A new infinite class of 6 d , N = (1 , 0) supergravities

We present a new infinite class of non-abelian, 6d supergravities with eight supercharges. These theories not only satisfy all known low-energy consistency conditions, such as being free of anomalies, but also evade the constraints arising from the consistency of string probes, even after assuming BPS completeness. This demonstrates that some additional UV input or hitherto unknown IR condition is needed in order to be left with a finite landscape, as is generally anticipated from a theory of quantum gravity.


I. INTRODUCTION
In recent years there has been much progress in understanding to what extend string universality holds in higher dimensions.Together, supersymmetry and anomaly cancellation constrain the space of allowed lowenergy theories significantly and one can show that there is an exact match with the string theory landscape for d = 8, 9, 10 [1][2][3][4][5][6].For d = 7 there is also a match, up to a classification of three-dimensional N = 4 SCFTs [7].
The situation changes dramatically in six dimensions, the maximal dimension which allows for only eight supercharges.While it has been shown that the number of anomaly-free theories is finite provided the number of tensor multiplets is bounded as T < 9 [8], there do appear infinite families for T ≥ 9. Nevertheless, assuming BPS completeness and using the anomaly-inflow arguments developed in [2] for string probes has proven very effective at truncating all of the (previously) known infinite families to a finite subset [2,9].
Recently, in [10] it was shown that there is a wide class of infinite families which satisfy all (known) low-energy consistency conditions.These are built starting from a "seed" theory, which is very loosely constrained, and augmenting by a huge number of exceptional gauge factors.Since the resulting gauge groups are enormous, it was anticipated that this class of theories would similarly be restricted to a finite number upon considering the constraints imposed by string probes; the purpose of this short article is to show that this is not the case.
The remainder of this article is organized as follows.In §II we recall the consistency conditions required of 6d supergravities with minimal supersymmetry.Next, in §III we present a new class of infinite families of anomalyfree theories in which the gauge group and numbers of tensor-and hypermultiplets are all controlled by a single parameter m ∈ Z >0 and in §III B we prove that all of the constraints coming from the consistency of string probes derived in [2] are satisfied for infinitely many values of m.Finally, we conclude in §IV.* gloges@post.kek.jp

II. CONSISTENCY CONDITIONS
For our purposes, specifying a theory amounts to choosing the following data: the number of tensor multiplets T ≥ 0, the non-abelian gauge group G = i G i with V = dim G corresponding vector multiplets transforming in the adjoint representation, the total (generally reducible) representation H of G for H = dim H hypermultiplets, and the anomaly vectors b I ∈ R 1,T (I = 0, i) [11].There is one self-dual 2-form field from the gravity supermultiplet and T anti-self-dual 2-form fields, one from each tensor multiplet, and the vectors b I control their couplings to the graviton and gauge vectors.
These data cannot be chosen freely.From low-energy considerations alone, gauge and gravity anomalies must be fully cancelled and what we call positivity and unimodularity conditions must be satisfied.Assuming BPS completeness (e.g.see [12][13][14]), there must be BPS strings charged under the (anti-)self-dual 2-form fields.When these string probes cannot be coupled to gravity, this signals the theory should be discarded as inconsistent.We recall each of these conditions in turn.

A. Anomaly cancellation
Gauge and gravitational anomalies may be cancelled by means of the Green-Schwarz-West-Sagnotti mechanism [15] (see also [16][17][18][19][20]) wherein the usual 1-loop contributions from chiral fermions are balanced against the tree-level exchange of the (anti-)self dual 2-form fields.All together, we have where irreducible term in Î8 there is a corresponding constraint: tr R 4 : tr The remaining, reducible terms determine all inner products amongst the vectors b I : In relating traces in a representation R of G i to the trace in the fundamental, we have introduced the indices With this choice of normalization A i R , B i R and C i R are nearly always integers [21] and we have A i Adj = 2h ∨ i with h ∨ i the duel Coxeter number of G i .See Table II for the indices of some common irreducible representations with this normalization.For the simple groups SU(2), SU(3), E n , F 4 and G 2 there is no independent quartic Casimir invariant and thus B i R = 0 for all representations R. Finally, there are additional potential global anomalies for SU(2), SU(3) and G 2 gauge factors [22][23][24][25] (manifesting as constraints modulo 12, 6 and 3, respectively) which are easily avoided.It has been shown that the absence of global anomalies follows from the absence of local anomalies and that all of the inner products b I • b J are integers [8].That is, Λ :=  lattice and the eigenvalue bounds discussed above are no longer sufficient.

