REVIEW 3 major objections 5 minor 45 references
Argyres-Douglas Theories, Macdonald Indices and Arc Space of Zhu Algebra
T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The MacDonald index of a 4d N=2 SCFT is shown to be the Hilbert series of the arc space of its Zhu algebra, which yields a closed formula for the (A1, D2n+1) Argyres-Douglas family.
desk verdict A useful arc-space theorem and two proven q-series identities, but the headline index conjecture has a concrete n=1 inconsistency that needs fixing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the arc space of the Zhu algebra of the Schur VOA. The Zhu algebra $A(V) = V/C_2(V)$ is the commutative ring built from the zero-mode products of vertex operators; its arc space is the space of formal power series through the variety it defines, and the Hilbert series of that arc space counts arcs with two gradings, conformal weight and number of strong generators. The isomorphism (3.35) between the filtration pieces of the VOA and the bigraded pieces of the arc space is what turns the refined character into the Hilbert series, and formula (3.56) evaluates the count by inclusion–exclusion over the leading monomials of a Gröbner basis. The second structural ingredient is the projected free-hypermultiplet block $\frac{1}{(q;q)_l} \sum_{k=0}^{2l} {2l \brack k}_q$, singled out by the Higgs-branch flow as the seed of the (A1, D_{2n+1}) index, together with the polynomial corrections $f^n_{m,l}$, which are stacked in the ladder $l \le m \le nl$ and satisfy the hypergeometric relation (1.2) that forces the correct Schur limit.
What would settle it
Push the established class-S formula (2.18)–(2.22) for the MacDonald index of (A1, D_{2n+1}) one step beyond the orders tabulated in the paper (Tables 6–8 stop at $t^{8}$): isolate the coefficient of $t^{9}$ for, say, (A1, D7) or (A1, D9), and compare its q-expansion with the same coefficient of the conjectured closed form (5.25) using (5.3). These two expressions are independent, so the first mismatch at any order in q would refute the conjecture, while agreement at every order would confirm it.
Extended reading notes
Core claim
On the paper's own terms, the central claim is a reduction: the MacDonald index of a 4d N=2 SCFT is an algebraic-geometric counting problem. Theorem 3.1 proves that for any strongly finitely generated VOA, the refined character with the non-affine fugacities set to one is exactly the Hilbert series of the arc space of the Zhu algebra, the commutative zero-mode ring built from the VOA's generators; assuming the refined character reproduces the MacDonald index (Conjecture 1), the unflavoured MacDonald index is that Hilbert series, computable by the Gröbner-basis formula (3.56). The second, conjectural result is a closed form for the MacDonald index of the (A1, D_{2n+1}) family, $$I_M = 1 + \sum_{m\ge 1} t^m \sum_{l=1}^{m} \frac{f^n_{m,l}(q)}{(q;q)_l} \sum_{k=0}^{2l} {2l \brack k}_q ,$$ with the polynomials $f^n_{m,l}$ defined in (5.3)–(5.4); for (A1, D3) this reduces to $I_M = \sum_{m\ge 0} \frac{t^m}{(q;q)_m} \sum_{k=0}^{2m} {2m \brack k}_q$. The shape is seeded by the Higgs-branch flow (A1, D_{2n+1}) to (A1, A_{2(n-1)}) times a projected free hypermultiplet, which motivates writing the index as the hypermultiplet block times the correction polynomials. Matching the known Schur limit requires proving the new product-sum identity (1.6), and the formula is further checked against arc-space Hilbert series and against the RG-flow factorisation.
Load-bearing premise
The load-bearing premise is that the BPS-counting formula for these theories has exactly the guessed shape — a base block like the free-hypermultiplet count, times small polynomial corrections, with no extra operator sectors — a pattern read off from the first few known terms, where the projection used to select the base block is admitted to be bookkeeping rather than a derived symmetry of the full theory.
Editorial extensions
If this is right
- Unflavoured MacDonald indices of 4d N=2 SCFTs become algorithmically computable from the Zhu algebra of the Schur VOA once Conjecture 1 is granted, replacing a non-Lagrangian field-theory computation with a Gröbner-basis calculation.
- For every (A1, D_{2n+1}) theory, the conjectured closed form fixes the full Schur-ring spectrum, with all orders in t determined by the finitely many polynomials f^n_{m,l}.
- The proven q-series identity (1.6) guarantees the proposed formula has the correct Schur limit for all n, and identity (1.7) is a new result of independent interest in the theory of partitions.
