aa r X i v : . [ h e p - t h ] D ec Preprint typeset in JHEP style - HYPER VERSION
MIT-CTP-4077
A bound on 6D N = 1 supergravities Vijay Kumar and Washington Taylor
Center for Theoretical PhysicsDepartment of PhysicsMassachusetts Institute of Technology77 Massachusetts AvenueCambridge, MA 02139, USA vijayk at mit.edu , wati at mit.edu Abstract:
We prove that there are only finitely many distinct semi-simple gauge groupsand matter representations possible in consistent 6D chiral (1 ,
0) supergravity theorieswith one tensor multiplet. The proof relies only on features of the low-energy theory; theconsistency conditions we impose are that anomalies should be cancelled by the Green-Schwarz mechanism, and that the kinetic terms for all fields should be positive in someregion of moduli space. This result does not apply to the case of the non-chiral (1 , ontents
1. Introduction 12. Six-dimensional (1,0) supergravity and anomaly cancellation 23. Finiteness of anomaly-free models in 6D 54. Classification of models 115. Conclusions and future directions 14A. Proof of Claim 3.1 15
A.1 Exceptional groups 15A.2 SU ( N ) 16A.3 SO ( N ) 18A.4 Sp ( N ) 19 B. (1 , supersymmetry and the case of one adjoint hypermultiplet 19
1. Introduction
The requirement of anomaly cancellation can provide non-trivial consistency conditions forquantum field theories. In the context of the Standard Model, for example, vanishing ofgauge anomalies requires that the number of generations of leptons and quarks are equal.The anomalies in any given model are completely determined by the low-energy effectivetheory, and in particular, by massless matter fields that violate parity (chiral fermions, self-dual tensors, etc). Anomalies arise at one loop and receive no further quantum corrections;this makes them easy to compute given the matter content of the model (for a review, see[1]). When the dimension of space-time is of the form 4 k + 2, there are purely gravitional,purely gauge and mixed gravitional-gauge anomalies [2]. In the case of ten dimensional N = 1 (chiral) supergravity coupled to a vector multiplet, it was shown by Green andSchwarz [3] that anomalies can be cancelled only when the gauge group is SO (32), E × E , U (1) × E or U (1) . This is a powerful constraint on this class of theories. Our goalhere is to explore the analogous constraint in six-dimensional supergravities.In this paper, we consider the constraints from anomaly cancellation in the case of six-dimensional (1 ,
0) supergravity coupled to one tensor multiplet, and any number of vectormultiplets and matter hypermultiplets. We prove the following result –– 1 – he set of semi-simple gauge groups and matter representations appearing in consistentchiral, six-dimensional, (1 , supergravity models which have positive gauge-kinetic termsand one tensor multiplet is finite. We restrict to the case of one tensor multiplet because in this case the details ofthe anomaly cancellation are simpler, and the theory has a Lagrangian description . Wehave also assumed that the gauge group contains no U (1) factors. We expect that ageneralization of the proof given in this paper to include abelian gauge group factors andan arbitrary number of tensor multiplets may be possible. Since we restrict ourselvesto chiral (1 ,
0) supergravity theories, our conclusions do not apply to models with (1 ,
2. Six-dimensional (1,0) supergravity and anomaly cancellation
We are interested in six-dimensional theories with (1 ,
0) supersymmetry which are gaugetheories with matter coupled to gravity and a single tensor multiplet. The field content ofsuch a theory consists of single (1, 0) gravity and tensor multiplets, n v vector multiplets, and n h hyper multiplets. Table 1 summarizes the matter content of the (1 ,
0) supersymmetrymultiplets we consider in this paper. We assume that the gauge group G = Q i G i is semi-simple and contains no U (1) factors. The low-energy Lagrangian for such a model takesthe form (see [4, 5, 6, 7]) L = − e − φ ( dB − ω ) · ( dB − ω ) − B ∧ d ˜ ω − X i ( α i e φ + ˜ α i e − φ )tr( F i ) + . . . (2.1)The index i runs over the various factors in the gauge group. φ is the scalar in the tensormultiplet. Note that there are many terms in the Lagrangian, including the hypermultipletkinetic terms, gravitational couplings and fermion terms, which we have not included asthey are not relevant for our discussion (for details see [4, 5]). ω, ˜ ω are Chern-Simons formsdefined as dω = 116 π tr R ∧ R − X i α i tr F i ∧ F i ! (2.2) d ˜ ω = 116 π tr R ∧ R − X i ˜ α i tr F i ∧ F i ! (2.3)The B -field transforms under a gauge transformation Λ i in the factor G i as δB = − π X i α i tr(Λ i F i ) (2.4)– 2 –ultiplet Matter ContentSUGRA ( g µν , B − µν , ψ − µ )Tensor ( B + µν , φ, χ + )Vector ( A µ , λ − )Hyper (4 ϕ, ψ + ) Table 1:
Representations of (1 ,
0) supersymmetry in 6D. The + and - indicate the chirality forfermions and self-duality or anti-self-duality for the two index tensor.
