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A de Rham model for complex analytic equivariant elliptic cohomology

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Pith's one-line read Complex analytic equivariant elliptic cohomology admits a unique refinement of the string orientation: the twisted equivariant Euler class of V3, realized inside a sheaf of differential graded algebras on the moduli space of G-bundles…

desk verdict A genuinely new cocycle model for complex analytic equivariant elliptic cohomology, with explicit Euler/Thom cocycles from sigma functions; the headline uniqueness claim rests on a 'by inspection' proof and deferred transformation laws, so send to a serious referee but demand the missing derivation. read the letter →

arxiv 1908.02868 v2 pith:N3DTKSMC submitted 2019-08-07 math.AT math.AGmath.RT

Daniel Berwick-Evans , Arnav Tripathy This is my paper · ORCID
classification math.ATmath.AGmath.RT MSC 55N3455N9114H5222E67
keywords equivariantellipticcohomologycocyclemodeldeRhamstringorientationtheoremofthecubeLooijengalinebundlesloopgrouprepresentationsWeierstrasssigmafunction
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

Equivariant elliptic cohomology is meant to be the arena for elliptic character theory, where loop-group representations and the geometry of elliptic curves meet, but the complex-analytic versions that existed came in two separate regimes: Grojnowski's theory for connected groups and Devoto's for finite groups. This paper claims both are facets of a single object: a sheaf of commutative differential graded algebras on the stack $\mathrm{Bun}_G(E)$ of flat $G$-bundles over elliptic curves, whose local data are equivariant differential forms on fixed-point manifolds $M^{h}$ for commuting pairs $h=(h_1,h_2)$. The model yields explicit equivariant Euler and Thom cocycles from the Weierstrass $\sigma$ function, identified with level-one vacuum characters of loop groups. The paper's central result is that the string (MString) orientation of complex analytic elliptic cohomology, built from the theorem of the cube, has exactly one equivariant refinement, and that refinement is the twisted equivariant Euler class of the universal virtual bundle $V_3$.

What carries the argument

The object that carries the whole construction is the sheaf of elliptic cocycles, $\widehat{\mathrm{Ell}}_G^{\bullet}(M)$: a local section assigns to every commuting pair $h=(h_1,h_2)$ in a neighborhood in $\mathrm{Bun}_G(E)$ a $G^{h}_0$-equivariant differential form on the fixed-point manifold $M^{h}$, subject to a conjugation-invariance condition and an analyticity condition, $\alpha_{h'}(X)=\operatorname{res}\alpha_h(X+(X_1-\tau X_2))$, that lets the forms vary holomorphically as $h$ is deformed. The specific formula carrying the orientation result is the cubical structure of the line bundle $\mathcal{O}(-0)$ on the elliptic curve, the unique section, by the theorem of the cube, of $\Theta_3(\mathcal{O}(-0))$ on $E\times E\times E$, written in $\sigma$-function coordinates as $$ s=\frac{\$\sigma$(x+y)\$\sigma$(x+z)\$\sigma$(y+z)\$\sigma$(0)}{\$\sigma$(x+y+z)\$\sigma$(x)\$\sigma$(y)\$\sigma$(z)} , $$ which is formula (70) of the paper. The proof of uniqueness asserts that this same formula is the unique equivariant extension, and matches it with the twisted equivariant Euler class of $V_3$. The $\sigma$ functions $\sigma(\tau,z)$ and $\upsilon(\tau,z)$ also supply the Euler and Thom cocycles for $U(n)$ and $\mathrm{Spin}(2n)$, linking them to loop-group characters.

