Mirror Symmetry

Cumrun Vafa; Kentaro Hori

REVIEW

Mirror Symmetry

Kentaro Hori, Cumrun Vafa

High Energy Physics - Theory · Submitted 25 Feb 2000 · 105 references

Pith's verdict: The exact twisted superpotential and the vortex derivation are the real contribution and they hold up; the abstract's unqualified 'we prove' outruns what the paper establishes for positive first Chern class, and the authors say so themselves in §5.1 and §5.3.

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arXiv IDhep-th/0002222 (version 3)

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CitationCumrun Vafa, Kentaro Hori. "Mirror Symmetry." arXiv:hep-th/0002222 (2000). https://pith.science/paper/CEB7ROFG

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The thing to take away is equation (3.38): the exact twisted superpotential of the dualized U(1) theory, with the exponential terms generated by vortices. That is genuinely new and it is the engine of everything else. Once you have it, CP^{N-1} versus affine Toda, the Greene-Plesser construction at d = N, Givental's toric quantum cohomology, and the trilogarithm prepotential for O(-1)+O(-1) over CP^1 all drop out as cross-checks. The derivation in §3 is clean, the localization reduction to the single-flavor case is a sensible move, and the vortex zero-mode computation is done properly rather than waved through. The paper earns its reputation on that section alone. The soft spot is the word 'proof' in the abstract. §5.1 explicitly splits strong-sense from weak-sense proofs and says they coincide only under an assumption about cc/ac rings plus integral structure. For Calabi-Yau targets the IR fixed-point argument closes the gap. For positive c1 the theory is asymptotically free and flows to a massive theory, and the D-term protection is asserted, not shown — §5.3 says 'it is expected that the sine-Gordon theory is integrable and the D-term is protected.' That is a real gap, and the stress-test note is right to land on it. The reader's localization worry in §3.2.1 is secondary; superpotential independence from D-terms is standard non-renormalization, and the singularity-crossing caveat is likely benign. Two other honest admissions worth weighing: §5.3 concedes the critical-point count matching the Euler number is not known in general, and §7.2 says a needed implication is not proved and an extra condition may be required. §7.3 is presented as a proposal and Appendix A is a conjecture. The classifier record says no limitations were found; that is wrong, the paper states several, and they sit exactly where the conditional verdict belongs. One more: §6 leans on reference [79], Hori-Iqbal-Vafa, listed as to appear. That is a load-bearing unpublished citation and a referee should ask for it. Who should read it: anyone working on toric geometry, quantum cohomology, or LG orbifolds. It is a working tool, not just a result. Send it out.

Referee report

Significance

This is a landmark paper. It converts mirror symmetry from a case-by-case conjecture into a derivation from ordinary abelian R → 1/R duality plus a semiclassical vortex-instanton sum, and it does so uniformly across Calabi-Yau, non-compact, and positive-first-Chern-class targets. The derivation is cross-checked against several independent known results: the CP^{N−1}/affine Toda duality (§5.3), the Greene-Plesser construction at d = N (§7.1), Givental's toric quantum cohomology formula (Eq. 5.16), and the trilogarithm prepotential for O(−1)⊕O(−1) over CP^1 (Eq. 6.16). The paper is also unusually candid about what it does and does not establish: §5.1 defines the two senses of proof, and Appendix A is explicitly labelled a conjecture. That candour is a genuine strength and should be weighed in favour of the manuscript.

Major objections (3)

[Abstract vs. §5.1] The abstract states without qualification that 'We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds' and that this covers 'sigma models on manifolds with positive first Chern class' [abstract:s0, abstract:s3]. §5.1, however, distinguishes a strong sense of proof (equivalence up to D-term deformations) from a weak sense (agreement of cc/ac rings, BPS soliton structure, and D-brane structure), and for positive-first-Chern-class targets the paper establishes the weak sense only: §5.3 states that D-term protection for the sine-Gordon theory 'is expected' rather than proved. The body is careful; the abstract is not. I recommend that the abstract be qualified to say that the equivalence is established up to D-term deformations, or that for positive c_1 the matching is at the level of twisted F-terms, BPS spectra, and D-branes. This is a scope issue rather than an error, but it is load-bearing for the headline claim.
[§7.2] In the derivation of the complete-intersection mirror the manuscript states: 'We assume that this implies Φ_i = 0 for all i. (We do not attempt to prove it here. It is possible that in general an extra condition is required.)' This step is load-bearing for the claim that the two LG orbifolds with superpotentials (7.37) and (7.41) are mirror to each other, and hence for the complete-intersection case (iii) of §5.2. The authors acknowledge the gap, which is to their credit, but the scope of the resulting claim is left unclear. Please either supply the argument, or state precisely the class of charge matrices and degrees for which the implication holds, and confirm that the worked example of Eqs. (7.42)–(7.45) falls inside that class.
[§3.2.1] The localization argument reduces the general N-flavour theory to N copies of the single-flavour case by continuing the inverse gauge-coupling matrix from a diagonal form to the U(1) embedding, and the text states this is valid 'as long as there is no singularity in going from the diagonal one.' That caveat is not made precise: a reader cannot tell which charge assignments Q_i are covered, nor what would constitute a singularity in the deformation. Since Eq. (3.38) for general N and Q_i is one of the two main results of the paper, a concrete statement of the condition (or an example where it fails) would materially strengthen the derivation. The independence of the twisted superpotential from D-term variations is standard non-renormalization and is not in question here; the issue is only the singularity-crossing caveat.

