Quantum K-theory levels in physics and math

E. Sharpe; H. Zhang; I. Huq-Kuruvilla; L. Mihalcea

REVIEW 3 major objections 4 minor 60 references

Quantum K-theory levels in physics and math

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper identifies Chern-Simons levels of three-dimensional gauge theories with Ruan-Zhang levels of twisted quantum K-theory for projective spaces, Grassmannians, and flag manifolds.

desk verdict Solid dictionary for projective spaces and Grassmannians; the flag-manifold case is only half verified, and the most load-bearing technical results live in a companion paper. read the letter →

arxiv 2507.00116 v2 pith:WELKTTTU submitted 2025-06-30 hep-th math.AG

I. Huq-Kuruvilla , L. Mihalcea , E. Sharpe , H. Zhang This is my paper · ORCID

classification hep-thmath.AG MSC 14N3519L47

keywords quantumK-theoryChern-SimonslevelRuan-ZhangstructuregaugedlinearsigmamodelCoulombbranchI-functionabelian/nonabeliancorrespondencegeometricwindow

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 proposes a precise dictionary between two ways of twisting quantum K-theory: the Chern-Simons levels of a three-dimensional gauged linear $\sigma$ model (GLSM) and the Ruan-Zhang levels of twisted quantum K-theory in mathematics. For projective spaces, Grassmannians, and partial flag manifolds, the dictionary identifies a Ruan-Zhang level with a shifted U(1) Chern-Simons level, for flags given by $\ell_i = \kappa^{(i)}_{U(1)} + (k_{i-1}+k_{i+1})/2$. The authors verify this by showing that the physical Coulomb branch equations, including the Chern-Simons terms, are exactly the symbols of the $\tau$-difference operators that annihilate the abelianized twisted I-function. They also match the physical 'geometric window' of allowed Chern-Simons levels with the mathematical condition that the twisted I and J functions coincide. The dictionary matters because it turns a heuristic physics-mathematics relation into a checkable identity with explicit formulas for quantum K-theory rings.

What carries the argument

The load-bearing object is the abelianized twisted I-function $\hat{I}^\ell_{\mathrm{Ab}}$, the quasimap generating function of the abelianized GIT quotient with a Ruan-Zhang level structure plus an Euler-class twisting by the root bundle. The paper shows this function satisfies explicit $\tau$-difference equations $D^{(i)}_a \hat{I}^\ell_{\mathrm{Ab}} = \hat{I}^\ell_{\mathrm{Ab}}$, and taking the symbol (replacing $\tau q\partial_q$ by the corresponding K-theory line bundle class) yields the physical Coulomb branch equations after applying the abelian/nonabelian correspondence map $\phi$, which specializes $\lambda\to 1$ and identifies the quantum parameters. The mirror-triviality condition $I^\ell = J^\ell$, equality of the quasimap I-function and the stable-map J-function, is the mathematical avatar of the physical geometric window. The Ruan-Zhang level is the integer $\ell$ in the twisting factor $\det(\cdot)^{-\ell}$ of the virtual structure sheaf, and the dictionary fixes it in terms of the Chern-Simons levels.

What would settle it

Compute both sides of equation (1.3) for a partial flag manifold with at least two steps, using the explicit abelianized I-function (5.15) and the physical superpotential (5.4) for a choice of levels outside the range covered by Yan's window; if the symbol of the difference operator and $\exp(\partial W/\partial \ln X)$ differ for any allowed level, the dictionary fails. Alternatively, if the companion paper's $\phi$ does not commute with symbols on any example, the central identification collapses.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: after abelianization and the map $\phi$ that converts abelian data into the nonabelian gauge theory, the exponential of the logarithmic derivative of the twisted superpotential $W$ equals the symbol of the difference operator $D^{(i)}_a$ annihilating the abelianized I-function, namely $\exp(\partial W/\partial \ln X^{(i)}_a)=\phi(\sigma(D^{(i)}_a))$. Under the level dictionary (1.4), the Coulomb branch equations $\exp(\partial W/\partial \ln X^{(i)}_a)=1$ become the Bethe-Ansatz-type spectral equations coming from these difference operators. For projective spaces and Grassmannians the paper also proves that the physical geometric window of Chern-Simons levels coincides with the mathematical geometric window, characterized as mirror-triviality, the equality of the twisted I-function and J-function $I^\ell = J^\ell$, with explicit windows (3.19) and (4.18). The same operator-symbol identification is established for partial flags, with the window known only partially. This gives a mathematical meaning to the physical Coulomb branch equations and to the allowed range of Chern-Simons levels.

