embedded in 6d supergrav- k = Q b 0 , (26) ity theories must give rise· ≥ to unitary 2d SCFTs. For a uni- i · i ≥ tary 2d CFT, the central charges must be non-negative, i.e. for the strings we are interested in, where we used the con- cL,cR 0. If the central charges computed through the vention that left-movers have positive contributions to flavor anomaly≥ inflow for a string are negative, the corresponding central charges. In the F-theory viewpoint, the condition (26) anomalies cannot be cancelled by a unitary 2d worldsheet the- is the same as the condition that the curve class Q is effective ory. This results in that the 6d supergravity theory with such and irreducible within the Mori cone of the K¨ahler base B. strings is inconsistent hosting non-vanishing anomalies along Note that a 2d theory on instanton strings can have right- the 2d string worldsheet, and it thus belongs to the swamp- movers associated to bosonic zero modes parametrizing the land. So we can use the anomaly inflow on 2d strings to ana- moduli space of Gi instantons. These right-movers can pro- lyze the consistency of 6d supergravity theories. vide negative contributions to the flavor central charges. How- We remark that the strings in 6d SCFTs or little string the- ever, such instanton strings correspond to the strings in local ories (LSTs) contained in 6d supergravity theories in general 6d SCFTs or 6d LSTs. When a string degenerate to a prod- lead to 2d CFTs having a negative value for cR given in (20). uct of the instanton strings, the low energy theory will include For example, the unit string charge Q for a 2d string in the 6d 2d theories for the strings in local 6d SCFTs or LSTs which SO(8) non-Higgsable SCFT have the properties Q Q = 4 have the accidental SU(2) R-symmetry. As discussed above, · − I and Q K = +2. So the value for cR of this string with unit we are not interested in the worldsheet theories with SU(2)I charge·Q is 18. This seems to say that the theoryis inconsis- R-symmetry. So we shall only focus on strings and the assoc- − tent since its central charge is negative cR < 0 by the formula itated 2d SCFTs satisfying the condition (26) aswell as (25). 6

For such 2d SCFTs, we have G current algebra with level where q 3 q and q 9 q . In addition, the i 1:3 ≡ i=1 i 4:9 ≡ i=4 i ki. Using supersymmetry algebra in the context of BPS flavor central charges are restricted by the unitarity bound (27) strings, one can show that the current algebra is on the left- P P 2 2 movers in the (0, 4) SCFTs and its central charge contribution k1(N 1) k2(N 1) − + − cL , (30) is given in (11). Therefore, we find the following constraint k1 + N k2 + N ≤ on the 2d worldsheet SCFT in the 6d supergravity: where the left-moving central charges is

ki dimGi 9 · ∨ cL . (27) k + h ≤ 2 2 i i i c = 3(q q )+9(3q + q + q )+2 . (31) X L 0 − i 0 1:3 4:9 i=1 So the 2d SCFTs on strings satisfying the conditions in the X equations (25)and(26) must have central charges constrained As discussed above, if this bound is violated for any Q satis- by the equation (27). Otherwise, the 2d worldsheet theory fying (29), the anomaly inflow from the bulk 6d supergravity is non-unitary. In conclusion, we claim that a 6d supergravity theory cannot be cancelled by a unitary 2d CFT which renders theory embedding 2d strings whose worldsheet theory violates the 6d supergravity inconsistent at the quantum level. the condition (27) is inconsistent and it therefore belongs to The bound (30) gives the strongest constraint on N of the the swampland. 