The Lagrangian (2.1) is not gauge invariant, because of the B ∧ d ˜ ω coupling; this term isneeded for the six-dimensional analogue [8] of the ten-dimensional Green-Schwarz mecha-nism ∗ [3]. The tree-level gauge variation of the Lagrangian cancels the one-loop quantumanomaly, and as a result, the full quantum effective action is anomaly free. We discussonly some aspects of this mechanism here, referring the reader to [11, 12, 13] for furtherdetails.The gauge anomaly in a d -dimensional theory is related to the chiral anomaly in d + 2dimensions, where it is expressed as a d + 2-form [1]. Given a six-dimensional (1 ,
0) modelwith one tensor multiplet, the anomaly polynomial is an 8-form and takes the form [14] I = − n h − n v − R −
44 + n h − n v R ) −
196 tr R X i " Tr F i − X R x iR tr R F i + 124 " Tr F i − X R x iR tr R F i − X i,j,R,S x ijRS (tr R F i )(tr S F j ) , (2.5)where F i denotes the field strength for the simple gauge group factor G i , tr R denotes thetrace in representation R of the corresponding gauge group factor, and Tr denotes thetrace in the adjoint representation. x iR and x ijRS denote the numbers of hypermultiplets inrepresentation R of G i , and ( R, S ) of G i × G j . We can express the traces in terms of thetrace in the fundamental representation.tr R F = A R tr F tr R F = B R tr F + C R (tr F ) (2.6)Note that tr (without any subscript) will always denote the trace in the fundamentalrepresentation. Formulas for the group-theoretic coefficients A R , B R , C R can be found in[14] † . The polynomial I can be written in terms of tr R , tr F i , tr R , tr F i using (2.6). Werescale the polynomial so that the coefficient of the (tr R ) term is one. Anomalies can becancelled through the Green-Schwarz mechanism when this polynomial can be factorizedas I = (tr R − X i α i tr F i )(tr R − X i ˜ α i tr F i ) (2.7) ∗ In the case of six-dimensional models with multiple tensor multiplets, there is a generalized mechanismdue to Sagnotti [9, 10]. Since we only consider the case of one tensor multiplet here, this mechanism is notrelevant to the analysis of this paper. † Note that in [14], the coefficients v, t, u correspond to A R , B R , C R respectively. – 3 – necessary condition for the anomaly to factorize in this fashion is the absence of anyirreducible tr R and tr F i terms. This gives the conditionstr R : n h − n v = 244 (2.8)tr F i : B iAdj = X R x iR B iR (2.9)For groups G i which do not have an irreducible tr F i term, B iR = 0 for all representations R and therefore (2.9) is always satisfied. The sum in (2.9) is over all hypermultiplets thattransform under any representation R of G i . For example, a single hypermultiplet thattransforms in the representation ( R, S, T ) of G i × G j × G k contributes dim( S ) × dim( T ) to x iR . The condition (2.8) plays a key role in controlling the range of possible anomaly-freetheories in 6D. The anomaly cancellation conditions constrain the matter transforming un-der each gauge group so that the quantity n h − n v in general receives a positive contributionfrom each gauge group and associated matter, and the construction of models compatiblewith (2.8) thus has the flavor of a partition problem. While there are some exceptions tothis general rule, and some subtleties in this story when fields are charged under more thanone gauge group, this idea underlies several aspects of the proof of finiteness given here;we discuss this further in Section 4.After rescaling the anomaly polynomial so that the coefficient of (tr R ) is one, when(2.8) is satisfied (2.5) becomes I = (tr R ) + 16 tr R X i " Tr F i − X R x iR tr R F i − " Tr F i − X R x iR tr R F i + 4 X i,j,R,S x ijRS (tr R F i )(tr S F j ) (2.10)For a factorization to exist, in addition to (2.8) and (2.9), the following equations musthave a solution for real α i , ˜ α i α i + ˜ α i = 16 X R x iR A iR − A iAdj ! (2.11) α i ˜ α i = 23 X R x iR C iR − C iAdj ! (2.12) α i ˜ α j + α j ˜ α i = 4 X R,S x ijRS A iR A jS (2.13)Notice that the coefficients in the factorized anomaly polynomial α i , ˜ α i , are relatedto the coefficients in the ∗ B · ( F i ∧ F i ) and B ∧ F i ∧ F i terms in the Lagrangian (2.1).This was first observed in [9], and its consequences were discussed further in [6, 7, 10].The coefficients α i , ˜ α i are fixed by the anomaly polynomial, which in turn is determinedcompletely by the choice of the gauge group and the matter content. Moreover, due totheir origin in the anomaly, the coefficients α i , ˜ α i are immune to quantum corrections. The– 4 –oefficients α i , ˜ α i play a key role in the structure of consistent 6D supergravity theories.In particular, the signs of these terms affect the behavior of the gauge fields in the theoryat different values of the dilaton through the gauge kinetic terms in (2.1) [9]. When thegauge kinetic term controlled by α i e φ + ˜ α i e − φ becomes negative, the theory develops aperturbative instability. When the gauge kinetic term vanishes, the gauge field becomesstrongly coupled and the theory flows to an interacting superconformal field theory in theIR, with tensionless string excitations [7]. While there are potentially interesting featuresassociated with some of these superconformal theories [15, 16], in this work we focus ontheories where all gauge kinetic terms are positive for some value of the dilaton. Thisimposes the constraint that there exists some φ so that for each gauge group at least oneof α i , ˜ α i is positive.As stated in the introduction, the goal of this paper is to demonstrate that only a finite number of gauge groups and matter representations are possible in consistent 6Dchiral supergravity theories with one tensor multiplet. We prove this in the followingsection, based only on the following two assumptions regarding the theory:1 . The anomalies can be cancelled by the Green-Schwarz mechanism.2 . The kinetic terms for all fields are positive in some region of moduli space. (2.14)Note in particular that the statement and proof of the finiteness of this class of theoriesis purely a statement about the low-energy effective theory, independent of any explicit re-alization of a UV completion of the theories, such as string theory. Clearly, it is of interestto understand which low-energy theories admit a consistent UV completion as quantumgravity theories. In [17], we conjectured that all UV-consistent 6D supergravity theoriescan be realized in some limit of string theory. The constraints on 6D supergravity theoriesarising from anomaly cancellation have intriguing structural similarity to the constraintsassociated with specific string compactification mechanisms [18, 19, 20]. This correspon-dence was made explicit for a particular class of heterotic compactifications in [17], and hasbeen explored in the context of F-theory by Grassi and Morrison [21, 22], who showed thatthe anomaly cancellation conditions in 6D give rise to new nontrivial geometric constraintson Calabi-Yau compactifications. The proof of finiteness in this paper suggests a systematicapproach to classifying the complete set of consistent chiral 6D supergravity theories. Aswe discuss in Section 4, such a classification would help in making a more precise dictionarybetween the structure of low-energy theories and string compactifications.
