SciPost Submission Page
Schur Connections: Chord Counting, Line Operators, and Indices
by Oscar Lewis, Mark Mezei, Matteo Sacchi, Sakura Schafer-Nameki
Submission summary
| Authors (as registered SciPost users): | Oscar Lewis |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2506.17384v1 (pdf) |
| Date submitted: | Aug. 13, 2025, 2:39 p.m. |
| Submitted by: | Oscar Lewis |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Recently, an intriguing correspondence was conjectured in arXiv:2409.11551 between Schur half-indices of pure 4d $SU(2)$ $\mathcal{N}=2$ supersymmetric Yang-Mills (SYM) theory with line operator insertions and partition functions of the double scaling limit of the Sachdev-Ye-Kitaev model (DSSYK). Motivated by this, we explore a generalization to $SU(N)$ $\mathcal{N}=2$ SYM theories. We begin by deriving the algebra of line operators, $\mathcal{A}_{\text{Schur}}$, representing it both in terms of the $\mathfrak{q}$-Weyl algebra and $\mathfrak{q}$-deformed harmonic oscillators, respectively. In the latter framework, the half-index admits a natural description as an expectation value in the Fock space of the oscillators. This $\mathfrak{q}$-oscillator perspective further suggests an interpretation in terms of generalized colored chord counting, and maps the half-index to a purely combinatorial quantity. Finally, we establish a connection with the quantum Toda chain, which is an integrable model whose commuting Hamiltonians can be identified with the Wilson lines of the $SU(N)$ SYM, and their eigenfunctions correspond to the function basis appearing in the half-index.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2025-9-19 (Invited Report)
The referee discloses that the following generative AI tools have been used in the preparation of this report:
After I prepared a report in a text editor, I let ChatGPT 5 polish the whole report. Then I checked and edited the content.
Strengths
- Makes a number of intriguing connections between different subjects in mathematical physics.
- Derivations are detailed.
- Some claims are proved in general settings.
Weaknesses
- The connections made may be mere coincidences.
- Contains typos and grammatical errors.
Report
At present, it is unclear whether these connections are mere mathematical coincidences or the first indications of a deeper physical structure. Nonetheless, the range of concrete correspondences uncovered by the authors meets two of the journal’s criteria:
• providing a novel and synergetic link between different research areas, and
• opening a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work.
However, the current version falls short of meeting two other essential criteria:
• being written in a clear and intelligible way, free of unnecessary jargon, ambiguities and misrepresentations, and
• providing citations to relevant literature in a way that is as representative and complete as possible.
In addition, the manuscript contains numerous typographical and grammatical errors, which hinder readability. Some specific issues are:
• In (2.5), the character in terms of $z_a$ should be simply $\sum_{a=1}^N z_a$. It seems the authors have confused (2.5) with (3.54), which itself contains a typo (see below).
• In (3.3), $n_{i-1}$ should be $n_i - 1$.
• In (3.54), the factor should be $v_i v_{i-1}^{-1}$, not $v_i v_{i+1}^{-1}$.
• Four lines below (4.29), the phrase “so between points 3 and 4” is unclear and should be clarified.
• Substituting (4.39) into (4.38) produces a minus sign in front of \Delta in (4.40), which is currently missing.
• In (4.40), a comma is required between $\beta_1$ and $\beta_2$.
• In (4.40), there is a $\Delta$ in the exponent of $q$, but no corresponding $\Delta$ in (4.44) or in the expression involving $q$ below (4.44). Consistency needs to be checked.
Requested changes
- Correct the typos and grammatical errors listed.
- Carefully review the manuscript once again to eliminate further typographical errors and imprecise statements not listed.
- At the beginning of the paragraph containing (4.42), it is claimed that (4.41) coincides with a Schur half-index decorated by a domain wall. Some explanation is provided, but without appropriate references or a more detailed derivation, the claim cannot be verified. The physical interpretation of $\beta_1$ and $\beta_2$ in the domain wall setting should also be clarified.
Recommendation
Ask for minor revision