- The index factorises along the Higgs-branch flow: the product of the (A1, A_{2(n-1)}) and projected-free-hypermultiplet indices matches the conjectured (A1, D_{2n+1}) index up to an explicit correction factor.
Reading between the lines
- The arc-space algorithm is not tied to Argyres-Douglas theories: applied to any strongly finitely generated VOA with a known Zhu algebra, such as other W-algebras with conjectured presentations, the same Gröbner-basis computation would produce candidate MacDonald indices the paper does not attempt.
- The recursion relation (G.1) for f^n_{m,l} is proven here only for n = 2; upgrading it to all n would turn Conjecture 4 into a theorem, and the recursion's form hints at an underlying quasi-particle sum structure that the paper does not pursue.
- Footnote 9 treats the Z2 projection as bookkeeping rather than a derived symmetry of the full theory; a direct operator-level map between the Schur rings of (A1, D_{2n+1}) and (A1, A_{2(n-1)}) tensored with the hypermultiplet would either justify the ansatz or expose the extra sectors it omits.
- The same ansatz — a projected free-field block times polynomial corrections — could be tested on the even series (A1, D_{2n}), whose Higgs-branch flows end on (A1, A_{2n-1}) rather than (A1, A_{2n-2}); the paper's checks do not cover that family, so it is an immediate out-of-sample test.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a relation between the Macdonald index of a 4d N=2 SCFT and the Hilbert series of the arc space of the Zhu algebra of its Schur VOA, and uses this relation to conjecture a closed form for the Macdonald index of the (A1,D_{2n+1}) Argyres-Douglas theories. The main structural result is Theorem 3.1, which identifies Song's refined character, with non-affine fugacities set to one, with the arc-space Hilbert series of the Zhu algebra. The main conjectural result is Conjecture 4, giving I_M^{(A1,D_{2n+1})} as a finite sum over l of a projected free-hypermultiplet factor times polynomial corrections f^n_{m,l}(q), with f^n_{m,l} given by the explicit hypergeometric-type formula (5.3)/(H.19). The paper also proves new q-series identities used to match the Schur limit, and checks the formula against the known class-S index, the arc-space Hilbert series for the conjectured Zhu algebra, and the Higgs-branch RG flow to (A1,A_{2(n-1)}) times a projected free hypermultiplet.
Significance. If the main conjectures are correct, the paper offers a genuinely useful bridge between 4d superconformal indices and commutative algebra: the arc space of the Zhu algebra becomes a computational tool for Macdonald indices, and the conjectured closed form for (A1,D_{2n+1}) is remarkably simple. The paper has several concrete strengths: the proof of the q-series identities in Theorem 4.1 and Theorem 4.2 is explicit and self-contained; the arc-space Hilbert series computations in Section 4.2.2 are carried out in detail; and the relation to the RG flow in Section 5.3 gives a nontrivial consistency condition. However, the significance is limited by two issues: the proof of Theorem 3.1 is only sketched and depends on unproved statements about generation by singular vectors and on the acknowledged non-universality of the R-filtration weight assignment; and the central numerical validation of Conjecture 4 is partly fit-then-check, because the polynomials f^n_{m,l} are extracted from the same class-S data that are later used as checks.
major comments (3)
- [§1, Eqs. (5.1)-(5.3), (H.19), and (4.16)] The claimed universal closed form does not reduce to the (A1,D3) index at n=1. Under the natural empty-sum specialization of (H.19) for k=1, one obtains f^1_{2,1}(q)=1 rather than 0. Substituting this into Conjecture 4 (5.1) gives an extra l=1 contribution at t^2 whose q^1 coefficient is 4, so the t^2 q^1 coefficient becomes 12, whereas (4.16) gives 8 and Table 5 confirms 8. Thus the statement in the introduction that the closed form (1.3) has the special case I_{D3} = sum_m t^m/(q;q)_m sum_k [2m choose k]_q is false if (5.3) is meant to hold for all n. The paper should either explicitly restrict Conjecture 4 and the closed form (5.3) to n>=2, or correct the k=1 specialization so that f^1_{m,l}=delta_{m,l}.
- [§3.2.1 and §3.2.3] The proof of Theorem 3.1 is not complete as written. The identification of the Zhu algebra ideal with the ideal generated by singular-vector relations is essential for the isomorphism (3.37)-(3.38), but the argument assumes that the null ideal of V is generated by singular vectors and that the images of the corresponding polynomials generate A(N); the latter is asserted rather than proved. The two-bullet plausibility argument after (3.43) describes how constraints can arise on both sides, but it is not a rigorous bijection of the ideals. In addition, the refined character uses the assignment w(u_i)=1 for every strong generator (3.12), which the paper itself acknowledges is not correct in general; consequently Theorem 3.1 should be stated with this R-filtration caveat or with an explicit hypothesis, otherwise its range of applicability is unclear.
