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On a super-Virasoro group, a semigroup of annuli, and Gauss--Berezin integral operators

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Pith's one-line read This paper proves that unitary highest-weight representations of the Neveu–Schwarz super-Virasoro algebra extend to projective representations of a semigroup of contact super-annuli, provided the central charge and conformal weight satisfy

desk verdict A substantial superization with a real gap: the semigroup and Gauss–Berezin machinery are new and carefully done, but Theorem 8.3 is not proved for the irreducible quotients it names. read the letter →

arxiv 2607.17168 v1 pith:QGPYXLXY submitted 2026-07-19 math.RT math-phmath.FAmath.GRmath.MP

Yury A. Neretin This is my paper · ORCID
classification math.RTmath-phmath.FAmath.GRmath.MP MSC 17B6817B6722E6558A50
keywords Neveu–Schwarzsuperalgebrasuper-Virasorogroupsemigroupofsuper-annuliGauss–Berezinintegraloperatorshighestweightrepresentationssuper-Fockspacecontactstructuresunitary
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

This paper tries to establish that the unitary highest-weight representations of the Neveu–Schwarz super-Virasoro algebra can be continued from the group of contact transformations of a super-circle to a semigroup whose elements are contact super-annuli. The semigroup operation is gluing, and the paper shows that, for parameters c≥3/2 and h≥(c−3/2)/24, the representation operators extend to these glued surfaces. The construction works by realizing each super-annulus as a bounded Gauss–Berezin integral operator on a super-Fock space, so composition of gluing corresponds to composition of operators. If correct, this gives a concrete analytic model of super-Virasoro representation theory and a super-version of the classical semigroup of annuli.

What carries the argument

The central mechanism is the pairing of two constructions: first, a representation of the semigroup Γ•(A) by affine Lagrangian superrelations in an orthosymplectic space W[A], obtained from logarithmic densities on super-annuli; second, a representation of such affine Lagrangian relations by bounded Gauss–Berezin integral operators on a super-Fock space. The link is a fractional-linear coordinate transform on Lagrangian Grassmannians that turns composition of relations into matrix operations. The Gauss–Berezin operators are built from bosonic and fermionic creation-annihilation operators, and their boundedness on the smooth-vector space is controlled by norm estimates in terms of the eigenva

What would settle it

Compute the Potapov transform of the affine Lagrangian relation Δ_{μ,ν}(P) for a super-annulus P built from a contactomorphism with large winding and check the contraction inequalities of Proposition 9.2; alternatively, evaluate the Gauss–Berezin operator on a null vector in a reducible Fock module within the stated (h,c) range and test whether its image stays in the maximal submodule.

Watch

Extended reading notes

Core claim

Theorem 8.3 claims that the Neveu–Schwarz algebra representation defined by formulas (1.13)–(1.15) on the Fock space, with μ,ν∈R, integrates the modules L^{ns}(h,c) for c≥3/2 and h≥(c−3/2)/24 to the semigroup Γ•(A). The elements of Γ•(A) are annuli with an extra odd coordinate and a contact structure, and multiplication is gluing. The integration assigns to each super-annulus a Gauss–Berezin operator acting on smooth vectors of the Fock space, and the semigroup composition law becomes the product of these operators. The paper also embeds Γ•(A) into affine Lagrangian superrelations in an orthosymplectic space and uses that embedding to control the operators.

Load-bearing premise

The passage from the constructed operators on the Fock space to operators on the irreducible quotient modules L^{ns}(h,c) is assumed rather than proved; if a Gauss–Berezin operator fails to preserve the maximal submodule, the extension to Γ•(A) collapses.

Editorial extensions

If this is right

  • Every element of Γ•(A) with the stated parameters corresponds to a bounded Gauss–Berezin operator on the smooth-vector space, so gluing super-annuli is implemented by explicit integral operators.
  • The semigroup law (gluing) is compatible with operator composition, giving the Neveu–Schwarz super-Virasoro group a complex-analytic boundary analogous to the classical annulus semigroup.
  • For unitary highest-weight modules with c≥3/2 and h≥(c−3/2)/24, the projective representation of the contact super-diffeomorphism group extends to all of Γ•(A).
  • The affine Lagrangian relation attached to a super-annulus determines the corresponding operator up to a scalar, so two annuli producing the same relation yield proportional operators.

