Discrete Flavour Symmetries from the Heisenberg Group

Non-abelian discrete symmetries are of particular importance in model building. They are mainly invoked to explain the various fermion mass hierarchies and forbid dangerous superpotential terms. In string models they are usually associated to the geometry of the compactification manifold and more particularly to the magnetised branes in toroidal compactifications. Motivated by these facts, in this note we propose a unified framework to construct representations of finite discrete family groups based on the automorphisms of the discrete and finite Heisenberg group. We focus in particular in the $PSL_2(p)$ groups which contain the phenomenologically interesting cases.


Introduction
Non-abelian discrete symmetries play a prominent rôle in model building.Among other objectives, more than a decade ago, they have been widely used to interpret the neutrino data in various extensions of the Standard Mode [1,2,3].The ensuing years there were attempts to construct them in the context of string theory models.Within this framework, important activity has been focused on elaborating predictions for physically measurable quantities such as mass textures and CP-violating matrices.Indeed, (non-abelian) discrete symmetries in string theory emerge in the context of various compactifications and recently they have attracted considerable attention [4]- [10].In fact, it has been realised that they can act as family symmetries which restrict the arbitrary Yukawa parameters of the superpotential and lead to acceptable quark and lepton masses and mixing.Moreover, they can suppress undesired proton decay operators and various -yet unobserved-flavour violating interactions.
More recently, the implementation of the idea of discrete symmetries has also been considered in F-theory constructions.In F-theory [11] the elliptically fibred space consists of a K3 manifold with a torus attached at each point.The τ -modulus of the torus is defined in terms of the two scalar fields of the type IIB string theory and the fibration is described by the Weierstraß model.According to the standard interpretation, the associated geometric singularities (classified as ADE types) are linked to the gauge symmetries of the effective models.The highest singularity of the elliptic fibration is described by the E 8 exceptional symmetry, so that ordinary successful GUT symmetries such as SU (5) and SO (10) are easily embedded [12] in the maximal group E 8 and correspond to a particular divisor of the internal manifold.Hence, the remaining symmetry can in principle accommodate some suitable non-abelian discrete group which could act as a family symmetry.If, for example, the GUT model is SU (5) which is the minimal symmetry accommodating the Standard Model, then the commutant with respect to E 8 is also SU (5) (denoted usually as perpendicular, SU (5) ⊥ , to the GUT).The latter naturally incorporates phenomenologically viable non-abelian discrete groups [9], such as S n , A n where usually n ≤ 5 and more generally P SL 2 (p), where p ≤ 11.
The last couple of years, in the context of F-theory, several works focused also in the low energy implications of the two torus geometry, in a different approach.In general, discrete symmetries in these constructions are of Abelian nature.Such cases are the Torsion part of the Mordell-Weil group of rational points on elliptic curves and more generally the Tate-Shafarevich group which has been shown to determine the discrete symmetries arising in F-theory [6].
Furthermore, there exist cases in string theory [4] where non-abelian finite groups may emerge as well.In this context a class of non-Abelian discrete symmetries may arise from discrete isometries of the torus geometry, on which the Heiseberg group has a natural action.Discrete non-abelian symmetries are also realised in magnetised D-brane models in toroidal compactifications [4,8].Along these lines, explicit F-theory constructions have also recently appeared [10].Let us finally note that finite groups as subgroups of continuous non-abelian symmetries have been discussed and classified in an orbifold context [13].
Motivated by the above facts, in the present work we will develop a unified method for the construction of the smaller non-trivial representations of certain finite groups.Because their main rôle in particle physics is to discriminate the three fermion families we are focusing mainly on the P SL 2 (p) groups possessing triplet representations.
2 The non-abelian discrete groups SL 2 (p) and P SL 2 (p) Among the various discrete symmetries used to interpret the fermion mass hierarchy are the special linear groups SL 2 (p) [14,15] and their corresponding projective restrictions P SL 2 (p).Some of these groups coincide with the symmetries of regular polyedra in three or higher dimensional space dimensions.Not all the representations of these groups are relevant for model building.Because only three families of fermions exist in nature, only the groups with particular representations related to Yukawa and gauge couplings are considered in the literature [1,2,3].