- [§5.2, Tables 6-8, and Appendix G] The numerical evidence for Conjecture 4 is partly circular. The polynomials f^n_{m,l} in Appendix G are obtained by matching the ansatz (5.25) to the class-S index (2.18), so reproducing the coefficients in Tables 6-8 is not an independent test of the conjecture. The Schur-limit check is genuine as a q-series identity, but the defining relation (5.30) is exactly the condition that makes the Schur limit match, so it tests the identity (5.29) rather than the physical prediction beyond the Schur locus. To strengthen the conjecture, the paper should provide an out-of-sample check, for example computing (5.3) at orders beyond those used to fit the f^n_{m,l} and comparing with (2.18), or deriving the ansatz from a structural argument.
minor comments (5)
- [§2.3, last paragraph] The sentence 'Indeed, we show in Section 2.3 that such a factorization of MacDonald index holds' should refer to Section 5.3, where the factorization is actually demonstrated.
- [Proof of Theorem 4.1] The proof refers to 'the last line of (5.34)' twice, but the displayed equations being discussed are in Section 4; these references should be corrected to the appropriate equation numbers in Section 4.
- [Throughout] There are several typographical errors that should be fixed, including 'Summurize' (page 8), 'techincal' (page 29), and 'ccofficients' (page 30).
- [Footnote 9, §4.1.1] The paper correctly notes that the Z2 projection of the free hypermultiplet is a bookkeeping device rather than a symmetry of the full theory; this limitation should also be stated when the RG-flow factorization (5.36) is used in Section 5.3, since the correction factor is motivated by the projected operator map.
- [Conjectures 3 and 5] The arc-space computations in Section 5.1 rely on Conjecture 3 for the leading monomials of the Gröbner basis and on Conjecture 5 for the Zhu algebra presentation; these are stated as conjectures, so the Hilbert-series comparisons using them should be described as consistency conditions rather than as derivations of the Macdonald index formula.
Circularity Check
The central (A1,D2n+1) conjecture is validated by fit-then-check: the functions f^n are computed by matching the class S index (2.18)–(2.22), then the same tables are cited as checks; the Schur-limit and RG-flow checks are enforced by construction.
-
fitted input called prediction
[Section 5.1, after Eq. (5.24); repeated in the Section 5 preamble around Eq. (5.1)]
"Using Mathematica computations, and matching the results to the MacDonald index (2.18) – (2.22) obtained from class S description, we were able to compute the functions f^2_{m,l}(q) for high values of m, l. The result is recorded in Appendix G and are consistent with the closed formula (H.19). Numerical evidence suggests the general form of MacDonald index for (A1,D2n+1) theory to be ... It was checked on Mathematica that with f^k_{m,l} given above, the proposed MacDonald index (5.25) reproduces the known coefficients in Tables 6 – 8."
The central formula (5.1)/(5.25) is not obtained as an independent prediction: the polynomials f^n_{m,l} are computed by matching the ansatz to the class S index (2.18)–(2.22). The subsequent 'check' against Tables 6–8 and the statement that the Hilbert series 'reproduces the MacDonald index (5.1)' reuse the same class S data used for the fit. The agreement is therefore forced by construction, not an independent confirmation.
-
self definitional
[Section 5.2, around Eqs. (5.29)–(5.30); also the Section 5 opening text before Conjecture 4]
"As we explain in Section 5.2, the condition (5.2) on f^n_{m,l}(q) makes sure that our proposed formula satisfy the Schur limit. ... Closely examining the proof of Theorem 4.1, we notice that (5.29) holds iff f^n_{m,l} satisfies the hypergeometric relation: 1 + sum_{m>=1} q^m sum_{l=1}^m f^n_{m,l}(q)/(q;q)_l (q^N;q)_l(q^{1-N};q)_l = q^{nN(N-1)}."
The defining relation (5.2)/(5.30) is exactly the condition required for the Schur limit of the ansatz to match the known Schur index. The functions f are chosen and proved to satisfy this relation, so the Schur-limit identity (5.29) is guaranteed by construction rather than being an independent check. The q-series proof verifies internal consistency of the ansatz, not an independent physical prediction.
