The Standard Model Fermion Spectrum From Complex Projective Spaces

Abstract

It is shown that the quarks and leptons of the standard model, including a right-handed neutrino, can be obtained by gauging the holonomy groups of complex projective spaces of complex dimensions two and three. The spectrum emerges as chiral zero modes of the Dirac operator coupled to gauge fields and the demonstration involves an index theorem analysis on a general complex projective space in the presence of topologically non-trivial SU(n) × U(1) gauge fields. The construction may have applications in type IIA string theory and non-commutative geometry.

Dimensional reduction and vacuum structure of quiver gauge theory

“…The complete spectrum of the Dirac operator on CP 2 , coupled to arbitrary instanton and monopole backgrounds, was worked out in [16]. The eigenspinors for an arbitrary monopole background, without instantons, were constructed in the context of the fuzzy projective plane CP 2 F in [17], 1 while the number of zero modes in a rank two instanton background with arbitrary monopole charge was originally computed in [3]. The number of spinor harmonics in a generic instanton background and with arbitrary monopole number is computed in §C.…”

“…However, globally well-defined spinors can be constructed by twisting the Dirac operator on CP 2 with half-integer powers L e b , b ∈ Z + 1 2 of the monopole line bundle L. The index of the Dirac operator associated with this twisted complex is computed by the Atiyah-Singer index theorem to be [3] is an integer. 4 In the main text we use the index (2.54) for higher rank SU(3)-equivariant bundles over CP 2 , and we will now derive this formula here.…”

“…4 In the main text we use the index (2.54) for higher rank SU(3)-equivariant bundles over CP 2 , and we will now derive this formula here. From (A.3) the zero mode structure of the Dirac operator for spinor fields transforming under the holonomy group H = SU(2) × U(1), in the fundamental representation of SU (2) and in the background gauge field of Q⊗L e b , is easily evaluated [3]. Denoting this index index by ν b;1 we have, using (A.3), the formula There is a technical issue, however, because Q ⊗n is a bundle of rank 2 n which is reducible in terms of SU (2) representations and it will be more convenient for our purposes to decompose it into irreducible representations.…”

“…Hence for odd n taking b = b − 3 2 = − n+3 2 gives the index ν − n+3 2 ; n = − 1 8 (n + 1) 3 , (C.12) and corresponds to spinors coupling to pure anti-selfdual SU(2) gauge fields on CP 2 in the (n + 1)-dimensional irreducible representation with no U(1) component. Since n = 2k + 1 is odd this is necessarily a spinor representation of SU (2), though the magnitude of the index (C.12) corresponds to the dimension of a real representation C k,k of SU (3). At the opposite extreme, the integer (C.1) is the index for spinors coupling to a pure U(1) self-dual gauge field on CP 2 with no SU(2) component, and ν b equals the dimension of the SU(3) representation C b,0 for b ≥ 0 while −ν b equals the dimension of C 0,|b|−3 for b ≤ −3.…”

Noncommutative vector bundles over fuzzy Bbb CBbb PNand their covariant derivatives

“…Since, for our purposes, we will not need to diagonalizeL 0 it is more convenient to work in a basis where we do not distinguish between a † 1 and a † 2 , but rather we will leave the SU(2) symmetry manifest. Our preferred basis for F L is the set of vectors 10) which satisfy the orthogonality relation…”