Minor objections (9)

[Eq. (2.50)] The second line reads 'R^2δψ_− = i√2(∂_0 +∂_1)ηǫ_−', but comparison with the preceding line and with the structure of (2.49) suggests the derivative should be (∂_0 − ∂_1) and the parameter ǫ_+; please check the spinor index and sign conventions against Eq. (B.3).
[Eq. (7.53)] The range 'for j = d = 1,...,N' appears to be a typographical error for 'j = d+1,...,N', consistent with the split made in (7.51)–(7.53) and with the subsequent use of U_j for j > d.
[Eq. (6.16)] The final expression is printed as '−e^{−t}(1 −e^{−t})', but the Gaussian integral gives −e^{−t}/(1 −e^{−t}), which is what reproduces the series −∑_{n>0} e^{−nt} and hence the trilogarithm prepotential (6.10). Please correct the ratio.
[Eqs. (7.20)–(7.22)] The closed form (7.20) does not obviously follow from the stated recursion ♯(C_N) = (d−1)^{N−1} − ♯(C_{N−1}) with ♯(C_1) = 0: for N = 3, d = 3 the recursion gives 2 while (7.20) gives 10/3. Since (7.21) and (7.22) are built on (7.20), please add the intermediate step or correct the expression. Relatedly, the text notes that untwisted-sector ground states carry (−1)^F = (−1)^N, but that sign does not appear in (7.21); please confirm the two conventions are consistent.
[§4.0.2] The bosonic potential is written as '1/2|m̃_A V_A|^2 + 1/2|m̃_A V_A|^2', with the two terms identical; presumably the second is the complex conjugate 1/2|m̃_A V̄_A|^2. Please correct.
[§6, reference [79]] The construction of Lagrangian submanifolds in LG models and the statement that their image on the W-plane is a straight line are attributed to reference [79], listed as 'K. Hori, A. Iqbal and C. Vafa, to appear'. Since this construction underlies the period integrals (6.1)–(6.3) used in §6.2 and §7, either summarize the argument or note that the period results are independently verified by the known prepotential, as (6.16) indeed does.
[§5.2] The statement that 'the mass scale for the modes tangent to ∑_i Q_iY_i = t is of order μ√|r_0|' is not derived; the √|r_0| factor should be explained, since the natural scale from the Kähler potential (3.25) alone would appear to be different.
[§5.3] The observation that the SU(2) symmetry of the N = 2 sine-Gordon theory holds only in the limit of vanishing Kähler potential is interesting and is used to reconcile the CP^1 and sine-Gordon global symmetries, but it is stated without derivation. A sentence of explanation, or a pointer to where it is shown, would help the reader.
[Throughout] There are numerous typographical errors that should be corrected in proof: 'borken' (§1), 'electrodymanics' (§1), 'Thoeries' (§3 title), 'anamaly' and 'obviuosly' (§2.1), 'comlpexiﬁed' (§4), 'resepectively' (§3.2), 'indivisual' (§1), 'Thirrli ng' (§2.2.2), 'hupersurface' (§7.1.1), 'pertubed' and 'cancell' (§4.1).

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The paper

The paper claims

Mirror symmetry of two-dimensional sigma models follows from abelian gauge duality: the dual theory is a Landau-Ginzburg model with superpotential mu times the sum of exponentials e to the minus Y_i, generated by vortices.