B. Positivity and unimodularity
Scalars in the tensor multiplets parametrize the tensor branch of the moduli space, j ∈ R 1,T with j • j = 1.The gauge kinetic terms are −j • b i tr F 2 i , so we require that there exists some choice for j ∈ R 1,T with j • j > 0 that gives j • b i > 0 for all i.The analogous quantity j • b 0 controls the coefficient of the Gauss-Bonnet term, and although we do not demand that it is strictly positive we will see that in fact j • b 0 > holds for the class of theories constructed in §III.
As we saw above, the absence of anomalies implies that the lattice Λ is integral.Λ is a sub-lattice of the string charge lattice Γ and in [26] it was shown by reducing to four and two dimensions that Γ must be a unimodular (i.e.integral and self-dual) lattice of signature (1, T ).
For the examples discussed in §III, this condition will be manifestly satisfied since we will realize b I directly as elements of the odd unimodular lattice Z 1,T .We note that in general, even if the lattice Λ ⊂ R 1,T is completely fixed up to O(1, T ; R) transformations, there may be several inequivalent ways to realize Λ as a sub-lattice of Γ (i.e. with different Γ/Λ or equivalently no O(1, T ; Z) transformation relating them).

C. String probes and anomaly inflow
When a string probe charged under the 2-form fields is introduced into a background configuration, in general this induces anomalies on the worldsheet which can then be cancelled by the anomaly-inflow mechanism [2,[27][28][29][30].The central charges were computed in [2]: after subtracting off the center-of-mass contributions which decouple in the IR, these read c L and c R are the gravitational central charges for the left-and right-moving sectors and the levels k ℓ and k i correspond to SU(2) ℓ and the bulk gauge symmetry G i , respectively.When all quantities are non-negative, a non-trivial constraint arises from requiring that c L is large enough to accommodate a unitary representation of the current algebra, namely Showing that a theory is inconsistent amounts to an existential statement: In contrast, showing that a theory is not revealed to be inconsistent via anomaly inflow is a universal statement and can be considerably more difficult to establish: The computations of §III B amount to a proof of exactly this statement for the class of theories we will describe.
There, it will be convenient to refer to charges Q ∈ Γ satisfying Eqn.(7) as admissible.The region of the Q • Q vs. Q • b 0 plane carved out by three of the conditions, c L ≥ 0, c R ≥ 0 and k ℓ ≥ 0, is shown in Fig. 1; we will make repeated use of the following weaker bounds, the second clearly only being useful for Q • b 0 ≥ 1.

