Mersenne Primes, Polygonal Anomalies and String Theories Classification

It is pointed out that the Mersenne primes Mp = (2 p − 1) and associated perfect numbers Mp = 2 p−1 Mp play a significant role in string theory; this observation may suggest a classification of consistent string theories. Typeset using REVTEX 1 Anomalies and their avoidance have provided a guidepost in constraining viable particle physics theories. From the standard model to superstrings, the importance of finding models where the concelation of local and global anomalies that spoil local invariance properties of theories, and hence render them inconsistent, cannot be overestimated. The fact that anomalous thories can be dropped from contention has made progress toward the true theory of elementary particles proceed at an enormously accelerated rate. Here we take up a systematic search, informed by previous results and as yet partially understood connections to number theory, for theories free of leading gauge anomalies in higher dimensions. We will find new cases and be able to place previous results in perspective. In number theory a very important role is played by the Mersenne primes Mp based on the formula Mp = 2 p − 1 (1) where p is a prime number. Mp is sometimes itself a prime number. The first 33 such Mersenne primes correspond [1–3] to prime numbers below one million: p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433. (2) As a comparison to this remarkable sequence of the first 33 Mersenne primes, there are altogether 78498 primes below one million so that Eq.(1), although an invaluable source of large prime numbers, far more often generates a composite number than a prime. On the occasion that Eq.(1) does generate a prime, an immediate derivative thereof is the perfect number which we shall designate Mp given by Mp = 2 Mp. It is straightforward and pleasurable to prove in general that Mp is perfect, defined as Mp equalling the sum of all of its divisors. For example, M2 = 6 = 1 + 2 + 3, M3 = 28 = 1 + 2 + 4 + 7 + 14, and so on. The Mp are the only even perfect numbers; it is unknown if there is an odd perfect number but if there is one it is known [4] that it is larger than 10. 2 In the present Letter, we shall associate the perfect numbers derived from Mersenne primes with the polygonal anomalies whose cancellation underlies the successful string theories. For example, heterotic and type-I superstrings in ten dimensions are selected to have gauge groups O(32) and E(8) × E(8) on the basis of anomaly cancellation of the hexagon anomaly [5–8]. Equivalently, these two superstrings correspond to the only self-dual lattices in 16 dimensions: Γ8 ⊕ Γ8 and Γ16 [9]. The dimension of these two acceptable gauge groups in d = 10 is dim(G) = 496 = M5, indeed a perfect number of the Mersenne sequence. Further motivation in low dimensions for consideration of the perfect number comes from [10] M3 the SO(8) and G2 × G2 supergravities in 6 dimensions for M3 , from noting that 0(4) and SU(2)×SU(2) are anomaly free in four dimensions for M2 and from the existence of an N = 2 world sheet supersymmetric string theory in 2 dimensions [11]with gauge group SO(2) ∼ U(1) for M1. The appropriate polygon for spacetime dimension d is the l−agon where l = ( 2 + 1). One way to discover the significance of Mp and Mp in string theory is to recognize that the leading l−agon anomaly for a k-rank tensor of SU(N) or O(N) is given [5,7] by a generalized Eulerian number ( the Eulerian numbers are AN(N, k))