Reading between the lines

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

  • A natural next step, left implicit in the paper, is the Ramond algebra: replacing the two-sheeted covering by the trivial spin structure should give an analogous semigroup representation, since the paper's machinery is built for Neveu–Schwarz but notes the Ramond variant.
  • The boundedness estimates suggest a sharp phase boundary: outside the stated (h,c) range the contraction inequalities of Proposition 9.2 are expected to fail, so the Gauss–Berezin operators should become unbounded; checking this at small c would test whether the semigroup representation is a genuine extension.
  • Because multiplication of Gauss–Berezin operators is governed by a super-Gaussian integral, the construction may extend from annuli to more general super-Riemann surfaces with tubes, giving an analytic model for superconformal blocks.
  • The unproved quotient-descent step could be checked directly at boundary values such as c=3/2, h=0, where the Fock module is reducible; a successful check there would likely fill the proof gap.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a DeWitt-style supergroup NS(A) for the Neveu–Schwarz Lie superalgebra, describes it explicitly as a group of contactomorphisms of the supercircle S^{1|1}_•, embeds it into an affine orthosymplectic group, and constructs a semigroup Γ•(A) whose elements are contact super-annuli and whose multiplication is gluing. The central claim, Theorem 8.3, is that for c ≥ 3/2 and h ≥ 1/24(c − 3/2) the unitary highest-weight modules L^{ns}(h,c) of the Neveu–Schwarz algebra integrate to projective representations of Γ•(A). The proof in §10–§11 introduces Gauss–Berezin integral operators on a super-Fock space and attempts to match products of these operators with products of Lagrangian super-affine relations associated to super-annuli.

Significance. If the claimed theorem is fully established, the paper gives a genuine super-analog of Neretin's semigroup of annuli for the Neveu–Schwarz super-Virasoro group, with an explicit model via Gauss–Berezin integral operators and Potapov transforms. The paper contains detailed and valuable technical material: the explicit description of contact super-annuli, the norm estimates in Lemma 6.1, and the systematic use of super-affine Lagrangian relations. However, the proof of the central theorem is incomplete at two load-bearing points: the descent from Fock-space representations to irreducible quotients is not shown, and multiplicativity of the Gauss–Berezin correspondence is not fully proved. These gaps affect exactly the statement of Theorem 8.3, so the paper needs substantial revision before the main result can be regarded as established.