3. Finiteness of anomaly-free models in 6D
We now proceed to prove the finiteness result stated in the introduction, using constraints1 and 2 above. We will prove finiteness by contradiction. Assume there exists an infinitefamily F of distinct models M , M , · · · . Each model M γ has a nonabelian gauge group G γ which is a product of simple group factors, and matter hypermultiplets that transformin arbitrary representations of the gauge group. The number of hypermultiplets, the matter– 5 –epresentations and the gauge groups themselves are constrained by anomaly cancellation.We show that these constraints are sufficient to demonstrate that no infinite family ofanomaly-free models exists, with distinct combinations of gauge groups and matter.We first examine some simple consequences of the anomaly constraints. The absenceof purely gravitational anomalies requires that n h − n v + 29 n t = 273, and for the case ofone tensor multiplet (2.8) n h − n v = 244. If every model in an infinite family F has thesame gauge group G γ = G , but with distinct matter representations, we immediately arriveat a contradiction. If { R i } denotes the irreducible representations of G , then we must have P i x i dim( R i ) = dim( G ) + 244. Since x i ≥ ∀ i , and the number of representations of G below a certain dimension is bounded, there are only a finite number of solutions to(2.8). Therefore, there exists no such infinite family. If we assume that model M γ hasa gauge group G γ , such that the dimension of the gauge group is always bounded fromabove (dim( G γ ) < M, ∀ γ ), we again arrive at a contradiction. This is because there areonly finitely many gauge groups below a certain dimension, and again the gravitationalanomaly condition only has a finite number of solutions.For any infinite family of anomaly-free models, the dimension of the gauge group istherefore necessarily unbounded. There are two possibilities for how this can occur, andwe show that in each case the assumption of the existence of an infinite family leads to acontradiction.1. The dimension of each simple factor in G γ is bounded. In this case, the number ofsimple factors is unbounded over the family.2. The dimension of at least one simple factor in G γ is unbounded. For example, thegauge group is of the form G γ = SU ( N ) × ˜ G γ , where N → ∞ . Case 1: Bounded simple group factors
Assume that the dimension of each simple group factor is bounded from above by M . Inthis case, the number of factors in the gauge group necessarily diverges for any infinitefamily of distinct models {M γ } . Any model in the family has a gauge group of the form G γ = Q Nk =1 G k , where each G k is simple with dim( G k ) ≤ M and N is unbounded over thefamily.Since each model in the family is free of anomalies, the anomaly polynomial for eachmodel factorizes as I = (tr R − N X k =1 α k tr F k )(tr R − N X k =1 ˜ α k tr F ) (3.1)The positivity condition on the kinetic terms (2.14) requires that at least one of α k , ˜ α k ispositive, for each factor G k in the gauge group. The coefficient of tr F i tr F j in the anomalypolynomial (2.5) is related to the number of hypermultiplets charged simultaneously underboth G i and G j . Hence, for every pair of factors G i and G j in the gauge group, thecorresponding coefficients in the anomaly polynomial must satisfy α i ˜ α j + α j ˜ α i ≥ at most one α i is negative among all i , and at most one˜ α j is negative among all j . This condition does allow an arbitrary number of α i or ˜ α j tobe zero. There are three possibilities for a given factor G i –1. Type 0 : One of α i , ˜ α i is zero.2. Type N : One of α i , ˜ α i is negative.3. Type P : Both α i , ˜ α i are positive.For any model in the family, there can be at most two type N factors.We first show that the number of type 0 factors in the gauge group is unbounded for anyinfinite family. Assume the contrary is true, that is the number of such factors is boundedabove by K over the entire family. There are at most two type N factors, so at least N − K − G i and G j among these N − K − α i ˜ α j + α j ˜ α i is strictly positive, so there are matter hypermultiplets charged simultaneouslyunder G i × G j . If each of these matter hypermultiplets are charged under at most twogroups then the number of such hypermultiplets is ( N − K − N − K − /
2, whichgoes as O ( N ) for large N . This is an overcount of the number of hypermultiplets needed,however, because there could be a single hypermultiplet charged under λ > λ ( λ − times. Let λ denote the maximum number of gauge groupfactors that any hypermultiplet transforms under. The dimension of such a representation ≥ λ . The number of vector multiplets scales linearly with N , since each factor in thegauge group is bounded in dimension. As a result, λ ∼ O (log N ); if it scales faster with N , the gravitional anomaly condition n h − n v = 244 would be violated. In the worst case,if all the matter hypermultiplets transform under λ ∼ log N gauge group factors, n h stillgrows as O ( N / log N ). Therefore, n h − n v ∼ O ( N / log N ) as N → ∞ . This would clearlyviolate the gravitational anomaly condition (2.8) at sufficiently large N .The above argument shows that the number of type 0 factors with α or ˜ α equal tozero must be unbounded. It also shows that the number of type P factors with both α, ˜ α strictly positive can grow at most as fast as O ( √ N ) (dropping factors of log N ). Therefore,there are O ( N ) factors of type 0. For each factor G i of this type, the coefficient of the(tr F i ) term is zero. This implies that the third term in the anomaly polynomial (2.10)vanishes for each type 0 factor G i , so X R x R tr R F i = Tr F i (3.3)where x R is the number of hypermultiplets in representation R of the group G i .We now show that type 0 factors all give a positive contribution to n h − n v in thegravitational anomaly condition (2.8), so that the number of these factors is bounded. Claim 3.1.
