The author introduces the supersymmetric localization technique, a new
approach for computing path integrals in quantum field theory on curved
space (time) defined with interacting Lagrangian.
The author focuses on a particular quantity called the superconformal
index (SCI), which is defined by considering the theories on the product
space of two spheres and circles, in order to clarify the validity of
so-called three-dimensional mirror symmetry, one of the famous duality
proposals. In addition to a review of known results, the author presents
a new definition of SCI by considering theories on the product space of
real-projective space and circles. In this book, he explains the concept
of SCI from the point of view of quantum mechanics and gives
localization computations by reducing field theoretical computations to
many-body quantum mechanics. He applies his new results of SCI with
real-projective space to test three-dimensional mirror symmetry, one of
the dualities of quantum field theory. Real-projective space is known to
be an unorientable surface like the Mobius strip, and there are many
exotic effects resulting from Z2 holonomy of the surface.
Thanks to these exotic structures, his results provide completely new
evidence of three-dimensional mirror symmetry.
The equivalence expected from three-dimensional mirror symmetry is
transformed into a conjectural non-trivial mathematical identity through
the new SCI, and he performs the proof of the identity using a
q-binomial formula.