major comments (3)
  1. [§11.3, Theorem 8.3] The proof constructs bounded Gauss–Berezin operators on the smooth Fock space F∞, but it never proves that these operators descend to the irreducible quotient L^{ns}(h,c). For c>3/2 and h>0 the Fock module is irreducible, as the paper notes in §5.3, but Theorem 8.3 also covers c=3/2, h=0, where the Fock module is reducible (M_{-1/2}vac has norm 2h=0). The only relevant sentence in §11.3, 'Left and right multiplications by surplace contactomorphisms preserves these properties,' concerns trivial kernels and dense images, not invariance of the maximal proper submodule. Consequently the theorem is proved only for the Fock representation, not for the modules L^{ns}(h,c) named in the statement.
  2. [§11.1 and §1.2] The central charge is misstated in §11.1 as c=1+12ν², whereas the representation defined by (1.13)–(1.15) has central charge c=3/2+12ν², as stated in §1.2. With the erroneous value, the inequality h≥1/24(c−3/2) is not equivalent to the real-parameter condition μ,ν∈R; it becomes h≥ν²/2−1/48. The theorem's parameter range and the subsequent derivation in §11 depend on the correct value c=3/2+12ν². This must be corrected explicitly.
  3. [§10.10(c), §11.3] The proof of Theorem 8.3 requires the multiplicativity identity B[PQ]=B[P]B[Q] for all semigroup elements, but this is not established. Section 10.10(c) explicitly states that a product of Gauss–Berezin operators may fail to be Gauss–Berezin and that the authors is 'not sure' that this is the only obstacle. Corollary 11.4 proves the product statement only for Gauss–Berezin operators in the narrow sense. The final passage of §11.3 invokes 'Proposition' without a number; Proposition 11.3 as stated assumes that a Q with eL(Q)=eL(P)◦eL(R) exists and concludes Q=PR, which does not prove the existence of Q as the product PR. The reduction by D_i does not close this gap. Thus the semigroup representation is not rigorously constructed.
minor comments (5)
  1. [§1.2, formulas (1.13)–(1.15)] The linear terms contain an undefined index n: in (1.13) the term should presumably be (µ+iνα)T_α, and in (1.15) the last term should presumably be (µ+iνr)A_r. The same issue appears in formula (1.4). Please correct these formulas, since they define the representation used in Theorem 8.3.
  2. [§11.3] 'By Proposition' is missing a number and the surrounding argument is compressed to the point of being unverifiable. Please state precisely which proposition is being cited and fill in the steps from transversality of the relations to multiplicativity of the operators.
  3. [§11.1] The sentence 'Denote by L_A the operator L, where the boxed unit is replaced by A>0' refers to a 'boxed unit' that is not displayed; the notation should be clarified.
  4. [§1.4 and elsewhere] There are several typographical errors such as 'S^{1]1}' instead of 'S^{1|1}', and similar OCR/garbled symbols in the proof of Proposition 11.3 (e.g., ℸ, ℶ, ג). A careful editing pass will be needed.
  5. [§10.10(c)] The explicit admission of uncertainty about products of Gauss–Berezin operators is not acceptable as a final statement in a proof of a theorem. Either the product is proved to be Gauss–Berezin under the required hypotheses, or the definition must be broadened.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the semigroup representation is constructed from the Fock representation via independently proven Gauss–Berezin machinery; flagged gaps are correctness issues, not circular reductions.

full rationale

The central theorem (8.3) is not obtained by renaming an input or by fitting. The representation of ns on Fock space is given explicitly by (1.13)–(1.15); the semigroup Γ•(A) is defined geometrically in §8.4; the map from annuli to affine Lagrangian relations is built from logarithmic densities (§9), and the correspondence to Gauss–Berezin operators is proved in Theorem 10.9 and Proposition 11.3. None of these steps assumes the conclusion. The parameter range c≥3/2, h≥1/24(c−3/2) is exactly the image of real μ,ν via c=3/2+12ν², h=½(μ²+ν²), so it is not a fitted input passed off as prediction. The heavy use of the author's prior work ([43],[44],[45],[47],[50]) supplies the Potapov transform, Gaussian/Berezin operators, and Lagrangian super-Grassmannians; these are independent published results with proofs, and none is a uniqueness theorem invoked to forbid alternatives, so self-citation does not reduce the argument to itself. Two genuine gaps are present but are not circularity: §11.1 states 'c=1+12ν²' although §1.2 gives '3/2+12ν²' (a central-charge typo), and §11.3's descent to the irreducible quotient L^ns(h,c) is asserted rather than proved—'Left and right multiplications by surplace contactomorphisms preserves these properties' addresses kernels/images, not invariance of null submodules. These affect boundary cases in Theorem 8.3, but they are missing proofs/errata, not equivalence of conclusion to input by construction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 3 invented entities

No data fitting is present; μ and ν are representation labels. The construction depends on a large body of prior published mathematics, including several of the author's own works, but those results have independent proofs and are not tailored to force the target conclusion.

