4d index to 3d index and 2d topological quantum field theory

100

We study the matrix model for N M2-branes wrapping a Lens space L(p, 1) = S 3 /Z p . This arises from localization of the partition function of the ABJM theory, and has some novel features compared with the case of a three-sphere, including a sum over flat connections and a potential that depends non-trivially on p. We study the matrix model both numerically and analytically in the large N limit, finding that a certain family of p flat connections give an equal dominant contribution. At large N we find the same eigenvalue distribution for all p, and show that the free energy is simply 1/p times the free energy on a three-sphere, in agreement with gravity dual expectations.

Section: Introductionmentioning

confidence: 99%

“…Zeta function regularizing ∞ r=1 (pr) 4 zeta = (2π) 2 /p 2 and using the infinite product formula for sinh, we obtain [5,7] …”

Section: Vector Multipletmentioning

confidence: 99%

The large N limit of M2-branes on Lens spaces

Alday

Fluder

Sparks

2012

100

“…Using the path-integral representation of the index, one can relate the index to partition functions on other manifolds. For example, taking small thermal circle S 1 limit, the index calculation can be reduced to the calculation of partition function on S 3 [35][36] [37] (see also [38]). …”

Section: Superconformal Index Compatible With Line Operatorsmentioning

confidence: 99%

Line Operator Index on S 1 × S 3

Gang

Koh

Lee

2012

We derive a general formula of an index for N = 2 superconformal field theories on S 1 × S 3 with insertions of BPS Wilson line or 't Hooft line operator at the north pole and their anti-counterpart at the south pole of S 3 . One-loop and monopole bubbling effects are taken into account in the computation. As examples, we calculate the indices for N = 4 theories and N = 2 SU (2) theory with N f = 4, and find good agreements between indices of line operators related by S-duality. The relation between Verlinde loop operators and the indices is explored. The holographic correspondence between the fundamental (anti-symmetric) Wilson line operator and the fundamental string (D5 brane) in AdS 5 ×S 5 is confirmed by the index comparison.arXiv:1201.5539v2 [hep-th]

“…It would be interesting to understand purely from gauge theory why the simplification coming from combining (3.25) with the results of [32] occurs in this limit. 14 An exhaustive classification of N = 2 boundary conditions and how S-duality acts on them is not the purpose of this paper and therefore we leave it for future work. Nevertheless we see that the relation with Liouville theory with boundaries might be helpful as a guiding principle.…”

Section: Jhep12(2017)099mentioning

confidence: 99%

Agt/ℤ2

Floch

Turiaci

2017

We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional N = 2 gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different Z 2 quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a Z 2 quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the RP 4 partition function of four-dimensional N = 2 theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known RP 2 partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.