Schur Half-Indices in 4d N=2 SYM

• Schur half-indices are supersymmetric partition functions defined on a hemisphere or half-space in 4d N=2 theories, counting protected boundary operators.
• They admit exact closed forms derived from Macdonald identities, uniformly encoding Lie-theoretic data for arbitrary simple gauge groups.
• Recent developments reformulate them as q-oscillator matrix elements, generalized chord counting models, and DSSYK-like transition amplitudes in theories with matter.

Searching arXiv for papers on Schur half-indices and closely related exact formulas. Using arXiv search to verify the supplied papers and identify nearby Schur half-index literature. Schur half-indices are supersymmetric partition functions on a hemisphere or half-space that furnish the boundary analogue of the 4d $\mathcal N=2$ Schur index. In the formulation emphasized for pure super Yang–Mills, the relevant geometries are $HS^3\times \mathbb R$ and, after holomorphic-topological twisting, $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$, with a supersymmetric boundary condition at $\mathbb R_{\ge 0}=0$. They count protected local operators supported on a boundary or interface, and in the presence of Wilson lines they count local operators at the corresponding decorated boundary. For pure $\mathcal N=2$ SYM with simple gauge group $G$, the Neumann Schur half-index admits a uniform exact closed form for arbitrary $G$, derived from Macdonald identities for untwisted affine Lie algebras (Lü, 11 Nov 2025). In more recent developments, Schur half-indices have also been reformulated as $q$-oscillator matrix elements, generalized colored chord-counting problems, and, for $SU(2)$ theories with matter, DSSYK-like transition amplitudes in non-vacuum sectors (Lewis et al., 20 Jun 2025, Berkooz et al., 16 Jul 2025).

1. Definition and physical setting

For 4d $\mathcal N=2$ pure SYM with simple gauge group $HS^3\times \mathbb R$0, rank $HS^3\times \mathbb R$1, root system $HS^3\times \mathbb R$2, positive roots $HS^3\times \mathbb R$3, Weyl group $HS^3\times \mathbb R$4, Weyl vector $HS^3\times \mathbb R$5, and dual Coxeter number $HS^3\times \mathbb R$6, the full Schur index is given by a gauge integral with vector-multiplet factor

$HS^3\times \mathbb R$7

The corresponding Neumann half-index uses a single copy of the vector contribution,

$HS^3\times \mathbb R$8

and with a Wilson line in representation $HS^3\times \mathbb R$9, or highest weight $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$0, is written as

$\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$1

Here

$\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$2

is the Weyl-invariant Macdonald denominator (Lü, 11 Nov 2025).

Two equivalent physical realizations are emphasized. One is as a Witten index on $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$3, with $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$4 the hemisphere. The other is as an observable in the holomorphic-topological twist on

$\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$5

In the twisted operator-counting description, the full Schur index counts cohomology classes of local operators preserved by the Schur supercharge, while the Neumann half-index counts local operators on an interface between the empty theory and the 4d $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$6 SYM theory, possibly with a Wilson line insertion (Lü, 11 Nov 2025).

This boundary interpretation is structurally important. The half-index is not “half of the full index” in a naive arithmetic sense; it is a distinct supersymmetric observable associated with a space with boundary and with a specified boundary condition. In the holomorphic-topological twist, the full Schur index is identified with a gauged partition function of complex chiral $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$7 ghosts, whereas the Neumann half-index corresponds to gauged real chiral $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$8 ghosts. This distinction is the conceptual origin of later bilinear factorization statements for the full index (Lü, 11 Nov 2025).

2. Exact formulas for pure super Yang–Mills

The central exact formula for the Neumann Schur half-index in pure $\mathbb C\times \mathbb R\times \mathbb R_{\ge 0}$9 SYM is

$\mathbb R_{\ge 0}=0$0

with

$\mathbb R_{\ge 0}=0$1

This formula is uniform for arbitrary simple $\mathbb R_{\ge 0}=0$2 (Lü, 11 Nov 2025).

The coefficient $\mathbb R_{\ge 0}=0$3 is the character evaluated on Kostant’s principal element of type $\mathbb R_{\ge 0}=0$4. Using the Weyl character and denominator formulas, it can be written as

$\mathbb R_{\ge 0}=0$5

Hence the half-index may also be expressed as

$\mathbb R_{\ge 0}=0$6

A striking feature is the sparsity of the coefficient. The paper states that $\mathbb R_{\ge 0}=0$7, so only dominant weights in

$\mathbb R_{\ge 0}=0$8

contribute. In particular, the undecorated Neumann half-index is trivial: $\mathbb R_{\ge 0}=0$9 The interpretation given is that with Neumann boundary condition and no line insertion, only the identity operator contributes (Lü, 11 Nov 2025).

For $\mathcal N=2$0, with $\mathcal N=2$1, one has

$\mathcal N=2$2

Consequently,

$\mathcal N=2$3

This makes the selection rule completely explicit in the rank-one case (Lü, 11 Nov 2025).