Most of the phenomenologically viable cases, are included in the projective linear groups P SL 2 (p) for p = 3, 5, 7 and 11.All of them support triplets which are suitable representations to accommodate the three fermion generations.More specifically, for p = 3 we obtain P SL 2 (3) which is isomorphic to the alternating group A 4 .Furthermore, the group P SL 2 (5) is isomorphic to the smallest non-abelian simple group A 5 .The case of P SL 2 (7) is also phenomenologically interesting [16,17,18].
The P SL 2 (p) groups for values p ≤ 11 can be naturally incorporated in an Ftheory context.Indeed, for the most common grand unified (GUT) models such as SU (5), SO(10), E 6 embedded in the E 8 singularity, the possible gauge groups which could act as family symmetries can be read off from the embedding formula Therefore, we have the following cases 2) 3) From the above, we see that the corresponding flavour discrete groups should be embedded in SU (3) ⊥ , SU (4) ⊥ and SU (5) ⊥ .Indeed, A 4,5 are subgroups of SO(3) ∼ SU (2), P SL 2 (7) is a subgroup of SU (3) and P SL 2 (11) is contained in SU (5).
In this work, we present a unified approach for constructing the explicit relevant representations of these groups.The reason is that in the literature, up to our knowledge, while only explicit ad-hoc constructions have been presented, a systematic use of the theory of the representation of these particular groups does not exist yet.
3 Definition of SL 2 (p) and P SL 2 (p) groups The SL 2 (p) group is defined in the simplest way as a group of 2 × 2 matrices with elements integers modulo p, where p is a prime integer, and determinant one modulo p.These groups usually are generated by two elements which obey certain conditions and these define what is called a presentation of the group.These conditions depend on p but there is a universal presentation given by two generators which are conjugate one to another under the matrix These universal elements (for any value of p) are the following two matrices We can readily observe that each one of them generate an abelian group of order p and that they satisfy the Braid relation In the literature another basis of generators is used which is called Artin's presentation.These are defined through the Braid generators, as and so they are They satisfy the relations We can invert the relation of the two sets of generators as Apart from equations (3.2), the presentation in terms of a, b contains additional relations depending on the value of p.
The group SL 2 (p) has a normal subgroup of order two Z 2 = {1, −1}, so the coset space SL 2 (p)/{1, −1} is a group which is called the projective group P SL 2 (p).
Our aim is to construct some basic non-trivial irreducible representations of P SL 2 (p) out of which all others -physically relevant-are generated by tensor products.To this end, we are going to use a particular representation of SL 2 (p) which is known as the Weil metaplectic representation.In the physics literature is has been introduced by the work of Balian and Itzykson [19].Many details of this particular unitary representation with various applications has been presented in [20,21].As it will be shown in the next section, this representation is reducible in two irreducible unitary representations of dimensions p+1 2 and p−1 2 .Thus, we obtain discrete subgroups of the unitary groups SU ( p±1 2 ).For example, when p = 3 we obtain discrete subgroups of SU (2) and U (1), for p = 5 we get discrete subgroups of SU (3) and SU (2) and so on.The Finite Heisenberg group HW p [22], is defined as the set of 3 × 3 matrices of the form where r, s, t belong to Z p (integers modulo p), where the multiplication of two elements is carried modulo p.
When p is a prime integer there is a unique p-dimensional unitary irreducible and faithful representation of this group, given by the following matrices where ω = e 2πi/p , i.e. the p th primitive root of unity and the matrices P, Q are defined as where k, l = 0, . . ., p − 1.
It is to be observed that, if ω is replaced with ω k , for k = 1, 2, ..., p − 1 all the relations above remain intact.Since p is prime all the resulting representations are p-dimensional and inequivalent.
The matrices P, Q satisfy the fundamental Heisenberg commutation relation of Quantum Mechanics in an exponentiated form In the above, Q represents the position operator on the circle Z p of the p roots of unity and P the corresponding momentum operator.These two operators are related by the diagonalising unitary matrix F of P , so F is the celebrated Discrete Fourier Transform matrix An important subset of HW p consists of the magnetic translations J r,s = ω rs/2 P r Q s (4.8) with r, s = 0, . . ., p − 1.These matrices are unitary (J † r,s = J −r,−s ) and traceless, and they form a basis for the Lie algebra of SL(p, C).They satisfy the important relation This relation implies that the magnetic translations form a projective representation of the translation group Z p × Z p .The factor of 1/2 in the exponent of (4.9) must be taken modulo p.
