**Table of Contents**

*.......The Elegant Universe*

**THE ELEGANT UNIVERSE,****Brian Greene,**1999, 2003

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 10 - Tearing the Fabric of Space**

Mirror Symmetry

Through general relativity, Einstein forged a link between the physics of gravity and the geometry of spacetime. At first blush,

In 1988, Lance Dixon of the Stanford Linear Accelerator Center made a pivotal observation in this regard that was further amplified by Wolfgang Lerche of CERN, Vafa at Harvard, and Nicholas Warner, then of the Massachusetts Institute of Technology. Based upon aesthetic arguments rooted in considerations of symmetry, these physicists made the audacious suggestion that it might be possible for two different Calabi-Yau shapes, chosen for the extra curled-up dimensions in string theory, to give rise to identical physics.

To give you an idea of how this rather far-fetched possibility might actually occur, recall that the number of holes in the extra Calabi-Yau dimensions determines the number of families into which string excitations will arrange themselves. These holes are analogous to the holes one finds in a torus or its multihandled cousins, as illustrated in Figure 9.1. One deficiency of the two-dimensional figure that we must show on the printed page is that it cannot show that a six-dimensional Calabi-Yau space can have holes of a variety of dimensions. Although such holes are harder to picture, they can be described with well-understood mathematics. A key fact is that the number of families of particles arising from string vibrations is sensitive only to the total number of holes, not to the number of holes of each particular dimension (that's why, for instance, we did not worry about drawing distinctions between the different types of holes in our discussion in Chapter 9). Imagine, then, two Calabi-Yau spaces in which the number of holes in various dimensions differs, but in which the total number of holes is the same. Since the number of holes in each dimension is not the same, the two Calabi-Yaus have different shapes. But since they have the same total number of holes, each yields a universe with the

In the fall of 1987, 1 joined the physics department at Harvard as a postdoctoral fellow and my office was just down the hall from Vafa's. As my thesis research had focused on the physical and mathematical properties of curled-up Calabi-Yau dimensions in string theory, Vafa kept me closely apprised of his work in this area. When he stopped by my office in the fall of 1988 and told me of the conjecture that he, Lerche, and Warner had come upon, I was intrigued but also skeptical. The intrigue arose from the realization that if their conjecture was true, it might open a new avenue of research on string theory; the skepticism arose from the realization that guesses are one thing, established properties of a theory are quite another.

During the following months, I thought frequently about their conjecture arid, frankly, half convinced myself that it wasn't true. Surprisingly, though, a seemingly unrelated research project I had undertaken in collaboration with Ronen Plesser, then a graduate student at Harvard and now on the faculty of the Weizmann Institute and Duke University, was soon to change my mind completely. Plesser and I had become interested in developing methods for starting with an initial Calabi-Yau shape and mathematically manipulating it to produce hitherto unknown Calabi-Yau shapes. We were particularly drawn to a technique known as

After a few months of intensive pursuit of this idea we came to a surprising realization. If we glued particular groups of points together in just the right way, the Calabi-Yau shape we produced differed from the one we started with in a startling manner: The number of

Excited by the apparent contact we had made with the Dixon-Lerche-Vafa-Warner guess, Plesser and I pressed on to the linchpin question: Beyond the number of families of particles, do the two different Calabi-Yau spaces agree on the rest of their physical properties? After a couple more months of detailed and arduous mathematical analysis during which we received valuable inspiration and encouragement from Graham Ross, my thesis advisor at Oxford, and also from Vafa, Plesser and I were able to argue that the answer was, most definitely, yes. For mathematical reasons having to do with the even-odd interchange, Plesser and I coined the term

The weeks after finding this result were an extremely anxious time. Plesser and I knew that we were sitting on an important new piece of string physics. We had shown that the tight association between geometry and physics originally set down by Einstein was substantially modified by string theory: Drastically different geometrical shapes that would imply different physical properties in general relativity were giving rise to identical physics in string theory. But what if we had made a mistake? What if their physical implications did differ in some subtle way that we had missed? When we showed our results to Yau, for example, he politely but firmly claimed that we must have made an error; he asserted that from a mathematical standpoint our results were far too outlandish to be true. His assessment gave us substantial pause. It's one thing to make a mistake in a small or modest claim that attracts little attention. Our result, though, was suggesting an unexpected step in a new direction that would certainly engender a strong response. If it were wrong, everyone would know.

