Segal-Sugawara vectors for orthosymplectic Lie superalgebras

Alexander Molev; Madeline Nurcombe

REVIEW 3 major objections 3 minor 22 references

Segal-Sugawara vectors for orthosymplectic Lie superalgebras

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper constructs an explicit family of Segal–Sugawara vectors—central elements of the affine vertex algebra—for the orthosymplectic Lie superalgebra $\mathfrak{osp}_{M|2n}$ at the critical level.

desk verdict A genuinely new explicit construction of Segal–Sugawara vectors for osp_{M|2n} with a clever integral-form trick; the proof is coherent but leans on a few sign-sensitive 'straightforward' checks that deserve close referee scrutiny. read the letter →

arxiv 2507.22738 v1 pith:LG4S2ZQ4 submitted 2025-07-30 math.RT math-phmath.MP

Alexander Molev , Madeline Nurcombe This is my paper · ORCID

classification math.RTmath-phmath.MP MSC 17B6917B2517B67

keywords Segal-SugawaravectorsFeigin-FrenkelcentreaffinevertexalgebracriticallevelorthosymplecticLiesuperalgebraBrauerextendedBrauer-typecommutativesubalgebra

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to construct, in explicit closed form, a family of central elements of the affine vertex algebra associated with the orthosymplectic Lie superalgebra $\mathfrak{osp}_{M|2n}$ at the critical level. These elements, called Segal–Sugawara vectors, form the Feigin–Frenkel centre of the vertex algebra, a commutative superalgebra inside the enveloping algebra of the negative loop algebra. The main theorem gives a sum-over-partitions formula for vectors $\Phi_m$ and proves that they satisfy the annihilation condition that characterises the centre. The construction matters because explicit central elements are the raw material for commutative subalgebras, Gaudin-model Hamiltonians, and quantisation problems, and it supplies a new proof of the known central-element formulas for the orthogonal and symplectic Lie algebras.

What carries the argument

The carrying object is a new extended Brauer-type algebra $\widehat B_{2m+1}(\omega)$, generated by the Brauer algebra $B_{2m+1}(\omega)$ together with mode generators $f[r]_a$, a derivation $\tau$, and a central element $K$ whose relations mirror the matrix form of the affine $\mathfrak{osp}_{M|2n}$ commutation relations. In this algebra the authors form abstract Segal–Sugawara vectors $\phi_m=\gamma_m(\omega)q^{(m)}s^{(m)}f_1\cdots f_m q^{(m)}$, where $s^{(m)}$ is the Brauer symmetriser, $q^{(k)}=\epsilon_{1,m+1}\cdots\epsilon_{k,m+k}$, and $f_a=\tau+f[-1]_a$. Two cyclic properties show that $f[0]_0\phi_m$ and $f[1]_0\phi_m$ vanish modulo the left ideal generated by nonnegative modes, reproducing abstractly the annihilation condition for the centre. The decisive step is the integral form of Proposition 4.7, which uses the congruence $\gamma_\ell(\omega)s^{(\ell)}\equiv h^{(\ell)}\pmod{J_m^{(\ell)}}$ to replace the rational symmetriser by the symmetric-group symmetriser $h^{(\ell)}$, cancelling the poles at $\omega=M-2n$ and producing the explicit formula that survives evaluation at that point.

What would settle it

Compute the image of $\Phi_2$ in a small nontrivial case (for example $M=1$, $n=1$) directly from formula (2.6), apply $F_{ij}[0]$ and $F_{ij}[1]$ for all $i,j$, and look for nonzero products in the vacuum module; any nonzero result would disprove Theorem 2.1. A cheaper check is to verify every parity sign in the homomorphism of Proposition 3.3 against the explicit bracket of $\widehat{\mathfrak{osp}}_{M|2n}$.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 2.1: for every $m\ge 2$, the element \[ \Phi_m=\sum_{\$\lambda$\vdash m,\ \ell(\$\lambda$)\text{ even}} Y_{m,\ell}(M-2n-1)\, c_\$\lambda$\, \operatorname{str}_{1,\ldots,\ell} $H^{{(\ell)}}$ F[-\$\lambda$] \] belongs to the Feigin–Frenkel centre $z(\widehat{\mathfrak{osp}}_{M|2n})$ of the affine vertex algebra at the critical level. The sum is over even-length partitions of $m$, $c_\lambda$ is the number of permutations of cycle type $\lambda$, $H^{(\ell)}$ is the image of the symmetric-group symmetriser on tensor copies, $F[-\lambda]$ is a product of current generators, and the supertrace is taken over the first $\ell$ tensor factors. The authors prove the annihilation identities $F_{ij}[0]\Phi_m=F_{ij}[1]\Phi_m=0$ that define the centre, and Corollary 2.2 concludes that the $\Phi_m$ generate a commutative subalgebra of $U(t^{-1}\mathfrak{osp}_{M|2n}[t^{-1}])$. They conjecture that for odd $M$ the even-indexed vectors generate the full centre as a differential superalgebra, while for even $M$ extra super-Pfaffian elements are expected. Setting $M=0$ or $n=0$ recovers the earlier formulas for the symplectic and orthogonal Lie algebras, giving a new proof of those results.