What would settle it

Compute the space of $\mathrm{SL}_2(\mathbb{Z})$-equivariant holomorphic sections of the cubical line bundle $\Theta_3(\mathcal{O}(-0))$ over the universal triple $E^{\vee}\times E^{\vee}\times E^{\vee}$: Theorem 7.11 predicts this space is one-dimensional, spanned by formula (70). Exhibiting a second linearly independent section, for instance one differing from (70) by a nonconstant modular function or by a character of the elliptic curve, would falsify the claimed uniqueness; so would a direct failure of the rigid, symmetric, or cocycle conditions for any putative alternative section at the stalk level.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that complex analytic equivariant elliptic cohomology is carried by a single sheaf of cdgas: for a $G$-manifold $M$, the sheaf $\widehat{\mathrm{Ell}}_{G}^{\bullet}(M)$ on $\mathrm{Bun}_G(E)$ whose local section at a commuting pair $h=(h_1,h_2)$ is a $G^{h}_0$-equivariant differential form on the fixed-point manifold $M^{h}$, satisfying conjugation invariance and an analyticity condition, $\alpha_{h'}(X)=\operatorname{res}\alpha_h(X+(X_1-\tau X_2))$, that encodes the additive structure of the elliptic curve. Restricting the cohomology sheaf to a fixed elliptic curve and a maximal torus reproduces Grojnowski's connected-group theory (Theorem 4.1), and for finite groups the global sections reproduce Devoto's theory over $\mathbb{C}$ (Theorem 5.1). The paper then constructs Mathai–Quillen type equivariant elliptic Euler and Thom classes for $U(n)$ and $\mathrm{Spin}(2n)$ from $\theta$ functions, identifies them with supercharacters of level-one vacuum representations of loop groups, and proves that the $MO\langle 8\rangle$ (MString) orientation of complex analytic elliptic cohomology has a unique equivariant refinement, equal to the twisted equivariant Euler class of $V_3=(L_1-1)\otimes(L_2-1)\otimes(L_3-1)$; uniqueness is a consequence of the theorem of the cube for the line bundle $\mathcal{O}(-0)$.

Load-bearing premise

The headline uniqueness theorem rests on the proof's assertion, made by inspection without derivation, that the non-equivariant cubical structure formula (70) has exactly one equivariant extension given by the same formula; the construction also imports sigma-function transformation laws from the companion preprint [BET19] without reproducing them.

Editorial extensions

If this is right

  • One construction covers both regimes: connected groups recover the earlier cocycle-level theory and finite groups recover the earlier finite-group theory, so results proved in one formalism now transfer to the other.
  • The Euler and Thom cocycles give explicit, computable representatives for the $MU\langle 6\rangle$- and MString-orientations of complex analytic elliptic cohomology, upgrading those orientations from existence statements to formulas.
  • Because the Euler cocycles are supercharacters of level-one vacuum representations, characteristic classes in this theory come with a representation-theoretic reading: loop-group characters are the new Chern roots.
  • Complex orientations of elliptic cohomology admit no equivariant refinement at all, so the twisted equivariant setting is not a convenience but a necessity, and the uniqueness from the theorem of the cube removes the ambiguities that exist at the level of formal groups.
  • Global sections of the sheaf are 24-periodic rather than 2-periodic, and the theory's derived global sections are nontrivial, matching known computations of equivariant topological modular forms for $U(1)$.

Reading between the lines

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

  • If the uniqueness claim holds, a natural expectation is that the same equivariant rigidity applies to any elliptic cohomology theory over $\mathbb{C}$ whose formal group carries the standard coordinate: the phenomenon would be a feature of elliptic curves themselves, not of this particular model.
  • The paper leaves open the derived global sections of the sheaf for general $G$; computing them for examples such as $U(1)$ acting on $S^2$ would test whether the cocycle model is the right input for elliptic Springer theory and elliptic stable envelope constructions.
  • The asserted 'by inspection' uniqueness in the proof of Theorem 7.11 deserves a direct check: computing the space of $\mathrm{SL}_2(\mathbb{Z})$-equivariant holomorphic sections of $\Theta_3(\mathcal{O}(-0))$ over the universal family would either confirm the rigidity or expose hidden moduli of equivariant orientations.
  • The identification of Thom cocycles with loop-group characters suggests an elliptic index theorem: pushing forward equivariant elliptic operators along these Thom classes should yield modular forms whose $q$-expansions are governed by the corresponding characters, a testable analogue of the index-theoretic reading of equivariant K-theory.
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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