6d supergravity theory when the left-hand side is maximized, namely ki’s are minimized, while the right-hand side is min- imized. This implies the strongest bound can be given by a string with q2 q2 = 1 and k = 0, k = 1. This oc- C. Examples 0 i i 1 2 curs for Q = (1−, 1, 0, 0, −1, 05). The central charge bound for the string configurationP− − being unitary is The basic structure of our examples is as follows. For each 2 2 one we have the Ω,a,bi given by anomaly cancellation condi- k2(N 1) N 1 tions. We use this to find the allowed ranges for J and choose − cL − 8 N 9 . (32) k2 + N ≤ → 1+ N ≤ → ≤ a particular J in the allowed region. We then use this to re- strict the allowed string charges Q’s and use that to compute Therefore the 6d supergravity theory with N > 9 belongs to central charges cR,cL and kl, ki and see if we have any con- the swampland containing non-unitary string configurations. tradictions with unitarity. This bound is stronger than the bound N 12 from the Ko- ≤ Let us first consider the 6d supergravity theory coupled to daira condition in F-theory [8]. It is interesting that we can T =9 tensors with SU(N) SU(N) gaugegroupand two bi- thus rule out would be purely geometric constructions which fundamental hypermultiplets× introduced in [8] (See also [20] could have in principle realized this modelfor N = 10, 11, 12. for T = 1 models). The anomaly polynomial of this model In other words our arguments can be used to teach us some factorizes for an arbitrary N and hence it seems that they pro- facts about the geometry of elliptic Calabi-Yau threefolds! vide an infinite family of consistent 6d supergravity theories. Also, it is reassuring that this bound does not rule out the It was however shown in [8] that these models have no F- string theory realization for N =8 givenin [36, 37]and all the theory realization at large enough N. N 8 theories which one can obtain from it by partial Hig- ≤ Let us examine these models with 2d strings to see if the gsing. Remarkably, our worldsheet analysis provides a new consistency conditions of the worldsheet theory on the strings bound on the rank of gauge groups in the 6d bulk supergravity can provide any bound on N. theory and the result is consistent with the F-theory argument We can always choose a tensor basis such that the bilinear and also the known string theory realization. It would be in- teresting to see if one can construct the N =9 case which we form Ω and the vectors a,b1,b2 are given as follows [8]: were not able to rule out. Ω = diag(+1, ( 1)9) , a = ( 3, (+1)9) , The second example is the 6d supergravity with T =1 and − − SU(N) gauge group coupled to one symmetric and N 8 b = (1, 1, 1, 1, 06) , b = (2, 0, 0, 0, ( 1)6) . (28) − 1 − − − 2 − fundamental hypermultiplets first introduced in [8, 38]. The rank of the gauge group is bounded as N 30 from the 6d In this basis, one can easily see that a K¨ahler form chosen as anomaly cancellation conditions. For this model,≤ we are free 9 2 J = (1, 0 ) satisfies the conditions J > 0,J b > 0 and to choose a tensor basis giving J a< 0. · · Consider a string of a generic charge Q = (q0, q1, , q9) Ω = diag(1, 1) , a = ( 3, 1) , b = (0, 1) . (33) ··· − − − with qi Z. This string with q0 > 0 has a positive tension with respect∈ to J. The conditions (25) and (26) on the IR The K¨ahler form can always be chosen as J = (n, 1) with SCFT for this string can be summarized as n2 > 1 and n > 0. This theory has no F-theory realization because, when we identify the base B with a Hirzebruch sur- 9 9 face F1, the tensor for b cannot be mapped to any effective 2 2 2 2 q q 1 , q q 3q0 q1:3 q4:9 2 , curve class [8]. 0 − i ≥− 0 − i − − − ≥− i=1 i=1 X X We shall now see if the consistency conditions on string k = q + q 0 , k =2q + q 0 , (29) configurations of this 6d theory can provide a stronger bound 1 0 1:3 ≥ 2 0 4:9 ≥ 7 on the rank N. Consider a generic string with Q = (q1, q2) for k =1 and satisfying the conditions (25),(26), namely 25 25 2 2 2 2 Ω = diag(1, ( 1) ) , a = ( 3, 1 ) , q1 q2 1 , q1 q2 3q1 q2 2 , − − − ≥− − − − ≥− 11 13 13 11 (41) k = Q b = q 0 . (34) b1 = (0, 1, ( 1) , 0 ) , b2 = (0, 0 , 1, ( 1) ) , · 2 ≥ − − Also, nq1 > q2 from J Q> 0. These conditions can be then for k =2. Thus the above analysis does not apply to the k = simplified, for the strings· interacting with the gauge group, as 1, 2 cases. We do not find any string configuration showing inconsistencies for these cases. Indeed, the 6d gravity theory q 3 q 2 q > 0 . (35) 1 ≥ 1 − ≥ 2 with k = 2 can be realized by the compactification of M- 1 Z The constraint on the central charges theory on K3 (S / 2), where we place 24 M5 branes on the interval [39].