free parameters (2)
  • μ = arbitrary real (representation label)
    Parameter in the Fock representation (1.13)–(1.15); not fitted to data. It determines the highest weight h. The theorem assumes μ,ν∈R.
  • ν = arbitrary real; c = 3/2 + 12ν²
    Parameter in the Fock representation determining the central charge c; not fitted. Listed for exhaustiveness.
assumptions (7)
  • standard math PBW theorem for countable-dimensional Lie superalgebras
    Used in §2.9 to construct universal enveloping algebras and surplace groups.
  • domain assumption Berezin–Kats Grassmannization produces the Lie algebra g(A) and surplace groups with exponential/log
    Foundational framework in §2; cited to [8] and [6].
  • domain assumption Goodman–Wallach integration theorem for Virasoro highest-weight modules to SDiff(S1)
    Used in §5.4 and §6.4 to integrate the even part; cited to [22].
  • domain assumption Neeb–Salmasian integration theorem for ns to the supergroup SCont
    Theorem 5.2 follows from [38]; the paper gives an independent proof but still uses it as background.
  • standard math Nelson commutator theorem for essential self-adjointness
    Used in §6.3 to prove self-adjointness of e^{iπ/4}M(b).
  • domain assumption Properties of Potapov transforms and Gaussian/Berezin operators from the author's earlier work
    Bulk of §7, §9.5, and §10 assume results from [43], [44], [47], [50]; these are published proofs, not ad hoc assumptions.
  • domain assumption Removable singularity after gluing annuli and extension of complex/contact structures across the gluing curve
    Used in Theorem 9.1 and §8.1; the ordinary case is standard (Bers), but the supercase is stated without a fully detailed proof.
invented entities (3)
  • Semigroup Γ•(A) of contact super-annuli
    purpose: Complexification of the Neveu–Schwarz supergroup; elements are annuli with a contact structure and A-valued contactomorphisms on boundaries, multiplied by gluing.
    Definitional mathematical object; no external falsifiable handle.
  • Gauss–Berezin integral operators
    purpose: Integral operators on super-Fock space that realize the semigroup representation via Lagrangian relations.
    New class of operators defined in §10; definitional, no empirical evidence.
  • Super-affine Lagrangian relations
    purpose: Encoding of annulus elements as relations in an orthosymplectic space W[A], yielding Potapov transforms.
    Technical construct from §7–§9; definitional.

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Pith. "Pith review of On a super-Virasoro group, a semigroup of annuli, and Gauss--Berezin integral operators." pith.science (2026). https://pith.science/paper/QGPYXLXY

@misc{pith2026260717168,
  author       = {Pith},
  title        = {Pith review of: On a super-Virasoro group, a semigroup of annuli, and Gauss--Berezin integral operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QGPYXLXY}},
  note         = {Machine review of arXiv:2607.17168}
}
abstract

Denote by ${\mathcal A}$ the Grassmann algebra with a countable number of generators, by ${\mathcal A}_{\overline 0}$, ${\mathcal A}_{\overline 1}$ its even and odd parts. We consider supergroups as groups over $\mathcal A$. Consider the Neveu--Schwarz Lie superalgebra $\mathfrak{ns}= \mathfrak{ns}_{\overline 0}\oplus \mathfrak{ns}_{\overline 1}$. Consider its Grassmannization $\mathfrak{ns}(\mathcal{A}):= (\mathfrak{ns}_{\overline 0}\otimes {\mathcal A}_{\overline 0}) \oplus (\mathfrak{ns}_{\overline 1}\otimes {\mathcal A}_{\overline 1})$ and the corresponding supergroup $\mathrm{NS}(\mathcal {A})$. We describe this group explicitly in different ways. Next, we define a complexification $\Gamma({\mathcal A})$ of $\mathrm{NS}({\mathcal A})$. It is a semigroup, whose elements are superannuli of dimension $1|1$ equipped with contact structures; the multiplication is gluing of such superannuli. Under some inequalities for parameters, we show that unitary representations of $\mathrm{NS}({\mathcal A})$ admit extensions to representations of $\Gamma(\mathcal {A})$. We do not assume that the reader has prior knowledge of Lie superalgebras and supergroups.

