Boundary conditions and localization on AdS. Part II. General analysis

Abstract

We develop the method of Green's function to evaluate the one loop determinants that arise in localization of supersymmetric field theories on AdS spaces. The theories we study have at least N = 2 supersymmetry and normalisable boundary conditions are consistent with supersymmetry. We then show that under general assumptions the variation of the one loop determinant with respect to the localizing background reduces to a total derivative. Therefore it receives contributions only from the origin of AdS and from asymptotic infinity. From expanding both the Greens function and the quadratic operators at the origin of AdS and asymptotic infinity, we show that the variation of the one loop determinant is proportional to an integer. Furthermore, we show that this integer is an index of a first order differential operator. We demonstrate that these assumptions are valid for Chern-Simons theories coupled to chiral multiplets on AdS 2 × S 1 . Finally we use our results to show that U (N c ) Chern-Simons theory at level k coupled to N f chiral multiplets and N f anti-chiral multiplets in the fundamental obeys level-rank duality on AdS 2 × S 1 .

Free energy and defect C-theorem in free scalar theory

“…worthwhile to revisit the O(N ) model discussed in a recent paper [56] that admits various non-trivial boundary conditions and supersymmetric theories with defects [43,[89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104][105][106][107] from the viewpoint of this paper. (See also [108][109][110][111][112][113] for related works.) It should be possible to extend our analysis to fields with spin.…”

One-loop determinants for black holes in 4d gauged supergravity

“…Of course, due to our choice of rotated reality conditions (2.13), the action of Q is no longer (pseudo-)real. We now further split the fields into the following sets [30]: 15) and their Q-images QX I 0 , QX I 1 . This so-called cohomological split is particularly useful for the computation of one-loop determinants since it allows us to isolate the differential operator D 10 , as already explained around (1.13).…”

“…We will nevertheless go ahead and often use a given theorem under the extra assumption that for our purposes the statement of compactness can be replaced by a careful choice of boundary conditions, which are known to be crucial at the asymptotic boundary of Euclidean AdS spaces. A similar approach has already been advocated and successfully used for the computation of one-loop determinants in a number of interesting examples [9][10][11][12][13][14][15]. We focus our one-loop analysis mostly on the contributions from an arbitrary number n V of abelian vector multiplets and hypermultiplets (both the compensating one and a possible number n H of physical ones) in the conformal supergravity formalism.…”

Topological Correlators and Surface Defects from Equivariant Cohomology

“…Incorporating gauge theories in our language would also allow to investigate the way mirror symmetry manifests itself at the level of one-dimensional actions. Moreover, supersymmetric localization was successfully applied to theories defined on non-compact manifolds [20][21][22][23] and on manifolds with boundaries [24][25][26][27]. Generalizing (1.1) to such instances would be a natural direction to explore.…”