4 major / 4 minor

Summary. The paper constructs a sheaf of commutative differential graded algebras, denoted Ell_G(M), on the moduli stack Bun_G(E) of G-bundles over elliptic curves, with local sections given by compatible equivariant de Rham complexes on fixed-point loci. It claims that restricting to a fixed elliptic curve recovers Grojnowski's complex analytic equivariant elliptic cohomology for connected G, and that for finite G the global sections recover Devoto's theory. It then constructs explicit Mathai-Quillen type Euler and Thom classes using products of sigma functions, identifies the relevant twistings with Looijenga line bundles and level-1 loop-group characters, and gives an elliptic Chern-Weil map. The final section studies equivariant refinements of orientations: it shows that ordinary complex orientations do not refine equivariantly, constructs a twisted refinement via the function upsilon, and claims in Theorem 7.11 that there is a unique equivariant refinement of the MO<8>-orientation, equal to the twisted equivariant Euler class of the virtual bundle V3=(L1-1) otimes (L2-1) otimes (L3-1), obtained from the theorem-of-the-cube formula (70).

Significance. If the claims hold, the paper provides a useful and explicit differential-geometric model for complex analytic equivariant elliptic cohomology: it gives a uniform cocycle-level framework, recovers Grojnowski's and Devoto's theories, and produces concrete representatives of elliptic Euler and Thom classes tied to loop-group characters and the Atiyah-Segal completion perspective. The stack Bun_G(E), the holomorphic structure on it, and the treatment of twisting line bundles are carefully set up, and the examples involving U(1)-actions on spheres are informative. However, the headline uniqueness theorem is currently asserted rather than proved, and several load-bearing transformation laws are imported from the companion preprint [BET19], so the central statement needs substantial repair before the results can be regarded as fully established.

major comments (4)
  1. [§7.3, Theorem 7.11] The proof of Theorem 7.11 is not sufficient for the paper's headline claim. The assertion that formulas (70) 'by inspection' have a unique equivariant extension skips the verification that the displayed function defines a global section of Θ3(O(-0)) on (E∨)^3, or equivalently on Bun_{U(1)^3}(E): one must check quasi-periodicity in each variable under the cocharacter lattice, the rigid, symmetric, and cocycle conditions, and the SL2(Z)-equivariance and descent data required by Definition 3.3. The uniqueness statement is also asserted without proof: the paper does not show that the completion map from global sections of Θ3(O(-0)) to the formal power series ring is injective, nor does it otherwise rule out other equivariant sections with the same non-equivariant expansion. Since Theorem 7.11 is the central result advertised in the abstract, this step needs a real derivation rather than an inspection claim.
  2. [§6.4, Definition 6.5 and Proposition 6.10] The construction of the line bundles A_G and L_G and their identification with level-1 Looijenga line bundles depends on transformation formulas for σ and υ quoted as [BET19, Equations 146-149] and not reproduced in this paper. These transformation laws are needed for Proposition 6.8, for the gluing of the Thom forms in Proposition 6.13, and ultimately for the twisted equivariant Euler class used in Theorem 7.11(2). The paper should either include these formulas or prove them directly; delegating a load-bearing step to a companion preprint is not adequate for the claims made here.
  3. [§3, Proposition 3.5] The proof of Proposition 3.5(1) is a sketch of the key descent construction. It asserts that the data assemble via Proposition 2.14 and that conditions (C1)-(C2) follow from the invariance property (25), but it does not actually verify the equivariance of the restriction maps (28) under composition, the independence of choices of logarithms and maximal commuting subalgebras, or the sheaf condition on arbitrary open covers. Since this proposition defines the central object of the paper, the argument should be written out in enough detail to be checked.
  4. [§4, Theorem 4.1] The proof of Theorem 4.1 is compressed at a technically sensitive point: the passage from W-invariants of the sum over W/W_h to W_h-invariants of the identity summand is stated without justification, and the final stalk argument is asserted to be clear. This is not necessarily wrong, and the comparison is plausible, but given that the recovery of Grojnowski's theory is one of the paper's main advertised results, a short proof or a precise reference for this finite-group induction step should be supplied.
minor comments (4)
  1. [§4.1, Definition 4.2] The phrase "can be identifies" should read "can be identified."
  2. [§7.1, Definition 7.6] The letter E is used both for the universal elliptic curve and for the restricted family over U⊂H; this notational overlap should be resolved for clarity.
  3. [§3, proof of Proposition 3.5] The phrase "Wh-invariant elements of cdga associated with H×T^ε_h" is awkward; it would be clearer to say "Wh-invariant sections of the sheaf of cdgas on H×T^ε_h."
  4. [§7.3, Definition 7.10] The definition of an equivariant refinement would be clearer if it explicitly stated that the class is a global section of the Θ3(O(-0))-twisted sheaf on Bun_{U(1)^3}(E), rather than leaving this to the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is explicit and the self-citations are not load-bearing.