1 more flagged steps
-
other
[Section 5.3.2, Eqs. (5.46)–(5.47) and Eq. (5.55)]
"Taking a product of these two indices, we obtain, ... (5.46). We claim that the correction factor is q^{(n-1)^2 N_{n-1}^2 -(n-1)N_{n-1}^2 -2N_{n-1}(N_1+...+N_{n-2})} (q;q)_{N_{n-1}}(q;q)_{2l}/(q;q)_{(n-2)N_{n-1}-N_1}(q;q)_{l-(n-1)N_{n-1}}. To prove this, let us insert the correction factor in the expression (5.46). We get ... = I^{(A1,D_{2n+1})}_M(q,t)."
The 'consistency with RG flow' check is constructed as an identity: the correction factor (5.47) is defined so that the product of the (A1,A2(n-1)) index and the projected free-hypermultiplet index becomes exactly the conjectured (A1,D2n+1) index. Inserting this chosen factor and recovering I_M is an algebraic tautology; the correction factor is not computed independently from the RG flow or from any additional data. Thus this check does not provide independent support.
full rationale
The load-bearing parts of the paper are Theorem 3.1 and Conjectures 2 and 4. Theorem 3.1, the arc-space/Zhu-algebra equivalence, is a self-contained mathematical proof (modulo Song's externally cited conjecture), so it is not circular by itself. The D3 formula (4.16) is partly supported by the independent low-order arc-space Hilbert series computation in Section 4.2.2. However, the general (A1,D2n+1) conjecture is fitted rather than predicted: Section 5.1 explicitly states that the f^n functions were computed by matching the class S index (2.18)–(2.22), and the later 'reproduces Tables 6–8' check uses the same class S data. The Schur-limit check is enforced by the defining hypergeometric relation (5.2)/(5.30), and the RG-flow check is a correction-factor identity rather than an independent constraint. These make the central validation partially circular. The reader-flagged n=1 inconsistency is a correctness risk and is not counted as circularity; the present analysis is limited to circularity proper. Overall score 6: the main conjecture's support partly reduces to its own fitting data and self-imposed constraints, although Theorem 3.1 and the D3 arc-space computation retain independent content.
Assumptions & free parameters
free parameters (1)
- Polynomial corrections f^n_{m,l}(q) =
Tables for n=2,3 and m up to 12 (Appendix G); closed form (5.3)
assumptions (6)
- domain assumption Song's conjecture: the MacDonald index of a 4d N=2 SCFT equals the refined character of its Schur VOA
- domain assumption Schur VOAs of 4d N=2 SCFTs are strongly finitely generated
- ad hoc to paper Second grading wt(u_i) = 1 for all strong generators (R-filtration)
- standard math The Zhu algebra ideal is generated by singular vector relations
- ad hoc to paper Conjecture 3: explicit leading monomials of the arc-space Groebner bases
- ad hoc to paper Conjecture 5: Zhu algebra of su(2)_{-4n/(2n+1)} is R as in (5.26)
invented entities (1)
-
Z2-projected free hypermultiplet (HM/Z2)
Cite this review
Pith. "Pith review of Argyres-Douglas Theories, Macdonald Indices and Arc Space of Zhu Algebra." pith.science (2026). https://pith.science/paper/HRGGNSSZ
@misc{pith2026250706294,
author = {Pith},
title = {Pith review of: Argyres-Douglas Theories, Macdonald Indices and Arc Space of Zhu Algebra},
year = {2026},
howpublished = {\url{https://pith.science/paper/HRGGNSSZ}},
note = {Machine review of arXiv:2507.06294}
}
abstract
In this paper, we relate the MacDonald index of a 4d $\mathcal{N}=2$ SCFT with the Hilbert series of the arc space of the Zhu algebra of the corresponding Schur VOA. Using this, we conjecture a simple formula for the MacDonald index of $(A_1,D_{2n+1})$ Argyres-Douglas theory. We perform checks of the formula against the known Schur limits and RG flows. To match the Schur limit, we prove new $q$-series identities.