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Reference graph

Works this paper leans on

65 extracted references

  1. [1]

    Banyaga A.The structure of classical diffeomorphism groups.Kluwer, Dordrecht, 1997

    1997

  2. [2]

    Bargmann, V.On unitary ray representations of continuous groups.Ann. of Math. (2) 59 (1954), 1-46

    1954

  3. [3]

    Barron K.The moduli space ofN= 1superspheres with tubes and the sewing operation. Mem. Amer. Math. Soc.162 (2003), no.772,

    2003

  4. [4]

    Barron K.The moduli space ofN= 2super-Riemann spheres with tubes.Commun. Contemp. Math. 9(2007), no.6, 857-940

    2007

  5. [5]

    A.The method of second quantization.(Russian) Nauka, Moscow 1965; English transl.: Academic Press, New York-London, 1966

    Berezin F. A.The method of second quantization.(Russian) Nauka, Moscow 1965; English transl.: Academic Press, New York-London, 1966

    1965

  6. [6]

    A.Introduction to algebra and analysis with anticommuting variables

    Berezin F. A.Introduction to algebra and analysis with anticommuting variables. (Russian), edited by A. A. Kirillov and V. P. Palamodov, Moscow State University, Moscow, 1983; English transl.:Introduction to superanalysis. Edited and with a foreword by A. A. Kirillov. With an appendix by V. I. Ogievetsky. Translated from the Russian by J. Niederle and R. ...

    1983

  7. [7]

    A., Le ˘ ıtes D

    Berezin F. A., Le ˘ ıtes D. A.Supermanifolds.Sov. Math. Dokl. 16 (5) (1975) 1218-1222

    1975

  8. [8]

    A., Kats G

    Berezin F. A., Kats G. I.Lie groups with commuting and anticommuting parameters.Math. USSR-Sb. 11:3 (1971), 311–325

    1971

Show all 65 references
  1. [9]

    Bers L.Simultaneous uniformization.Bull. Amer. Math. Soc. 66 (1960), 94-97

    1960

  2. [10]

    Boucher W., Friedan D., Kent A.Determinant formulae and unitarity for theN= 2super- conformal algebras in two dimensions or exact results on string compactification.Phys. Lett. B172 (1986), no.3-4, 316-322

    1986

  3. [11]

    DeWitt B.Supermanifolds.Cambridge University Press, 1984; 2nd ed. 1992

    1984

  4. [12]

    Dixmier J.Enveloping algebras.North-Holland, Amsterdam-New York-Oxford, 1977

    1977

  5. [13]

    K.Invariant differential operators

    Dobrev V. K.Invariant differential operators. Vol. 4. AdS/CFT, (super-)Virasoro, affine (super-)algebras.De Gruyter, Berlin, 2019

    2019

  6. [14]

    L., Leites D

    Feigin B. L., Leites D. A.,New Lie superalgebras of string theories, inGroup theoretical methods in physics(eds. M. A. Markov, V. I. Man’ko and A. E. Shabad), Vol. 1-3 (Zvenigorod, 1982), 623-629, Harwood Academic Publishers, Chur, 1985

    1982

  7. [15]

    Physical Review Letters

    Friedan D., Qiu Z., Shenker S.Conformal invariance, unitarity and critical exponents in two dimensions. Physical Review Letters. 52 (18): 1575-1578

  8. [16]

    Friedan D., Qiu Z., Shenker S.Superconformal invariance in two dimensions and the tricrit- ical Ising model.Phys. Lett. 151B, 37-43 (1985)

    1985

  9. [17]

    O.Mathematical aspects of the quantum theory of fields.Interscience Publishers, New York, 1953

    Friedrichs K. O.Mathematical aspects of the quantum theory of fields.Interscience Publishers, New York, 1953. in any formal sense. On the other hand, there is a nontrivial theory of ‘infinite dimensional Lie groups’, which is interesting for PDE and global analysis. 65

    1953

  10. [18]