full rationale

The derivation chain is self-contained in the relevant sense. Definition 3.3 constructs the sheaf Ell^•_G(M) from explicit equivariant de Rham data with analyticity conditions, and Theorems 4.1 and 5.1 verify compatibility with Grojnowski and Devoto by stalkwise comparisons rather than by assuming those theories. The Euler and Thom cocycles (Propositions 6.8, 6.13) are built from sigma-function formulas and verified locally (invariance, analyticity, nonvanishing), so their twisted global-section status is not obtained by renaming an input. The headline uniqueness in Theorem 7.11 is asserted 'by inspection' (Section 7.3) and the transformation laws are deferred to the companion preprint [BET19] in places such as Definition 6.5 and Proposition 6.10; however, these are classical sigma-function transformation laws, not results that assume the target theorem, and the uniqueness claim is an omitted derivation rather than a circular reduction: no equation is defined in terms of the conclusion, and no fitted parameter is later called a prediction. The comparisons with Grojnowski and Devoto are external benchmarks. Hence no circular step; score 0.

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

No free parameters appear in the construction; the theory is parameter-free. The auxiliary objects, the sheaf Ell_G and the stack Bun_G(E), are explicit definitions rather than postulated entities with independent physical content. The axioms above are standard facts invoked without proof.

assumptions (5)
  • standard math The theorem of the cube: the line bundle O(-0) on an elliptic curve has a unique cubical structure.
    Invoked in §7.3 to establish uniqueness of the MU(6)/MO(8) orientation and its equivariant refinement; the proof is not given, with citations to [Hop94] and [AHS01].
  • domain assumption Sections of the level l Looijenga line bundle are spanned by super characters of positive energy representations of the loop group at level l.
    Used in Proposition 3.17 to identify global sections with the Verlinde algebra; the paper cites Ando [And00, Corollary 10.9].
  • standard math Atiyah-Bott localization holds for the equivariant cohomology of fixed point sets.
    Used in the review of Grojnowski's theory in §4.1 and in the comparison Theorem 4.1; cited to Atiyah-Bott [AB84].
  • domain assumption Every G-manifold embeds equivariantly into a finite-dimensional G-representation.
    Stated in the notation section (§1) and used in Lemma 3.1 to ensure fixed-point loci are constant for small deformations.
  • domain assumption The Weierstrass sigma function transformation laws define the line bundles A_G and L_G on Bun_G(E).
    Used in Definition 6.5 and Proposition 6.10; the explicit transformations are cited to the companion paper [BET19, Equations 140-141, 146-149] rather than proved here.

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Pith. "Pith review of A de Rham model for complex analytic equivariant elliptic cohomology." pith.science (2026). https://pith.science/paper/N3DTKSMC

@misc{pith2026190802868,
  author       = {Pith},
  title        = {Pith review of: A de Rham model for complex analytic equivariant elliptic cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N3DTKSMC}},
  note         = {Machine review of arXiv:1908.02868}
}
abstract

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai--Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique equivariant refinement of Hopkins' "theorem of the cube" construction of the ${\rm MString}$-orientation of elliptic cohomology.