Finally, after much checking and rechecking, our confidence grew and we sent our paper off for publication. A few days later, I was sitting in my office at Harvard and the phone rang. It was Philip Candelas from the University of Texas, and he immediately asked me if I was seated. I was. He then told me that he and two of his students, Monika Lynker and Rolf Schimmrigk, had found something that was going to knock me off of my chair. By carefully examining a huge sample set of Calabi-Yau spaces that they had generated by computer, they found that almost all came in pairs differing precisely by the interchange of the number of even and odd holes. I told him that I was still seated—that Plesser and I had found the same result. Candelas's and our work turned out to be complementary; we had gone one step further by showing that all of the resulting physics in a mirror pair was identical, whereas Candelas and his students had shown that a significantly larger sample of Calabi-Yau shapes fell into mirror pairs. Through the two papers, we had discovered the

**string theory strengthens and broadens the link between physics and geometry**, since the properties of vibrating strings—their mass and the force charges they carry—are largely determined by the properties of the curled-up component of space.**(Epsilon=One: The properties of strings are dependent on all the forms of oscillation; not just vibration. Such oscillations determine the "force fields." The phenomenon of "mass" is a resultant of various harmonic oscillations resonating as Resoloids.)**We have just seen, though, that quantum geometry—the geometry-physics association in string theory—has some surprising twists. In general relativity, and in "conventional" geometry, a circle of radius*R*is different from one whose radius is 1/*R*, pure and simple; yet, in string theory they are physically indistinguishable. This leads us to be bold enough to go further and ask whether there might be geometrical forms of space that differ in more drastic ways—not just in overall size, but possibly also in shape—but that are nevertheless physically indistinguishable in string theory.In 1988, Lance Dixon of the Stanford Linear Accelerator Center made a pivotal observation in this regard that was further amplified by Wolfgang Lerche of CERN, Vafa at Harvard, and Nicholas Warner, then of the Massachusetts Institute of Technology. Based upon aesthetic arguments rooted in considerations of symmetry, these physicists made the audacious suggestion that it might be possible for two different Calabi-Yau shapes, chosen for the extra curled-up dimensions in string theory, to give rise to identical physics.

To give you an idea of how this rather far-fetched possibility might actually occur, recall that the number of holes in the extra Calabi-Yau dimensions determines the number of families into which string excitations will arrange themselves. These holes are analogous to the holes one finds in a torus or its multihandled cousins, as illustrated in Figure 9.1. One deficiency of the two-dimensional figure that we must show on the printed page is that it cannot show that a six-dimensional Calabi-Yau space can have holes of a variety of dimensions. Although such holes are harder to picture, they can be described with well-understood mathematics. A key fact is that the number of families of particles arising from string vibrations is sensitive only to the total number of holes, not to the number of holes of each particular dimension (that's why, for instance, we did not worry about drawing distinctions between the different types of holes in our discussion in Chapter 9). Imagine, then, two Calabi-Yau spaces in which the number of holes in various dimensions differs, but in which the total number of holes is the same. Since the number of holes in each dimension is not the same, the two Calabi-Yaus have different shapes. But since they have the same total number of holes, each yields a universe with the

*same number of families.*This, of course, is but one physical property. Agreement on*all*physical properties is a far more restrictive requirement, but this at least gives the flavor of how the Dixon-Lerche-Vafa-Warner conjecture could possibly be true.In the fall of 1987, 1 joined the physics department at Harvard as a postdoctoral fellow and my office was just down the hall from Vafa's. As my thesis research had focused on the physical and mathematical properties of curled-up Calabi-Yau dimensions in string theory, Vafa kept me closely apprised of his work in this area. When he stopped by my office in the fall of 1988 and told me of the conjecture that he, Lerche, and Warner had come upon, I was intrigued but also skeptical. The intrigue arose from the realization that if their conjecture was true, it might open a new avenue of research on string theory; the skepticism arose from the realization that guesses are one thing, established properties of a theory are quite another.

