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By Root 2023 0

tell you the resulting shape of spacetime. Different physical environments (different configurations of mass and energy) yield differently shaped spacetimes; different spacetimes correspond to physically distinct environments. What would it feel like to fall into a black hole? Calculate with the spacetime geometry that Karl Schwarzschild discovered in his study of spherical solutions to Einstein’s equations. And if the black hole is rapidly spinning? Calculate with the spacetime geometry found in 1963 by the New Zealand mathematician Roy Kerr. In general relativity, geometry is the yin to physics’ yang.

String theory provides a twist to this conclusion by establishing that there can be different shapes for spacetime that nevertheless yield physically indistinguishable descriptions of reality.

Here’s one way to think about it. From antiquity to the modern mathematical era, we’ve modeled geometrical spaces as collections of points. A Ping-Pong ball, for example, is the collection of points that constitute its surface. Prior to string theory, the basic constituents making up matter were also modeled as points, point particles, and this commonality of basic ingredients spoke to an alignment between geometry and physics. But in string theory, the basic ingredient is not a point. It’s a string. This suggests that a new kind of geometry, based not on points but rather on loops, should be linked to string physics. The new geometry is called stringy geometry.

To get a feel for stringy geometry, picture a string moving through a geometrical space. Notice that the string can behave much like a point particle, innocently gliding from here to there, bumping into walls, navigating chutes and valleys, and so on. But in certain situations, a string can also do something novel. Imagine that space (or a piece of space) is shaped like a cylinder. A string can wrap itself around such a piece of space, much like a rubber band stretched around a can of soda, realizing a configuration that’s simply unavailable to a point particle. Such “wrapped” strings, and their “unwrapped” cousins, probe a geometrical space in different ways. Should a cylinder grow fatter, a string encircling it will respond by stretching, while an unwrapped string sliding on its surface won’t. In this way, wrapped and unwrapped strings are sensitive to different features of a shape through which they’re moving.

This observation is of great interest because it gives rise to a striking and thoroughly unexpected conclusion. String theorists have found special pairs of geometrical shapes for space that have completely different features when each is probed by unwrapped strings. They also have completely different features when each is probed by wrapped strings. But—and this is the punch line—when probed both ways, with wrapped and unwrapped strings, the shapes become indistinguishable. What the unwrapped strings see on one space, the wrapped strings see on the other, and vice versa, rendering identical the collective picture gleaned from the full physics of string theory.

Shapes that form such pairs provide a powerful mathematical tool. In general relativity, if you’re interested in one or another physical feature, you must complete a mathematical calculation using the unique geometrical space relevant to the situation being studied. But in string theory, the existence of pairs of physically equivalent geometrical shapes means that you have a newfound choice: you can choose to perform the necessary calculation using either shape. And the extraordinary thing is that while you’re guaranteed to get the same answer using either shape, the mathematical details en route to the answer can be vastly different. In a variety of situations, overwhelmingly difficult mathematical calculations on one geometrical shape translate into exceedingly easy calculations on the other. And any framework that makes hard mathematical calculations easy is, clearly, of great value.

Over the years, mathematicians and physicists have leveraged this hard-to-easy dictionary to make headway on a number