- String theory and number theory have common ground in a specific kind of elliptic curve, called a torus.
- Toruses aren't just for compressing atoms: these curves represent torus cross-sections.
- The secret to the link is in modularity and invariance in the equations representing the curves.

Scientists have found what they say is a link between __complex number theory and string theory__. A mathematician and physicist working together found the connection, which they say is part of plotting elliptic curves in complex space. This is a two-axis plot where instead of *x* and *y* axes, there are imaginary and real axes.

In __their paper__, mathematician Satoshi Kondo and physicist Taizan Watari join forces to explore an overlap they found between their two fields.

“It is known that the L-function of an elliptic curve defined over ℚ is given by the Mellin transform of a modular form of weight 2. Does that modular form have anything to do with string theory?” they wonder in the abstract.

If so, this could open doors toward new understandings of what counts as a number or not. As one of the universities said in a statement: “The precise details of how the concept of numbers can be extended has been considered as one of the important themes in number theory.”

In other words, an L-function over the set of rational numbers, which is one of the ground-floor functions studied in complex space, ends up being a particular kind of integral complement that can be plotted more easily. Most problems in complex space are based on intensive and extensive theory, but are high-level abstract and hard to conceptualize. And a modular form is invariant: “This function has been expected to be a modular form, a function that remains invariant under certain operations,” the statement says.

In this case, invariant means that different transformations don’t affect the outcome. Invariance is critical to parts of superstring theory, and it’s in this shared definition that the researchers began to find a real link. “Watari and [...] Kondo dared to ask why such functions are invariant under certain operations,” the statement says. For specific applications of the L-function they describe, they found that their results correspond to another specific result from string theory.

This shared space, where invariant modular functions represent ways the definition of “numbers” can continue to expand and grow nuanced, forms kind of a logic gateway that links the two sets of functions from number theory and string theory. In the complex plane, the resulting mapped shapes are called __elliptic curves__, which behave like sections of a torus—a mathematical 3D shape that’s often shown to be donut-shaped, but which can have many more “donut holes” than just the one.

If these specifically defined elliptic curves embody the overlapping space between number theory and superstring theory in particular, other geometric shapes with specific qualities could fit the same logic gateway shared by the two fields, leaving a whole world of functions and special cases to explore in future research.

“The question remains open to whether the functions for more general class of geometric objects are expressed in terms of observables in superstring theory,” the statement concludes.

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