During the following months, I thought frequently about their conjecture arid, frankly, half convinced myself that it wasn't true. Surprisingly, though, a seemingly unrelated research project I had undertaken in collaboration with Ronen Plesser, then a graduate student at Harvard and now on the faculty of the Weizmann Institute and Duke University, was soon to change my mind completely. Plesser and I had become interested in developing methods for starting with an initial Calabi-Yau shape and mathematically manipulating it to produce hitherto unknown Calabi-Yau shapes. We were particularly drawn to a technique known as

*orbifolding,*which was pioneered by Dixon, Jeffrey Harvey of the University of Chicago, Vafa, and Witten in the mid-1980s. Roughly speaking, this is a procedure in which different points on an initial Calabi-Yau shape are glued together according to mathematical rules that ensure that a new Calabi-Yau shape is produced. This is schematically illustrated in Figure 10.4. The mathematics underlying the manipulations illustrated in Figure 10.4 is formidable, and for this reason string theorists had thoroughly investigated this procedure only as applied to the simplest of shapes—higher-dimensional versions of the doughnut shapes shown in Figure 9.1. Plesser and I realized, though, that some beautiful new insights of Doron Gepner, then of Princeton University, might give a powerful theoretical framework for applying the orbifolding technique to full-fledged Calabi-Yau shapes, such as the one in Figure 8.9.**Figure 10.4**Orbifolding is a procedure in which a new Calabi-Yau shape is produced by gluing together various points on an initial Calabi-Yau shape.

*odd*-dimensional holes in the new Calabi-Yau shape equaled the number of*even*-dimensional holes in the original, and vice versa. In particular, this means that the total number of holes—and therefore the number of particle families—in each is the*same*even though the even-odd interchange means that their shapes and fundamental geometrical structures are quite different.*5*Excited by the apparent contact we had made with the Dixon-Lerche-Vafa-Warner guess, Plesser and I pressed on to the linchpin question: Beyond the number of families of particles, do the two different Calabi-Yau spaces agree on the rest of their physical properties? After a couple more months of detailed and arduous mathematical analysis during which we received valuable inspiration and encouragement from Graham Ross, my thesis advisor at Oxford, and also from Vafa, Plesser and I were able to argue that the answer was, most definitely, yes. For mathematical reasons having to do with the even-odd interchange, Plesser and I coined the term

*mirror manifolds*to describe the physically equivalent yet geometrically distinct Calabi-Yau spaces.**The individual spaces in a mirror pair of CalabiYau spaces are not literally mirror images of one another, in the sense of everyday usage. But even though they have different geometrical properties, they give rise to one and the same physical universe when used for the extra dimensions in string theory.***6*The weeks after finding this result were an extremely anxious time. Plesser and I knew that we were sitting on an important new piece of string physics. We had shown that the tight association between geometry and physics originally set down by Einstein was substantially modified by string theory: Drastically different geometrical shapes that would imply different physical properties in general relativity were giving rise to identical physics in string theory. But what if we had made a mistake? What if their physical implications did differ in some subtle way that we had missed? When we showed our results to Yau, for example, he politely but firmly claimed that we must have made an error; he asserted that from a mathematical standpoint our results were far too outlandish to be true. His assessment gave us substantial pause. It's one thing to make a mistake in a small or modest claim that attracts little attention. Our result, though, was suggesting an unexpected step in a new direction that would certainly engender a strong response. If it were wrong, everyone would know.

Finally, after much checking and rechecking, our confidence grew and we sent our paper off for publication. A few days later, I was sitting in my office at Harvard and the phone rang. It was Philip Candelas from the University of Texas, and he immediately asked me if I was seated. I was. He then told me that he and two of his students, Monika Lynker and Rolf Schimmrigk, had found something that was going to knock me off of my chair. By carefully examining a huge sample set of Calabi-Yau spaces that they had generated by computer, they found that almost all came in pairs differing precisely by the interchange of the number of even and odd holes. I told him that I was still seated—that Plesser and I had found the same result. Candelas's and our work turned out to be complementary; we had gone one step further by showing that all of the resulting physics in a mirror pair was identical, whereas Candelas and his students had shown that a significantly larger sample of Calabi-Yau shapes fell into mirror pairs. Through the two papers, we had discovered the

*mirror symmetry*of string theory.*7*