Reference graph
Works this paper leans on
-
[1]
P. C. Argyres and M. R. Douglas,New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93–126, [hep-th/9505062]
arXiv 1995
-
[2]
P. C. Argyres, M. R. Plesser, N. Seiberg, and E. Witten,New N=2 superconformal field theories in four-dimensions, Nucl. Phys. B461 (1996) 71–84, [hep-th/9511154]
arXiv 1996
-
[3]
Witten,Solutions of four-dimensional field theories via M-theory, Nucl
E. Witten,Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B500 (1997) 3–42, [hep-th/9703166]
arXiv 1997
-
[4]
S. Cecotti, A. Neitzke, and C. Vafa,R-Twisting and 4d/2d Correspondences, arXiv:1006.3435
-
[5]
Xie,General Argyres-Douglas Theory, JHEP 01 (2013) 100, [arXiv:1204.2270]
D. Xie,General Argyres-Douglas Theory, JHEP 01 (2013) 100, [arXiv:1204.2270]
arXiv 2013
-
[6]
R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi,Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031, [arXiv:0807.0004]
arXiv 2008
-
[7]
F. A. Dolan and H. Osborn,On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41–89, [hep-th/0209056]
arXiv 2003
-
[8]
C. Cordova, T. T. Dumitrescu, and K. Intriligator,Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP 03 (2019) 163, [arXiv:1612.00809]
arXiv 2019
Show all 45 references
-
[9]
P. C. Argyres, M. R. Plesser, and N. Seiberg,The Moduli space of vacua of N=2 SUSY QCD and duality in N=1 SUSY QCD, Nucl. Phys. B471 (1996) 159–194, [hep-th/9603042]
1996 arXiv
-
[10]
Maruyoshi and J
K. Maruyoshi and J. Song,N = 1deformations and RG flows ofN = 2SCFTs, JHEP 02 (2017) 075, [arXiv:1607.04281]
2017 arXiv
-
[11]
Maruyoshi and J
K. Maruyoshi and J. Song,Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index, Phys. Rev. Lett.118 (2017), no. 15 151602, [arXiv:1606.05632]
2017 arXiv
-
[12]
Kinney, J
J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju,An Index for 4 dimensional super conformal theories, Commun. Math. Phys.275 (2007) 209–254, [hep-th/0510251]
2007 arXiv
-
[13]
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees,Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys.336 (2015), no. 3 1359–1433, [arXiv:1312.5344]
2015 arXiv
-
[14]
Gadde, L
A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan,Gauge Theories and Macdonald Polynomials, Commun. Math. Phys.319 (2013) 147–193, [arXiv:1110.3740]
2013 arXiv
-
[15]
Buican and T
M. Buican and T. Nishinaka,Argyres-Douglas Theories, the Macdonald Index, and an RG Inequality, JHEP 02 (2016) 159, [arXiv:1509.05402]
2016 arXiv
-
[16]
Cordova and S.-H
C. Cordova and S.-H. Shao,Schur Indices, BPS Particles, and Argyres-Douglas Theories, JHEP 01 (2016) 040, [arXiv:1506.00265]
2016 arXiv
-
[17]
Song,Macdonald Index and Chiral Algebra, JHEP 08 (2017) 044, [arXiv:1612.08956]
J. Song,Macdonald Index and Chiral Algebra, JHEP 08 (2017) 044, [arXiv:1612.08956]
2017 arXiv
-
[18]
Loustaunau,An Introduction to Grobner Bases
W. Loustaunau,An Introduction to Grobner Bases. American Mathematical Soc., 1994
1994
-
[19]
Gaiotto, G
D. Gaiotto, G. W. Moore, and A. Neitzke,Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013) 239–403, [arXiv:0907.3987]
2013 arXiv
-
[20]
Gaiotto,N=2 dualities, JHEP 08 (2012) 034, [arXiv:0904.2715]
D. Gaiotto,N=2 dualities, JHEP 08 (2012) 034, [arXiv:0904.2715]
2012 arXiv
-
[21]
A. D. Shapere and Y. Tachikawa,Central charges of N=2 superconformal field theories in four dimensions, JHEP 09 (2008) 109, [arXiv:0804.1957]
2008 arXiv
-
[22]
Chacaltana and J
O. Chacaltana and J. Distler,Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099, [arXiv:1008.5203]
2010 arXiv
-
[23]
Tachikawa,N=2 supersymmetric dynamics for pedestrians, vol
Y. Tachikawa,N=2 supersymmetric dynamics for pedestrians, vol. 890. 12, 2013. – 59 –
2013
-
[24]
Le Floch,A slow review of the AGT correspondence, J
