Deformed Schur Index Overview

• The deformed Schur index is a refined version of the Schur index, enriched by defect insertions, flavor/mass deformations, and modified boundary conditions in 4D supersymmetric theories.
• Methodologies include infrared trace formulas, matrix integral computations using single-letter contributions, and q-difference operator techniques for analyzing line and surface defects.
• These deformations establish connections to 2D chiral algebras, modular invariance, and quasi-modular forms, enabling precise defect enumeration and S-duality checks.

to=arxiv_search 天天中彩票公众号json {"^^^^^^^^3query3 Schur index^^^^^^^^3\3 OR ^^^^^^^^3\3 Schur indices^^^^^^^^3\3 to=arxiv_search 盈立json {"^^^^^^^^3query3 to=arxiv_search qq天天中彩票json {"^^^^^^^^3query3 index defects infrared computations (&&&^^^^3query3^^^^&&&) line defects boundary conditions","max_results":^^^^3\3query3^^^^,"sort_by":"relevance"} The deformed Schur index denotes several closely related refinements of the Schur index of four-dimensional supersymmetric theories. In one usage, originating in infrared formulations, the deformation is produced by boundary conditions, line defects, or surface defects, and the resulting quantity is an infrared trace of quantum monodromies with defect insertions (&&&^^^^3query3^^^^&&&, Fluder et al., 2019). In another usage, common in PRESERVED_PLACEHOLDER_^^^^3query3^^^^ super Yang–Mills and PRESERVED_PLACEHOLDER_^^^^3\3^^^^ theories, it denotes flavor or mass deformations of the Schur limit, including the two-parameter family PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^ and its B/C/D-type analogues (Hatsuda, 5 Mar 2025, Ren et al., 15 Jul 2025). These constructions share the undeformed Schur index as their starting point: a protected trace over the Hilbert space on $S^3$, equal to the vacuum character of the associated two-dimensional chiral algebra (&&&^^^^3query3^^^^&&&).

^^^^3\3^^^^. Baseline Schur index and its normalizations

For a $4$d $\mathcal N=2$ SCFT with flavor symmetry of rank $n_f$, the Schur index is

$$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$

with the unconventional fermion number $(-1)^F=e^{2\pi iR}$. The relevant short multiplets obey

$$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$

The fugacities PRESERVED_PLACEHOLDER_^^^^3\3query3^^^^ refine the index by flavor charges PRESERVED_PLACEHOLDER_^^^^3\3\3^^^^ (&&&^^^^3query3^^^^&&&).

For Lagrangian theories, the ultraviolet computation uses single-letter indices and a plethystic exponential. In the convention PRESERVED_PLACEHOLDER_^^^^3\3 OR \3^^^^,

PRESERVED_PLACEHOLDER_^^^^3\33^^^^

and the full index is obtained by integrating over the gauge group with the Haar measure and the appropriate gauge and matter characters. In infrared formulas, normalization factors appear as

PRESERVED_PLACEHOLDER_^^^^3\3^^^^4

where PRESERVED_PLACEHOLDER_^^^^3\3^^^^5 is the Coulomb-branch rank; the factor PRESERVED_PLACEHOLDER_^^^^3\3^^^^6 precisely accounts for the PRESERVED_PLACEHOLDER_^^^^3\3^^^^7 vector multiplets in the low-energy description (&&&^^^^3query3^^^^&&&).

This baseline definition underlies later deformations. In particular, later PRESERVED_PLACEHOLDER_^^^^3\3^^^^8 constructions continue to compute Schur-limit indices by matrix integrals built from the PRESERVED_PLACEHOLDER_^^^^3\3^^^^9 vector and hypermultiplet single-letter contributions, rather than replacing the Schur sector by a different BPS limit (Nakanishi et al., 2024).