Load-bearing premise

Everything rests on the claim, verified only by a 'straightforward' check in Proposition 3.3, that the abstract relations of the extended algebra exactly match the parity-corrected commutation relations of the orthosymplectic loop algebra; a single wrong sign there would make the constructed elements fail the annihilation test.

Editorial extensions

If this is right

The elements $\Phi_m$ are explicit even central elements of the affine vertex algebra at the critical level, so each can be written down directly from a partition sum.
The $\Phi_m$ generate a commutative associative subalgebra of $U(t^{-1}\mathfrak{osp}_{M|2n}[t^{-1}])$ that is invariant under the derivation $\tau$.
Specialising $M=0$ or $n=0$ gives a new proof of the known Segal–Sugawara vectors for $\mathfrak{sp}_{2n}$ and $\mathfrak{o}_M$.
Evaluating at $t=z$ sends $\Phi_m$ to central elements of the finite universal enveloping algebra $U(\mathfrak{osp}_{M|2n})$.
If the paper's conjecture is correct, the even vectors for odd $M$ give a complete explicit description of the Feigin–Frenkel centre for the orthosymplectic family, and for even $M$ the centre requires additional super-Pfaffian elements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One testable extension the paper leaves implicit is to verify the generation conjecture in small ranks by comparing the subalgebra generated by $\Phi_{2k}$ with known orthosymplectic invariants.
The integral-form device—trading a rational Brauer symmetriser for a symmetric-group symmetriser before specialising the parameter—looks applicable to other superalgebra settings where the Brauer symmetriser has poles, though the authors do not claim this.
Because explicit Segal–Sugawara vectors are the input to shift-of-argument constructions, these formulas are natural candidates for higher Gaudin Hamiltonians with $\mathfrak{osp}$-symmetry; the paper points towards the Gaudin connection but does not construct the Hamiltonians.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: this is a solid, genuinely new result. The authors give the first explicit family of Segal–Sugawara vectors for the orthosymplectic superalgebra osp_{M|2n}, using a new extended Brauer-type algebra that is more than a routine generalization of the classical Brauer method. The main formula (2.6) is explicit and concrete, and the integral-form trick—replacing the singular symmetriser s(m) by the symmetric-group symmetriser h(m) via the cyclic properties— is the right way to handle the pole at ω = M−2n. Recovering the classical formulas for types B, C, D as special cases is a good sanity check, and the authors are honest about what remains conjectural: Conjecture 2.3 is clearly labeled, and the note about even M and super-Pfaffians shows they know the boundary of their result.

Now the soft spots, in proportion. The proof transfers abstract annihilation in B̂_{2m+1}(ω) to the vacuum module through the homomorphism ρ of Proposition 3.3, and that transfer is load-bearing. The verification of ρ is delegated to a 'straightforward' check, and the crucial supertrace relation (3.6) is also stated as easily verified. In this superalgebra setting, where parity signs multiply, that is exactly where a small error would kill the theorem. Lemma 4.4 is likewise compressed into 'tedious but straightforward.' I have not mechanically checked every sign, so I cannot certify the proof with 100% confidence. But—and this matters—the relations are written out explicitly, the overall structure is independent and coherent, and the diagrammatic arguments for the cyclic properties look sound. These are not black boxes; they are verifiable checks that take time, not conceptual gaps.

