Pith. sign in

REVIEW 3 major objections 6 minor 44 references

A topological Chern character for matrix factorizations

T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read For a Landau–Ginzburg model whose critical points lie in the zero fiber, there is a canonical Chern character from the Grothendieck group of matrix factorizations to critical cohomology, and it satisfies Grothendieck–Riemann–Roch.

desk verdict The core construction is solid, but the abstract overstates the scope: all main theorems require Crit(w)⊂w^{-1}(0), which is not stated. read the letter →

arxiv 2607.21788 v1 pith:LMLNC3SI submitted 2026-07-23 math.AG math-phmath.ATmath.MP

Mark Shoemaker This is my paper · ORCID
classification math.AGmath-phmath.ATmath.MP MSC 14F0819L1014J33
keywords matrixfactorizationsCherncharactertopologicalK-theorycriticalcohomologyLandau–GinzburgmodelsGrothendieck–Riemann–Rochvanishingcyclessingularitycategory
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 constructs, for a Landau–Ginzburg model (a smooth variety Y with a regular function w whose critical points lie in the zero fiber), a canonical homomorphism from the Grothendieck group of matrix factorizations of w to the critical cohomology of the pair. The map is built in two steps: first a topological K-theoretic lift that factors through relative K-theory of neighborhoods of the zero fiber, then a localized Chern character with Todd class. The main theorem is a Grothendieck–Riemann–Roch statement: pushforward along proper maps of Landau–Ginzburg models commutes with this Chern character. This matters because matrix factorizations serve as the B-model D-brane category in mirror symmetry, and critical cohomology is the state space of the gauged linear sigma model; the new map gives a candidate integral lattice and central charge. The construction is shown to be compatible with pullbacks, shifts, tensor products, and Sebastiani–Thom sums.

What carries the argument

The load-bearing object is the critical topological K-theory K0(Y,w), defined as the direct limit of K0_top(U, U∩{Re(w)<0}) over Euclidean neighborhoods of the zero fiber X=w^{-1}(0). The map alpha is induced by sending a matrix factorization to the associated two-term complex of vector bundles, which is exact off X, and then pulling back to relative K-theory of neighborhoods. A second key input is the equivalence identifying matrix factorizations with the singularity category of X, which reduces the construction to showing that the map kills the image of perfect complexes. The singular Riemann–Roch theorem for quasi-projective varieties (applied to the zero fiber) provides the pushforward c

What would settle it

Take Y = C and w = x^2 − 1. The critical point 0 lies outside X = {±1}. Compute the two direct limits in Proposition 2.8: over neighborhoods of X, H^0(U,U<0) is nonzero (one generator for each component of X), while over neighborhoods of Z = {0}, H^*(V,V<0) = 0 because V<0 = V for small V. Since the limits differ, the identification of critical cohomology with a limit over the critical locus fails outside Assumption 2.1, and the construction of tau collapses on such examples.

Watch

Extended reading notes

Core claim

The central claim is that the assignment sending a matrix factorization E=(E0,E1,phi0,phi1) to the class of the two-term complex E1 -> E0 in relative topological K-theory of (Y, w^{-1}(0)) depends only on the class of E in the Grothendieck group of matrix factorizations. Passing to the direct limit over neighborhoods of the zero fiber gives a homomorphism alpha(Y,w): K0(MF(Y,w)) -> K0(Y,w), and composing with the localized Chern character and the Todd class defines tau = Td_Y ∪ ch ∘ alpha, valued in critical cohomology. The paper proves tau is functorial for proper maps (Grothendieck–Riemann–Roch), natural under pullback, shift, and tensor products, and compatible with the Sebastiani–Thom su

Load-bearing premise

The paper assumes the critical locus of w is contained in the zero fiber w^{-1}(0); if a critical point occurs at a nonzero critical value, the target cohomology defined via neighborhoods of the zero fiber need not capture the vanishing-cycle contribution at that point, and the construction loses its meaning.