    InEssays on supersymmetry (ed

    Fronsdal C., Hirai T.,Unitary representations of supergroups. InEssays on supersymmetry (ed. C. Fronsdal), 15–66. D. Reidel Publishing Co., Dordrecht, 1986

    1986

  11. [19]

    Furutsu H., Hirai T.Representations of Lie superalgebras. I. Extensions of representations of the even part.J. Math. Kyoto Univ.28 (1988), no.4, 695–749

    1988

  12. [20]

    Furutsu H., Nishiyama K.Classification of irreducible super-unitary representations of su(p, q/n). Comm. Math. Phys. 141 (1991), no. 3, 475-502

    1991

  13. [21]

    Goddard P., Kent A., Olive, D.Unitary representations of the Virasoro and super-Virasoro algebras.Comm. Math. Phys. 103 (1986), no.1, 105-119

    1986

  14. [22]

    R.Projective unitary positive-energy representations ofDiff(S 1)

    Goodman R., Wallach N. R.Projective unitary positive-energy representations ofDiff(S 1). J. Funct. Anal.63 (1985), no.3, 299–321

    1985

  15. [23]

    Grozman P., Leites D., Shchepochkina I.Lie superalgebras of string theories.Acta Math. Vietnam. 26:1 (2001), 27–63

    2001

  16. [24]

    Iohara, K., Koga, Y.,Representation theory of Neveu-Schwarz and Ramond algebras. II. Fock modules.Ann. Inst. Fourier (Grenoble) 53 (2003), no. 6, 1755-1818

    2003

  17. [25]

    Iohara K., Koga Y.Representation theory ofN= 2super Virasoro algebra: twisted sector. J. Funct. Anal. 214(2004), no.2, 450-518

    2004

  18. [26]

    P.The full set of unitarizable highest weight modules of basic classical Lie superalgebras.Mem

    Jakobsen H. P.The full set of unitarizable highest weight modules of basic classical Lie superalgebras.Mem. Amer. Math. Soc. 111 (1994)

    1994

  19. [27]

    Kac V.G.Contravariant form for infinite-dimensional Lie algebras and superalgebras, in Group theoretical methods in Physics(eds. J. Ehlers, K. Hepp, R. Kippenhahn, H. A. Wei- denm¨ uller, and J. Zittartz), Lecture Notes in Physics, 94 (1979), 441-445

    1979

  20. [28]

    G., van de Leur, J

    Kac V. G., van de Leur, J. W.On classification of superconformal algebras.In collection Strings’88 (eds. S. J. Gates, Jr., C. R. Preitschopf, and W. Siegel), 77–106, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989

    1989

  21. [29]

    G., Raina A

    Kac V. G., Raina A. K.,Bombay lectures on highest weight representations, World Scientific, Singapore, 1987

    1987

  22. [30]

    Kac V. G., Wakimoto M.Unitarizable highest weight representations of the Virasoro, Neveu- Schwarz and Ramond algebras, inConformal groups and related symmetries: physical re- sults and mathematical background (Clausthal-Zellerfeld, 1985)(eds. H. Araki, A. O. Barut, H. D. Doebn...

    1985

  23. [31]

    A.The orbits of the group of diffeomorphisms of the circle, and local Lie super- algebras.Funct

    Kirillov A. A.The orbits of the group of diffeomorphisms of the circle, and local Lie super- algebras.Funct. Anal. Appl., 15:2 (1981), 135–137

    1981

  24. [32]

    A.,Introduction to the theory of supermanifolds, Russian Math

    Leites D. A.,Introduction to the theory of supermanifolds, Russian Math. Surveys, 35:1 (1980), 1–64

    1980

  25. [33]

    Meurman A., Rocha-Caridi A.,Highest weight representations of the Neveu–Schwarz and Ramond algebras.Comm. Math. Phys. 107 (1986), no.2, 263–294

    1986

  26. [34]

    Michel J.-P., Duval C,On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative.Int. Math. Res. Not. IMRN 2008, no. 14, Art. ID rnn054

    2008

  27. [35]