III. A NEW CLASS OF INFINITE FAMILIES A. Preliminaries
Our starting point will be a "seed" infinite family with simple, non-abelian group G seed and charged hypermultiplets H ch seed chosen so that is constant.This requires H ch seed to be of the form where we take both H ch 0 and H ch 1 to be some fixed (nontrivial) representations of G seed .The hypermultiplets in H ch 0 determine δ and must satisfy the constraint of Eqn.(3), while the hypermultiplets in H ch 1 must satisfy so that x ≥ 0 can be chosen freely.For each such seed, we can construct the following infinite family depending only on m ≥ 1, where we choose x to grow with m in such a way that The (integer) constants γ δ and C δ have been introduced for later convenience: the reason for having different cases based on the value of δ modulo four is to allow for b I ∈ Z 1,T , as we will see shortly.Let us immediately check that the gravitational anomaly can be cancelled for arbitrarily large m.Using Eqn. ( 14), if we separate the contributions from H ch 0 and H ch 1 and write then from Eqn. (17) we have x = (const.)+ 24m (b•b)1 and Therefore is required so that H ch − V + 29T decreases with m.By taking m large enough we can ensure that H ch − V + 29T ≤ 273 for any choice of H ch 0 and then by adding in an appropriate number of neutral hypermultiplets, the constraint of Eqn. ( 2) can be met exactly.However, the simple group G seed is indirectly restricted to have small rank via Eqn.( 20): a complete list of possibilities for G seed and H ch 1 is given in Table III.From Eqns. ( 13), ( 16) and ( 17), the inner products are These can be realized by the integer vectors where r ∈ {0, 1, . . ., m − 1} and t ∈ {1, 2, 3, 4}.We have introduced both and the following 12-component vectors: Also, it is straightforward to check that gives time-like j with j •b 0 > 0, j •b seed > 0 and j •b E8 i > when 24m + 4C δ γ δ > max{0, 2δ} and ϵ > 0 is small enough.
In summary, we conclude that all of the anomaly cancellation, positivity and unimodularity conditions are met, provided Eqn. ( 20) is satisfied and m is taken large enough to ensure that H ch − V + 29T ≤ 273 and 24m + 4C δ γ δ > max{0, 2δ}.In the next section we will make use of one additional mild lower bound on m, which is non-trivial only for δ = −1, 1, 3, 7 or δ ≤ −3.Also, for the most part we will only need the average E anomaly-vector, This must satisfy k E8 avg := Q • b E8 avg ≥ 0 as well, but k E8 avg is clearly no longer necessarily integer-valued.

B. Constraints from string probes are satisfied
It only remains to show that the constraints imposed by string probes are satisfied for all admissible Q.While we have in mind the situation where m is large (as must ultimately occur if this is to be an infinite family), this will only serve to guide the analysis and suggest a line of attack: we make no approximations or large-m expansions and all of the inequalities we derive, while not necessarily sharp, are exact.
By design, the vectors b I have the following three key features: (i) there are no "free" components of b 0 which could be leveraged to decrease 4r+t are all of opposite sign (or zero), and (iii) since on average the contributions to Q•b E8 i from each of these groups of 12 components are proportional to the corresponding contributions to Q • b 0 .This is the reason why the number of E 8 factors was chosen to be a multiple of four.These three features together lead to two mechanisms which will ensure that c L on the right-hand side of Eqn. ( 8) outpaces the left-hand side as m increases.The first is that the cone described by i ≥ 0 is restricted to be quite narrow thanks to (ii), and due to charge quantization the smallest non-zero admissible Q is therefore necessarily large.The second is that together (i) and (iii) will ultimately force Q • b 0 to be positive which, as we saw in Eqn.(12), also provides a non-trivial lower bound c L ≥ 12Q • b 0 − 4.
We now set out to prove that (10) holds for all largeenough m.To begin, write the string charge as Q := 1 2 (q + + q − ); 1 2 (q + − q − ), q 1 , . . ., q 12m , (29) where the "light-cone charges" q ± must have the same parity.The inner products with Q are where in order to reduce clutter we will leave all sums over a ∈ {1, 2, . . ., 12m} unadorned, other than primes such as in Q•b seed above which indicate that for δ even the indices a ≡ 1, 2, 3, 4 mod 12 are omitted: cf.Eqn.(24).Notice that if G seed and the corresponding vector b seed were absent, then the charge for which c L = 8, c R = 0, k ℓ = 0 and k E8 i = 1, would immediately reveal the theories with G = E 4m 8 and no charged hypermultiplets to be inconsistent for all m ≥ 1 [31].However, the requirement of having non-negative level, Q • b seed ≥ 0, presents an obstacle to choosing such a charge.