× 2 q2(N 1) 2 2 The last example is the 6d supergravity theory with T = 0 − 3(q1 q2)+9(3q1 + q2)+2 (36) q2 + N ≤ − and gaugegroup SU(8) coupled to an exotic hypermultipletin the ‘box’ representation, which was introduced in [23]. This can provide the strongest bound on N when Q = (3, 1), and theory cannot be realized in F-theory. The 6d anomaly can- the bound is N 117. This bound is weaker than the bound cellation sets the vectors as a = 3 and b =8. N 30 coming≤ from the 6d anomaly cancellation conditions. − ≤ The 2d SCFTs on a string with charge Q > 0 in this the- This may imply, unless another inconsistency is revealed by ory satisfy the conditions (25) and (26). The strongest con- any other means, that these 6d supergravity models with N straint on the left-moving central charge is given by the mini- are all consistent theories though they do not seem to admit≤ 30 mal string with Q = 1. The central charge constraint for this an F-theory realization. model is marginally satisfied as The anomaly inflow consideration can provide a new bound on a family of models with T = 8k + 9 and gauge group k k 63 G = (E8) for arbitrary large k, which was introduced in × c 31.5 32 for k = Q b =8 . (42) k +8 ≤ L → ≤ · [8]. The vectors a and bi in the anomaly polynomial satisfy a b = 10, b b = 2δ with i, j =1, , k. When k 3, i i j ij Thereforeat least as far as the unitarity constraint is concerned one· can choose· a basis− for tensors in [8] that··· gives rise to≥ this theory is not ruled out and the strings can consistently Ω = diag(1, ( 1)8k+9) , a = ( 3, 18k+9) , couple to this 6d supergravity theory. − − b = ( 1, 1, 04(i−1), ( 1)3, 3, 08k+8−4i) , (37) i − − − − The K¨ahler form in this basis can be chosen as IV. CONCLUSIONS 4k+1 4k+8 J = ( j0, 0 , 1 ) , (4k + 8)/3 > j0 > √4k +8 . − (38) In summary, we have discussed the consistencies of 10d and 6d = (1, 0) supergravity theories as seen from 2d strings Now consider a string with charge Q = ( q, 08k+9) in thatN couple to the 2-forms in the bulk. We have identified the − this 6d model. This string has a positive tension if q > 0. central charges of the worldsheet SCFTs on the strings using Moreover, the conditions kl 0,cR 0 and ki 0 can be the anomaly inflow from the bulk supergravity theory. The ≥ ≥ ≥ satisfied if q > 2. However, the bound on the levels of flavor unitarity of the worldsheet SCFTs associated to the central current algebras ki = Q bi = q: charges leads to novel constraints on the allowed supergravity · models, that are not visible from the particle viewpoint. k 248k 248q i c k 3q(q 9)+2 (39) In this paper, we analyzed only a handful of 6d supergravity k + 30 ≤ L → q + 30 ≤ − i=1 i models. A large class of would-be consistent 6d supergravity X theories has been discussed in the literature, for example [8, cannot be satisfied by, for example, strings with charge 3 ≤ 23, 38]. It might be possible to similarly rule out many such q 14 for any k 3. This result demonstrates that all models using more detailed constraints from string probes that these≤ 6d supergravity≥ models for k 3 endowed with the ≥ we considered in this paper. We leave this for future work. bilinear form Ω and vectors a,bi given in (37) reveal non- It would be straightforward to generalize the anomaly in- vanishing anomalies on the 2d strings, and therefore they are flow consideration discussed in this paper to other type of in the swampland. branes coupled to the supergravity theories. Our discussion Note however that the 6d supergravity theories of this type in this paper is merely a starting point of a bigger program to for k 2 are not ruled out by this analysis. When k = 1, 2, ≤ understand the consistency of quantum gravitational theories there exists another solutions of Ω and a,bi cancelling the in various dimensions by coupling them to all possible branes anomalies, like this: and defects of the theories. We hope this program ultimately Ω = diag(1, ( 1)17) , a = ( 3, 117) , provides a complete classification of consistent supergravity − − theories in six- and perhaps also other dimensions, and more b = (0, 1, ( 1)11, 05) , (40) 1 − broadly deepens our understanding of the swampland criteria. 8