Editorial extensions

If this is right

  • For any Landau–Ginzburg model satisfying the critical-locus assumption, the Chern character gives a canonical homomorphism from the Grothendieck group of matrix factorizations to rational critical cohomology.
  • Proper morphisms of Landau–Ginzburg models satisfy a Grothendieck–Riemann–Roch formula: pushforward in matrix-factorization K-theory commutes with the Chern character after multiplying by the Todd class.
  • The construction is compatible with pullback along maps of Landau–Ginzburg models, with the shift functor, and with tensoring by classes on the zero fiber, giving a package of standard operations on D-brane charges.
  • The Chern character respects Sebastiani–Thom sums of potentials, so the external tensor product of matrix factorizations matches the Künneth product in critical K-theory.
  • For geometric phases, the map is identified with the classical map from algebraic to topological K-theory, so the topological invariant of the Landau–Ginzburg model recovers the K-theory of the complete intersection it represents.

Reading between the lines

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

  • If the Riemann–Roch compatibility extends to the equivariant setting, it would provide a way to compute quantum-corrected D-brane charges in gauged linear sigma models directly from matrix-factorization K-theory.
  • The assumption that critical points lie in the zero fiber holds automatically for homogeneous potentials of positive degree, but generic perturbations of such potentials violate it; the behavior of the map under deformation (wall-crossing) is a natural next question.
  • Because tau factors through topological K-theory, pairing it with gamma classes or other cohomology operations could define richer invariants, possibly refining the integral lattice of flat sections of the A-model connection.
  • The compatibility with adding a quadratic term suggests an infinite ladder of identities relating the Chern characters of w and w ⊞ (x^2+y^2); iterating gives a checkable family of commutative diagrams that any candidate topological Chern character must satisfy.
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.

Referee Report

3 major / 6 minor

Summary. The paper constructs, for a smooth quasi-projective complex variety Y and a regular function w (subject to the standing assumption Crit(w) ⊂ w^{-1}(0)), a homomorphism α(Y,w): K0(MF(Y,w)) → K0(Y,w) from the Grothendieck group of matrix factorizations to a critical topological K-theory group, and a Chern character τ(Y,w) = Td_Y ∪ ch(Y,w) ∘ α(Y,w) landing in critical cohomology. It proves functoriality of α under pullback and proper pushforward, a Grothendieck–Riemann–Roch theorem for proper morphisms of LG models, and compatibility with Sebastiani–Thom sums. The main construction reduces to the Baum–Fulton–MacPherson map on the zero locus X=w^{-1}(0), with Theorem 3.2 proved by a local calculation.

Significance. If correct, the paper provides a canonical functorial topological invariant for LG models, generalizing Brown's affine result and offering a potential link to GLSM state spaces and central charges. The proof of Theorem 3.2 is a clean and convincing reduction to the classical BFM map; the GRR square (Theorem 0.1/Corollary 4.3) follows from known covariance of α_* and ch, and the geometric-phase compatibility (Corollary 4.2) is a nice application. The paper is carefully organized and the main technical constructions are explicit. However, the scope is substantially narrower than the abstract: all main results depend on Assumption 2.1, which is omitted from the abstract and from the introductory Theorem 0.1. The reference [CFKS08] is also mismatched. These issues are correctable but need attention before the claims as stated are justified.