Every gauge group factor G i that satisfies (3.3) also satisfies h i − v i > ,where h i = P R x R dim( R ) denotes the number of hypermultiplets charged under G i , and v i = dim( G i ) . – 7 –he proof of this claim is in Appendix A. Using the statement of the claim, we wishto show that n h − n v is positive and grows without bound over the family. We must becareful since there could be hypermultiplets charged under multiple groups, which will beovercounted if we simply add h i − v i for each factor G i . It is easily checked that there is nomatter charged simultaneously under three or more type 0 factors ‡ For two factors G i , G j ,hypermultiplets can be charged under G i × G j if ( α i , ˜ α i ) = (+ ,
0) and ( α j , ˜ α j ) = (0 , +)(or the other way around). For large enough N , however, all the type 0 factors must haveeither α i = 0 or ˜ α i = 0, and hence, there cannot be any hypermultiplet charged under twotype 0 factors G i and G j . To prove this assertion, assume there is one factor G of theform (0 , +) and O ( N ) factors are of the other type (+ , R of G is large and scales as O ( N ).This is impossible, since this factor must satisfy P R x R tr R F = Tr F , while the RHS isfixed and tr R F > R . Thus, for large enough N , n h − n v = X type 0 ( h i − v i ) + P/N-type contribution (3.4)Since the dimension of each gauge group factor is bounded from above by M , the contri-bution of each factor is bounded from below n h − n v ≥ − M . So, even if the O ( √ N ) typeP factors contributed negatively, n h − n v ≥ O ( N ) − M O ( √ N ) (3.5) n h − n v is positive and grows as O ( N ), in contradiction with the gravitional anomalycondition (2.8).This shows that given a finite list of simple groups, there is no infinite family ofanomaly-free models, each of which has a gauge group consisting of an arbitrary productof simple groups in the list and matter in an arbitrary representation. Case 2: Unbounded simple group factor
The only other way in which a gauge group could grow unbounded over a family of modelsis if the gauge group contains a classical group H ( N ) (either SU ( N ), SO ( N ) or Sp ( N/ F necessarily contains aninfinite sub-family of models with gauge group H ( N ) × G N with N → ∞ . We now showthat this case also leads to a contradiction. Brief outline of the proof for case 2 : We examine the tr F conditions for models in thefamily with N > ˜ N . This allows us to show that every infinite family with an unboundedfactor, at large enough N , has an infinite sub-family with gauge group and matter contentparameterized in one of a few different ways. For each of these possible parameterizationsof gauge group and associated matter fields, the contribution to n h − n v diverges with N . This would violate the gravitational anomaly condition, unless there is a sufficiently ‡ If there was such a hypermultiplet charged under G × G × · · · × G n , for n >
2, then for all pairs x ij ∝ α i ˜ α j + α j ˜ α i = 0. This is impossible, since for at least 2 factors, ˜ α i = ˜ α j = 0 or α i = α j = 0 ⇒ x ij = 0for every such pair. – 8 –roup Representation Dimension A R B R C R SU ( N ) N N − N N N ( N − N − N − N ( N +1)2 N + 2 N + 8 3 N ( N − N − N − N +62 N − N +542 N − N ( N − N − N −
27 6 N N ( N +1)( N +2)6 N +5 N +62 N +17 N +542 N + 12 SO ( N ) , Sp ( N ) N N ( N − N − N − N ( N +1)2 N + 2 N + 8 3 Table 2:
Values of the group-theoretic coefficients A R , B R , C R for some representations of SU ( N ), SO ( N ) and Sp ( N/ SO ( N ) and Sp ( N/
2) are given bythe 2-index antisymmetric and the 2-index symmetric representation respectively. negative contribution from the rest of the gauge group G N and the associated chargedmatter. This, however, requires dim( G N ) to grow without bound in such a way that thecontribution to n h − n v is negative and divergent, which we show is impossible.Consider first the case when the gauge group is of the form SU ( N ) × G N , where G N is an arbitrary group. Anomaly cancellation (2.9) for the SU ( N ) gauge group componentrequires that B Adj = 2 N = X R x R B R (3.6)where R is a representation of SU ( N ). x R denotes the total number of hypermultipletsthat transform under representation R of SU ( N ); it also includes hypermultiplets that arecharged under both SU ( N ) and G N . For example, in a model with gauge group SU ( N ) × SU (4) with matter content 1( N,
1) + 1(1 ,
6) + 2( N,
4) + c.c , the number of hypermultipletsin the N of SU ( N ) would be counted as x N = 1 + 2 × A R , B R , C R for various representations of SU ( N ) are shown in Table 2.The values of B R for all representations of SU ( N ) other than the fundamental, adjoint,and two-index symmetric and antisymmetric representations grow at least as fast as O ( N )as N → ∞ . For any given infinite family, there thus exists an ˜ N such that for all N > ˜ N ,the models in the family have no matter in any representations other than these. Thisfollows because the LHS of equation (3.6) scales as O ( N ), and each x R ≥ N , we only need to consider the fundamental, adjoint, symmetricand anti-symmetric representations. For these representations, (3.6) reads2 N = x + 2 N x + ( N − x + ( N + 8) x (3.7)The only solutions to this equation when N is large are shown in Table 3. We havediscarded solutions ( x , x , x , x ) = (0 , , ,
0) and (0 , , , α = ˜ α = 0 so the– 9 –roup Matter content n h − n v α, ˜ αSU ( N ) 2 N N + 1 2 , − N + 8) + 1 N + N + 1 2 , − N −
8) + 1 N − N + 1 − ,
116 + 2 15 N + 1 2 , SO ( N ) ( N − N − N − , Sp ( N/
2) ( N + 8) N + N , −
116 + 1 15 N − , Table 3:
Allowed charged matter for an infinite family of models with gauge group H ( N ). Thelast column gives the values of α, ˜ α in the factorized anomaly polynomial. kinetic term for the gauge field is identically zero. We repeat the same analysis for thegroups SO ( N ) and Sp ( N/ N , we only need to consider the fundamen-tal, anti-symmetric and symmetric representations. The allowed matter hypermultipletsfor models with non-vanishing kinetic term are shown in Table 3.We have shown that for any infinite family of models where the rank of one of thegauge group factors increases without bound, the matter content is restricted to one ofthose in Table 3. Notice that for each of these possibilities the contribution to n h − n v frommatter charged under H ( N ) diverges as N → ∞ (either as O ( N ) or O ( N )). Since thegauge group is H ( N ) × G N , the contribution to n h − n v from G N and associated mattercharged under this group must be negative and unbounded (at least −O ( N ) or −O ( N ))in order to satisfy the gravitational anomaly condition. If we assume that the dimensionof each simple factor in G N is bounded, and that the number of simple factors diverges,we arrive at a contradiction in exactly the same way as we did in Case 1 above: n h − n v ispositive and scales at least linearly with the number of factors. G N must therefore containanother classical group factor ˆ H ( M ), whose dimension also increases without bound overthe infinite family.Therefore, any given infinite family must have an infinite sub-family, with gauge groupof the form ˆ H ( M ) × H ( N ) × ˜ G M,N , with both
