**Table of Contents**

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

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

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 11 - Tearing the Fabric of Space**

Inching Forward

Off and on during 1992, Plesser and I tried to show that the fabric of space can undergo space-tearing flop transitions. Our calculations yielded bits and pieces of supporting circumstantial evidence, but we could not find definitive proof. Sometime during the spring, Plesser visited the Institute for Advanced Study in Princeton to give a talk, and privately told Witten about our recent attempts to realize the mathematics of space-tearing flop transitions within the physics of string theory. After summarizing our ideas, Plesser waited for Witten's response. Witten turned from the blackboard and stared out of his office window. After a minute of silence, maybe two, he turned back to Plesser and told him that if our ideas worked out, "it would be spectacular." This rekindled our efforts. But after a while, with our progress stalled, each of us turned to working on other string theory projects.

Even so, I found myself mulling over the possibility of

Batyrev had become very interested in mirror symmetry, especially in the wake of the success of Candelas and his collaborators in using it to solve the sphere-counting problem described at the end of Chapter 10. With a mathematician's perspective, though, Batyrev was unsettled by the methods Plesser and I had invoked to find mirror pairs of Calabi-Yau spaces. Although our approach used tools familiar to string theorists, Batyrev later told me that our paper seemed to him to be "black magic." This reflects the large cultural divide between the disciplines of physics and mathematics, and as string theory blurs their borders, the vast differences in language, methods, and styles of each field become increasingly apparent. Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it's not that one approach is right and the other wrong—the methods one chooses to use are largely a matter of taste and training.

Batyrev set out to recast the construction of mirror manifolds in a more conventional mathematical framework, and he succeeded. Inspired by earlier work of Shi-Shyr Roan, a mathematician from Taiwan, he found a systematic mathematical procedure for producing pairs of Calabi-Yau spaces that are mirrors of one another. His construction reduces to the procedure Plesser and I had found in the examples we had considered, but offers a more general framework that is phrased in a manner more familiar to mathematicians.

The flip side is that Batyrev's papers invoked areas of mathematics that most physicists had never previously encountered. I, for example, could extract the gist of his arguments, but had significant difficulty in understanding many crucial details. One thing, however, was clear: The methods of his paper, if properly understood and applied, could very well open a new line of attack on the issue of space-tearing flop transitions.

By late summer, energized by these developments, I decided that I wanted to return to the problem of flops with full and undistracted intensity. I had learned from Morrison that he was going on leave from Duke to spend a year at the Institute for Advanced Study, and I knew that Aspinwall would also be there, as a postdoctoral fellow. After a few phone calls and e-mails,

Even so, I found myself mulling over the possibility of

**space-tearing flop transitions**. As the months went by, I felt increasingly sure that they**had to be part and parcel of string theory.****(Epsilon=One: Yes! The concept is analogous to that of the emergence of the Emergent Ellipsoid (EEd), which is essentially the Fundamental Postulate of**The preliminary calculations Plesser and I had done, together with insightful discussions with David Morrison, a mathematician from Duke University, made it seem that this was the only conclusion that mirror symmetry naturally supported. In fact, during a visit to Duke, Morrison and I, together with some helpful observations from Sheldon Katz of Oklahoma State University, who was also visiting Duke at the time, outlined a strategy for proving that flop transitions can occur in string theory. But when we sat down to do the required calculations, we found that they were extraordinarily intensive. Even on the world's fastest computer, they would take more than a century to complete. We had made progress, but we clearly needed a new idea, one that could greatly enhance the efficiency of our calculational method. Unwittingly, Victor Batyrev, a mathematician from the University of Essen, revealed such an idea through a pair of papers released in the spring and summer of 1992.*Reality*(FPR))Batyrev had become very interested in mirror symmetry, especially in the wake of the success of Candelas and his collaborators in using it to solve the sphere-counting problem described at the end of Chapter 10. With a mathematician's perspective, though, Batyrev was unsettled by the methods Plesser and I had invoked to find mirror pairs of Calabi-Yau spaces. Although our approach used tools familiar to string theorists, Batyrev later told me that our paper seemed to him to be "black magic." This reflects the large cultural divide between the disciplines of physics and mathematics, and as string theory blurs their borders, the vast differences in language, methods, and styles of each field become increasingly apparent. Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it's not that one approach is right and the other wrong—the methods one chooses to use are largely a matter of taste and training.

Batyrev set out to recast the construction of mirror manifolds in a more conventional mathematical framework, and he succeeded. Inspired by earlier work of Shi-Shyr Roan, a mathematician from Taiwan, he found a systematic mathematical procedure for producing pairs of Calabi-Yau spaces that are mirrors of one another. His construction reduces to the procedure Plesser and I had found in the examples we had considered, but offers a more general framework that is phrased in a manner more familiar to mathematicians.

The flip side is that Batyrev's papers invoked areas of mathematics that most physicists had never previously encountered. I, for example, could extract the gist of his arguments, but had significant difficulty in understanding many crucial details. One thing, however, was clear: The methods of his paper, if properly understood and applied, could very well open a new line of attack on the issue of space-tearing flop transitions.

By late summer, energized by these developments, I decided that I wanted to return to the problem of flops with full and undistracted intensity. I had learned from Morrison that he was going on leave from Duke to spend a year at the Institute for Advanced Study, and I knew that Aspinwall would also be there, as a postdoctoral fellow. After a few phone calls and e-mails,

**I arranged to take leave from Cornell University**and spend the fall of 1992**at the Institute**as well.**(Epsilon=One: A personal note: Cornell to the IAS brings back memories of a similar, earlier venture. With much melancholy and some sadness, I vividly remember the pleasant, late-spring morning when I left Ithaca for Princeton with a feeling of**

The gift I bore the old-man was a first postulate that was essentially utmost complexity from utmost simplicity; the simplicity*joie de vivre*until I heard a jolting news announcement on the car radio that Albert Einstein had died early that morning. I was on a journey that Philip Morrison had suggested a few days earlier when the essence of string theory was born along with the phrase "Paradigm*Shift!.*" I was expecting to bring instant, great joy to an old man; sort of a crowning reward for his lifetime of courageous, unrelenting search for the benefit of a suffering humanity chained to superstition; I often, now, refer to—what was then the "Unified Concept" or simply a Paradigm-*Shift!*—the overall string theory concept as Oscillation Theory based upon the the philosophical logic of Philogic and the algebraic geometry of Pulsoid Theory. My personal preference for all humanity is simply: ONE*ness*.The gift I bore the old-man was a first postulate that was essentially utmost complexity from utmost simplicity; the simplicity

*began*its perpetual, cyclic evolution in accordance with the*Bjerknes phenomenon.*I figured "Al" could take it from there. It never occurred to me that I'd be the old-man writing this some 59 years later.)