Load-bearing premise

The identification assumes that the map converting the abelianized computation into the full gauge-theory statement can be extended to the quantum parameters in a way that commutes with taking symbols, a construction whose proof is postponed to a companion paper.

Editorial extensions

If this is right

For any level pair satisfying the dictionary, the Coulomb branch equations with Chern-Simons terms are exactly the symbol equations of the difference operators annihilating the abelianized I-function, so the Bethe Ansatz equations of the GLSM and the spectral problem of twisted quantum K-theory coincide.
For projective spaces and Grassmannians, the physical geometric window (no extra topological vacua) is the same set of levels for which $I^\ell = J^\ell$, giving explicit ranges: $0 \leq \ell \leq N+1$ for $\mathbb{P}^N$ and $-k < \ell \leq n-k+1$ for $\mathrm{Gr}(k,n)$.
Within the window, the paper conjectures that symmetrizations of the Coulomb branch equations generate the relations of the twisted quantum K-ring $QK^\ell_T(X)$, extending the known $\ell=0$ results.
The I-function level duality $I^\ell_{\mathrm{Gr}} = \mu^* I^{-\ell}_{\mathrm{Gr}^*}$ for Grassmannians (and the analogous flag statement) reproduces the physical IR duality between the corresponding three-dimensional Chern-Simons-matter theories.
For gerbes, the paper predicts that ordinary quantum K-theory decomposes into $k^2$ copies of the base, while a generic level twist yields only $k$ copies, a testable topological prediction.

Reading between the lines

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

Beyond the paper: if the symbol/operator identification holds for every Fano GIT quotient, then the Coulomb branch equations provide a physical algorithm for computing twisted quantum K-rings beyond projective spaces, Grassmannians, and flags, with the geometric window determined by testing mirror-triviality computationally.
Beyond the paper: the dictionary predicts that Chern-Simons levels are not arbitrary but constrained to the window where the twisted I and J functions match; this could be tested numerically for small flag manifolds by evaluating the pole conditions from the explicit formula (5.15).
Beyond the paper: the gerbe conjecture implies a sharp dichotomy, anomaly-free levels give $k^2$-fold decomposition while generic levels give $k$-fold, which could be checked by Coulomb-branch computations in the gerbe GLSMs of Section 6 rather than by stack-theoretic quantum K-theory.
Beyond the paper: because the dictionary identifies the bare Chern-Simons level with a Ruan-Zhang level, it suggests that level-rank dualities in Chern-Simons-matter theories should correspond to a symmetry $\ell \to -\ell$ in twisted quantum K-theory across all Fano GIT quotients, not just Grassmannians.

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

First thing to know: this is a real contribution, not a bubble. The dictionary between Chern-Simons levels and Ruan-Zhang levels is made explicit, and the checks for projective spaces and Grassmannians are concrete and internally consistent: Coulomb branch equations become symbols of difference operators annihilating the abelianized I-function, and the geometric window matches the mirror-triviality range I=J. That part I trust.

What's actually new: the operator-symbol interpretation (Theorem 1.1), the characterization of geometric windows via I=J, the extension from projective spaces to Grassmannians and flag manifolds on the Coulomb/symbol side, and the gerbe decomposition conjectures for nonzero levels. The paper gives Ueda-Yoshida credit for the projective space seed, which is fine. The computations of difference equations, symbols, and window ranges are detailed enough to follow, and the equivariant parameters make the matching less accidental.

Soft spots, in proportion. The flag-manifold window is not computed. Section 5.2.2 says the general range 'remains unknown,' Theorem 5.4 gives only a non-sharp bound for a single nonzero level, and the physics side itself only conjectures the window shape. So the abstract's claim of verifying the dictionary for flag manifolds by comparing geometric windows is stronger than what the paper supports. The Coulomb/symbol match for flags is done, so the dictionary is partially checked, not fully.

Second, the quantum extension of the abelian/nonabelian map phi, which Theorem 1.1(b) hinges on, is deferred to the companion paper [33]. The explicit examples use a direct specialization, so there's no sign of an actual failure, but a reader can't verify the general statement from this preprint. The same is true for Theorems 2.5 and 2.6. These aren't gaps in the examples, but they do make the central theorem conditional.

Third, the gerbe section is speculative, as the authors say. The BZ_k computation is a nice sanity check; the rest is a conjecture, clearly labeled. That's fine.