^^^^3 OR \3^^^^. Infrared formulation and defect enrichment

In a generic Coulomb-branch vacuum, the infrared theory is a PRESERVED_PLACEHOLDER_^^^^3 OR \3query3^^^^ gauge theory with charge lattice PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^, Dirac pairing PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^, and central charges PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^. The quantum torus variables PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^4 satisfy

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^5

The flavor sublattice PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^6 consists of charges with trivial Dirac pairing, and the trace annihilates nonzero gauge charge: PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^7 The basic one-particle building block is the quantum dilogarithm

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^8

from which one forms the refined Kontsevich–Soibelman factors and the ordered spectrum generator PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^9 (&&&^^^^3query3^^^^&&&).

The infrared Schur index is then

$S^3$^^^^3query3^^^^

A UV line defect $S^3$^^^^3\3^^^^ of phase $S^3$^^^^3 OR \3^^^^ is represented in the IR by a framed BPS generating function

$S^3$3

and the defect-deformed Schur index becomes

$S^3$4

For half-BPS boundary conditions with IR expansion

$S^3$5

the Schur half-index is

$S^3$6

and with a line defect

$S^3$7

These expressions are framed-wall-crossing invariant because the jumps of $S^3$8, $S^3$9, and $4$^^^^3query3^^^^ are compensated by conjugations with $4$^^^^3\3^^^^ inside the trace (&&&^^^^3query3^^^^&&&).

In this infrared literature, “deformed” means enriched by defect insertions, not by changing the fugacity specialization as in Macdonald or Hall–Littlewood limits. The deformation is implemented by inserting $4$^^^^3 OR \3^^^^ or $4$3 into the quantum-monodromy trace, thereby counting defect-compatible Schur operators (&&&^^^^3query3^^^^&&&).

3. Line and surface defect deformations

For class-$4$4 theories of type $4$5 with at least one regular puncture, vortex surface defects of vorticity $4$6 act on the Schur index by explicit $4$7-difference operators localized at the puncture. Writing $4$8, the operator is

$4$9

The trace of the defect BPS monodromy coincides with the action of this operator on the pure Schur trace, and the resulting infrared bootstrap reproduces the ultraviolet gauging-and-Higgsing description of surface defects (Fluder et al., 2019).

Line-operator deformations appear in several complementary forms. In $\mathcal N=2$^^^^3query3^^^^ $\mathcal N=2$^^^^3\3^^^^ SYM on $\mathcal N=2$^^^^3 OR \3^^^^, a Wilson or Polyakov loop in representation $\mathcal N=2$3 is inserted by multiplying the matrix integrand by $\mathcal N=2$4,

$\mathcal N=2$5

In the Fermi-gas treatment, the normalized fundamental-loop expectation value has leading large-$\mathcal N=2$6 behavior

$\mathcal N=2$7

and symmetric and antisymmetric towers admit a compact grand-canonical generating function (&&&^^^^3\3^^^^4&&&).

A further reformulation is available for pure $\mathcal N=2$8d $\mathcal N=2$9 $n_f$^^^^3query3^^^^ SYM. There, the Schur half-index with half-BPS line insertions is represented in terms of a non-commutative algebra $n_f$^^^^3\3^^^^, a $n_f$^^^^3 OR \3^^^^-Weyl algebra, and an equivalent system of $n_f$3-deformed harmonic oscillators. In this language the half-index becomes a vacuum expectation value,

$n_f$4

which leads to a colored-chord combinatorial model and identifies the commuting Wilson operators with the Hamiltonians of the relativistic open Toda chain (&&&^^^^3\3^^^^5&&&).

4. Fugacity and mass deformations in $n_f$5 and $n_f$6

A distinct use of the term appears in recent work on $n_f$7 $n_f$8 SYM. Starting from the full index with constraint $n_f$9, setting

$$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$^^^^3query3^^^^

defines a two-parameter deformation with reduced single-letter index

$$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$^^^^3\3^^^^

The resulting matrix integral,

$$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$^^^^3 OR \3^^^^

admits an exact evaluation by Macdonald polynomials: $$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$3 This form interpolates between the Schur limit, the Hall–Littlewood limit, half-index limits, and $$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$4-Whittaker-type limits, and it extends directly to line-operator indices, particularly for antisymmetric representations (Hatsuda, 5 Mar 2025).