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Works this paper leans on

62 extracted references · 61 canonical work pages

  1. [1]

    Atiyah and R

    M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984)

    work page 1984

  2. [2]

    Atiyah, R

    M. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl. 1, 3--38

    work page 1964

  3. [3]

    Ando and C

    M. Ando and C. French, Discrete torsion for the supersingular orbifold sigma genus, London Math. Soc. Lecture Note Ser. Elliptic cohomology (2007)

    work page 2007

  4. [4]

    M. Ando, M. Hopkins, and C. Rezk, Multiplicative orientations of KO -theory and the spectrum of topological modular forms , Draft (2010)

    work page 2010

  5. [5]

    M. Ando, M. Hopkins, and N. Strickland, Elliptic spectra, the theorem of the cube, and the Witten genus , Invent. math. 146 (2001)

    work page 2001

  6. [6]

    Ando, Power operations in elliptic cohomology and representations of loop groups, Transactions of the American Mathematical Society 352 (2000)

    M. Ando, Power operations in elliptic cohomology and representations of loop groups, Transactions of the American Mathematical Society 352 (2000)

    work page 2000

  7. [7]

    , The sigma orientation for analytic circle-equivariant elliptic cohomology, Geom. Topol. 7 (2003), 91--153

    work page 2003

  8. [8]

    Aganagic and A

    M. Aganagic and A. Okounkov, Elliptic stable envelope, Arxiv preprint (2016)

    work page 2016

Show all 62 references
  1. [9]

    Bullimore, T

    M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn, and H. Kim, Vortices and V ermas , Arxiv preprint (2016)

    2016

  2. [10]

    Bullimore, T

    M. Bullimore, T. Dimofte, D. Gaiotto, and J. Hilburn, Boundaries, mirror symmetry, and symplectic duality in 3d N =4 gauge theory , Arxiv preprint (2016)

    2016

  3. [11]

    Berwick-Evans, Equivariant elliptic cohomology, gauged sigma models, and discrete torsion, preprint (2014)

    D. Berwick-Evans, Equivariant elliptic cohomology, gauged sigma models, and discrete torsion, preprint (2014)

    2014

  4. [12]

    , Supersymmetric field theories and the elliptic index theorem with complex coefficients, to appear in Geometry and Topology (2020)

    2020

  5. [13]

    Berwick-Evans and A

    D. Berwick-Evans and A. Tripathy, A model for complex analytic equivariant elliptic cohomology from quantum field theory, preprint (2019)

    2019

  6. [14]

    Block and E

    J. Block and E. Getzler, Equivariant cyclic homology and equivariant differential forms, Ann. Sci. Ecole Normale Sup. 4 (1994)

    1994

  7. [15]

    Berline, E

    N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators , Springer, 1992

    1992

  8. [16]

    Borel, Sous-groupes commutatifs et torsion des groupes de L ie compactes , T\^ohoku Math

    A. Borel, Sous-groupes commutatifs et torsion des groupes de L ie compactes , T\^ohoku Math. Jour. 13 (1962)

    1962

  9. [17]

    Baker and C

    A. Baker and C. Thomas, Classifying spaces, Virasoro equivariant bundles, elliptic cohomology and Moonshine , Glasgow University Mathematics Department preprint (1999)

    1999

  10. [18]

    Behrend and P

    K. Behrend and P. Xu, Differentiable stacks and gerbes, J. Symplectic Geom. 9 (2011), no. 3

    2011

  11. [19]

    Ben-Zvi and D

    D. Ben-Zvi and D. Nadler, Elliptic S pringer theory , Compos. Math. 151 (2015), no. 8, 1568--1584

    2015

  12. [20]

    Carnahan, Generalized moonshine I : Genus-zero functions , Algebra Number Theory 4 (2010), no

    S. Carnahan, Generalized moonshine I : Genus-zero functions , Algebra Number Theory 4 (2010), no. 6

    2010

  13. [21]

    Devoto, Equivariant elliptic homology and finite groups, Michigan Math

    J. Devoto, Equivariant elliptic homology and finite groups, Michigan Math. J. 43 (1996)

    1996

  14. [22]

    , An algebraic description of the elliptic cohomology of classifying spaces, Journal of Pure and Applied Algebra 130 (1998)