Who should read this: anyone working on 3d GLSMs and quantum K-theory, or on level structures in Gromov-Witten theory. It deserves a serious referee. My recommendation: send it out, but with a request to fix the abstract so the flag-manifold claim is stated accurately, and to either include the deferred proofs or make the dependence on [33] explicit in each theorem statement. I would accept it after those changes.

Referee Report

3 major / 4 minor

Summary. The paper proposes a dictionary between Chern-Simons levels in three-dimensional gauged linear sigma models and Ruan-Zhang levels in twisted quantum K-theory, for projective spaces, Grassmannians, and partial flag manifolds. The dictionary is checked by matching the physical Coulomb branch equations with symbols of difference operators annihilating a twisted abelianized I-function, and by comparing the physical geometric window of Chern-Simons levels with the mathematical window in which the I-function equals the J-function. The paper also contains conjectures on the twisted quantum K-theory of gerbes, including a predicted k-fold rather than k^2-fold decomposition for generic levels. Several key structural results, including the quantum extension of the abelian/nonabelian map and the I=J mirror-triviality criterion, are stated with proofs deferred to a companion paper [33].

Significance. If the dictionary holds, the paper gives a genuinely useful mathematical interpretation of physical Coulomb branch equations and of the geometric window, with explicit and detailed computations for projective spaces and Grassmannians. The recurrence-relation proofs of the difference equations, the matching of windows for P^N and Gr(k,n), and the level-rank duality checks are concrete and mutually consistent, and the gerbe conjecture is a falsifiable prediction. However, the advertised verification for flag manifolds is incomplete, and the central nonabelian step relies on results deferred to [33]. The paper is therefore valuable as a conditional proposal backed by strong special cases, rather than as a complete, self-contained proof of the dictionary.

major comments (3)

[§5.2.2, Conjecture 5.1, and Abstract] The abstract and introduction state that the dictionary is verified for flag manifolds, but the mathematical geometric window for flags is not actually computed. Section 5.2.2 says the general window 'remains unknown,' and Theorem 5.4 covers only the case of a single nonzero level and gives a non-sharp bound. Consequently, Conjecture 1.1, which identifies the physical window with mirror triviality, is not tested for flags, and the claim that the dictionary is verified for flag manifolds should be narrowed or the missing computation supplied.
[§2.6, Theorem 1.1(b)] Theorem 1.1(b) is the central statement that Coulomb branch equations equal symbols of difference operators after the abelian/nonabelian map, but the map's 'natural extension to quantum parameters' is deferred to the companion paper [33]. The explicit examples, such as Section 4.2.1 and Theorem 4.3, work by direct coordinate specialization, yet no general proof is given that this extension exists and commutes with taking symbols. Since that compatibility is exactly what converts the abelianized computation into a statement about the nonabelian gauge theory, the main theorem is conditional on an external result.
[§2.6, Theorems 2.5 and 2.6] The criterion that I=J is equivalent to the pole conditions on the coefficients I_d, stated as Theorem 2.5, is used in all the window computations (Theorems 3.2, 4.4, and 5.4), but the proof is deferred to [33]. Likewise, Theorem 2.6, which identifies the abelianized I-function with the I-function of the abelianized space with specified twistings, is stated without proof. These are load-bearing for the mathematical window verification, so the paper as submitted does not give a complete proof of the window comparisons it advertises.

minor comments (4)

[§5.2.1, equation (5.21)] The symbol v_i in the exponent (-1)^{v_i-1} is not defined anywhere; from comparison with equation (5.7) it should be k_i.
[Abstract and throughout] There are several typos, including 'identitifies' in the abstract, 'appropiate' in the introduction and Section 2, and 'unil' in Section 2; these should be corrected.
[§2.3, Definition 2.1] The text attributes the quasimap twistings to 'Ruan-Zhang in [44]', but reference [44] is by M. Zhang and Y. Zhou; the attribution should be made consistent with the bibliography.
[§3.2.2, Theorem 3.2] The range '-1 < ℓ ≤ N+1' is equivalent to '0 ≤ ℓ ≤ N+1' only because Ruan-Zhang levels are integers; stating the integrality explicitly next to the theorem would prevent the apparent mismatch with the proof's condition 'ℓ ≥ 0'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the proposed dictionary is checked by independent Coulomb-symbol matching and external window theorems.