3. Macdonald identities and bilinear factorization

The exact evaluation is derived by expanding the Macdonald denominator in the Kostant–Fegan form,

$\mathcal N=2$4

and then using character orthogonality,

$\mathcal N=2$5

This immediately yields the closed half-index formula above. The same structure may also be written in terms of the skew affine denominator $\mathcal N=2$6, with the torus integral picking out the unique lattice term satisfying $\mathcal N=2$7 (Lü, 11 Nov 2025).

All dependence on the gauge group enters through standard Lie-theoretic data: $\mathcal N=2$8, $\mathcal N=2$9, $G$0, $G$1, $G$2, $G$3, $G$4, and $G$5. This is why the result is completely uniform for arbitrary simple $G$6. The relevant Macdonald identities are those for untwisted affine Lie algebras, including the Weyl-invariant expansion

$G$7

and its Kostant–Fegan form quoted above (Lü, 11 Nov 2025).

The same technology produces a bilinear structure for the full Schur index: $G$8 This follows from the reproducing kernel

$G$9

which acts as a Dirac-delta kernel identifying boundary gauge fugacities across the interface. Physically, this is the gluing of two Neumann hemispheres. In the ghost description, complex chiral $G$0 ghosts split into two real chiral $G$1 systems, and gluing the two real systems reproduces the full complex system (Lü, 11 Nov 2025).

The same paper also records the corresponding full-index $G$2-series and eta-quotient formulas. For the full Schur index,

$G$3

and in particular

$G$4

For Schur half-indices, this full-index background matters because the coefficient $G$5 is precisely the same principal specialization at Kostant’s element that underlies the product formulas for the full index (Lü, 11 Nov 2025).

4. Line operators, $G$6-oscillators, chord counting, and Toda systems

For pure $G$7 SYM, Schur half-indices with line insertions admit a second, operator-algebraic description. The half-index is again the partition function on $G$8 with Neumann boundary conditions for the 4d $G$9 vector multiplet, while half-BPS line operators lie along the half-equator. Their ordering defines a noncommutative $q$0-algebra $q$1. For Wilson lines, however, the generated subalgebra is commutative and fuses by tensor product of $q$2 representations (Lewis et al., 20 Jun 2025).

The Wilson-line half-index is given by the matrix integral

$q$3

In the same work, $q$4 is represented first by a $q$5-Weyl algebra and then by $q$6 decoupled $q$7-oscillators satisfying

$q$8

The Schur half-index becomes a vacuum expectation value,

$q$9

This realizes Schur half-indices as matrix elements in an ordinary Fock space (Lewis et al., 20 Jun 2025).

For $SU(2)$0, the fundamental Wilson operators are explicitly

$SU(2)$1

with $SU(2)$2 and $SU(2)$3. Their moments reproduce half-indices such as

$SU(2)$4

and mixed moments such as

$SU(2)$5

are equally explicit (Lewis et al., 20 Jun 2025).

The oscillator representation has a combinatorial interpretation as generalized colored chord counting. For $SU(2)$6, the relevant transfer matrix is the familiar

$SU(2)$7

recovering the ordinary bivalent chord picture. For $SU(2)$8, the fundamental transfer matrix defines a model with $SU(2)$9 colors and $\mathcal N=2$0-valent elementary vertices. Same-color crossings carry weight $\mathcal N=2$1, while different-color crossings carry weight $\mathcal N=2$2. The paper states that

$\mathcal N=2$3

so the half-index is exactly the partition function of this colored chord ensemble (Lewis et al., 20 Jun 2025).

For Wilson lines, the same operators are also the commuting Hamiltonians of the relativistic open $\mathcal N=2$4 Toda chain. Their common eigenfunctions are $\mathcal N=2$5-Whittaker polynomials, and the Schur half-index measure is the corresponding spectral measure. This yields a proof of the oscillator/half-index formula for generic $\mathcal N=2$6 (Lewis et al., 20 Jun 2025).

5. Theories with matter and DSSYK-like reformulations

Schur half-indices also exist for $\mathcal N=2$7 gauge theories with matter. In this setting the theory is placed on

$\mathcal N=2$8

or equivalently on $\mathcal N=2$9, with Neumann boundary conditions for gauge fields and compatible half-BPS boundary conditions for hypermultiplets, so that only one $HS^3\times \mathbb R$00 chiral inside each $HS^3\times \mathbb R$01 multiplet contributes. Operationally, the half-index is obtained by taking the square root of the full Schur-integrand (Berkooz et al., 16 Jul 2025).

For $HS^3\times \mathbb R$02 with $HS^3\times \mathbb R$03 fundamental half-hypermultiplets and $HS^3\times \mathbb R$04 insertions of a fundamental Wilson line, the half-index is

$HS^3\times \mathbb R$05

For one adjoint hypermultiplet,

$HS^3\times \mathbb R$06

The paper notes the convention shift $HS^3\times \mathbb R$07 relative to some earlier literature (Berkooz et al., 16 Jul 2025).