    W., Moore J

    Milnor J. W., Moore J. C.On the structure of Hopf algebras.Ann. of Math. (2) 81 (1965), 211–264

    1965

  28. [36]

    Nonlinear Math

    Molotkov V,Infinite-dimensional and colored supermanifolds.J. Nonlinear Math. Phys. 17 (2010), suppl. 1, 375-446

    2010

  29. [37]

    M.Lie superalgebras and enveloping algebras.Amer

    Musson I. M.Lie superalgebras and enveloping algebras.Amer. Math. Soc., Providence, RI, 2012

    2012

  30. [38]

    Z., 275 (2013), 419-451

    Neeb K.-H., Salmasian H.Differentiable vectors and unitary representations of Fr´ echet –Lie supergroups.Math. Z., 275 (2013), 419-451

    2013

  31. [39]

    A.Unitary representations with highest weight of the group of diffeomorphisms of a circle.Functional Anal

    Neretin Yu. A.Unitary representations with highest weight of the group of diffeomorphisms of a circle.Functional Anal. Appl. 17 (1983), no. 3, 235–237

    1983

  32. [40]

    A.Unitary highest weight representations of Virasoro algebra.(Russian) Ph

    Neretin Yu. A.Unitary highest weight representations of Virasoro algebra.(Russian) Ph. D., Moscow State University, 1983, 72p

    1983

  33. [41]

    A.On a complex semigroup containing the group of diffeomorphisms of the circle

    Neretin Yu. A.On a complex semigroup containing the group of diffeomorphisms of the circle. Funct. Anal. Appl., 21:2 (1987), 160-161

    1987

  34. [42]

    A.,Representations of Virasoro algebra and affine algebras(Russian), inRepre- sentations theory and noncommutative harmonic analysis

    Neretin Yu. A.,Representations of Virasoro algebra and affine algebras(Russian), inRepre- sentations theory and noncommutative harmonic analysis. 2.(A. A. Kirillov, Yu. A. Neretin), Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 22, VINITI, Moscow, 1988, 163-224...

    1988

  35. [43]

    A.Spinor representation of an infinite-dimensional orthogonal semigroup and the Virasoro algebra.Funct

    Neretin Yu. A.Spinor representation of an infinite-dimensional orthogonal semigroup and the Virasoro algebra.Funct. Anal. Appl. 23:3 (1990), 196–207

    1990

  36. [44]

    A.On a semigroup of operators in the bosonic Fock space.Funct

    Neretin Yu. A.On a semigroup of operators in the bosonic Fock space.Funct. Anal. Appl. 24:2 (1990), 135–144

    1990

  37. [45]

    A.Holomorphic continuations of representations of the group of diffeomorphisms of the circle(Russian) Mat

    Neretin Yu. A.Holomorphic continuations of representations of the group of diffeomorphisms of the circle(Russian) Mat. Sb. 180 (1989), no. 5, 635–657; English transl.: Math. USSR-Sb. 67:1 (1990), 75–97

    1989

  38. [46]

    A.Extension of representations of classical groups to representations of cate- gories.St

    Neretin Yu. A.Extension of representations of classical groups to representations of cate- gories.St. Petersburg Math. J., 3:1 (1992), 147-169

    1992

  39. [47]

    A.Categories of symmetries and infinite-dimensional groups.Oxford University Press, New York, 1996

    Neretin Yu. A.Categories of symmetries and infinite-dimensional groups.Oxford University Press, New York, 1996

    1996

  40. [48]

    A.Integral operators with Gaussian kernels and symmetries of canonical com- mutation relations.InContemporary mathematical physics

    Neretin, Yu. A.Integral operators with Gaussian kernels and symmetries of canonical com- mutation relations.InContemporary mathematical physics. Berezin memorial volume(eds. R. L. Dobrushin, R. A. Minlos, M. A. Shubin and A. M. Vershik), 97–135, American Mathe- matical Society...