major comments (3)
  1. [Abstract and §0.1] The abstract states the construction and GRR theorem for an arbitrary regular function on a quasi-projective variety. The body imposes Assumption 2.1 (Crit(w) ⊂ w^{-1}(0)) 'throughout the paper', and the target groups K0(Y,w), H(Y,w) are defined as limits over neighborhoods of X=w^{-1}(0). This assumption is load-bearing: the critical cohomology defined in (0.3) is not the usual vanishing-cycle cohomology without it. For example, Y=C, w=z^2+1 has X={±i} and Z=Crit(w)={0}. A sufficiently small neighborhood U of X has U_<0 a disjoint union of half-disks, so H^*(U,U_<0)=0 and the limit over X is zero, while the limit over neighborhoods of Z is nonzero. Thus the claim in the abstract is too broad. The abstract and Theorem 0.1 must explicitly include Assumption 2.1 (and smoothness of Y), or the target groups must be redefined over Crit(w).
  2. [§2.1, Proposition 2.8] Proposition 2.8 asserts that the critical cohomology and critical K-theory can be computed as limits over neighborhoods of Z=Crit(w). The proof identifies the limit over X with H^*(i^{-1}RΓ_{Y≥0}(Z)) and then uses that the vanishing-cycle sheaf is supported on Z. This argument silently uses Assumption 2.1: the vanishing-cycle sheaf is supported on Z, but i^{-1} of it is zero on X when Z is not contained in X, so the two limits cannot coincide. The explicit counterexample w=z^2+1 on C shows the proposition is false without the assumption. The proof needs to state where the inclusion Z⊂X is used, and the statement must be restricted accordingly.
  3. [§4.1, Theorem 4.1 and §5, Proposition 5.1] Several compatibility statements are given with very brief justifications. In Theorem 4.1(5), the claimed module compatibility uses (3.3) after identifying K0_top(X) with K0_top(U) for a neighborhood U of X; this requires the existence of a homotopy-equivalent Euclidean neighborhood, which is not automatic for arbitrary neighborhoods in the direct limit. Proposition 5.1 defines the external product using the alternative description of K0(Y,w) as a limit over neighborhoods of Z, which depends on Proposition 2.8 and thus on Assumption 2.1. Please expand these arguments or state the needed hypotheses so the reader can verify the squares rigorously.
minor comments (6)
  1. [Abstract and Definition 1.1] The abstract says 'quasi-projective complex variety', but Definition 1.1 requires Y smooth for an LG model. The abstract should say 'smooth quasi-projective' or align with the body.
  2. [§0.1, Theorem 0.1] The statement of Theorem 0.1 in the introduction does not mention the standing assumption Crit(w)⊂w^{-1}(0). It should be added there, not only in Assumption 2.1, to avoid misleading readers who read the introduction independently.
  3. [§0.4 and References] The citation [CFKS08] is incorrect. The reference listed is Ciocan-Fontanine–Kim–Sabbah, 'The abelian/nonabelian correspondence and Frobenius manifolds', which is not a GRR theorem for matrix factorizations by 'Choa–Kim–Sreedhar'. This should be corrected or replaced by the intended work.
  4. [§2.1, Proposition 2.8] The notation is confusing: Z is used both for the critical locus and for the constant sheaf (in RΓ_{Y≥0}(Z)). Use an underline or a different symbol for the constant sheaf.
  5. [§5] In Proposition 5.1, the proof should explicitly note that the alternative description over Z is only valid under Assumption 2.1, and that the assumption is inherited by the Sebastiani–Thom sum when each factor satisfies it.
  6. [References] There is a typo in [ST23]: 'Spetember' should be 'September'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: α(Y,w) is constructed from Orlov equivalence and BFM topological K-theory; τ-GRR is a formal consequence, not an input.

full rationale

The paper's main construction α(Y,w) is not assumed or fitted: it is defined by the explicit composition i_<0^* h_top^{-1} α_* on K_0(D^b(X)) (Theorem 3.2), and the nontrivial factorization through the singularity quotient is proved using Orlov's equivalence (Theorem 1.8), the BFM GRR theorem and its module compatibility (3.3), and the elementary fact K_0(C,{Re z<0})=0. The map τ(Y,w) is defined as Td_Y ∪ ch_{Y,w} ∘ α(Y,w) (Definition 3.3), so the Grothendieck–Riemann–Roch statement for τ is an immediate consequence of the functoriality of α proved in Theorem 4.1 and covariance of the localized Chern character; it is a derived theorem, not a separate prediction forced by a fitted parameter. There are no fitted inputs, no load-bearing self-citations, and no uniqueness theorem imported from the author's own prior work; self-citations appear only in motivational applications and are not used in any proof. Two non-circular caveats should be noted: Assumption 2.1 (Crit(w) ⊂ w^{-1}(0)) is stated at the start of Section 2 and is genuinely needed for Proposition 2.8 and hence for the external product, but the abstract omits it, so the abstract's 'regular function' claim is broader than the theorems; and the Section 0.4 citation '[CFKS08]' is a mismatch—it points to Ciocan-Fontanine–Kim–Sabbah, not a GRR theorem for matrix factorizations by 'Choa–Kim–Sreedhar'. These are scope/reference issues, not circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

Pure theory paper: no free parameters or fitted constants. The load-bearing input is a set of established theorems plus one domain restriction (Assumption 2.1) that is under-advertised in the abstract.