The same program has been generalized from type $$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$5 to B/C/D gauge groups. For $$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$6, $$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$7, and $$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$8, the deformed Schur indices are written as $$\mathcal{I}(q,\vec{z})=\mathrm{Tr}_{\mathcal{H}_{S^3}}\!\left[e^{2\pi i R}\,q^{\Delta-R}\prod_{i=1}^{n_f}z_i^{f_i}\right],$$9-Pochhammer integrals whose natural symmetric-function description uses Koornwinder polynomials together with Macdonald polynomials. The defining integrals are symmetric under $(-1)^F=e^{2\pi iR}$^^^^3query3^^^^, and special limits provide explicit S-duality checks; for example,

$(-1)^F=e^{2\pi iR}$^^^^3\3^^^^

Low-rank Schur-limit expansions of $(-1)^F=e^{2\pi iR}$^^^^3 OR \3^^^^ and $(-1)^F=e^{2\pi iR}$3 agree term by term to the displayed orders (Ren et al., 15 Jul 2025).

For $(-1)^F=e^{2\pi iR}$4 SYM, the deformed Schur index is the mass-deformed Schur index obtained from the flavored $(-1)^F=e^{2\pi iR}$5 index. With

$(-1)^F=e^{2\pi iR}$6

one may write $(-1)^F=e^{2\pi iR}$7 as a Fermi-gas partition function with kernel given by the Kronecker theta function. The canonical answer is a sum over Young diagrams built from spectral zeta functions $(-1)^F=e^{2\pi iR}$8, and the normalized deformed index lies in the polynomial ring generated by the Kronecker theta function and twisted Weierstrass functions, hence in the ring of quasi-Jacobi forms. In the unflavored limit $(-1)^F=e^{2\pi iR}$9, these indices reduce to explicit quasi-modular expressions, including large-$$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$^^^^3query3^^^^ generating functions for overpartitions and $$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$^^^^3\3^^^^-colored partitions (&&&^^^^3\3^^^^8&&&).

5. Modularity, dimensional reduction, and large-$$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$^^^^3 OR \3^^^^ expansions

The small-circle reduction of the Schur index provides another deformation-theoretic interpretation. For $$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$3d $$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$4 SYM with gauge group $$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$5, the Schur limit is parameterized by

$$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$6

and the leading small-$$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$7 behavior is

$$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$8

For $$\tfrac{1}{2}\big(\Delta-j_1-j_2\big)-R=0,\qquad r+j_1-j_2=0.$$9 and PRESERVED_PLACEHOLDER_^^^^3\3query3query3^^^^, this agrees, up to an overall normalization, with the mass-deformed PRESERVED_PLACEHOLDER_^^^^3\3query3\3^^^^ partition function of ABJM theories at levels PRESERVED_PLACEHOLDER_^^^^3\3query3 OR \3^^^^ and PRESERVED_PLACEHOLDER_^^^^3\3query33^^^^, respectively; the divergence is interpreted in terms of flat directions inherited from Wilson lines around PRESERVED_PLACEHOLDER_^^^^3\3query3^^^^4 (Nakanishi et al., 2024).

A modular analysis of flavored and defect Schur indices shows that their modular orbits span finite-dimensional spaces, and the dimension of the flavored orbit predicts the minimal order of the unflavored modular differential equation. For genus-zero PRESERVED_PLACEHOLDER_^^^^3\3query3^^^^5 class-PRESERVED_PLACEHOLDER_^^^^3\3query3^^^^6 theories, the high-temperature limit PRESERVED_PLACEHOLDER_^^^^3\3query3^^^^7 identifies the flavored or defect Schur index with the PRESERVED_PLACEHOLDER_^^^^3\3query3^^^^8-partition function of the PRESERVED_PLACEHOLDER_^^^^3\3query3^^^^9 star-shape quiver, with defect indices mapping to Wilson-line insertions. An explicit feature of this identification is the relation between the linear independence of defect indices and the convergence of the corresponding Wilson-line matrix integrals (&&&^^^^3 OR \3query3^^^^&&&).