    1996

  41. [49]

    Neretin Yu.On topologies on Malcev completions of braid groups.Mosc. Math. J.12(2012), no.4, 803-824

    2012

  42. [50]

    A.Gauss-Berezin integral operators, spinors over orthosymplectic supergroups, and Lagrangian super-Grassmannians

    Neretin Yu. A.Gauss-Berezin integral operators, spinors over orthosymplectic supergroups, and Lagrangian super-Grassmannians. InSymmetry in geometry and analysis. Vol. 2. Festschrift in honor of Toshiyuki Kobayashi(eds. M. Pevzner and H. Sekiguchi), 383-455, Birkh¨ auser/Sprin...

    2025

  43. [51]

    H.Factorizable dual model of pions.Nucl

    Neveu P., Schwarz J. H.Factorizable dual model of pions.Nucl. Phys. B31, 86-112 (1971)

    1971

  44. [52]

    Puk´ anszky, L.The Plancherel formula for the universal covering group ofSL(2,R). Math. Ann. 156 (1964), 96-143

    1964

  45. [53]

    I.The Weil representation and the norms of Gaussian operators.Funct

    Olshanski, G. I.The Weil representation and the norms of Gaussian operators.Funct. Anal. Appl. 28 (1994), no. 1, 42-54

    1994

  46. [54]

    O.Superstring Schwarz derivative and the Bott cocycle

    Radul A. O.Superstring Schwarz derivative and the Bott cocycle. InIntegrable and superin- tegrable systems,In (B. A. Kupershmidt, ed.) 336-351. World Scientific, Teaneck, NJ, 1990

    1990

  47. [55]

    Ramond P.Dual theory for free fermions.Phys. Rev. D (3)3 (1971), 2415–2418

    1971

  48. [56]

    Simon B.Methods of modern mathematical physics

    Reed M. Simon B.Methods of modern mathematical physics. II. Fourier analysis, self- adjointness. Academic Press, New York-London, 1975

    1975

  49. [57]

    Reutenauer C.Free Lie algebras.Oxford University Press, New York, 1993

    1993

  50. [58]

    Theory and applications.World Scientific Publishing, Hackensack, NJ, 2007

    Rogers A.Supermanifolds. Theory and applications.World Scientific Publishing, Hackensack, NJ, 2007

    2007

  51. [59]

    Schmitt T.Supergeometry and quantum field theory, or: what is a classical configuration?. Rev. Math. Phys. 9 (1997), no.8, 993-1052

    1997

  52. [60]

    Berezin memorial volume (eds

    Schwarz A.Superanalogs of symplectic and contact geometry and their applications to quan- tum field theory,in:Topics in Statistical and Theoretical Physics. Berezin memorial volume (eds. R. L. Dobrushin, R. A. Minlos, M. A. Shubin, and A. M. Vershik.), 203–218, Amer. Math. Soc...

    1996

  53. [61]

    Segal G.Unitary representations of some infinite-dimensional groups.Comm. Math. Phys. 80:3 (1981), 301–342

    1981

  54. [62]

    B.The definition of conformal field theory

    Segal G. B.The definition of conformal field theory. InDifferential geometrical methods in theoretical physics (Como, 1987)(eds K. Bleuler, M. Werner), 165–171, Kluwer, Dordrecht, 1988

    1987

  55. [63]

    Lectures given at Harvard University, 1964.W

    Serre J.-P.Lie algebras and Lie groups. Lectures given at Harvard University, 1964.W. A. Benjamin, Inc., New York-Amsterdam, 1965

    1964

  56. [64]

    Vaintrob A.Conformal Lie superalgebras and moduli spaces.J. Geom. Phys.15(1995), no.2, 109-122

    1995

  57. [65]

    S., Volovich, I

    Vladimirov V. S., Volovich, I. V.,Superanalysis. I. Differential calculus.Theoret. and Math. Phys. 59 (1984), no. 1, 317-335. University of Graz, Department of Mathematics and Scientific computing; Higher School of Modern Mathematics MIPT, Moscow State University, MechMath. De...

    1984

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