assumptions (9)
  • standard math Orlov equivalence MF(Y,w) ≅ D_Sg(X) (Theorem 1.8)
    Used to identify K0(MF(Y,w)) with K0(D_Sg(X)) in Corollary 1.9, which is the starting point for the Chern character.
  • standard math BFM topological K-theory/Riemann-Roch: α_* is covariant and τ factors through topological K-theory (equations 3.2, 3.3)
    The proof of Theorem 4.1(1) is a composition of BFM covariance squares.
  • standard math Localized Chern character existence and covariance (Ive76, BFM75)
    Defines ch(Y,w) on critical K-theory in Section 3.3.
  • standard math Atiyah-Hirzebruch spectral sequence and exactness of direct limits
    Used in Proposition 2.8 to compare limits over neighborhoods of X and Z.
  • standard math Lojasiewicz triangulation of semialgebraic sets
    Establishes CW-finiteness of neighborhoods in Lemma 2.6.
  • standard math Dimca's identification of i^{-1}RΓ_{Y≥0}(Z) with the sheaf of vanishing cycles
    Used in Proposition 2.8 to identify critical cohomology over X with cohomology over Z.
  • domain assumption Assumption 2.1: Crit(w) ⊂ w^{-1}(0)
    Needed for Proposition 2.8 and all later results; omitted from the abstract.
  • domain assumption Y is smooth quasi-projective and w ≠ 0
    LG model definition; ensures Orlov equivalence and that the two-term complex is exact off the zero fiber.
  • standard math Isik-Shipman-Hirano equivalence D^b(Z) ≅ MF^{C*}(Y,w)
    Used in the geometric phase Corollary 4.2 and Proposition 2.11.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A topological Chern character for matrix factorizations." pith.science (2026). https://pith.science/paper/LMLNC3SI

@misc{pith2026260721788,
  author       = {Pith},
  title        = {Pith review of: A topological Chern character for matrix factorizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMLNC3SI}},
  note         = {Machine review of arXiv:2607.21788}
}
abstract

For $Y$ a quasi-projective complex variety and $w \colon Y \to \mathbb C$ a regular function, we construct a Chern character from the Grothendieck group of the category of matrix factorizations of $w$ to the critical cohomology of $w$, and show that it factors through a certain topological $K$-theory group. We prove a Grothendieck-Riemann-Roch theorem with respect to this Chern character, and verify several functorial properties.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 3 linked inside Pith

  1. [1]

    Aspinwall

    Paul S. Aspinwall. Exoflops in two dimensions. Journal of High Energy Physics , 2015(7):104, Jul 2015

    2015

  2. [2]

    Brown and Tobias Dyckerhoff

    Michael K. Brown and Tobias Dyckerhoff. Topological K -theory of equivariant singularity categories. Homology Homotopy Appl. , 22(2):1--29, 2020

    2020

  3. [3]

    A category of kernels for equivariant factorizations and its implications for H odge theory

    Matthew Ballard, David Favero, and Ludmil Katzarkov. A category of kernels for equivariant factorizations and its implications for H odge theory. Publ. Math. Inst. Hautes \'Etudes Sci. , 120:1--111, 2014

    2014

  4. [4]

    Variation of geometric invariant theory quotients and derived categories

    Matthew Ballard, David Favero, and Ludmil Katzarkov. Variation of geometric invariant theory quotients and derived categories. J. Reine Angew. Math. , 746:235--303, 2019

    2019

  5. [5]

    Riemann- R och for singular varieties

    Paul Baum, William Fulton, and Robert MacPherson. Riemann- R och for singular varieties. Inst. Hautes \'Etudes Sci. Publ. Math. , (45):101--145, 1975

    1975

  6. [6]

    Riemann- R och and topological K \ theory for singular varieties

    Paul Baum, William Fulton, and Robert MacPherson. Riemann- R och and topological K \ theory for singular varieties. Acta Math. , 143(3-4):155--192, 1979

    1979

  7. [7]

    Topological K -theory of complex noncommutative spaces

    Anthony Blanc. Topological K -theory of complex noncommutative spaces. Compos. Math. , 152(3):489--555, 2016

    2016

  8. [8]

    Quantum cohomology of the springer resolution

    Alexander Braverman, Davesh Maulik, and Andrei Okounkov. Quantum cohomology of the springer resolution. Advances in Mathematics , 227(1):421--458, 2011

    2011

Show all 44 references
  1. [9]

    Michael K. Brown. Kn\"orrer periodicity and B ott periodicity. Doc. Math. , 21:1459--1501, 2016