The reading

Mirror symmetry is a conjectured equivalence between two Calabi-Yau manifolds whose complex and Kahler structures are exchanged. Hori and Vafa argue it is not a new symmetry but a consequence of a duality that is already known. Their starting point is the gauged linear sigma model: a two-dimensional U(1) gauge theory with N charged chiral fields whose low-energy limit is a sigma model on a toric variety. They dualize the phase of each charged field, the same radius-inversion step that maps a circle of radius R to one of radius 1/R. The charged fields become neutral periodic fields Y_i coupled to the gauge field strength Sigma through a term Sigma times (sum of Q_i Y_i minus t). The key step is that vortices, the instantons of this theory, generate extra terms in the twisted superpotential. A vortex around which the i-th scalar winds is dual to the insertion of e to the minus Y_i. Summing the vortex gas gives the exact twisted superpotential: Sigma times (sum Q_i Y_i minus t) plus mu times the sum of e to the minus Y_i. Integrating out Sigma imposes sum Q_i Y_i equals t, leaving N minus 1 periodic variables with superpotential mu times the sum of e to the minus Y_i. That is the mirror: a Landau-Ginzburg model on an algebraic torus. For CP^{N-1} the constraint turns this into the affine A_{N-1} Toda superpotential, reproducing a known mirror pair. For a degree-d hypersurface in CP^{N-1} the mirror is an orbifold of the Landau-Ginzburg model with superpotential sum X_i^d plus e^{t/d} times the product of X_i. The paper extends this to complete intersections, computes D-brane periods and Picard-Fuchs equations, and recovers the trilogarithm prepotential for the O(-1) plus O(-1) bundle over CP^1.

Extended reading notes

Core claim

The exact twisted superpotential of the dualized U(1) gauge theory is W-tilde = Sigma (sum_i Q_i Y_i - t) + mu sum_i e^{-Y_i}, with the exponential terms generated by vortex instantons, and integrating out Sigma yields the Landau-Ginzburg mirror of the sigma model on the toric variety. The paper states this in section 3.2 as equation (3.38) and calls it one of the main results. If correct, mirror symmetry for toric targets, CP^{N-1} (affine Toda), line bundles over projective space, and hypersurfaces follows from ordinary abelian duality plus a semiclassical instanton sum, rather than from case-by-case conjecture.

Load-bearing premise

The localization argument in section 3.2.1, which reduces the general N-flavor theory to N copies of the single-flavor case. It assumes the twisted superpotential is independent of D-term variations, so it can be computed with a diagonal gauge coupling matrix and then continued to the U(1) embedding, and it assumes no singularity is crossed along that deformation. The paper states this directly: the superpotential is valid for all values of the inverse coupling matrix as long as there is no singularity in going from the diagonal one. If a singularity is crossed, or if the twisted superpotential picks up D-term dependence, the general-N result does not follow from the single-flavor computation.

Original abstract

We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation. This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,667 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.13 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation. This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions.	result 0.53	0.95	0.95	asserted 1.00
s1	The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type.	method 0.99	0.88	0.05	asserted 1.00
s2	Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation.	method 1.00	0.74	0.04	asserted 1.00
s3	This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries.	result 0.73	0.89	0.44	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Showing 60 of 105 Pith papers that cite this

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math.AG 2026-10 conditional novelty 8.0 of 10

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hep-th 2015-06 conditional novelty 8.0 of 10

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hep-th 2012-10 conditional novelty 8.0 of 10

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hep-th 2010-06 accept novelty 8.0 of 10

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Two-Sphere Partition Functions and Gromov-Witten Invariants
hep-th 2012-08 accept novelty 7.0 of 10

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hep-th 2012-06 accept novelty 7.0 of 10

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hep-th 2012-04 conditional novelty 7.0 of 10

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Quivers, YBE and 3-manifolds
hep-th 2012-03 conditional novelty 7.0 of 10

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hep-th 2011-06 conditional novelty 7.0 of 10

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hep-th 2006-06 conditional novelty 7.0 of 10

Strings on gerbes are T-dual to strings on disconnected unions of ordinary spaces, resolving the apparent cluster-decomposition violation.
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hep-th 2005-11 conditional novelty 7.0 of 10

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GLSM's for gerbes (and other toric stacks)
hep-th 2005-02 accept novelty 7.0 of 10

Gauged linear sigma models with nonminimal U(1) charges describe toric stacks, and their infrared physics depends only on the stack, not on the chosen presentation.
Quantum Foam and Topological Strings
hep-th 2003-12 conditional novelty 7.0 of 10

The melting-crystal partition function equals a sum over Kähler geometries and topologies, which the authors identify with a topologically twisted U(1) gauge theory of ideal sheaves on the Calabi-Yau.
Matrix Models, Geometric Engineering and Elliptic Genera
hep-th 2003-10 accept novelty 7.0 of 10

For N=2* theory compactified from six to four dimensions on a torus, matrix models, geometric engineering via the topological vertex, and elliptic-genus instanton calculus all yield the same prepotential, encoded in a...
T-Duality: Topology Change from H-flux
hep-th 2003-06 accept novelty 7.0 of 10

T-duality of a circle bundle exchanges the first Chern class with the fiber integral of the H-flux, and this exchange is an isomorphism of twisted K-theory and twisted cohomology.
Disk Instantons, Mirror Symmetry and the Duality Web
hep-th 2001-05 accept novelty 7.0 of 10