full rationale

The paper proposes a dictionary between Chern-Simons levels and Ruan-Zhang levels rather than fitting parameters to data and relabeling the result as prediction. The Coulomb-branch/symbol equalities in Sections 3.2.1, 4.2.1, and 5.2.1 are computed from independently defined objects: the I-functions from Wen / Givental-Yan / Dong-Wen, and the physical twisted superpotential. The dictionary parameters are then read off by comparing exponents and prefactors; this is standard dictionary-building, and the subsequent agreement is a consistency check rather than a forced identity, especially because the geometric windows come from external literature ([27], [48]) and, for Grassmannians, the dictionary maps the physics interval exactly onto the Ruan-Zhang interval (4.18). The flag-manifold window is explicitly left open in Section 5.2.2, so that part of the advertised verification is incomplete but not circular. The companion-paper citations [33] supply proofs of technical results such as the natural extension of the map phi and are a conditionality on the general form of Theorem 1.1(b), but the explicit examples are computed directly and do not reduce to those citations. A minor correctness issue, not circularity, is that the projective-space range comparison in Section 3.3 claims equality with (3.19), while the mathematics range includes -1 < ell < 0 and the physics range does not; this does not affect the circularity assessment.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The main assumptions are standard GIT/quantum K-theory machinery plus two deferred results from the companion paper [33]. The dictionary itself introduces no free parameters except the chosen CS level combination; the RZ levels are the variables being mapped.

free parameters (1)

Second Chern-Simons level combination = κ_U(1) - κ_SU(k) = -k
The dictionary between CS and RZ levels for Grassmannians and flags requires the additional condition (κ_U(1)-κ_SU(k))/k = -1 (equations 4.24 and 5.25). This combination is not derived; it is chosen so that the ratio term in the Coulomb branch equations matches the product ∏ P_a/P_b arising from the symbol of the difference operator. The abstract formula (1.1) omits this constraint.

assumptions (5)

domain assumption GIT quotient assumptions: stable and semistable loci coincide, stable locus smooth and without isotropy.
Used throughout Section 2 to define quasimap moduli spaces and their virtual structure sheaves.
ad hoc to paper Existence of quantum extension of abelian/nonabelian correspondence map ϕ with ϕ(q_a)=q and ϕ(λ)=1.
Invoked in Theorem 1.1 and Sections 4-5; construction deferred to [33].
ad hoc to paper Theorem 2.5: I=J mirror triviality is equivalent to pole conditions on coefficients I_d.
Used to identify geometric windows mathematically; full proof deferred to [33].
ad hoc to paper Theorem 2.6: bI^ℓ_Ab is the I-function of the abelianized space with RZ level structure and Euler twisting.
Used to compute abelianized I-functions; proof deferred to [33].
standard math Wen's theorem [26] relating the abelianized I-function to the nonabelian I-function.
Quoted as Theorem 2.7 and used to equate ϕ(bI^ℓ_Ab) with I^ℓ_{V//G}.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantum K-theory levels in physics and math." pith.science (2026). https://pith.science/paper/WELKTTTU

@misc{pith2026250700116,
  author       = {Pith},
  title        = {Pith review of: Quantum K-theory levels in physics and math},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WELKTTTU}},
  note         = {Machine review of arXiv:2507.00116}
}

read the original abstract

The purpose of this paper is to describe the basics of a dictionary between Chern-Simons levels in three-dimensional gauged linear sigma models (GLSMs) and the (coincidentally-named) Ruan-Zhang levels for twisted quantum K-theory in mathematics. Each defines a twisting of quantum K-theory, and our proposed dictionary identifies these two twistings, in the cases of projective spaces, Grassmannians, and flag manifolds. We verify the dictionary by realizing the Coulomb branch equations as symbols of certain differential operators annihilating a twisted version of the I function associated to the abelianized GLSM theory, and also by comparing the geometric window for Chern-Simons levels to an analogous window for the Ruan-Zhang levels. In the process, we interpret the geometric window for the Chern-Simons levels in terms of equalities of I and J functions. This provides a fuller mathematical understanding of some special cases in the physics literature. We also make conjectures for twisted quantum K-theory of gerbes, following up earlier conjectures on ordinary quantum K-theory of gerbes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 20 canonical work pages

[33]

Huq-Kuruvilla, L

I. Huq-Kuruvilla, L. Mihalcea, E. Sharpe, H. Zhang, to appear

work page
[1]

Defects and quantum Seiberg-Witten geometry,

M. Bullimore, H. C. Kim and P. Koroteev, “Defects and quantum Seiberg-Witten geometry,” JHEP 05 (2015) 095, arXiv:1412.6081 [hep-th]

arXiv 2015
[2]

