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Topology is a mathematical concept that embodies the concept of those properties of a geometrical space which does not change if the space is stretched, twisted, bent or torn. A doughnet and a sphere are distinct from the topological view point because there is no way to defrom one into the other smoothly. ie.,with out tearing either object. Say a doughnut and a tea-cup which has a hole each can be continuously deformed into each other and hence have the same topology. General Theory predicts that the space time deform its shape and size very smoothly to the presence of matter and energy. A familiar manifestation is the stretching of the universe (the expansion of the universe already predicted by hubble and proved beyong doubt that the universe is expanding). The topology of the universe is fixed. A long standing question is that there must be a physical process other than what we encountered with Theory of Relativity where the topology of universe changes. This heuristic reason comes from the simple application of Quantum Mechanics. As per Quantum Mechanics even the most quiescent systems undergo quantum jitter. ie., the value of quantities characterizing the system fluctuate, cometimes violently, averaging out to their mesaured values on larger dustance scales. This notion, when applied to the fabric of Space-time yeilds the image of a frothing, undulating structure on small distnace scales which averages out on larger scales to the smooth grometrical description of general relativity. It can be conceived that behing the veil of quantum-jitter the fabric of spacetime could momentarily tear and subsequently reconnect in a manner resulting in a change in the topology of the universe. Prior to the String theory, the incompatibility of general theory and Quantum mechanics made it impossible to address this possibility in a qunatitative manner.
Considering the facts above String theory is considered as the theory which can unite both Quantum Theory and Theory of Relativity. The possibility of spacetime topology change was suggested as a novel characteristic of the union of both the theories. String theory helps in this union by adding atleast 6 extra dimensions. Consider the mathematical implication of the operation called flop. It is a systematic procedure for changing the topology of a geometrical space in a minimal manner. The procedure involves singling out a sphere in space. Now shrink it till it volume becomes zero, leaving the rest of space fully intact, then bowl its volume back up, but in an orthogonal direction. The point at which the volume is zero is the singularity which may be considered as a minimal tear. The result of this operation is a new Geometrical Space whose topology is different from the original. Considering this mathematically, the solution is a very complex rigorously defined and well studied operation. For these theorems to be physically checked it requires the power equivalent to the energy generated around the world for a year as per current requirements. So its a dauting taks which cannot be visualized in the current scenario.
The following years after the K-K theory it was shown that the interpretation of string theory using the Klauza-Klein idea if curled up dimensioans comes with a remarkable twist. Two completely different possiblities for the curled space(different sizes, shapes and number of holes) can, if properly chosen, give rise to identical boservable physics. This is completely unexpected from a paticle point of view. The reason for this is that the in point particle theories the physical and mathematical descriptions of a geometric space are both based on considering it to be a collection of an infinite number of points grouped together in a particular manner. In String theory, the physical model is based on tiny loops and hence differs markedly from the mathematical description. This, in turn, allows two mathematically distinct curled up spaces to yield physically identical string models. This is a purely string theoretic phenomenon which relies profoundly on the extended nature of a string. Although either member of a mirror pair gives rise to the same physical theory, the technical description of a given physical process very often differs drastically between the two constructions. For certain processes when analyzed with an addition of curled up space became complex and profound, became very easy to analyze when mirrors were used. This mirror analysis is applied to the topology of the flop operation. It resulted in remarkable simplification of the string equations governing this process. An analysis of the simplified equations has revealed that there are no catastrophic physical consequences of this topology changing process. In fact, the mirror analycis makes it clear that such topology changing events are not only physically realizable, but commonplace as well. Thus, using the tool of Mirro Manifolds, it has been shown that the long suspected possibility of topology changing processes can be explicitly realized in string theory.
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