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observation implies that a three-brane wrapping around this three-dimensional sphere—which appears to us as a black hole—will have ever smaller mass until, at the final point of collapse, it will be massless. But, as we asked above, what does this mean? The answer became clear to us by invoking the second observation. Our work showed that the new massless pattern of string vibration arising from the space-tearing conifold transition is the microscopic description of a massless particle into which the black hole has transmuted. We concluded that as a Calabi-Yau shape goes through a space-tearing conifold transition, an initially massive black hole becomes ever lighter until it is massless and then it transmutes into a massless particle—such as a massless photon—which in string theory is nothing but a single string executing a particular vibrational pattern. In this way, for the first time, string theory explicitly establishes a direct, concrete, and quantitatively unassailable connection between black holes and elementary particles.

"Melting" Black Holes

The connection between black holes and elementary particles which we found is closely akin to something we are all familiar with from day-to-day life, known technically as a phase transition. A simple example of a phase transition is the one we mentioned in the last chapter: water can exist as a solid (ice), as a liquid (liquid water), and a gas (steam). These are known as the phases of water, and the transformation from one form to another is called a phase transition. Morrison, Strominger, and I showed that there is a tight mathematical and physical analogy between such phase transitions and the space-tearing conifold transitions from one Calabi-Yau shape to another. Again, just as someone who has never before encountered liquid water or solid ice would not immediately recognize that they are two phases of the same underlying substance, physicists had not realized previously that the kinds of black holes we were studying and elementary particles are actually two phases of the same underlying stringy material. Whereas the surrounding temperature determines the phase in which water will exist, the topological form—the shape—of the extra Calabi-Yau dimensions determines whether certain physical configurations within string theory appear as black holes or elementary particles. That is, in the first phase, the initial Calabi-Yau shape (the analog of the ice phase, say), we find that there are certain black holes present. In the second phase, the second Calabi-Yau shape (the analog of the liquid water phase), these black holes have gone through a phase transition—they have "melted" so to speak—into fundamental vibrational string patterns. The tearing of space through conifold transitions takes us from one Calabi-Yau phase to the other. In so doing, we see that black holes and elementary particles, like water and ice, are two sides of the same coin. We see that black holes snugly fit within the framework of string theory.

We have purposely used the same water analogy for these drastic space-tearing transmutations and for the transmutations from one of the five formulations of string theory to another (Chapter 12) because they are deeply connected. Recall that we expressed through Figure 12.11 that the five string theories are dual to one another and thereby are unified under the rubric of a single overarching theory. But does the ability to move continuously from one description to another—to set sail from any point on the map of Figure 12.11 and reach any other—persist even after we allow the extra dimensions to curl up into some Calabi-Yau shape or another? Prior to the discovery of the drastic topology-changing results, the anticipated answer was no, since there was no known way to continuously deform one Calabi-Yau shape into any other. But now we see that the answer is yes: Through these physically sensible space-tearing conifold transitions, we can continuously change any given Calabi-Yau space into any other. By varying coupling constants and curled-up Calabi-Yau geometry,