Exact partition functions on $ \mathbb{R}{{\mathbb{P}}^2} $ and orientifolds

Abstract

We consider gauged linear sigma models (GLSM) on RP 2 , obtained from a parity projection of S 2 . The theories admit squashing deformation, much like GLSM on S 2 , which allows us to interpret the partition function as the overlap amplitude between the vacuum state and crosscap states. From these, we extract the central charge of Orientifold planes, and observe that the Gamma class makes a prominent appearance as in the recent D-brane counterpart. We also repeat the computation for the mirror Landau-Ginzburg theory, which naturally brings out the θ-dependence as a relative sign between two holonomy sectors on RP 2 . We also show how our results are consistent with known topological properties of D-brane and Orientifold plane world-volumes, and discuss what part of the wrapped D-brane/Orientifold central charge should be attributed to the quantum volumes.

Agt/ℤ2

“…On the other hand, boundary conditions typically affect one-loop determinants by selecting some modes of the Laplace/Dirac operators without altering the eigenvalues for these modes. It is thus plausible that our non-constant vector multiplet only affects chiral multiplet one-loop determinants through its value near the poles and the gauge holonomy along the equator (this is confirmed by the fact that we reproduce RP 2 results [63]). Both coincide with the constant η = η(0) background studied in [84].…”

“…From the Möbius strip bootstrap we derive the ADE Toda cross-cap wavefunction (A.40), which appears to be new. In appendix B we motivate the gluing procedure we use for the RP 4 b partition function by checking that the same technique reproduces the RP 2 partition function of [63], and we check Seiberg dualities. In appendix C we describe quotients of S 4 b consistent with localization; this suggests possible generalizations of our work.…”

“…We naturally obtain a "Higgs branch" expression of the RP 2 partition function (with a sum over vortices), which must be equal to the "Coulomb branch" integral of [63]. We show the equality for a class of theories by summing up the residues in the Coulomb branch integral.…”

Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone

“…Following the work of Pestun [4], the method has been applied to a large number of theories in two [5][6][7][8], three [9][10][11][12][13][14][15] four [16,17], and five dimensions, [18][19][20][21][22][23][24]. This development went hand-in-hand with an increased interest in theories with rigid supersymmetry on curved manifolds [25][26][27][28][29].…”

Some recent advances in gauged linear sigma models

“…[30,31,32,36,37,38,39,40,41,42,43]. More recently, localization was applied to compute partition functions for hemispheres [44,45,46,47]. One of the results of that work was an expression for central charges of D-branes involving a new multiplicative characteristic class Γ [48,49,50,51], which has been applied to systematically generate arbitrarily-high order loop corrections to higher-dimensional Calabi-Yau geometries [52].…”