V. ACKNOWLEDGMENTS To cancel the 1-loop anomaly, we add to the action the Green-Schwarz term We would like to thank E. Bergshoeff, M. Roˇcek and SGS = B X . (A5) W. Taylor for useful and informative discussions. H.K. and 2 ∧ 8 G.S. would like to thank Harvard University for hospital- Z ity during part of this work. H.K. is supported by the Here the 2-form field B2 in the 10d theory transforms under POSCO Science Fellowship of POSCO TJ Park Foundation local gauge and Lorentz group as and the National Research Foundation of Korea (NRF) Grant 1 2018R1D1A1B07042934. G.S. is supported in part by the B2 B2 Tr(ΛF ) + tr(ΘR) , (A6) DOE grant de-sc0017647 and the Kellett Award of the Uni- → − 4 versity of Wisconsin. The research of CV is supported in part where Λ and Θ are the transformation parameters. It then by the NSF grant PHY-1719924 and by a grant from the Si- follows that the Green-Schwarz term induces anomalies un- mons Foundation (602883, CV). der the gauge and Lorentz transformations which may cancel the 1-loop anomalies. We normalize ‘Tr’ such that the inte- 1 2 gral of 4 TrF over a 4-manifold gives the instanton number Appendix A: Q Z. Note that the gauge transformation of B2 is fixed by Anomalies in 10d supergravity theories supersymmetry∈ and the gauge invariance of the 3-form field strength H3 [16, 17]. The Lorentz transformation of B2 is on We adopt the normalization used in [40], but a factor of the other hand fixed by the higher order correction on H3 in 1/4π is included in the curvature 2-form R and the field the derivative expansion. strength F includes a factor of 1/2π. An = 1 super- gravity theory in ten dimensions contains aN Majorana-Weyl 1 Appendix B: Anomaly inflows from 6d to 2d gravitino, some spin 2 fermions with negative chirality, and gauginos with positive chirality. The gravitino contributes to the anomaly as Let us briefly review the anomaly inflow computation in 6d theories in the presence of 2d strings discussed in [28, 29]. 3/2 11 5 7 I = trR6 + trR4 trR2 (trR2)3 , (A1) When Q strings are located at x1,2,3,4 =0, the Bianchi iden- 12 −126 96 − 1152 i tity for the 2-form fields is modified as while the contribution from a spin 1 fermion is 2 4 α α α a a 1/2 1 6 1 4 2 dH = X4 + Q δ(x )dx . (B1) I = (trR1) trR + trR trR 12 5670 4320 a=1  Y 1 2 3 1 2 1 4 1 2 2 The shift in the right-hand side in the Bianchi identity applies + (trR ) trRF trR + (trR ) 10368 − 2 360 288 to for the anomaly contribution from the Green-Schwarz term    as 1 4 2 1 6 + (trRF )trR trRF , (A2) 4 4 288 − 720 GS 1 α α a a β α a a I8 =− Ωαβ X4 +Q δ(y )dy X +Q δ(y )dy . 2 4 where denotes the representation of the fermion under the a=1 ! a=1 ! gauge algebra.R The total 1-loop anomaly of the theory is given Y Y (B2) by the sum over all fermion contributions as As a result, a non-trivialanomaly inflow is induced toward the 1−loop 3/2 1/2 1/2 string worldsheet. The anomaly inflow can be computed by I = I I R=1 + I R=adj 12 12 − 12 | 12 | integrating the 8-form anomaly polynomial over the 4 trans- dimG 496 dimG + 224 = − trR6 + trR4 trR2 verse directions to the strings. One computes 5670 4320 dimG 64 2 3 inflow α α 1 β + − (trR ) I = ΩαβQ X + Q χ4(N4) . (B3) 10368 4 − 4 2   1 1 1 tr F 2 trR4 + (trR2)2 This inflow must be cancelled by the anomalies arising from − 2 adj 360 288   the worldsheet degrees of freedom on the 2d strings. Hence 1 1 + tr F 4 trR2 tr F 6 , (A3) the anomaly polynomial of the 2d worldsheet SCFT must be 288 adj − 720 adj 1 where dimG is the dimension of the gauge group. When this I = Iinflow =Ω Qα Xα + Qβχ (N ) . (B4) 4 − 4 αβ 4 2 4 4 1-loop anomaly factorizes as   1−loop Here χ4(N4) is the Euler class of the SO(4) = SU(2)l I12 = X4 X8 , (A4) 4 × ∧ SU(2)R normal bundle for the transverse R directions and it can be cancelled by the Green-Schwarz mechanism [12]. it can also be written as χ (N ) = c (l) c (R) in terms 4 4 2 − 2 This factorization condition allows only 4 choices of gauge of the second Chern-classes c2(l) and c2(R) for SU(2)l and groups: SO(32), E E , E U(1)248, U(1)496. SU(2) . 8 × 8 8 × R 9

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