A 3d gauge theory/quantum K-theory correspondence,

H. Jockers and P. Mayr, “A 3d gauge theory/quantum K-theory correspondence,” Adv. Theor. Math. Phys. 24 (2020) 327-457, arXiv:1808.02040 [hep-th]

arXiv 2020
[3]

Quantum K-theory of Calabi-Yau manifolds,

H. Jockers and P. Mayr, “Quantum K-theory of Calabi-Yau manifolds,” JHEP 11 (2019) 011, arXiv:1905.03548 [hep-th]

arXiv 2019
[4]

Wilson loop algebras and quantum K-theory for Grassmannians,

H. Jockers, P. Mayr, U. Ninad and A. Tabler, “Wilson loop algebras and quantum K-theory for Grassmannians,” JHEP 10 (2020) 036, arXiv:1911.13286 [hep-th]

arXiv 2020
[5]

BPS indices, modularity and pertur- bations in quantum K-theory,

H. Jockers, P. Mayr, U. Ninad and A. Tabler, “BPS indices, modularity and pertur- bations in quantum K-theory,” JHEP 02 (2022) 044, arXiv:2106.07670 [hep-th]

arXiv 2022
[6]

3d N = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians,

K. Ueda and Y. Yoshida, “3d N = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians,” JHEP 08 (2020) 157, arXiv:1912.03792 [hep-th]

arXiv 2020
[7]

Quantum K- theory of quiver varieties and many-body systems,

P. Koroteev, P. P. Pushkar, A. V. Smirnov and A. M. Zeitlin, “Quantum K- theory of quiver varieties and many-body systems,” Selecta Math. 27 (2021) 87, arXiv:1705.10419 [math.AG]. 50

arXiv 2021

Show all 60 references

[8]

Comments on twisted indices in 3d supersymmetric gauge theories,

C. Closset and H. Kim, “Comments on twisted indices in 3d supersymmetric gauge theories,” JHEP 08 (2016) 059, arXiv:1605.06531 [hep-th]

2016 arXiv
[9]

Supersymmetric partition functions and the three- dimensional A-twist,

C. Closset, H. Kim and B. Willett, “Supersymmetric partition functions and the three- dimensional A-twist,” JHEP 03 (2017) 074, arXiv:1701.03171 [hep-th]

2017 arXiv
[10]

Seifert fibering operators in 3d N = 2 theories,

C. Closset, H. Kim and B. Willett, “Seifert fibering operators in 3d N = 2 theories,” JHEP 11 (2018) 004, arXiv:1807.02328 [hep-th]

2018 arXiv
[11]

Three-dimensional N = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review,

C. Closset and H. Kim, “Three-dimensional N = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review,” Int. J. Mod. Phys. A 34 (2019) 1930011, arXiv:1908.08875 [hep-th]

2019 arXiv
[12]

Twisted indices, Bethe ideals and 3d N = 2 infrared dualities,

C. Closset and O. Khlaif, “Twisted indices, Bethe ideals and 3d N = 2 infrared dualities,” JHEP 05 (2023) 148, arXiv:2301.10753 [hep-th]

2023 arXiv
[13]

Grothendieck lines in 3d N = 2 SQCD and the quantum K-theory of the Grassmannian,

C. Closset and O. Khlaif, “Grothendieck lines in 3d N = 2 SQCD and the quantum K-theory of the Grassmannian,” JHEP 12 (2023) 082, arXiv:2309.06980 [hep-th]

2023 arXiv
[14]

Quantum K theory of symplectic Grass- mannians,

W. Gu, L. Mihalcea, E. Sharpe and H. Zou, “Quantum K theory of symplectic Grass- mannians,” J. Geom. Phys. 177 (2022) 104548, arXiv:2008.04909 [hep-th]

2022 arXiv
[15]

Quantum K theory rings of partial flag manifolds,

W. Gu, L. Mihalcea, E. Sharpe, W. Xu, H. Zhang and H. Zou, “Quantum K theory rings of partial flag manifolds,” J. Geom. Phys. 198 (2024) 105127, arXiv:2306.11094 [hep-th]

2024 arXiv
[16]

Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles,

W. Gu, L. C. Mihalcea, E. Sharpe and H. Zou, “Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles,” arXiv:2208.01091 [math.AG]

arXiv
[17]

Quantum K Whitney relations for partial flag varieties,

W. Gu, L. C. Mihalcea, E. Sharpe, W. Xu, H. Zhang and H. Zou, “Quantum K Whitney relations for partial flag varieties,” arXiv:2310.03826 [math.AG]

arXiv
[18]