An upper bound
To warm up, let us first bound the left-hand side of Eqn.(8) from above.Using the inequality which holds for any set of N non-negative real numbers [32], we have Therefore ≤ min 1, It will be important that this quantity grows as quickly as 992m only if k E8 avg ≳ 30 is appreciable.For small k E8 avg the growth with m is greatly reduced.

A bound on q±
Next, we claim that q ± > 0 is required for a nonzero charge to be admissible.The alternatives are each quickly ruled out in turn: and |q ± | = 1.However, using W δ (m) > C δ (which follows easily from Eqn. ( 26)) in ( 32) and (33) we see that no choice of signs for q ± allows for both one of the q a is nonzero, say q b , and that avg ≥ 0 the only possibility is q + = 0 and q − ≥ 0. However, k ℓ ≥ 0 now gives and since q − ≡ q + mod 2 we must have q − = 0 as well.
If q b = 0, then Q = 0 and we are done.Otherwise, if but then from the form of ⃗ v t it is clear that we cannot choose q b so that all Q • b E8 i are non-negative.
and so (26).The second line above follows from the Cauchy-Schwarz inequality.

A lower bound on cL
Continuing with q ± > 0, we immediately have that Q • b 0 is positive by combining Eqns.( 31) and ( 33): If we use the crude bounds q ± ≥ 1 then we find Q • b 0 ≥ 2m(1 + k E8 avg ) + 3 and c L ≥ 24m(1 + k E8 avg ) + 32 after using Eqn.(12).However, this is not sufficient to ensure that Eqn. ( 8) is satisfied since it is well below the upper bound from Eqn. (37).By finding better bounds on q ± we will be able to improve the lower bound on c L .
Inspired by Fig. 1, we should expect that the most constraining bound comes from k ℓ ≥ 0 since we have already established that Q•b 0 ≥ 1.Using Eqns.( 30), ( 31) and (33) in k ℓ ≥ 0, we find again using Cauchy-Schwarz for the second inequality.
For large m we must have q − ≳ O(mq + ) which means that Q roughly aligns with b seed and thus also j in Eqn.(25), as expected: note that together b seed •b seed > 0, Q•Q > 0, j •j > 0, Q•b seed > 0 and j •b seed > 0 imply that the string's tension j • Q is automatically positive.The right-hand side above is manifestly positive and therefore we must have q + ≥ 3 and q − ≥ 2. Already this improves the earlier bounds to Q • b 0 ≥ 2m(3 + k E8 avg ) + 7 and c L ≥ 24m(3 + k E8 avg ) + 80, but still this is insufficient.We can clearly do much better since the right-hand side above is ≥ 9m; dividing through by q + − 2 > 0 to bound q − and using Eqn.(41), we find A quick calculation shows that for fixed k E8 avg ≥ 0 and q + ≥ 3, C(q + , k E8 avg ) is minimized at and that This gives us our final lower bound on c L :

Summary
In summary, we have shown that for all non-zero charges Q ∈ Γ satisfying Eqn.(7), for all k E8 avg ≥ 0 so that, given that dim G seed is fixed, we can always ensure Eqn. ( 8) is satisfied by taking m large enough.Therefore we conclude that, provided only that Eqn. ( 20) is met, the theories of Eqn. ( 16) satisfy all of the consistency conditions, including those stemming from the consistency of string probes, for infinitely many values of m.

