100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Space Time in String Theory 321an orientation reversing transformation on space-time with an orientationreversing transformation on M2 brane world volumes. Instead of simplyidentifying points under a discrete translation group, we combine it withsuch an orientation reversal, insisting that the combined operation haveno fixed points j . This leads to the moduli space of CHL 14 strings and itsgeneralizations 15 . Only half of the supercharges survive the projection ontoobjects invariant under the discrete translation-reflection group.Another way to find models with only 16 supercharges is to compactifyon manifolds, which have a number of covariantly constant spinors equalto half the flat space maximum. These manifolds have the form K3 × T k ,k ≤ 3. The K3 surfaces are the 2 (complex) dimensional Ricci-flat Kahlermanifolds. They are a single topological manifold, with a 19 parametermoduli space of Ricci flat metrics. The second Betti number of K3 is 22.The intersection form on H 2 (K3) has signature (3, 19). It is the direct sumof three copies of a (1, 1) signature lattice and two copies of the Cartanmatrix of E 8 .If we shrink the volume of K3 to zero, we get a new low tension, stringby wrapping four directions of the M5 brane on K3. This is the heteroticstring 16 . For small K3 volume, the heterotic string model is compactified ona large three torus, whose geometry arises in much the same way as that ofthe IIB circle. Momentum and heterotic string winding number around thethree toroidal directions are dual to membrane wrapping numbers aroundtwo cycles of K3, associated with the (1, 1) 3 lattice. Membrane wrappingnumbers around the E82 cycles are realized as charges of a U(1)16 gaugegroup in the 7 asymptotically flat dimensions. The U(1) gauge fields themselvesare three form gauge fields of the eleven dimensional theory, whichhave the form ∑ a I ⊗ ω I , where ω I are the harmonic two forms on K3,dual to the cycles with E8 2 intersection matrix. These charges are quantizedin the way that would be realized in the Higgs phase of an E 8 × E 8 gaugegroup, with Higgs field in the adjoint.Mathematicians have long known that the singularities of K3 are foundat places in the moduli space where the volume of some set of two cyclesshrinks to zero. The shrinking cycles are associated with the Cartan matrixof some ADE Dynkin diagram. M-theory allows us to understand thesesingularities as the result of new massless particles (M2 branes wrapped onthe shrinking cycle), which appear in the spectrum at these special points inj A simple case with fixed points is the Horava-Witten description of the strongly coupledheterotic string 13 .