M, N → ∞ . Note that values of ( α, ˜ α ) foreach unbounded classical factor are restricted to the values ( ± , ∓ ± , ∓ ± , ∓ , , F denotes the field strength of the ˆ H ( M ) factor and F that of H ( N ), thecoefficient of the tr F tr F term in the anomaly polynomial is 4 P R,S A R A S x RS , where x RS is the number of hypermultiplets in the ( R, S ) representation of ˆ H ( M ) × H ( N ). Comparingcoefficients, we have α ˜ α + α ˜ α = 4 X R,S A R A S x RS ∈ Z (3.8)From Table 3, the coefficients α i , ˜ α i do not diverge with N and M , and therefore theLHS also does not diverge. The group-theoretic factors A R and A S on the other hand,have positive leading terms divergent with N and M for all representations except thefundamental. Hence, R and S are restricted to be the fundamental representations of thegroups ˆ H ( M ) and H ( N ) respectively. The RHS of the above equation is, then, just fourtimes the number of bi-fundamentals and therefore non-negative. Since at most one α i , and– 10 –t most one ˜ α i can be negative, we can without loss of generality fix α > α ≥ ˜ α .If we require that α ˜ α + α ˜ α be divisible by four and non-negative, the possible valuesfor ( α , ˜ α , α , ˜ α ) are –1. (2 , − , , , − , , , , , H ( M ) × H ( N ). As a consequence, the number of hy-permultiplets charged under the fundamental representation of H ( N ) scales linearlywith M , which diverges. This is in contradiction with Table 3 where the number ofhypermultiplets in the fundamental is fixed at 16.2. (2 , , , n h − n v from ˆ H ( M ) × H ( N ) diverges with M, N . If such an infinite family were toexist, the same argument as before implies that ˜ G M,N must contain a classical groupfactor that grows without bound and cancels the divergent contribution to n h − n v .As this analysis shows, this factor must have ( α, ˜ α ) = (2 , n h − n v . Therefore, thereis no way that the divergent contribution from ˆ H ( M ) × H ( N ) can be compensatedby a negative divergence from factors in ˜ G M,N and so we have a contradiction in thiscase as well.3. (2 , − , − , , (2 , − , − , α = − α , ˜ α = − ˜ α . Thus, at any given value of the dilaton, the sign of thegauge kinetic terms is opposite for the two gauge group factors, and most be negativefor one gauge group. Thus, these models all have a perturbative instability. While itis possible that there is some way of understanding these tachyonic theories, they donot satisfy our condition of positive gauge kinetic terms, so we discard them. Notethat the first two of these five infinite families were discovered by Schwarz [23], andthe third was found in [17]. The argument presented here shows that this list of fivefamilies is comprehensive.Note that there are no families with three or more gauge groups of unbounded ranksatisfying anomaly factorization. The analysis above gives the values allowed of the α, ˜ α ’sfor each possible pair of groups. With three groups, additional bifundamental matter fieldsbetween each pair would be needed so that the matter content for any component groupcould not be among the possibilities listed in Table 3.This completes the proof of Case 2, showing that there are no infinite families con-taining a simple group factor of unbounded rank. This in turn completes the proof thatthere exists no infinite family of models with anomalies that can be cancelled by the Green-Schwarz mechanism and with kinetic terms positive in some region of moduli space. (cid:3)
4. Classification of models
In proving the finiteness of the number of possible gauge fields and matter representationswhich are possible in chiral 6D SUSY gauge theories, we have developed tools which could– 11 –auge Group Matter content Anomaly polynomial SU ( N ) × SU ( N ) 2( , ¯) ( X − Y + 2 Z )( X + 2 Y − Z ) SO (2 N + 8) × Sp ( N ) ( , ) ( X − Y + Z )( X + 2 Y − Z ) SU ( N ) × SO ( N + 8) ( , ) + ( ,
1) ( X + Y − Z )( X − Y + 2 Z ) SU ( N ) × SU ( N + 8) ( , ) + ( ,
1) + (1 , ) ( X − Y + 2 Z )( X + Y − Z ) Sp ( N ) × SU (2 N + 8) ( , ) + (1 , ) ( X − Y + 2 Z )( X + Y − Z ) Table 4:
Infinite families of anomaly-free 6D models, where the anomaly polynomial factorizesas shown.
X, Y, Z denote tr R , tr F , tr F respectively, where F is the field strength of the firstgauge group factor and F that of the second. In each of the above models, the number of neutralhypermultiplets is determined from the n h − n v = 244 condition. Note that the (cid:3) of SU ( N ) canbe exchanged for the ¯ (cid:3) , to generate a different model. lead directly to a systematic enumeration of all possible consistent low-energy models ofthis type. A previous enumeration of some of these models with one and two gauge groupfactors was carried out in [24].In particular, it is possible to use the gravitational anomaly and anomaly factorizationconditions to analyze possible gauge groups in a systematic way by considering each simplefactor of the gauge group separately. The anomaly conditions (2.9), (2.11) and (2.12)constrain the possible sets of matter fields which transform under any given simple factorin the gauge group, independent of what other factors appear in the full gauge group. Thegravitational anomaly (2.8) places a strong limit on the number of hypermultiplet matterfields which can be included in the theory. Since, as we found in several places in the proofin the preceding section, the contribution of components of the gauge group to n h − n v is generally positive, we can think of the problem of enumerating all possible gauge andmatter configurations for chiral 6D supergravity theories as like a kind of partition problem.This problem is complicated by the matter fields charged under more than one gauge group,which contribute to n h only once. Nonetheless, by analyzing individual simple factors andassociated allowed matter representations, we can determine a set of building blocks fromwhich all chiral 6D theories may be constructed. The contribution to n h − n v is reasonablylarge for most possible blocks (it seems that only a small number of gauge group factors andassociated matter configurations give negative contributions [which can only appear once]or positive contributions of much less than 30 or 40 to n h − n v ). Furthermore, the numberof