Special High Energy Theory Seminar
Indices, Localization, Rational Invariants, and H-Saddles
In recent years, varieties of index-like quantitites have been computed by exact path integral, a.k.a. the localization. For gauge theories, the computations reduces to contour integrals, famously riddled with subtleties. However, the interpretation of the results, which should be really called the twisted partition functions rather than the indices, seem to require even more
care.After a cursory review of recent derivations and accompanying subtleties with empasis on 1d and 2d, we consider theories with noncompact Coulomb phases. Rational nature of the 1d twisted partition functions is observed and physically explained, which gives us an unexpcted tool for extracting the integral refined
index out of the twisted partition functions. Applied to pure Yang-Mills, this solves an old problem of counting D0 bound states in orbifolded M-theory, and, along the way, we resolve a critical conflict, circa 1999-2002, between Kac/Smilga and
Staudacher/Pestun by isolating the notion of H-saddles. The latter proves to be a universal feature of partition functions in the high "temperature" limit, in any spacetime dimensions, with serious ramifications on recent proposals on limits of 4d partition functions.
care.After a cursory review of recent derivations and accompanying subtleties with empasis on 1d and 2d, we consider theories with noncompact Coulomb phases. Rational nature of the 1d twisted partition functions is observed and physically explained, which gives us an unexpcted tool for extracting the integral refined
index out of the twisted partition functions. Applied to pure Yang-Mills, this solves an old problem of counting D0 bound states in orbifolded M-theory, and, along the way, we resolve a critical conflict, circa 1999-2002, between Kac/Smilga and
Staudacher/Pestun by isolating the notion of H-saddles. The latter proves to be a universal feature of partition functions in the high "temperature" limit, in any spacetime dimensions, with serious ramifications on recent proposals on limits of 4d partition functions.