S-duality and the N=2 Lens Space Index

Masahito Yamazaki; Shlomo S. Razamat

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S-duality and the N=2 Lens Space Index

Shlomo S. Razamat, Masahito Yamazaki

High Energy Physics - Theory · Submitted 6 Jun 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects.

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We discuss some of the analytic properties of lens space indices for 4d N=2 theories of class S. The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects. We explicitly identify the difference operators to be a matrix-valued generalization of the elliptic Ruijsenaars-Schneider model. In a special limit these difference operators can be expressed naturally in terms of Cherednik operators appearing in the double affine Hecke algebras, with the eigenfunctions given by non-symmetric Macdonald polynomials.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss some of the analytic properties of lens space indices for 4d N=2 theories of class S.	contribution 0.85	0.67	0.07	asserted 0.73
s1	The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects.	result 0.60	0.36	0.40	asserted 1.00
s2	We explicitly identify the difference operators to be a matrix-valued generalization of the elliptic Ruijsenaars-Schneider model.	result 0.53	0.91	0.32	asserted 1.00
s3	In a special limit these difference operators can be expressed naturally in terms of Cherednik operators appearing in the double affine Hecke algebras, with the eigenfunctions given by non-symmetric Macdonald polynomials.	result 0.84	0.75	0.18	asserted 0.90

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Original abstract

We discuss some of the analytic properties of lens space indices for 4d N=2 theories of class S. The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects. We explicitly identify the difference operators to be a matrix-valued generalization of the elliptic Ruijsenaars-Schneider model. In a special limit these difference operators can be expressed naturally in terms of Cherednik operators appearing in the double affine Hecke algebras, with the eigenfunctions given by non-symmetric Macdonald polynomials.

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References

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Cited by 16 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Surface Defects in the $\mathcal{N}=2$ Lens Space Index and Elliptic Integrability
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Surface-defect operators in the N=2 lens space index are shown to factorize into a product of two elliptic Ruijsenaars-Schneider operators, implying commutativity and enabling a systematic analysis.
Root-of-unity limit of the $A_{n-1}$ lens elliptic gamma function solution of the star-star relation
math-ph 2026-10 conditional novelty 7.0 of 10

In the root-of-unity limit, the A_{n-1} lens elliptic gamma star-star solution yields, under unproven saddle point assumptions, a (Z_{rN})^{n-1} star-star solution that specializes to established discrete models, incl...
Surface defects on lens space and transfer matrices
hep-th 2026-10 conditional novelty 7.0 of 10

On L(r,1) times S^1, all class-S surface defects are products and fusion polynomials of two commuting elementary half-holonomy-shift operators, which are traces of lens Sklyanin L-operators; class S_k defects are prop...
Monopoles, Center Vortices, Confinement in (3+1)d, and the Lens-Space Twisted Partition Function
hep-th 2026-06 unverdicted novelty 7.0 of 10

Proposes torus and lens-space twisted partition functions as criteria for center-vortex and monopole condensation and proves vortex condensation implies monopole condensation in gapped phases.
Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
hep-th 2017-01 conditional novelty 7.0 of 10

The Coulomb branch index of Argyres-Douglas theories on a lens space reproduces wild Hitchin characters and, at a root of unity, matrix elements of ST^kS in the associated chiral algebra.

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N=1 theories of class S_k
hep-th 2015-03 conditional novelty 7.0 of 10

A new class of N=1 superconformal theories is conjectured, labeled by punctured Riemann surfaces, with its supersymmetric index built from eigenfunctions of an elliptic integrable model.
Equivariant Verlinde formula from fivebranes and vortices
hep-th 2015-01 conditional novelty 7.0 of 10

A mass-deformed complex Chern-Simons theory on a Riemann surface times a circle yields a t-deformed Verlinde formula whose coefficients are graded dimensions of finite-dimensional pieces of an otherwise infinite-dimen...
Complex Chern-Simons theory at level k via the 3d-3d correspondence
hep-th 2014-09 accept novelty 7.0 of 10

A state-integral model is constructed that computes SL(n,C) Chern-Simons partition functions at arbitrary level k from ideal triangulations, generalizing the known k=0 and k=1 cases.
4d N=1 from 6d (1,0)
hep-th 2016-10 conditional novelty 6.0 of 10

Compactifying 6d (1,0) SCFTs on a Riemann surface yields 4d N=1 theories whose conformal manifold is enlarged by flat connections and quantized fluxes for the 6d flavor symmetry, verified for two M5 branes on a Z2 sin...
Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT
hep-th 2015-09 conditional novelty 6.0 of 10