B. Le Floch,A slow review of the AGT correspondence, J. Phys. A55 (2022), no. 35 353002, [arXiv:2006.14025]
2022 arXiv
-
[25]
C. Beem, M. Martone, M. Sacchi, P. Singh, and J. Stedman,Simplifying the TypeA Argyres-Douglas Landscape, arXiv:2311.12123
-
[26]
Tachikawa,A review of theTN theory and its cousins, PTEP 2015 (2015), no
Y. Tachikawa,A review of theTN theory and its cousins, PTEP 2015 (2015), no. 11 11B102, [arXiv:1504.01481]
2015 arXiv
-
[27]
C. Beem, W. Peelaers, L. Rastelli, and B. C. van Rees,Chiral algebras of class S, JHEP 05 (2015) 020, [arXiv:1408.6522]
2015 arXiv
-
[28]
D. Adamović,Realizations of simple affine vertex algebras and their modules: the cases sl (2) sl (2)ˆ and osp (1, 2) osp (1, 2)ˆ, Communications in mathematical physics366 (2019) 1025–1067
2019
-
[29]
Deb and S
A. Deb and S. S. Razamat,Generalized Schur partition functions and RG flows, arXiv:2506.13764
-
[30]
C. Beem, C. Meneghelli, and L. Rastelli,Free Field Realizations from the Higgs Branch, JHEP 09 (2019) 058, [arXiv:1903.07624]
2019 arXiv
-
[31]
C. Beem, A. Deb, M. Martone, C. Meneghelli, and L. Rastelli,Free field realizations for rank-one SCFTs, JHEP 12 (2024) 004, [arXiv:2407.01674]
2024 arXiv
-
[32]
Arakawa and K
T. Arakawa and K. Kawasetsu,Quasi-lisse vertex algebras and modular linear differential equations, arXiv:1610.05865
-
[33]
Zhu,Modular invariance of characters of vertex operator algebras, Journal of the American Mathematical Society9 (1995) 237–302
Y. Zhu,Modular invariance of characters of vertex operator algebras, Journal of the American Mathematical Society9 (1995) 237–302
1995
-
[34]
Xie and W
D. Xie and W. Yan,Schur sector of Argyres-Douglas theory andW-algebra, SciPost Phys. 10 (2021), no. 3 080, [arXiv:1904.09094]
2021 arXiv
-
[35]
I. B. Frenkel and Y. Zhu,Vertex operator algebras associated to representations of affine and virasoro algebras, Duke Mathematical Journal66 (1992), no. 1 123–168
1992
-
[36]
Bruschek, H
C. Bruschek, H. Mourtada, and J. Schepers,Arc spaces and rogers-ramanujan identities, Discrete Mathematics & Theoretical Computer Science(2011), no. Proceedings
2011
-
[37]
Infinite chiral symmetry in four and six dimensions
C. Beem and L. Rastelli, “Infinite chiral symmetry in four and six dimensions.” Seminar at Harvard University, Nov., 2014. Presented by L. Rastelli
2014
-
[38]
Beem and L
C. Beem and L. Rastelli,Vertex operator algebras, Higgs branches, and modular differential equations, JHEP 08 (2018) 114, [arXiv:1707.07679]
2018 arXiv
-
[39]
Bhargava, M
C. Bhargava, M. Buican, and H. Jiang,Exact Operator Map from Strong Coupling to Free Fields: Beyond Seiberg-Witten Theory, Phys. Rev. Lett.132 (2024), no. 3 031602, [arXiv:2306.05507]
2024 arXiv
-
[40]
Y. Bai, E. Gorsky, and O. Kivinen,Quadratic ideals and rogers–ramanujan recursions, The Ramanujan Journal 52 (2020) 67–89
2020
-
[41]
Buican and T
M. Buican and T. Nishinaka,On the superconformal index of Argyres–Douglas theories, J. Phys. A 49 (2016), no. 1 015401, [arXiv:1505.05884]
2016 arXiv
-
[42]
Buican and T
M. Buican and T. Nishinaka,Argyres–Douglas theories, S1 reductions, and topological symmetries, J. Phys. A 49 (2016), no. 4 045401, [arXiv:1505.06205]
2016 arXiv
-
[43]
Andrews,The Theory of Partitions
G. Andrews,The Theory of Partitions. Cambridge mathematical library. Cambridge University Press, 1998
1998
-
[44]
Gasper and M
G. Gasper and M. Rahman,Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2004
2004
-
[45]
Buchberger,Gröbner bases: An algorithmic method in polynomial ideal theory, inMultidimensional Systems Theory(N
B. Buchberger,Gröbner bases: An algorithmic method in polynomial ideal theory, inMultidimensional Systems Theory(N. K. Bose, ed.), pp. 184–232. D. Reidel Publishing Company, 1985. – 60 –
1985
Reviewed August 6, 2026 · model on record in the stance chip above.
Discussion (0). Sign in to comment.