bifundamental fields grows as the square of the number of type P factors in the gaugegroup, so it seems that the combinatorial possibilities for combining blocks are not too vast.A very crude estimate suggests that the total number of models may be under a billion,and that a complete tabulation of all consistent models is probably possible. Certainlywe do not expect anything like the ∼ distinct models which can arise from 4D typeIIB flux compactifications. It is also worth mentioning that as n t increases, the allowedcontribution to n h − n v decreases, so that the total number of possibilities may decreaserapidly for larger n t despite the more complicated anomaly-cancellation mechanism. Wewill give a more systematic discussion of the block-based construction of models in a futurepaper. – 12 –s an example of the limitations on this type of “building blocks”, consider SU ( N ) with x f hypermultiplet fields in the fundamental representation and x a in the antisymmetrictwo-index representation. The condition (2.9) relates x a and x f through x f = 2 N − x a ( N − n h − n v = 1 + N ( x f + 7 x a ) /
2. This quantity is necessarily positive, and can be usedto bound the allowed values of
N, x f , and x a either for a group with one factor SU ( N )or with several factors of which one is SU ( N ). A similar analysis can be carried outfor larger representations of SU ( N ). For example, if we allow the 3-index antisymmetricrepresentation , with a gauge group U ( N ) and no additional factors, we find that nomodel has matter in the representation unless N ≤
8. There are some examples of thistype of model, such as the SU(7) theory with matter 24 + 2 + , which satisfy anomalycancellation and which are (we believe) not yet identified as string compactifications § .A complete classification of allowed 6D theories would have a number of potentialapplications. In analogy to the story in 10D, where such a classification led to the discoveryof the heterotic E × E string, discovery of novel consistent low-energy models in sixdimensions may suggest new mechanisms of string compactification. Or, finding a setof apparently consistent theories which do not have string realizations might lead eitherto a discovery of new consistency conditions which need to be imposed on the low-energytheory, or to a clearer understanding of a 6D “swampland” [25, 26] of apparently consistenttheories not realized in string theory. If all theories satisfying the constraints we areusing here can be either definitively realized as string compactifications or shown to beinconsistent, it would prove the conjecture of “string universality” stated in [17] for chiral6D supersymmetric theories, at least for the class of models with one tensor multiplet andno U (1) gauge factors.The proof given here has shown that there are a finite number of distinct gauge groupsand matter content which can be realized in chiral 6D supergravity theories with one tensormultiplet. We have not, however, addressed the question of whether a given gauge groupand matter content can be associated with more than one UV-complete supergravity theory(by “theory” here, meaning really a continuous component of moduli space). In [27], weshowed that for one class of gauge groups, almost all anomaly-free matter configurationsare realized in a unique fashion in heterotic compactifications, but that some models canbe realized in distinct fashions characterized by topological invariants described in thatcase by the structure of a lattice embedding. It would be interesting to understand moregenerally the extent to which the models considered here have unique UV completions.A complete classification of allowed theories in six dimensions could lead to a bet-ter understanding of how various classes of string compactifications populate the stringlandscape. Such lessons might be helpful in understanding the more complicated case offour-dimensional field theories with gravity. As mentioned in Section 2, in some situationsthe anomaly constraints in six dimensions correspond precisely to the constraints on string § As this work was being completed we learned that Grassi and Morrison have found a local constructionof this kind of 3-index antisymmetric representation through string compactification in the language ofF-theory [22] – 13 –ompactifications; in both cases these constraints follow from various index theorems. Mak-ing this connection more precise in six dimensions could shed new light on the relationshipbetween string theory and low-energy effective theories in any dimension.
5. Conclusions and future directions
We have shown that for 6D (1 ,
0) supersymmetric theories with gravity, one tensor multi-plet, a semi-simple gauge group and hypermultiplet matter in an arbitrary representation,the conditions from anomaly cancellation and positivity of the kinetic terms suffice to provefiniteness of the set of possible gauge groups and matter content.In this analysis we have only considered semi-simple gauge groups. When there are U (1) factors in the gauge group, there is a generalized Green-Schwarz mechanism discussedin [28], which involves the tree-level exchange of a 0-form. Addressing the question offiniteness of theories including U (1) factors would require further analysis.We have also restricted attention here to theories with one tensor multiplet, whichadmit a low-energy Lagrangian description [29]. There are many string compactificationswhich give rise to 6D models with more than one tensor multiplet [30, 31, 32, 33, 34].While a proof of finiteness for models with more tensor multiplets is probably possible,the exchange of multiple anti-self-dual tensor fields in the Green-Schwarz mechanism asdescribed by Sagnotti [9] makes this analysis more complicated, and we leave a treatmentof such cases to future work.A comment may also be helpful on non-chiral theories with (1, 1) supersymmetry;this issue is addressed in further detail in Appendix B. A (1 ,
1) model in six dimensionsalso has (1 ,
0) supersymmetry, but contains an additional (1, 0) gravitino multiplet beyondthe gravity, tensor, hyper and vector (1, 0) multiplets in the theories we have consideredhere. It seems that one cannot include the gravitino multiplet without having (1 ,
1) localsupersymmetry, and we have restricted our attention here to models with (1 ,
0) supersym-metry without this gravitino multiplet. Since (1 ,