The main conceptual claim is a DSSYK-like reformulation. In ordinary DSSYK the transfer matrix is

$HS^3\times \mathbb R$08

with $HS^3\times \mathbb R$09-oscillator algebra

$HS^3\times \mathbb R$10

For pure $HS^3\times \mathbb R$11, the half-index with $HS^3\times \mathbb R$12 Wilson lines matches the vacuum amplitude

$HS^3\times \mathbb R$13

With matter, the same ordinary DSSYK Hamiltonian is retained, but the initial and final states become non-vacuum states: $HS^3\times \mathbb R$14 Thus the matter content is encoded in the boundary states rather than in a modified bulk Hamiltonian (Berkooz et al., 16 Jul 2025).

The chord-diagram interpretation is modified accordingly. Matter introduces “special segments” or “reservoir segments,” whose Hilbert-space avatars are coherent states of the $HS^3\times \mathbb R$15-oscillator. For one reservoir,

$HS^3\times \mathbb R$16

For $HS^3\times \mathbb R$17, the half-index is an amplitude between two coherent states,

$HS^3\times \mathbb R$18

The same object is also identified with the partition function of a particle on the quantum disk, whose noncommutative coordinate algebra is

$HS^3\times \mathbb R$19

After analytic continuation $HS^3\times \mathbb R$20, the quantum-disk density becomes exactly the $HS^3\times \mathbb R$21 Schur half-index density (Berkooz et al., 16 Jul 2025).

The polynomial family governing the transfer matrices varies with matter content: continuous $HS^3\times \mathbb R$22-Hermite for $HS^3\times \mathbb R$23, generalized/continuous big $HS^3\times \mathbb R$24-Hermite for $HS^3\times \mathbb R$25, Al-Salam–Chihara for $HS^3\times \mathbb R$26, continuous dual $HS^3\times \mathbb R$27-Hahn for $HS^3\times \mathbb R$28, Askey–Wilson for $HS^3\times \mathbb R$29, and continuous $HS^3\times \mathbb R$30-ultraspherical for the adjoint case. This situates Schur half-indices within the $HS^3\times \mathbb R$31-Askey scheme of orthogonal polynomials (Berkooz et al., 16 Jul 2025).

6. Terminology, neighboring constructions, and scope

The phrase “Schur half-index” does not have a single use across the broader literature. The supplied papers distinguish several nearby notions.

Notion	Meaning in the supplied literature
Boundary or hemisphere Schur half-index	Partition function on $HS^3\times \mathbb R$32 or half-space with supersymmetric boundary conditions
Schur index with a half line defect	Schur index counting endpoint operators for a $HS^3\times \mathbb R$33-BPS line supported on a ray
Interface line defect half-index	Half-index counting BPS local operators at the junction of an interface and a line operator

In the Argyres–Douglas context, “half” can mean a half-line rather than a hemisphere. The index

$HS^3\times \mathbb R$34

counts endpoint operators for half line defects on $HS^3\times \mathbb R$35. The paper is explicit that this is not a boundary or hemisphere half-index; the “half” refers to the defect being a half-line (Neitzke et al., 2017).

A different neighboring construction appears for $HS^3\times \mathbb R$36 SYM interfaces. There the half-index counts BPS local operators at the junction of a codimension-1 interface and a codimension-2 line operator. In the Higgs limit, normalized D5-interface Wilson-line one-point functions become principal specializations of Schur polynomials,

$HS^3\times \mathbb R$37

but this construction is not the conventional 4d Schur limit of the superconformal index (Hatsuda et al., 29 Oct 2025).

Class-$HS^3\times \mathbb R$38 line-operator technology is also adjacent. The supplied excerpt of the network/skein-relation paper does not define Schur half-indices, hemisphere indices, boundary conditions, or gluing formulas explicitly. A plausible implication is that its puncture-local skein relations may constrain defect-decorated half-index building blocks, but that is an inference rather than an explicit statement of the excerpt (Watanabe, 2017).

Within the boundary/hemisphere meaning of the term, however, the main structural picture is now clear. In pure $HS^3\times \mathbb R$39 SYM, the Neumann Schur half-index is an exactly solvable Macdonald-denominator integral with a one-term closed form (Lü, 11 Nov 2025). With Wilson lines in pure $HS^3\times \mathbb R$40 SYM it is equivalently a $HS^3\times \mathbb R$41-oscillator vacuum expectation value, a generalized colored chord partition function, and, for the Wilson subalgebra, a relativistic Toda spectral observable (Lewis et al., 20 Jun 2025). With matter in $HS^3\times \mathbb R$42 theories it admits a DSSYK-like interpretation in non-vacuum sectors, and for $HS^3\times \mathbb R$43 also a quantum-disk realization (Berkooz et al., 16 Jul 2025). These descriptions define the current core of the subject.