The level structure in quantum K-theory and mock theta func- tions,

Y. Ruan, M. Zhang, “The level structure in quantum K-theory and mock theta func- tions,” arXiv:1804.06552 [math.AG]

arXiv
[19]

Quantum K-theory of toric varieties, level structures, and 3d mirror symmetry,

Y. Ruan, Y. Wen, Z. Zhou, “Quantum K-theory of toric varieties, level structures, and 3d mirror symmetry,” Adv. Math. 410 (2022) 108770, arXiv:2011.07519 [math.AG]

2022 arXiv
[20]

One-form symmetries and the 3dN = 2 A-model: Topologically twisted indices for any G,

C. Closset, E. Furrer and O. Khlaif, “One-form symmetries and the 3dN = 2 A-model: Topologically twisted indices for any G,” arXiv:2405.18141 [hep-th]

arXiv
[21]

Quantum K-rings of partial flags, Coulomb branches, and the Bethe ansatz,

I. Huq-Kuruvilla, “Quantum K-rings of partial flags, Coulomb branches, and the Bethe ansatz,” arXiv:2409.15575 [math.AG]
[22]

Relations in twisted quantum K-rings,

I. Huq-Kuruvilla, “Relations in twisted quantum K-rings,” arXiv:2406.00916 [math.AG]. 51
[23]

The K-theory of abelian versus nonabelian symplectic quotients,

M. Harada and G. Landweber, “The K-theory of abelian versus nonabelian symplectic quotients,” arXiv:math/0612660 [math.SG]

arXiv
[24]

Symplectic quotients by a nonabelian group and by its maximal torus,

S. Martin, “Symplectic quotients by a nonabelian group and by its maximal torus,” arXiv:math/0001002 [math.SG]

arXiv
[25]

The abelian/nonabelian corre- spondence and Frobenius manifolds,

I. Ciocan-Fontanine, B. Kim and C. Sabbah, “The abelian/nonabelian corre- spondence and Frobenius manifolds,” Invent. Math. 171 (2008), no. 2, 301–343, arXiv:math/0610265 [math.AG]

2008 arXiv
[26]

K-theoretic I-function of V //θG and application,

Y. Wen, “K-theoretic I-function of V //θG and application,” arXiv:1906.00775 [math.AG]

1906 arXiv
[27]

Quantum K-theory of Grassmannians and non-Abelian localization,

A. Givental and X. Yan, “Quantum K-theory of Grassmannians and non-Abelian localization,” SIGMA 17 (2021) 018, arXiv:2008.08182 [math.AG]

2021 arXiv
[28]

Quantum K-theory of flag varieties via non-abelian localization,

X. Yan, “Quantum K-theory of flag varieties via non-abelian localization,” arXiv:2106.06281 [math.AG]

arXiv
[29]

Quantum K-theory I: foundations,

Y. P. Lee, “Quantum K-theory I: foundations,” Duke Math. J. 121 (2004) 389-424, arXiv:math/0105014 [math.AG]

2004 arXiv
[30]

On the WDVV equation in quantum K-theory,

A. Givental, “On the WDVV equation in quantum K-theory,” Michigan Math. J. 48 (2000) 295-304, arXiv:math/0003158 [math.AG]

2000 arXiv
[31]

Permutation-equivariant quantum K-theory III. Lefschetz’ formula on M0,n/Sn and adelic characterization,

A. Givental, “Permutation-equivariant quantum K-theory III. Lefschetz’ formula on M0,n/Sn and adelic characterization,” arXiv:1508.06697 [math.AG]

arXiv
[32]

Wall-crossing in genus zero quasimap theory and mirror maps,

I. Ciocan-Fontanine and B. Kim, “Wall-crossing in genus zero quasimap theory and mirror maps,” Algebr. Geom. 1 (2014) no. 4, 400–448, doi:10.14231/age.2014.16 arXiv:1304.7056 [math.AG]

2014 arXiv
[35]

On phases of 3d N=2 Chern-Simons-matter theories,

W. Gu, D. Pei and M. Zhang, “On phases of 3d N=2 Chern-Simons-matter theories,” Nucl. Phys. B 973 (2021) 115604, arXiv:2105.02247 [hep-th]

2021 arXiv
[36]

Vacuum structures revisited,

W. Gu, “Vacuum structures revisited,” arXiv:2110.13156 [hep-th]

arXiv
[37]

