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General Relativity is written in the language of tensors. A tensor is a mathematical object that maps a collection of real vectors and real covectors onto the real numbers.

What would be the physical significance of a complex tensor?

How about an integer-valued tensor?

In other words, can such an object be used to describe gravity, for example?

Using complex tensors is a rather standard procedure, and it usually reveals a very rich structure of the underlying theory. For example, in QFT you can continue S-matrix amplitudes into the complex plane, where the momenta effectively become complex; you can use the information to relate different physical processes (crossing symmetry) and to deduce analiticity properties of the scattering amplitudes (which are closely related to unitarity). Moreover, complex manifolds are used a lot in supersymmetry and other branches of physics.

On the other hand, tensors that map tensors into integers is rather useless, because you lose the ability to use differential equations.

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