    2016

  2. [10]

    Fundamental factorization of a GLSM P art I : C onstruction

    Ionut Ciocan-Fontanine, David Favero, J\'er\'emy Gu\'er\'e, Bumsig Kim, and Mark Shoemaker. Fundamental factorization of a GLSM P art I : C onstruction. Mem. Amer. Math. Soc. , 289(1435):iv+96, 2023

    2023

  3. [11]

    The abelian/nonabelian correspondence and frobenius manifolds

    Ionu t Ciocan-Fontanine, Bumsig Kim, and Claude Sabbah. The abelian/nonabelian correspondence and frobenius manifolds. Inventiones mathematicae , 171(2):301--343, 2008

    2008

  4. [12]

    The crepant transformation conjecture for toric complete intersections

    Tom Coates, Hiroshi Iritani, and Yunfeng Jiang. The crepant transformation conjecture for toric complete intersections. Advances in Mathematics , 329:1002--1087, 2018

    2018

  5. [13]

    Landau- G inzburg/ C alabi- Y au correspondence, global mirror symmetry and O rlov equivalence

    Alessandro Chiodo, Hiroshi Iritani, and Yongbin Ruan. Landau- G inzburg/ C alabi- Y au correspondence, global mirror symmetry and O rlov equivalence. Publ. Math. Inst. Hautes \'Etudes Sci. , 119:127--216, 2014

    2014

  6. [14]

    Algebraic virtual cycles for quantum singularity theories

    Huai-Liang Chang, Young-Hoon Kiem, and Jun Li. Algebraic virtual cycles for quantum singularity theories. Comm. Anal. Geom. , 29(8):1749--1774, 2021

    2021

  7. [15]

    K -theoretic pullbacks for L agrangians on derived critical loci

    Yalong Cao, Yukinobu Toda, and Gufang Zhao. K -theoretic pullbacks for L agrangians on derived critical loci. arXiv:2503.06025, March 2025

    2025 arXiv

  8. [16]

    Sheaves in topology

    Alexandru Dimca. Sheaves in topology . Universitext. Springer-Verlag, Berlin, 2004

    2004

  9. [17]

    Alexander I. Efimov. Cyclic homology of categories of matrix factorizations. Int. Math. Res. Not. IMRN , (12):3834--3869, 2018

    2018

  10. [18]

    Jarvis, and Yongbin Ruan

    Huijun Fan, Tyler J. Jarvis, and Yongbin Ruan. A mathematical theory of the gauged linear sigma model. Geom. Topol. , 22(1):235--303, 2018

    2018

  11. [19]

    General GLSM invariants and their cohomological field theories

    David Favero and Bumsig Kim. General GLSM invariants and their cohomological field theories. arXiv:2006.12182, 2020

    2006 arXiv

  12. [20]

    Categorical framework for the study of singular spaces

    William Fulton and Robert MacPherson. Categorical framework for the study of singular spaces. Mem. Amer. Math. Soc. , 31(243):vi+165, 1981

    1981

  13. [21]

    Gamma classes and quantum cohomology of F ano manifolds: gamma conjectures

    Sergey Galkin, Vasily Golyshev, and Hiroshi Iritani. Gamma classes and quantum cohomology of F ano manifolds: gamma conjectures. Duke Math. J. , 165(11):2005--2077, 2016

    2005

  14. [22]

    Cohomologie l-adique et Fonctions L

    Alexander Grothendieck and Luc Illusie. Cohomologie l-adique et Fonctions L. SGA 5 , volume 589 of Lecture Notes in Mathematics . Springer, 1977

    1977

  15. [23]

    A mirror theorem for toric complete intersections

    Alexander Givental. A mirror theorem for toric complete intersections. In Topological field theory, primitive forms and related topics ( K yoto, 1996) , volume 160 of Progr. Math. , pages 141--175. Birkh\"auser Boston, Boston, MA, 1998

    1996

  16. [24]

    Derived K n \"o rrer periodicity and O rlov's theorem for gauged L andau-- G inzburg models

    Yuki Hirano. Derived K n \"o rrer periodicity and O rlov's theorem for gauged L andau-- G inzburg models. Compositio Mathematica , 153(5):973--1007, 2017

    2017

  17. [25]

    Mirror symmetry

    Kentaro Hori and Cumrun Vafa. Mirror symmetry . 2000

    2000

  18. [26]