Decomposition squared,

E. Sharpe and H. Zhang, “Decomposition squared,” JHEP 10 (2024) 168, arXiv:2405.12269 [hep-th]

2024 arXiv
[38]

Stable quasimaps to GIT quotients,

I. Ciocan-Fontanine, B. Kim, B. and D. Maulik, “Stable quasimaps to GIT quotients,” J. Geom. Phys 75 (2014) 17-47, arXiv:1106.3724 [math.AG]. 52

2014 arXiv
[39]

Quantum Riemann-Roch, Lefschetz and Serre,

T. Coates and A. Givental, “Quantum Riemann-Roch, Lefschetz and Serre,” Ann. Math. 165 (2007) 15-53, arXiv:math/0110142 [math.AG]

2007 arXiv
[40]

Notes on stable maps and quantum cohomol- ogy,

W. Fulton and R. V. Pandharipande, “Notes on stable maps and quantum cohomol- ogy,” in Algebraic geometry—Santa Cruz 1995 , 45–96, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, arXiv:alg-geom/9608011 [math.AG]

1995 arXiv
[41]

The connectedness of the moduli space of maps to homogeneous spaces,

B. Kim and R. V. Pandharipande, “The connectedness of the moduli space of maps to homogeneous spaces,” in Symplectic geometry and mirror symmetry (Seoul, 2000) , 187–201, World Sci. Publ., River Edge, NJ, arXiv:math/0003168 [math.AG]

2000 arXiv
[42]

K-theoretic quasimap invariants and their wall-crossing,

H. Tseng and F. You, “K-theoretic quasimap invariants and their wall-crossing,” arXiv:1602.06494 [math.AG]

arXiv
[43]

Twisted K-theoretic Gromov-Witten invariants,

V. Tonita, “Twisted K-theoretic Gromov-Witten invariants,” Math. Ann 372 (2018) 489-526, arXiv:1309.3750 [math.AG]

2018 arXiv
[44]

K-theoretic quasimap wall-crossing,

M. Zhang, Y. Zhou, “K-theoretic quasimap wall-crossing,” arXiv:2012.01401 [math.AG]

2012 arXiv
[45]

Zakharevich, K-theory and characteristic classes: A homotopical perspective, Novem- ber 4, 2024, available at https://pi.math.cornell.edu/~zakh/book.pdf

I. Zakharevich, K-theory and characteristic classes: A homotopical perspective, Novem- ber 4, 2024, available at https://pi.math.cornell.edu/~zakh/book.pdf

2024
[46]

On the finiteness of quantum K-theory of a homogeneous space,

D. E. Anderson, L. Chen, H. Iritani, and H.-H. Tseng, “On the finiteness of quantum K-theory of a homogeneous space,” Int. Math. Res. Not. IMRN 2022, no. 2, 1313– 1349, arXiv:1804.04579 [math.AG]

2022 arXiv
[47]

Aspects of 3d N=2 Chern-Simons-matter theories,

K. Intriligator and N. Seiberg, “Aspects of 3d N=2 Chern-Simons-matter theories,” JHEP 07 (2013) 079, arXiv:1305.1633 [hep-th]

2013 arXiv
[48]

On the Witten index of 3d N = 2 unitary SQCD with general CS levels,

C. Closset and O. Khlaif, “On the Witten index of 3d N = 2 unitary SQCD with general CS levels,” SciPost Phys. 15 (2023) 085, arXiv:2305.00534 [hep-th]

2023 arXiv
[49]

Loop structure on equivariant K-theory of semi-infinite flag manifolds,

S. Kato, “Loop structure on equivariant K-theory of semi-infinite flag manifolds,” arXiv:1805.01718 [math.AG]

arXiv
[50]

Level correspondence of K-theoretic I-function in Grassmann duality,

H. Dong and Y. Wen, “Level correspondence of K-theoretic I-function in Grassmann duality,” arXiv:2004.10661 [math.AG]

2004 arXiv
[51]

Quantum K–theory of Grassmannians from a Yang-Baxter algebra,

V. Gorbounov, C. Korff, and L. C. Mihalcea, “Quantum K–theory of Grassmannians from a Yang-Baxter algebra,” arXiv:2503.08602 [math.AG]

arXiv
[52]

Closset, private communication

C. Closset, private communication
[53]

GLSM’s for partial flag manifolds,

R. Donagi and E. Sharpe, “GLSM’s for partial flag manifolds,” J. Geom. Phys. 58 (2008) 1662-1692, arXiv:0704.1761 [hep-th]. 53