1) models are non-chiral, they cannotbe constrained by anomaly cancellation. Some further mechanism would be needed toconstrain the set of (1, 1) supersymmetric models in six dimensions. It is possible thatsuch constraints could be found by demonstrating that string-like excitations of the the-ory charged under the tensor field must be included in the quantum theory; anomalies inthe world-volume theory of these strings would then impose constraints on the 6D bulktheory, as suggested in [20]. It may also be that understanding the dictionary betweenanomaly constraints and constraints arising from string compactification for chiral theoriesmay suggest a new set of constraints even for non-chiral 6D theories.The result of this paper ties into the question of the number of topologically distinctCalabi-Yau manifolds, since 6D supergravity theories can be realized by compactificationof F-theory on elliptically fibered Calabi-Yau manifolds [35]. It has been shown by Gross[36] that there are only finitely many Calabi-Yau manifolds that admit an elliptic fibration,up to birational equivalence. If we can prove that the set of (1 ,
0) models with any numberof tensor multiplets is finite then this would constitute a “physics proof” of the theorem.– 14 –he total number of consistent 6D models of the type we consider does not seem to beenormous. It is not hard to imagine that these theories could be completely enumerated ina systematic manner. This programme would be very useful to understand the structureof the landscape and swampland [25] in this special case of six dimensions with (1 , E × E heterotic string, we areoptimistic that further study of the set of consistent 6D supergravity theories will helpus better understand the rich structure of string compactifications, perhaps giving lessonswhich will be relevant to the more challenging case of compactifications to four dimensions. Acknowledgements : We would like to thank Allan Adams, Michael Douglas, DanielFreedman, Ken Intriligator, John McGreevy, and David Morrison for helpful discussions.This research was supported by the DOE under contract
A. Proof of Claim 3.1
In the proof of finiteness in Section 3, we claimed that for any group G if X R x R tr R F = Tr F , (A.1)for x R ∈ Z , x R ≥ R runs over all the representations of G , then h − v = X R x R dim( R ) − dim( G ) ≥ c > x R = 1 for the adjoint representation and0 for all other representations. In this case, however, the kinetic terms for the gaugegroup factor G would be zero, and we are not considering this case (See Appendix Bfor a discussion of this situation in the context of (1 ,
1) supersymmetry). If x R > R with dim( R ) > dim( G ), equation (A.2) is automatically satisfied. We onlyneed to consider situations where x R = 0 for all representations R with dim( R ) > dim( G ). A.1 Exceptional groups
We first consider the exceptional groups G , F , E , E , E . The only representation thatsatisfies dim( R ) ≤ dim( G ) in all these cases is the fundamental. Equation (A.1) is satisfiedfor these groups, if the number of hypermultiplets are – • G : x f = 10 ⇒ h − v = 70 −
14 = 56 > • F : x f = 5 ⇒ h − v = 130 −
52 = 78 > • E : x f = 6 ⇒ h − v = 162 −
78 = 84 > • E : x f = 4 ⇒ h − v = 224 −
133 = 91 > • E : There are no solutions because the fundamental representation of E is theadjoint representation. – 15 – .2 SU ( N )We first consider the case when N ≥
4, and then the cases SU (2) and SU (3). The set ofirreducible representations are in one-one correspondence with the set of Young diagramswith up to N − X R x R C R = 6 (A.3)Since x R ≥
0, we only need to consider representations which have C R ≤
6, where thecoefficients C R are defined in (2.6). For a given representation R of SU ( N ), choose F = F T R + F T R . The generators T and T are given (in the fundamental representation)by ( T ) ab = δ a δ b − δ a δ b (A.4)( T ) ab = δ a δ b − δ a δ b , a, b = 1 , , · · · , N (A.5)Substituting this form of F into the definition (2.6) for C R , we havetr R ( F T R + F T R ) = (2 B R + 4 C R )( F + F ) + 8 C R F F (A.6)Comparing coefficients of F F on both sides, we have the following formula for C R – C R = 34 tr R [( T R ) ( T R ) ] (A.7)Using the above formular for C R , we can show that the only representations of SU ( N )that satisfy C R ≤ , , for all N, and ( C R = 6 for N = 6) (A.8) Examples :1. : We must compute the trace in (A.7). The only states that give a non-zerocontribution are 1 3 , , , C = 3 for the two-index symmetric representation.2. : The only states that contribute to the trace are –13 , , ,
24 (A.9)Again, here C = 3 – 16 –. : The following states contribute to the trace –1 3 i , i , i , i , i , i , i , i , , , , , , , , i denotes any of the remaining N − C = 34 (4( N −
4) + 4( N −
4) + 4 ×
8) = 6 N (A.10)Any representation of SU ( N ) can be represented as a Young diagram with at most N − C R >
6, by explicitly enumerating states that contributeto C R as in the examples above. In proving this generally, it is useful to note that addingboxes to any nonempty horizontal row only increases the value of C , so it is sufficient toconsider only the totally antisymmetric representations and case 3 above.The only solutions to (A.1) are –1. 1 + 1 : h − v = 1 >
0. In this case α = ˜ α = 0, so we discard this solution.2. 16 + 2 : h − v = 15 N + 1 >
03. 1 + 24 , ( N = 6): h − v = 20 + 144 −
35 = 129 > SU (2). Since B R = 0, (A.1) for SU (2) becomes X R x R C R = 8 (A.11)We therefore must enumerate all representations where C R ≤
8. The formula for C R issimpler in the SU (2) case — C R = 14 tr R [( T R ) ] (A.12)Young diagrams for SU (2) only have one row. For a diagram with c columns, the states · · · and · · · show that C R ≥ c / > c >
2. For c = 2, we have the adjointrepresentation with C R = 8, which would lead to α = ˜ α = 0. The only possible remainingsolution to (A.11) is 16 hypermultiplets, which has h − v = 29 > SU (3), (A.1) becomes X R x R C R = 9 (A.13)We can compute C R using formula (A.12) for the SU (3) case as well. The Young diagramsfor SU (3) contain at most two rows. For a diagram with c columns in the first row, thefollowing states 1 1 · · · · · · · · · · · · Matter h − v −
21 = 64 >
08 8 spinor + 4 64 + 32 −
28 = 68 >
09 4 spinor + 5 64 + 45 −
36 = 73 >
010 4 spinor + 6 64 + 60 −
45 = 79 >
011 2 spinor + 7 64 + 77 −
55 = 86 >
012 2 spinor + 8 64 + 96 −
66 = 94 >
013 1 spinor + 9 64 + 117 −
78 = 113 >
014 1 spinor + 10 64 + 126 −
91 = 99 > Table 5:
Matter hypermultiplets that solve (A.1) for SO ( N ), 7 ≤ N ≤
14. All solutions for
N > h − v . give a lower bound C R ≥ c /
2, with C R > c >
2. The only representations (besides theadjoint) with C R ≤ , with C R = 1 / , /
2. The only combination of matterrepresentations that satisfies (A.13) with one of α, ˜ α positive is 18 with h − v = 46 > A.3 SO ( N )For N ≤
6, these are related to other simple Lie groups, so we only consider N ≥