The paper conjectures closed-form expressions for the Schur, Hall-Littlewood, and Macdonald indices of generalized Argyres-Douglas theories by assigning wave functions to irregular punctures in the class S TQFT.
Supersymmetric Casimir Energy and the Anomaly Polynomial
hep-th 2015-07 conditional novelty 6.0 of 10

The supersymmetric Casimir energy of an even-dimensional superconformal field theory equals the equivariant integral of its anomaly polynomial, checked in two, four and six dimensions.
Global Properties of Supersymmetric Theories and the Lens Space
hep-th 2013-07 conditional novelty 6.0 of 10

The lens space index distinguishes gauge groups differing by global structure and discrete theta angle, and matches across the Seiberg and S-duality pairs proposed by Aharony, Seiberg, and Tachikawa.
Higher Form Symmetries of Argyres-Douglas Theories
hep-th 2020-07 conditional novelty 5.0 of 10

The defect groups of all (g,g') Argyres-Douglas theories are computed from the intersection form of the singularity and tabulated, with independent agreement against proposed N=1 Lagrangians.
Down the rabbit hole with theories of class S
hep-th 2014-03 conditional novelty 4.0 of 10

The paper reviews and extends the correspondence between 4d class S indices and 3d N=4 partition functions, showing that T[SU(2)] blocks are eigenfunctions of the reduced difference operators.
The supersymmetric index in four dimensions
hep-th 2016-08 accept novelty 2.0 of 10

A review of the four-dimensional N=1 supersymmetric index covering its trace and partition-function definitions, its matrix-integral computation, its spectroscopic content, and the elliptic hypergeometric identities t...
The superconformal index of theories of class $\cal S$
hep-th 2014-12 accept novelty 2.0 of 10

The superconformal index of any class S theory of type A can be written as a sum over representations of a 2d TQFT correlator on the UV curve, with structure constants and puncture wavefunctions fixed by S-duality plu...

How the literature uses it

37 arXiv papers cite this in 43 sentences

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Papers citing it · 27 sentences without a confident label

Uses its method or tool 2 papers

Using the formula [ref], the single letter contributions from each field to the index [ref], and the projection condition [ref], the contributions to the Casimir energy were computed in [ref] , and the results are [equation] where $[x]_k$ means $xk$, $𝐬=(±1,±1,±1)$ is a vector of possible weights of the trifundamental fields, and the formula for $I_0^H$ contains a sum over all $8$ possibilities for $𝐬$.

Ground States of Class S Theory on ADE Singularities and dual Chern-Simons theory · Counting holonomy sectors for the Lens space · 0.96
We take the same approach, as one takes when computing, e.g. the supersymmetric Lens space $L(1,r)×𝕊^1$ indices [ref] .

Instanton counting in Class $\mathcal{S}_k$ · Orbifolding to 4d $=2$/$=1$ circular/toroidal quivers · 1.00

Compares with it 2 papers

We thus see that the $1$-cocycle condition leads to a concrete formula for the lens index in terms of the superconformal and $S^2×T^2$ indices, apparently avoiding difficulties with a direct definition of the lens index for $q>1$.[See a Appendix [ref] for a direct definition when $q=1$ and also [ref] .] We also note that the formula is structurally similar to a proposed formula for the $L(p,q)$ partition function of three-dimensional $𝒩=2$ theories [ref] .

Modular factorization of superconformal indices · A general formula · 0.91
Our definition of the supersymmetric Casimir energy follows that of [ref] and [ref] but not that of [ref] .

Exact Degeneracy of Casimir Energy for $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory on ADE Singularities and S-Duality · Introduction · 0.82

Cites it as background 12 papers

In this appendix, we collect the known contour integral formulae for the superconformal index $ℐ_(1,0)(ρ̂)$ [ref] , the lens space index $ℐ_(p,1)(ρ̂)$ [ref] and the $S^2×T^2$ index $ℐ_(0,-1)(ρ̂)$ [ref] of general $𝒩=1$ gauge theories.

Modular factorization of superconformal indices · Lens indices for general gauge theories · 0.83
and it is already very reminiscent of the setting of Lens space superconformal indices of class $$ theories [ref] : L(k,1)×S^1 × Σ ∩ L(k,1)×S^1 × T^* Σ × ^3 ↻ ↻ ↻ SO(4)_E U(1)_N SU(2)_R .

Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality · Introduction · 0.91
In this Appendix, we carefully define the $=2$ superconformal lens index [ref] (see also [ref] ).

Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra · Definition and review of the lens index · 0.93

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