The SL 2 (p) appears here as the automorphism group of magnetic translations and this defines the Weil's metaplectic representation.If we consider the action of an element A = a b c d on the coordinates (r, s) of the periodic torus Z p × Z p , this induces a unitary automorphism U (A) on the magnetic translations, since the representation of Heisenberg group is unitary and irreducible, This relation determines U (A) up to a phase and in the case of A ∈ SL 2 (p), the phase can be fixed to give an exact (and not projective) unitary representation of SL 2 (p).
The detailed formula of U (A) has been given by Balian and Itzykson [19].Depending on the specific values of the a, b, c, d parameters of the matrix A, we distinguish the following cases: ) where δ = 2 − a − d and σ(a) is the Quadratic Gauss sum given by while the Legendre symbol takes the values (a|p) = ±1 depending whether a is or is not a square modulo p.
It is possible to perform explicitly the above Gaussian sums noticing that where all indices take the values k, l, r, s = 0, . . ., p − 1.This has been done in [20,21].
In the case δ = 2 − a − d = 0 and c = 0, the result is If c = 0, then we transform the matrix A to one with c = 0.The other cases δ = 0 can be worked out easily using the matrix elements of J r,s given in (4.16).
It is interesting to notice that redefining ω to become ω k for k = 1, 2, ..., p − 1, the matrix U (A) transforms to the matrix U (A k ), where A k is the 2 × 2 matrix A k = is -up to a phase-the Discrete Finite Fourier Transform (4.7) where n = 0 for p = 4k + 1 and n = 1 for p = 4k − 1.
The Fourier Transform matrix generates a fourth order abelian group with elements The matrix S represents the element a 2 = −1 0 0 −1 . Its matrix elements are Because the action of S on J r,s changes the signs of r, s, while ∀A ∈ SL 2 (p) the unitary matrix U (A) depends quadratically on r, s in the sum (4.11), it turns out that S commutes with all U (A).Moreover, S 2 = I and we can construct two projectors with dimensions of their invariant subspaces p+1 2 and p−1 2 correspondingly.So the Weil p-dimensional representation is the direct sum of two irreducible unitary representations To obtain the block diagonal form of the above matrices U ± (A), we rotate with the orthogonal matrix of the eigenvectors of S. This p-dimensional orthogonal matrix, dubbed here O p , can be obtained in a maximally symmetric form (along the diagonal as well as along the anti-diagonal) using the eigenvectors of S in the following order: In the first (p + 1)/2 columns we put the eigenvectors of S of eigenvalue equal to 1, and in the next (p − 1)/2 columns the eigenvectors of eigenvalue equal to −1 in the specific order given below: where k = 0, . . ., p − 1.
Different orderings of eigenvectors may lead to different forms of the matrices U ± (A).The so obtained orthogonal matrix O p has the property due to its symmetric form.
The final block diagonal form of U ± (A) is obtained through an O p rotation 5 The construction of the SL 2 (p) generators In this section we are going to give explicit expressions for the SL 2 (p) generators a, b for any value of p in the p±1 2 irreducible representations.We will also consider the corresponding matrix expressions of the projective group P SL 2 (p).
According to the above construction the two generators a, b have the following unitary matrix representations with matrix elements where, as noted previously, n = 0 for p = 4k + 1 and n = 1 for p = 4k − 1.
In order to bring this in the block diagonal form (4.24) we need to perform a rotation with O p : For the second generator, b, given in (3.1) we obtain (5.3) and so, where k, l = 0, . . ., p − 1.As previously, in order to get the block diagonal form we have to rotate the so obtained matrix with O p : Our final goal is to obtain some basic representations of P SL 2 (p) which will be used to build higher dimensional ones.We observe that the difference between SL 2 (p) and P SL 2 (p) in the defining relations of generators a and b is that, for SL 2 (p) one has to take a 2 = b 3 = −I while for P SL 2 (p) we have the relations a 2 = b 3 = I.This last requirement comes from the different action of SL 2 (p) and P SL 2 (p) which are linear and Möbius correspondingly.
We can obtain irreducible representations of P SL 2 (p) from irreducible representations of SL 2 (p) in the following way.Taking into consideration the above observation we must find the representations of SL 2 (p) for which (A We can easily check that this happens for the p+1 2 dimensional representation only when p = 4k + 1, and for the p−1 2 one only when p = 4k − 1.This way, we get (2k + 1) and (2k − 1)-dimensional irreps of P SL 2 (p) correspondingly.
6 Examples: The cases p = 3, 5, 7 In this section using the method described above, we will present examples, considering the cases p = 3, 5 and p = 7.It is straightforward to construct similar representations of higher values of p.