    1998

  15. [23]

    Dixon, J

    L. Dixon, J. Harvey, C. Vafa, and E. Witten, Strings on orbifolds, Nuclear Physics B (1985)

    1985

  16. [24]

    Duflo and M

    M. Duflo and M. Vergne, Cohomologie \'equivariante et descente, Ast\'erisque 215 (1993)

    1993

  17. [25]

    Freed and F

    D. Freed and F. Quinn, Chern-Simons theory with finite gauge group , Comm. Math. Phys. 156 (1993)

    1993

  18. [26]

    Ganter, Hecke operators in equivariant elliptic cohomology and generalized M oonshine , Groups and symmetries, CRM Proc

    N. Ganter, Hecke operators in equivariant elliptic cohomology and generalized M oonshine , Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 173--209

    2009

  19. [27]

    , The elliptic W eyl character formula , Compos. Math. 150 (2014), no. 7, 1196--1234

    2014

  20. [28]

    Ginzburg, M

    V. Ginzburg, M. Kapranov, and E. Vasserot, Elliptic algebras and equivariant elliptic cohomology, Arxiv preprint (1995)

    1995

  21. [29]

    Gepner and L

    D. Gepner and L. Meier, On equivariant topological modular forms, preprint (2020)

    2020

  22. [30]

    Grojnowski, Delocalised equivariant elliptic cohomology, Elliptic cohomology: Geometry, applications, and higher chromatic analogues ( H

    I. Grojnowski, Delocalised equivariant elliptic cohomology, Elliptic cohomology: Geometry, applications, and higher chromatic analogues ( H. Miller and D. Ravenel , ed.), London Mathematical Society, 2007

    2007

  23. [31]

    Hopkins, N

    M. Hopkins, N. Kuhn, and D. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000)

    2000

  24. [32]

    Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube , Proceedings of the ICM 1 (1994)

    M. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube , Proceedings of the ICM 1 (1994)

    1994

  25. [33]

    Theo Johnson-Freyd, The Moonshine anomaly , Communications in Mathematical Physics 365 (2019)

    2019

  26. [34]

    Kazhdan and G

    D. Kazhdan and G. Lusztig, Proof of the D eligne- L anglands conjecture for H ecke algebras , Invent. Math. 87 (1987), no. 1, 153--215

    1987

  27. [35]

    Liu, Modular forms and topology, In Proc

    K. Liu, Modular forms and topology, In Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996)

    1996

  28. [36]

    Lurie, A survey of elliptic cohomology, Algebraic Topology (N

    J. Lurie, A survey of elliptic cohomology, Algebraic Topology (N. Baas, E. Friedlander, J. Bj\"orn, and P. O st r , eds.), vol. 4, Springer Berlin Heidelberg, 2009

    2009

  29. [37]

    Lurie, Elliptic cohomology III : T empered cohomology , Preprint (2019)

    J. Lurie, Elliptic cohomology III : T empered cohomology , Preprint (2019)

    2019

  30. [38]

    Mason, Finite groups and modular functions, The Arcata Conference on Representations of Finite Groups, Proc

    G. Mason, Finite groups and modular functions, The Arcata Conference on Representations of Finite Groups, Proc. Sympos. Pure Math., Amer. Math. Soc. 47 (1987)

    1987

  31. [39]

    Meinrenken, Equivariant cohomology and the C artan model , Overview article for the Encyclopedia of Mathematical Physics (2006)

    E. Meinrenken, Equivariant cohomology and the C artan model , Overview article for the Encyclopedia of Mathematical Physics (2006)

    2006

  32. [40]

    Maulik and A

    D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, Arxiv preprint (2012)

    2012

  33. [41]

    Morava, Moonshine elements in elliptic cohomology, Groups and symmetries, CRM Proc

    J. Morava, Moonshine elements in elliptic cohomology, Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 247--257

    2009

  34. [42]

    Mostow, Equivariant embeddings in Euclidean space , Anals

    G. Mostow, Equivariant embeddings in Euclidean space , Anals. Math. 65 (1957)

    1957

  35. [43]