On balance, the central argument holds up as far as I can see. The paper deserves a careful referee, not a desk reject. The referee should be asked to expand the verifications of Proposition 3.3, relation (3.6), and Lemma 4.4, and to check the parity signs carefully. If those go through—and I suspect they do—this is a correct and useful contribution for anyone working on Feigin–Frenkel centers, Gaudin models with osp symmetry, or the structure of affine vertex superalgebras at critical level.

Referee Report

3 major / 3 minor

Summary. The paper constructs explicit Segal–Sugawara vectors Φ_m for the affine vertex algebra at the critical level associated with the orthosymplectic Lie superalgebra osp_{M|2n}. The main formula (2.6) expresses Φ_m as a sum over even-length partitions λ of m of Y_{m,ℓ}(M−2n−1) c_λ str_{1,...,ℓ} H^{(ℓ)} F[−λ]. The proof introduces an extended Brauer-type algebra B̂_{2m+1}(ω), in which abstract analogues ϕ_m are shown to satisfy annihilation properties (Propositions 4.1 and 4.6), and then an integral form (Proposition 4.7 and Lemma 4.8) makes evaluation at ω = M−2n possible. A homomorphism ρ (Proposition 3.3) transfers the abstract annihilation to the actual elements Φ_m. Corollary 2.2 asserts that the Φ_m generate a commutative subalgebra of U(t^{-1}osp_{M|2n}[t^{-1}]).

Significance. The main theorem provides the first explicit family of Segal–Sugawara vectors for orthosymplectic Lie superalgebras at the critical level, extending the classical results for types B, C, and D. The strategy of lifting the construction to a new Brauer-type algebra with indeterminate ω is elegant and avoids the singularities of the Brauer symmetriser at ω = M−2n. The formulas are explicit, the combinatorial data (partitions, cycle counts, symmetrisers) is concrete, and the corollary on commutative subalgebras is a useful consequence. The paper also gives a new proof of the classical formulas in the limiting cases. If the proof is correct, the result is a significant contribution to the theory of the Feigin–Frenkel centre for superalgebras, with potential applications to Gaudin models and shift-of-argument subalgebras. The main risk is that several sign-sensitive superalgebra computations are delegated to 'straightforward' checks, and these steps are load-bearing for the validity of Theorem 2.1.

major comments (3)

[§3.1 and §3.2, Eq. (3.6) and Proposition 3.3] The transfer from the abstract algebra to the concrete orthosymplectic action rests entirely on the homomorphism ρ defined in Proposition 3.3 and on the supertrace identity (3.6). Both are asserted with the comments 'the verification of the relations is straightforward' and 'easily verified property'. These are precisely the places where the parity signs in P_{ab}, Q_{ab}, and F[r]_a (from §2, especially Eq. (2.3) and the definitions of P_{ab}, Q_{ab}) enter. A single sign error would make ρ fail to be a homomorphism or would change the image of ϕ_m, invalidating Theorem 2.1. Please provide the detailed verification of at least the defining relation (3.7) under ρ, and a derivation of (3.6) for a general X, with all parity factors exhibited.
[§4.1, Lemma 4.4] Lemma 4.4 is a key step: it asserts s_{ab} f_1...f_k s_{(k)} = f_1...f_k s_{(k)} and is used in Corollary 4.5, which in turn feeds into the central Proposition 4.6. The proof is delegated to a 'tedious but straightforward' calculation with only a brief sketch. Given the sign-sensitive nature of the algebra B̂_{2m+1}(ω) and the fact that Lemma 4.4 involves both the affine generators f_a and the Brauer symmetriser, the full calculation should be written out or included in an appendix so that the proof of Theorem 2.1 is independently verifiable.
[§4.2, proof of Theorem 2.1 after Proposition 4.6] The final passage from annihilation of ϕ_m in B̂_{2m+1}(ω)_{cri} to annihilation of Φ_m in the vacuum module requires that the elements of the left ideal be defined over C[ω] and remain well-behaved after evaluation at ω = M−2n. The text states that the first and third terms in the expansion of f[1]_0 ϕ_m are combined using cyclic property 2 and the listed relations, and then says that 'the argument used in the proof of Proposition 4.7 applies ... with obvious modifications'. This is a load-bearing step because it is where the integral form is used to remove the singularities of the symmetriser. Please spell out the modifications in detail, at least for one representative term, and explain why the C[ω]-integrality is preserved for all four terms.

