In 1953 the mathematician Eugenio Calabi was searching for a flat complex topology. His studies completely disconnected by the simultaneous physicists’ investigations and the same term String Theory still to be coined. Calabi conjectured that starting with the case of one complex dimension and two real dimensions, if the general Topology has average curvature zero, then it is possible to encounter a geometry (or, metric) where the curvature is zero everywhere.
For dimensions superior to these, his conjecture refers to Ricci curvature and the condition of average Ricci curvature zero is replaced by the condition of first Chern class being zero. He considered that if the topologic condition of first Chern class zero is met, then it exists a Kaehler metric with zero Ricci curvature. In 1973 the mathematician Shing-Tung Yau after years trying to disprove Calabi's Conjecture, discovered the way to prove it was ...correct !
This discovery reached physicists who integrated it as new metric (a new Geometry) for the 6-dimensional inner space imagined existing in each one point of the 4-dimensional space-time we perceive directly (3-D space) or indirectly (1-D time). But, why to add dimensions, when to have less should apparently mean to have also less complications ? Several excellent reasons. One of them being the constant accumulation of the varieties of particles, discovered along decades of High Energy Physics experiments. Their associated wide spectrum of properties (mass, energy, electric charge, spin, etc.) to be described in a unified mathematical model need… more space than what provided by total four dimensions. Below a table where they appear seventeen elementary particles.
Widespread consensus exists that the table is actually missing at least part of the tens of named supersymmetric partners, theorized by other studies and others. Each one time a new and more precise determination is made for e.g., the energy of a known particle or a new particle discovered, say each one time we proceed forward in the knowledge, this has effects also out of the experimental ambit. Experimental results compared with theories, implying e.g., the refutation of some theories’ underlying assumptions and/or confirmation of other theories predictions, in what since three centuries is the process of scientific discovery.
Key concept to such new scenario is the compactness of the 6-D manifold: in Geometry it is possible to have additional compact dimensions of infinite or finite size, lying in an infinitesimal space. The new category of geometric Calabi-Yau manifolds, allowed a rapid success of String Theory in terms of consistency with the experimental discoveries of accumulated along decades by the High Energy Physics (HEP).
The objects of String Theory, namely are strings, unidimensional objects (1-D), vibrating on tones and overtones. As an example, it was imagined that electrons were microscopic closed vibrating strings. String theorists, whose confidence was enforced by the high level of consistency of their logic construction with respect to Quantum Mechanics, were feeling the still missing inclusion of the gravitational force. The last decisive step toward the unification of all known forces.
This unification came in 1995: we are living an epoch when the most modern theory of all is no more String Theory, rather its successor M-Theory where the “M“ means Membranes.
Extended objects with one or more compact spatial dimensions named p-branes, objects of p-dimensional spatial extent:
0-brane is a point particle;
1-brane is a string;
2-brane may be imagined the Ocean's surface, wrapping the Earth and propagating in the 4-D spacetime of the Solar System.
Describing a brane as an object propagating through a spacetime we’d be placing the spacetime on a primary, and the brane on a secondary footing. In the reality, the String Field Theory (M-theory) on the brane is the primary concept, whereas the spacetime is a derived concept.
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