Large-PRESERVED_PLACEHOLDER_^^^^3\3\3query3^^^^ expansions expose further structure. For the flavored Schur index of PRESERVED_PLACEHOLDER_^^^^3\3\3\3^^^^ PRESERVED_PLACEHOLDER_^^^^3\3\3 OR \3^^^^ SYM, finite-PRESERVED_PLACEHOLDER_^^^^3\3\33^^^^ corrections are organized by a giant graviton expansion in wrapped D3-brane sectors. The brane index PRESERVED_PLACEHOLDER_^^^^3\3\3^^^^4 is an analytic continuation of the flavored PRESERVED_PLACEHOLDER_^^^^3\3\3^^^^5 Schur index, and away from the unflavored limit the resulting brane contributions are characterized by quasimodular forms built from Eisenstein series. This same machinery extends to the unflavored Schur indices of the non-Lagrangian theories PRESERVED_PLACEHOLDER_^^^^3\3\3^^^^6 with PRESERVED_PLACEHOLDER_^^^^3\3\3^^^^7 (&&&^^^^3 OR \3\3^^^^&&&).

6. Chiral algebra, PRESERVED_PLACEHOLDER_^^^^3\3\3^^^^8-deformed Yang–Mills, and Verlinde-type structures

The Schur sector is controlled by a two-dimensional chiral algebra. In defect-free situations the Schur index is its vacuum character; with line defects, the index decomposes into sums of non-vacuum characters. In examples such as PRESERVED_PLACEHOLDER_^^^^3\3\3^^^^9 PRESERVED_PLACEHOLDER_^^^^3\3 OR \3query3^^^^, the line-defect index is expressed in affine PRESERVED_PLACEHOLDER_^^^^3\3 OR \3\3^^^^ characters, while for Argyres–Douglas theories the line-defect OPE reduces in the index to the Verlinde algebra. The paper “Infrared Computations of Defect Schur Indices” exhibits this explicitly for the PRESERVED_PLACEHOLDER_^^^^3\3 OR \3 OR \3^^^^, PRESERVED_PLACEHOLDER_^^^^3\3 OR \33^^^^, and PRESERVED_PLACEHOLDER_^^^^3\3 OR \3^^^^4 theories. For instance, in the PRESERVED_PLACEHOLDER_^^^^3\3 OR \3^^^^5 model one finds PRESERVED_PLACEHOLDER_^^^^3\3 OR \3^^^^6 and PRESERVED_PLACEHOLDER_^^^^3\3 OR \3^^^^7, consistent with PRESERVED_PLACEHOLDER_^^^^3\3 OR \3^^^^8 (&&&^^^^3query3^^^^&&&).

A complementary deformation is produced by inserting Schur operators rather than extended defects. For class-PRESERVED_PLACEHOLDER_^^^^3\3 OR \3^^^^9 theories, current insertions correspond to torus correlators in the associated VOA and can be expressed in PRESERVED_PLACEHOLDER_^^^^3\33query3^^^^-deformed Yang–Mills language by replacing ordinary puncture wave functions with current wave functions. For a full puncture at critical level,

PRESERVED_PLACEHOLDER_^^^^3\33\3^^^^

These deformed Schur indices reduce, in the high-temperature limit, to topological Coulomb-branch correlators in PRESERVED_PLACEHOLDER_^^^^3\33 OR \3^^^^, and surface defects reduce to Wilson loops in the PRESERVED_PLACEHOLDER_^^^^3\333^^^^d mirror description (&&&^^^^3 OR \33^^^^&&&).

Taken together, these developments show that the phrase “deformed Schur index” names a coherent but multi-branched body of constructions. The common core is the Schur sector itself; the deformation may be implemented by defect insertions, by flavor or mass fugacities, or by local Schur-operator insertions. In each case the resulting object preserves strong structural control, whether through infrared wall-crossing invariants, symmetric-function technology, modularity, or chiral-algebra representation theory.