minor comments (3)

[§4.1, proof of Proposition 4.1] The summations in the final displayed computation use capital M instead of lowercase m: both 'M∑_{a=1}' should be 'm∑_{a=1}'.
[§2.1 and §2.3, Eq. (2.3)] The parity factor in the definition of F[r]_a involves (-1)^{ī j̄ + ī + j̄}, while the basis elements F_{ij} are defined with (-1)^{ī j̄ + j̄} ε_i ε_j. The relationship between these two conventions is not explained; a short remark reconciling them would help readers who want to check signs.
[§4.1, Proposition 4.6] The displayed formula for f[1]_0 q(m) s(m) f_1...f_m q(m) contains many terms with signs that are asserted after a long but compressed calculation. While the general structure of the proof is clear, a brief indication of how the coefficients (e.g., the factor m and the scalar (ω+2m−2)/(ω+2m−4)) arise would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Φ_m is explicit, annihilation is proved in an auxiliary algebra before transfer, and cited Brauer identities are independent published lemmas.

full rationale

The derivation chain is self-contained in the relevant sense. The elements Φ_m in (2.6) are given by an explicit closed formula, not by fitting or by renaming a prior result, and the claim that they belong to the Feigin–Frenkel centre is proved by first constructing an abstract vector ϕ_m in the new algebra B̂_{2m+1}(ω) and then verifying annihilation identities (Propositions 4.1 and 4.6) before any evaluation of ω. The homomorphism ρ of Proposition 3.3 is not used to define Φ_m; it transfers the annihilation statement from the abstract generators f[0]_0, f[1]_0 to the concrete operators F_ij[0], F_ij[1], and the identification ρ(ϕ_m) = Φ_m Q(m) is a check performed via Proposition 4.7 and the supertrace property (3.6), not an assumed equivalence. The 'straightforward' verification of Proposition 3.3 and of Lemma 4.4 is a correctness or sign-risk concern, not a circularity, since a failure there would invalidate the theorem rather than make it true by definition. The paper's reliance on the same author's earlier works [14] and [15] supplies parameter-free Brauer-algebra identities, such as the symmetriser expansion and the congruence γ_m(ω)s(m) ≡ h(m) mod J_m used in (3.2)–(3.4); these are independent published lemmas with stated assumptions that do not include the target central-elements theorem, so they are legitimate external support. No fitted parameter is renamed as a prediction, and no target claim is assumed in its own construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The proof rests on standard affine vertex algebra theory, Schur-Weyl duality for the orthosymplectic superalgebra, Brauer symmetriser formulas from the literature, and the new extended Brauer algebra constructed in the paper. No free parameters are fitted to data; the parameter ω is an indeterminate specialized to M-2n.

assumptions (5)

standard math The Brauer algebra B_m(ω) with ω = M-2n acts on (C^{M|2n})^{⊗m} via the homomorphism (3.5) sending s_ab to P_ab and ϵ_ab to Q_ab (Schur-Weyl duality for the orthosymplectic superalgebra).
This is the standard representation of the Brauer algebra underlying the whole construction; cited in Sec. 3.1 and used implicitly throughout.
standard math The explicit factorization of the Brauer symmetriser s_{(k)} given in equation (4.6), imported from [14, proof of Lemma 1.3.2], holds in the new algebra B̂_{2m+1}(ω).
Used in Lemmas 4.2 and 4.3 to compute expressions involving ϵ_{k,m+k} s_{(k)}; a failure of this identity would break the integral-form argument.
standard math The congruence γ_k(ω) s_{(k)} ≡ h_{(k)} mod J^{(k)}_m, imported from [15, Lemma 2.4], holds inside B̂_{2m+1}(ω) for the embeddings B_k ↪ B_m.
Load-bearing for Lemma 4.8, which converts the Brauer symmetriser s^{(ℓ)} to the symmetric group symmetriser h^{(ℓ)} in the integral form (4.10).
domain assumption The map ρ defined in (3.9), sending s_ab to P_ab, ϵ_ab to Q_ab, f[r]_a to F[r]_a, τ to τ, and K to K, is a well-defined homomorphism from B̂_{2m+1}(M-2n) to (End C^{M|2n})^{⊗(2m+1)} ⊗ U.
Proposition 3.3 states this with verification delegated to a 'straightforward' check of the matrix relations for the affine superalgebra; a parity-sign error here would invalidate Theorem 2.1.
standard math The critical level identification: the dual Coxeter number for osp_{M|2n} is h^∨ = M-2n-2, so that the abstract central element ω+K-2 matches K+h^∨ at ω = M-2n.
Used in the final paragraph of Sec. 4.2 to translate the abstract critical level condition into the vacuum module condition for the Feigin-Frenkel center.