7. Wechoose the commuting generators T and T of SO ( N ), defined as( T ) ab = iδ a δ b − iδ a δ b (A.15)( T ) ab = iδ a δ b − iδ a δ b (A.16)Notice that the squared SO ( N ) generators are identical to the squared SU ( N ) generatorswe used in the previous section. Thus, the formulae for any Young diagram carry over. Inthe case of SO ( N ), the Young diagrams are restricted so that the total number of boxesin the first two columns does not exceed N [37]. The antisymmetric representation is theadjoint, and so we need to find all diagrams with C R ≤
3. The only diagrams that satisfythis requirement are – fundamental, 2-index antisymmetric and the 2-index symmetricrepresentation.In addition to these, we also have the spinor representations of SO ( N ). The irreduciblespinor representation of SO ( N ) has dimension 2 ⌊ ( N − / ⌋ . It is smaller than the adjointonly for N ≤
14, and it can be checked (using the tables in [38]) that the only spinorrepresentation that is smaller in dimension than the adjoint is the Weyl/Dirac spinor. Thetrace formula for these is [14]tr s F = 2 ⌊ ( N +1) / ⌋− tr F (A.17)tr s F = − ⌊ ( N +1) / ⌋− tr F + 3 · ⌊ ( N +1) / ⌋− (tr F ) (A.18)For the spinor representation C ≤ ≤ N ≤
14. For each of these cases, we can solvefor representations that solve (A.1), and check whether h − v is positive. This is the casefor all the solutions, and these are shown in Table 5.– 18 – .4 Sp ( N )By Sp ( N ), we mean the group of 2 N × N matrices that preserve a non-degenerate,skew-symmetric bilinear form. We choose as generators T , T in the fundamental rep-resentation as ( T ) ab = δ a δ b + δ a δ b (A.19)( T ) ab = δ a δ b + δ a δ b , a, b = 1 , · · · N (A.20)Again, these generators have been chosen so that their squares are equal to the squares ofthe generators of SU (2 N ). The Young diagrams for Sp ( N ) are similar to those of SU (2 N ),except that only diagrams with less than or equal to N rows need to be considered [37].The adjoint of Sp ( N ) is the symmetric representation. The analysis for the SU ( N ) casecarries through, except that we only need to consider representations with C ≤
3. Theonly representations with this property are – fundamental, 2-index antisymmetric (tracelessw.r.t skew-symmetric form) and the 2-index symmetric. The only solution is 1 + 16 ,and for this solution, h − v = 30 N + 1 >
0. This proves Claim 3.1 in Section 3. (cid:3) B. (1 , supersymmetry and the case of one adjoint hypermultiplet In this paper we have focused on (1 ,
0) theories with chiral matter content. In this appendixwe discuss how these theories differ from (1 ,
1) non-chiral 6D supergravity theories ¶ .One can imagine a (1 ,
0) theory where the matter content consists of precisely onehypermultiplet in the adjoint representation for each factor in the gauge group. Together,these (1 ,
0) multiplets combine to form the (1 ,
1) vector multiplet. The field content of thismultiplet is [13] A µ + 4 φ + ψ + + ψ − Since this matter content is non-chiral, there are no gauge anomalies or mixed gravitational-gauge anomalies; there is still a purely gravitational anomaly, which is cancelled in the usualway. In the framework of the discussion here, (2.9), (2.11), and (2.12) are all satisfied with α i = ˜ α i = 0. Thus, the gauge kinetic terms vanish and we do not consider models of thistype in the analysis here.It may seem that this contradicts the straightforward observation that all (1 ,
1) super-gravity theories can be thought of as (1 ,
0) gravity theories while the gauge fields in (1 , ,
1) (local) supersymmetry. To see this, note that the (1 ,
1) gravity multipletconsists of g µν + B + µν + B − µν + 4 A µ + φ + ψ + µ + ψ − µ + χ + + χ − =( g µν + B − µν + ψ − µ ) + ( B + µν + φ + χ + ) + (4 A µ + ψ + µ + χ − ) ¶ Thanks to Ken Intriligator for discussions on this issue. – 19 –n the (1 ,
0) language, this corresponds to the gravity multiplet, tensor multiplet andan additional gravitino multiplet. The (1 ,
0) gravitino multiplet consists of four abelianvectors, two Weyl fermions and one gravitino, and is scarcely even mentioned in the vastliterature on 6D models. The reason for this is that if we consider a (1 ,
0) model with onegravitino multiplet, it seems that the model must have (1 ,
1) supergravity. This is certainlythe case in string theories, where the vertex operator for the supercharge is the same as thatof the gravitino. More generally, the common lore [12] states that a massless, interactingspin-3/2 field must couple to a local conserved supercurrent. This fact is compatible withthe anomaly conditions. The anomaly polynomial in the case of one gravitino multiplet is I = − n h − n v R − n h − n v R ) −
196 tr R X i " Tr F i − X R x iR tr R F i + 124 " Tr F i − X R x iR tr R F i − X i,j,R,S x ijRS (tr R F i )(tr S F j ) (B.1)The gravitational anomaly can be cancelled only if n h − n v = 0. One way to cancel theanomaly is to have a single hypermultiplet transforming in the adjoint of the gauge group G . In this case, the matter content is that of a (1 ,
1) 6D model with gravity and one vectormultiplet. It would be nice to have a simple proof directly from the anomaly cancellationconditions that this is the only way to have n h − n v = 0, which would amount to a proofthat any (1 ,
0) theory with a gravitino multiplet would have (1 ,
1) supersymmetry. In thecase of two gravitino multiplets, it seems that the only consistent solution is the non-chiral,maximal (2 ,
2) 6D supergravity.We can now understand the apparent discrepancy alluded to above. A (1 ,
0) modelwith one hypermultiplet in the adjoint of the gauge group G has α = ˜ α = 0, which impliesthat the gauge kinetic terms are zero. However, in a (1 ,
1) model with a vector multipletcorresponding to G , the gauge kinetic term is positive. (An easy way to see this is to considera T compactification of the heterotic string, which gives (1 ,
1) supergravity in 6D). Thisapparent inconsistency is due to the fact that in the case of the (1 ,
1) theory, α = 0 , ˜ α = 0,and this is sufficient to ensure that the anomaly polynomial vanishes. The B -field hasthe usual Chern-Simons coupling that makes it transform under gauge transformations.However, there is no B ∧ d ˜ ω term; the theory is non-anomalous and therefore there is noneed for a Green-Schwarz counterterm.Thus, we see that the conditions we impose in this paper on the field content of the(1 ,
0) theory exclude the gravitino multiplet needed to complete the (1 ,
0) graviton multipletto a (1 ,
1) graviton multiplet. As a result, the class of theories considered here does notinclude theories with (1 ,
1) supersymmetry; the (1 ,
1) theories are all anomaly free andwould need to be constrained by some alternative mechanism as discussed in 5.
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