2008 arXiv
[54]

Maeno, S

T. Maeno, S. Naito, and D. Sagaki, ”A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part I: The defining ideal”, J. Lond. Math. Soc. (2) 111 (2025), no. 3, Paper No. e70095, arXiv:2302.09485 [math.QA]

2025 arXiv
[55]

Toda-type presentations for the quantum K theory of partial flag varieties

K. Amini, I. Huq-Kuruvilla, L. Mihalcea, D. Orr, W. Xu, “Toda-type presentations for the quantum K theory of partial flag varieties”, arXiv:2504.07412 [math.AG]
[56]

Notes on gauging noneffective group actions,

T. Pantev and E. Sharpe, “Notes on gauging noneffective group actions,” arXiv:hep-th/0502027 [hep-th]

arXiv
[57]

GLSM’s for gerbes (and other toric stacks),

T. Pantev and E. Sharpe, “GLSM’s for gerbes (and other toric stacks),” Adv. Theor. Math. Phys. 10 (2006) no.1, 77-121, arXiv:hep-th/0502053 [hep-th]

2006 arXiv
[58]

String compactifications on Calabi-Yau stacks,

T. Pantev and E. Sharpe, “String compactifications on Calabi-Yau stacks,” Nucl. Phys. B 733 (2006) 233-296, arXiv:hep-th/0502044 [hep-th]

2006 arXiv
[59]

Quantum K-theory of toric stacks,

M. Zhang, “Quantum K-theory of toric stacks,” https://mathweb.ucsd.edu/~miz017/quantumk.pdf
[60]

Stringy K-theory and the Chern charac- ter,

T. J. Jarvis, R. Kaufmann and T. Kimura, “Stringy K-theory and the Chern charac- ter,” Invent. Math. 168 (2007) 23–81, arXiv:math/0502280 [math.AG]

2007 arXiv
[61]

Permutation-equivariant quantum K-theory IX. Quantum Hirzebruch-Riemann-Roch in all genera,

A. Givental, “Permutation-equivariant quantum K-theory IX. Quantum Hirzebruch-Riemann-Roch in all genera,” arXiv:1709.03180 [math.AG]. 54

arXiv

Pith tools

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

[1] [33]

Huq-Kuruvilla, L

I. Huq-Kuruvilla, L. Mihalcea, E. Sharpe, H. Zhang, to appear

work page

[2] [1]

Defects and quantum Seiberg-Witten geometry,

M. Bullimore, H. C. Kim and P. Koroteev, “Defects and quantum Seiberg-Witten geometry,” JHEP 05 (2015) 095, arXiv:1412.6081 [hep-th]

arXiv 2015

[3] [2]

A 3d gauge theory/quantum K-theory correspondence,

H. Jockers and P. Mayr, “A 3d gauge theory/quantum K-theory correspondence,” Adv. Theor. Math. Phys. 24 (2020) 327-457, arXiv:1808.02040 [hep-th]

arXiv 2020

[4] [3]

Quantum K-theory of Calabi-Yau manifolds,

H. Jockers and P. Mayr, “Quantum K-theory of Calabi-Yau manifolds,” JHEP 11 (2019) 011, arXiv:1905.03548 [hep-th]

arXiv 2019

[5] [4]

Wilson loop algebras and quantum K-theory for Grassmannians,

H. Jockers, P. Mayr, U. Ninad and A. Tabler, “Wilson loop algebras and quantum K-theory for Grassmannians,” JHEP 10 (2020) 036, arXiv:1911.13286 [hep-th]

arXiv 2020

[6] [5]

BPS indices, modularity and pertur- bations in quantum K-theory,

H. Jockers, P. Mayr, U. Ninad and A. Tabler, “BPS indices, modularity and pertur- bations in quantum K-theory,” JHEP 02 (2022) 044, arXiv:2106.07670 [hep-th]

arXiv 2022

[7] [6]

3d N = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians,

K. Ueda and Y. Yoshida, “3d N = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians,” JHEP 08 (2020) 157, arXiv:1912.03792 [hep-th]

arXiv 2020

[8] [7]

Quantum K- theory of quiver varieties and many-body systems,

P. Koroteev, P. P. Pushkar, A. V. Smirnov and A. M. Zeitlin, “Quantum K- theory of quiver varieties and many-body systems,” Selecta Math. 27 (2021) 87, arXiv:1705.10419 [math.AG]. 50

arXiv 2021