invented entities (2)

Affine extension of the Brauer algebra B^aff_m(ω)
purpose: Auxiliary algebra introduced in Definition 3.1 that packages the matrix relations of the affine superalgebra; its homomorphic image is the tensor product superalgebra used in the proof.
The algebra is a proof device. The paper notes in Remark 3.2 that additional relations would be needed for a reasonable structure theory, so it is not claimed to be canonical.
Extended Brauer-type algebra B̂_{2m+1}(ω)
purpose: Keeps ω as an indeterminate so the singular Brauer symmetriser s^{(m)} can be manipulated over C(ω); the integral form then allows evaluation at ω = M-2n.
This is the main new tool. It is validated internally by the homomorphism ρ to the orthosymplectic representation and by recovering known classical formulas as limit cases, but there is no external falsifiable prediction attached to the algebra itself.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Segal-Sugawara vectors for orthosymplectic Lie superalgebras." pith.science (2026). https://pith.science/paper/LG4S2ZQ4

@misc{pith2026250722738,
  author       = {Pith},
  title        = {Pith review of: Segal-Sugawara vectors for orthosymplectic Lie superalgebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LG4S2ZQ4}},
  note         = {Machine review of arXiv:2507.22738}
}

read the original abstract

We consider the centre of the affine vertex algebra at the critical level associated with the orthosymplectic Lie superalgebra. It is well-known that the centre is a commutative superalgebra, and we construct a family of its elements in an explicit form. In particular, this gives a new proof of the formulas for the central elements for the orthogonal and symplectic Lie algebras. Our arguments rely on the properties of a new extended Brauer-type algebra.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

[1]

Adamovi ´c, Realizations of simple affine vertex algebras and their modules: the cases ˆsl(2) and \osp(1, 2), Commun

D. Adamovi ´c, Realizations of simple affine vertex algebras and their modules: the cases ˆsl(2) and \osp(1, 2), Commun. Math. Phys. 366 (2019), 1025–1067

work page 2019
[2]

Adamovi ´c and S

D. Adamovi ´c and S. Nakatsuka, Center of affine sl2|1 at the critical level, arXiv:2412.04895

work page arXiv
[3]

A. V . Chervov and A. I. Molev,On higher order Sugawara operators, Int. Math. Res. Not. (2009), 1612–1635

work page 2009
[4]

Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, arXiv:hep-th/0604128

work page Pith review arXiv
[5]

Feigin and E

B. Feigin and E. Frenkel, Affine Kac–Moody algebras at the critical level and Gelfand– Dikii algebras, Int. J. Mod. Phys. A7, Suppl. 1A (1992), 197–215

work page 1992
[6]

Feigin, E

B. Feigin, E. Frenkel and N. Reshetikhin, Gaudin model, Bethe ansatz and critical level , Comm. Math. Phys. 166 (1994), 27–62

work page 1994
[7]

Feigin, E

B. Feigin, E. Frenkel and V . Toledano Laredo,Gaudin models with irregular singularities, Adv. Math. 223 (2010), 873–948

work page 2010
[8]

Frenkel, Langlands correspondence for loop groups , Cambridge Studies in Advanced Mathematics, 103

E. Frenkel, Langlands correspondence for loop groups , Cambridge Studies in Advanced Mathematics, 103. Cambridge University Press, Cambridge, 2007

work page 2007

Show all 22 references

[9]

Hu and Z

J. Hu and Z. Xiao, On tensor spaces for Birman–Murakami–Wenzl algebras , J. Algebra 324 (2010), 2893–2922. 22

2010
[10]