The case p = 3
The resulting group is SL 2 (3) which has 24 elements, while its projective subgroup P SL 2 (3) has 12 elements and is isomorphic to A 4 , the symmetry group of the even permutations of four objects, 3 or the symmetry group of the Tetrahedron T .The symmetry groups of the Cube and the Octahedron is S 4 which is isomorphic to P GL 2 (3), the automorphism group of SL 2 (3).

The cases p = 5
Next we elaborate the case of P SL 2 (5) which is isomorphic to the symmetry group I of the Dodecahedron and Icosahedron as well as to A 5 .The group SL 2 (5) is isomorphic to the symmetry group 2I of the Binary Icosahedron.The 60 elements of A 5 are generated by two generators a, b with the properties With the above method we find two representations of SL 2 (5), one of three and a second one of two dimensions.
The first generator is a unitary 3 × 3 matrix where in the last form, the matrix elements have been written in terms of the golden ratio, since The character of the representation is TrA [3] = −1, as expected from the character table of P SL 2 (5) .
The second generator has the following three dimensional representation while the character is TrB [3] ∝ 1 + η + η 2 + η 3 + η4 = 0.It can be readily checked that A [3] and B [3] satisfy the defining relations of the P LS 2 (5) group: [3] • B [3] )5 = I These generators correspond to 3 triplet.Indeed, in order to make contact with the form of generators given in recent literature we transform the above in the s 5 , t 5 basis 4 , setting s 5 ≡ A [3] , t 5 = A [3] • B [3] → B [3] = s 5 • t 5 Hence, the two new generators s 5 , t 5 are They satisfy the defining relations while their characters are It is possible to get the other triplet representation of SL 2 (5) (up to equivalence) given in [24], by redefining η to η 3 in eq.(6.12).As discussed in section 4 after eq.(4.17) this is equivalent to a rescaling of appropriate elements of SL 2 (5) which give the generators s 5 and t 5 .
The three and the two-dimensional representations of the generators constructed above, are unitary matrices and so they generate discrete subgroups of SU (3) and SU (2) Lie groups.

The cases p = 7
As a final example in this note, we consider the case p = 7.The associated groups are SL 2 (7) with 336 elements and its projective one P SL 2 (7) which has 168 elements and it is a discrete simple subgroup of SU (3).It is the group preserving the discrete projective geometry of the Fano plane realising the multiplication structure of the octonionic units.

Conclusions
In the present note, we have introduced an intriguing relation of the discrete flavour symmetries with the automorphisms of the magnetic translations of the finite and discrete Heisenberg Group.This relation is reminiscent of the discrete symmetries of the Quantum Hall effect, where in a toroidal two dimensional space the magnetic flux transforms the torus to a phase space and the Hilbert space of a charged particle becomes finite dimensional and the corresponding torus effectively discrete [27].Torii with fluxes in internal extra dimensions appear naturally in the framework of F-theory of elliptic fibrations over Calabi-Yau manifolds, where they generate the GUT gauge groups and other discrete symmetries at particular singularities of the fibration.Phenomenological explorations have shown that such discrete symmetries are particularly successful in predicting the fermion mass hierarchies and the flavour mixing.Inspired by these observations we made use of the discrete Heisenberg Group to develop a simple and unified method for the derivation of basic non-trivial representations of a large class of non-abelian finite groups relevant to the flavour symmetries.It will be important to construct explicit models of elliptic fibrations with fluxes, where the discrete magnetic translations appear naturally and the discrete flavour symmetries as their automorphisms.

4
Heiseberg-Weyl group HW p and the metaplectic representation of SL 2 (p) ,s where (r , s ) are given by (r , s ) = (r, s) a b c d (4.10) the same conjugacy class with A as long as k is a quadratic residue.If k = p − 1 we pass from the representation U (A) to the complex conjugate one U (A) * .The Weyl representation presented above, provides the interesting result that the unitary matrix corresponding to the SL 2 (p) element a = 0 −1 1 0