    Mathai and D

    V. Mathai and D. Quillen, Superconnections, Thom classes and equivariant differential forms , Topology 25 (1986)

    1986

  36. [44]

    Palais, Imbedding of compact differentiable transformation groups in orthogonal representations, J

    R. Palais, Imbedding of compact differentiable transformation groups in orthogonal representations, J. Math. Mech. 6 (1957)

    1957

  37. [45]

    Pressley and G

    A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986, Oxford Science Publications

    1986

  38. [46]

    Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull

    D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293--1298

    1969

  39. [47]

    Rosu, Equivariant elliptic cohomology and rigidity, Amer

    I. Rosu, Equivariant elliptic cohomology and rigidity, Amer. J. Math. 123 (2001), no. 4, 647--677

    2001

  40. [48]

    Rimanyi and A

    R. Rimanyi and A. Weber, Elliptic classes of Schubert cells via Bott-Samelson resolution , Arxiv preprint (2019)

    2019

  41. [49]

    Hautes \' E tudes Sci

    Graeme Segal, The representation ring of a compact L ie group , Inst. Hautes \' E tudes Sci. Publ. Math. (1968), no. 34, 113--128

    1968

  42. [50]

    Segal, Elliptic cohomology, S\'eminaire N

    G. Segal, Elliptic cohomology, S\'eminaire N. Bourbaki 695 (1988)

    1988

  43. [51]

    Sharpe, Discrete torsion in perturbative heterotic string theory, Phys.Rev

    E. Sharpe, Discrete torsion in perturbative heterotic string theory, Phys.Rev. D 68 (2003)

    2003

  44. [52]

    Stolz and P

    S. Stolz and P. Teichner, What is an elliptic object?, Topology, geometry and quantum field theory, London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247--343

    2004

  45. [53]

    Jur c o, H

    , Supersymmetric field theories and generalized cohomology, Mathematical Foundations of Quantum Field and Perturbative String Theory ( B. Jur c o, H. Sati, U. Schreiber , ed.), Proceedings of Symposia in Pure Mathematics, 2011

    2011

  46. [54]

    The Stacks Project Authors , Stacks Project , https://stacks.math.columbia.edu, 2018

    2018

  47. [55]

    Vafa, Modular invariance and discrete torsion on orbifolds, Nuclear physics B (1986)

    C. Vafa, Modular invariance and discrete torsion on orbifolds, Nuclear physics B (1986)

    1986

  48. [56]

    Vergne, Geometric quantization and equivariant cohomology, First European Congress of Mathematics (A

    M. Vergne, Geometric quantization and equivariant cohomology, First European Congress of Mathematics (A. Joseph, F. Mignot, F.. Murat, B. Prum, and R. Rentschler, eds.), Progress in Mathematics, vol 3. Birkh\"auser Basel, 1994

    1994

  49. [57]

    , Bouquets revisited and equivariant elliptic cohomology, preprint (2020)

    2020

  50. [58]

    Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids , Algebraic and Geometric Topology 8 (2008)

    S. Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids , Algebraic and Geometric Topology 8 (2008)

    2008

  51. [59]

    Witten, The index of the Dirac operator in loop space , Elliptic Curves and Modular Forms in Algebraic Topology (P

    E. Witten, The index of the Dirac operator in loop space , Elliptic Curves and Modular Forms in Algebraic Topology (P. Landweber, ed.), Springer Berlin Heidelberg, 1988

    1988

  52. [60]

    Yang and G

    Y. Yang and G. Zhao, Quiver varieties and elliptic quantum groups, Arxiv preprint (2017)

    2017

  53. [61]

    Zagier, Note on the Landweber-Stong elliptic genus , Elliptic curves and modular forms in Alg

    D. Zagier, Note on the Landweber-Stong elliptic genus , Elliptic curves and modular forms in Alg. Top. (1986)

    1986

  54. [62]

    Zhao and C

    G. Zhao and C. Zhong, Elliptic affine Hecke algebras and their representations , Arxiv preprint (2015)

    2015

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.