Kac,Vertex algebras for beginners, University Lecture Series, 10

V . Kac,Vertex algebras for beginners, University Lecture Series, 10. American Mathemat- ical Society, Providence, RI, 1997

1997
[11]

G. I. Lehrer and R. B. Zhang, Invariants of the orthosymplectic Lie superalgebra and super Pfaffians, Math. Z. 286 (2017), 893–917

2017
[12]

Luo and Y

Y . Luo and Y . Wang,The Schur–Weyl duality and invariants for classical Lie superalgebras, arXiv:2411.17093v1

arXiv
[13]

A. I. Molev, Feigin–Frenkel center in typesB,C andD, Invent. Math. 191 (2013), 1–34

2013
[14]

Molev, Sugawara operators for classical Lie algebras

A. Molev, Sugawara operators for classical Lie algebras . Mathematical Surveys and Monographs 229. AMS, Providence, RI, 2018

2018
[15]

A. I. Molev, On Segal–Sugawara vectors and Casimir elements for classical Lie algebras, Lett. Math. Phys. 111, 8 (2021), 23 pp

2021
[16]

A. I. Molev and E. E. Mukhin, Invariants of the vacuum module associated with the Lie superalgebra gl(1|1), J. Phys. A 48 (2015), 314001, 20 pp

2015
[17]

A. I. Molev and E. Ragoucy, The MacMahon Master Theorem for right quantum superal- gebras and higher Sugawara operators for ˆglm|n, Moscow Math. J. 14 (2014), 83–119

2014
[18]

A. I. Molev, E. Ragoucy and N. Rozhkovskaya,Segal–Sugawara vectors for the Lie algebra of typeG2, J. Algebra 455 (2016), 386–401

2016
[19]

Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra , J

M. Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra , J. Algebra 182 (1996), 664–693

1996
[20]

Riesen, Interpolating Feigin–Frenkel duality at the critical level to matrices of complex size, arXiv:2505.10439

A. Riesen, Interpolating Feigin–Frenkel duality at the critical level to matrices of complex size, arXiv:2505.10439

arXiv
[21]

L. G. Rybnikov, The shift of invariants method and the Gaudin model , Funct. Anal. Appl. 40 (2006), 188–199

2006
[22]

Yakimova, Symmetrisation and the Feigin–Frenkel centre, Compos

O. Yakimova, Symmetrisation and the Feigin–Frenkel centre, Compos. Math. 158 (2022), 585–622. School of Mathematics and Statistics University of Sydney, NSW 2006, Australia alexander.molev@sydney.edu.au madeline.nurcombe@sydney.edu.au 23

2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Adamovi ´c, Realizations of simple affine vertex algebras and their modules: the cases ˆsl(2) and \osp(1, 2), Commun

D. Adamovi ´c, Realizations of simple affine vertex algebras and their modules: the cases ˆsl(2) and \osp(1, 2), Commun. Math. Phys. 366 (2019), 1025–1067

work page 2019

[2] [2]

Adamovi ´c and S

D. Adamovi ´c and S. Nakatsuka, Center of affine sl2|1 at the critical level, arXiv:2412.04895

work page arXiv

[3] [3]

A. V . Chervov and A. I. Molev,On higher order Sugawara operators, Int. Math. Res. Not. (2009), 1612–1635

work page 2009

[4] [4]

Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, arXiv:hep-th/0604128

work page Pith review arXiv

[5] [5]

Feigin and E

B. Feigin and E. Frenkel, Affine Kac–Moody algebras at the critical level and Gelfand– Dikii algebras, Int. J. Mod. Phys. A7, Suppl. 1A (1992), 197–215

work page 1992

[6] [6]

Feigin, E

B. Feigin, E. Frenkel and N. Reshetikhin, Gaudin model, Bethe ansatz and critical level , Comm. Math. Phys. 166 (1994), 27–62

work page 1994

[7] [7]

Feigin, E

B. Feigin, E. Frenkel and V . Toledano Laredo,Gaudin models with irregular singularities, Adv. Math. 223 (2010), 873–948

work page 2010

[8] [8]

Frenkel, Langlands correspondence for loop groups , Cambridge Studies in Advanced Mathematics, 103

E. Frenkel, Langlands correspondence for loop groups , Cambridge Studies in Advanced Mathematics, 103. Cambridge University Press, Cambridge, 2007

work page 2007