IV. DISCUSSION
In this article we have demonstrated that the landscape of consistent 6d supergravities with eight supercharges and non-abelian gauge group is infinite, even after assuming BPS completeness and requiring the consistency of all string probes.This is in stark contrast to the situation in higher dimensions where supersymmetry and anomalies together lead to an exact match with the finite string landscape.
The examples we have constructed have very few objectionable features other than their large number of degrees of freedom and inclusion of many exceptional groups: all of the hypermultiplets can be chosen to be in standard representations (e.g.fundamental, two-index (anti-)symmetric, spinor and adjoint); we have the usual choice b 0 = (3; 1 T ) which is both a characteristic and primitive vector of the string charge lattice; although non-zero admissible charges must be large, there is no unnatural hierarchy since Q • Q ≳ O(m) arises from requiring Q have non-negative inner product with O(m) distinct vectors of norm O(1) and one vector of norm O(m).
For concreteness we have considered theories where H ch , V and T are tied together through the single parameter m.Given that not all of the inequalities are sharp and Eqn. ( 49) is satisfied by such a wide margin, it seems reasonable to expect that this is just the tip of the iceberg and more general examples of a similar nature could be found.There are a few obvious places for generalization: 1.The auxiliary E 8 gauge factors used here can likely be replaced by E 6 , E 7 or E 7 + 1 2 56 (with the vectors b I adjusted accordingly) without too much trouble: these are the four combinations identified in [10] as leading to infinite families with T unbounded.A key feature of the class of theories presented here, however, appears to be that H ch is unbounded: attempts to adapt Eqn.(16) to have H ch seed (and thus also b seed • b seed ) constant were unsuccessful, although perhaps an entirely different structure for the vectors b I which facilitates this could be engineered.
How can we recover a finite landscape, as generally expected from a theory of quantum gravity?Certainly there are no known ways to construct theories with an unbounded number of gauge factors or tensor multiplets from string theory.The class of theories constructed here provide very strong guidelines for any future attempts to definitively prove finiteness of the supergravity landscape.For example, it is not enough to bound the rank of individual gauge factors or limit the possible hypermultiplet representations since the examples above have rank G i ≤ 8 and hypermultiplets can be chosen to only appear in fundamental representations: the proposals of [9] are easily met.There appears to be two possible ways forward.(i) Some universal bound on one of T , V or H ch , perhaps in connection with the species scale [33][34][35] (which here clearly decreases rapidly with m), places an upper bound on m. (ii) Additional global anomalies kill these families at large T ∼ m, such as those of Dai-Freed type recently studied in [36] for T ≤ 1 [37].It may also be fruitful to consider the introduction of brane probes of other dimensionality, although their presence is then no longer guaranteed by the completeness hypothesis.We leave demoting this class of infinite families to the swampland for the bright future.
n i R gives the number of hypermultiplets in the representation R of the gauge factor G i .Of course, since the vectors b I live in R 1,T , the matrix of inner products b I • b J can have at most one positive eigenvalue and at most T negative eigenvalues.These bounds on the signature of b I • b J are actually necessary and sufficient to ensure that there exist vectors b I ∈ R 1,T which realize the inner products dictated by the massless spectrum.In cases where b I • b J has a positive eigenvalue or T negative eigenvalues, the vectors b I ∈ R 1,T are uniquely determined by their inner products up to O(1, T ; R) transformations.
the components of b seed and each quartet 4 t=1 b E8

2 . 3 .and by taking b seed 1 = b seed 2 the
We took b seed • b seed and b 0 • b seed to grow at the same rate with m (cf.Eqn.(13)); one could imagine relaxing this and allowing for different constants of proportionality in place of the "2" in Eqn.(15), which likely would change the requirement of Eqn.(20).We considered cases where G seed is a simple group, but it is possible to have G seed semi-simple.As a simple example, one can chooseG seed = Sp(2) × Sp(2) , H ch seed = (10, 1) ⊕ (1, 10) ⊕ x(4, 4) ⊕ (5, 1) ⊕ (1,analysis of §III continues to hold with essentially no changes.It may be possible to find more general families with semisimple G seed and vectors b I which realize the same mechanisms leveraged here.

TABLE I .
Normalization constants and dual Coxeter numbers for simple gauge factors.
1,T is a vector of 4-forms and the λ i are normalization constants given in TableI.The theory is free of local anomalies exactly when Î8 = 0.For each