    Equivalence of the derived category of a variety with a singularity category

    Mehmet Umut Isik. Equivalence of the derived category of a variety with a singularity category. International Mathematics Research Notices , 2013(12):2787--2808, 2012

    2013

  19. [27]

    Local C hern classes

    Birger Iversen. Local C hern classes. Ann. Sci. \'Ecole Norm. Sup. (4) , 9(1):155--169, 1976

    1976

  20. [28]

    D-branes in L andau- G inzburg models and algebraic geometry

    Anton Kapustin and Yi Li. D-branes in L andau- G inzburg models and algebraic geometry. J. High Energy Phys. , (12):005, 44, 2003

    2003

  21. [29]

    Atiyah class and C hern character for global matrix factorisations

    Bumsig Kim and Alexander Polishchuk. Atiyah class and C hern character for global matrix factorisations. J. Inst. Math. Jussieu , 21(4):1445--1470, 2022

    2022

  22. [30]

    Sheaves on manifolds , volume 292 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    Masaki Kashiwara and Pierre Schapira. Sheaves on manifolds , volume 292 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel

    1990

  23. [31]

    Lojasiewicz

    S. Lojasiewicz. Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) , 18:449--474, 1964

    1964

  24. [32]

    D. O. Orlov. Triangulated categories of singularities and D -branes in L andau- G inzburg models. Tr. Mat. Inst. Steklova , 246:240--262, 2004

    2004

  25. [33]

    Matrix factorizations for nonaffine LG -models

    Dmitri Orlov. Matrix factorizations for nonaffine LG -models. Math. Ann. , 353(1):95--108, 2012

    2012

  26. [34]

    Chern character for global matrix factorizations

    David Platt. Chern character for global matrix factorizations . ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)--University of Oregon

    2013

  27. [35]

    Matrix factorizations and singularity categories for stacks

    Alexander Polishchuk and Arkady Vaintrob. Matrix factorizations and singularity categories for stacks. Ann. Inst. Fourier (Grenoble) , 61(7):2609--2642, 2011

    2011

  28. [36]

    Chern characters and H irzebruch- R iemann- R och formula for matrix factorizations

    Alexander Polishchuk and Arkady Vaintrob. Chern characters and H irzebruch- R iemann- R och formula for matrix factorizations. Duke Math. J. , 161(10):1863--1926, 2012

    1926

  29. [37]

    Predictions for G romov- W itten invariants of noncommutative resolutions

    Eric Sharpe. Predictions for G romov- W itten invariants of noncommutative resolutions. J. Geom. Phys. , 74:256--265, 2013

    2013

  30. [38]

    A geometric approach to O rlov's theorem

    Ian Shipman. A geometric approach to O rlov's theorem. Compositio Mathematica , 148(5):1365--1389, 2012

    2012

  31. [39]

    Integral transforms and quantum correspondences

    Mark Shoemaker. Integral transforms and quantum correspondences. Adv. Math. , 374:107381, 59, 2020

    2020

  32. [40]

    Towards a mirror theorem for GLSM s

    Mark Shoemaker. Towards a mirror theorem for GLSM s. Compos. Math. , 162(1):125--164, 2026

    2026

  33. [41]

    Quantum spectrum and gamma structures for quasi-homogeneous polynomials of general type

    Shidhesh Supekar and Hsian-Hua Tseng. Quantum spectrum and gamma structures for quasi-homogeneous polynomials of general type. arXiv:2309.07446, Spetember 2023

    2023 arXiv

  34. [42]

    Correlation functions of gauged linear -model

    Gang Tian and Guangbo Xu. Correlation functions of gauged linear -model. Sci. China Math. , 59(5):823--838, 2016

    2016

  35. [43]

    o kova G eometry- T opology C onference 2016 , pages 86--111. G\

    Gang Tian and Guangbo Xu. The symplectic approach of gauged linear -model. In Proceedings of the G \" o kova G eometry- T opology C onference 2016 , pages 86--111. G\" o kova Geometry/Topology Conference (GGT), G\" o kova, 2017

    2016

  36. [44]

    Analysis of gauged W itten equation

    Gang Tian and Guangbo Xu. Analysis of gauged W itten equation. J. Reine Angew. Math. , 740:187--274, 2018

    2018

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.