Mirror symmetry computes disk instanton superpotentials for D-branes on non-compact Calabi-Yau threefolds, and the integer ambiguity in that computation maps to the framing choice of a knot in the dual Chern-Simons theory.
Mirror Symmetry, D-Branes and Counting Holomorphic Discs
hep-th 2000-12 accept novelty 7.0 of 10

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Evidence: substantial. 345 of the citing papers share no author with it, and 55 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 602 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

14 later papers by other authors are listed below: 2 built on it, 11 used its method or tool and 1 cited it as support. 1 more paper by the same authors is listed: 1 used its method or tool.

Builds on it 2 of 11 papers with this label

Generalized the physical approach taken in this paper to construct (2,2) mirror pairs for non-compact toric spaces.

Recent Developments in (0,2) Mirror Symmetry (2012) · 0.98 · paraphrased: the licence does not allow quoting
Generalized a related analysis of this paper.

Supersymmetric vortex defects in two dimensions (2017) · 0.96 · paraphrased: the licence does not allow quoting

Uses its method or tool 12 of 33 papers with this label · 1 by the same authors

Relied on the mirror formulation of this paper of the weighted CP model.

Non-Abelian Confinement in N=2 Supersymmetric QCD: Duality and Kinks on Confining Strings (2010) · 1.00 · paraphrased: the licence does not allow quoting
Computed the mirror ala this paper, finding Landau-Ginzburg models with superpotentials of the stated form.

Topological operators, noninvertible symmetries and decomposition (2021) · 0.99 · paraphrased: the licence does not allow quoting
Implemented the mirror description of kinks of the model of this paper.

String "Baryon'' in Four-Dimensional $\mathcal{N} = 2$ Supersymmetric QCD from 2D-4D Correspondence (2020) · 0.98 · paraphrased: the licence does not allow quoting
The vanilla anti-holomorphic involution hence acts as _v : Φ_i ↦Φ̅_i Σ_a ↦Σ̅_a We can now trace this through the dualisation procedure of [ref] to find the action of $_v$ on the mirror.

$G_2$ Mirrors from Calabi-Yau Mirrors (2023) · Anti-holomorphic Involutions and GLSMs · 0.97
Implemented the mirror description of kinks of the model of this paper.

Dynamics of non-Abelian strings in supersymmetric gauge theories (2020) · 0.97 · paraphrased: the licence does not allow quoting
Constructed the mirror geometry following this paper.

Topological recursion and mirror curves (2011) · 0.96 · paraphrased: the licence does not allow quoting
Used the change in measure proposed by this paper in its correlator sum.

Half-Twisted (0,2) Landau-Ginzburg Models (2007) · 0.95 · paraphrased: the licence does not allow quoting
Used the results of this paper to derive the Landau-Ginzburg model dual to the gauged sigma-model with target X.

The quantum equivariant cohomology of toric manifolds through mirror symmetry (2008) · 0.81 · paraphrased: the licence does not allow quoting
Followed the prescription proposed by this paper, in which vortex quantum corrections generate a non-perturbative twisted effective superpotential in dual field variables.

Dynamical Generation of the VY Superpotential in $N=1$ SYM: A Higher-Form Perspective (2026) · 0.93 · paraphrased: the licence does not allow quoting
Following this paper, first defined the mirror Landau-Ginzburg model on (C*)^N x C with the stated superpotential and Kähler potential.

Vacuum Structures Revisited (2021) · 0.90 · paraphrased: the licence does not allow quoting
Obtained the curve by considering the Hori-Vafa mirror of this paper.

Exponential BPS graphs and D-brane counting on toric Calabi-Yau threefolds: Part II (2020) · 0.82 · paraphrased: the licence does not allow quoting

By the same authors

Used the usual mirror symmetry arguments of this paper to arrive at an LG theory with the stated superpotential.

Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2 (2015) (same authors) · 0.90 · paraphrased: the licence does not allow quoting

Cites it as support 1 paper

In any case, the mirror symmetry result established in [ref] justifies our assumption a posteriori.

Refined instanton analysis of the 2D $\mathbb{C}P^{N-1}$ model: mass gap, theta dependence, and mirror symmetry (2023) · Vortex operators for the CPN-1 model · 1.00

How the literature uses it

359 arXiv papers cite this in 609 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 341 sentences without a confident label

Builds on it 11 papers

Employing the duality procedure of Roceck and Verlinde [ref] we have extended the mentioned exploration [ref] by constructing general T-dual models in GLSMs and in particular Non-Abelian T-dual models [ref] .

T-dualities in gauged linear sigma models · Introduction · 0.86
In this section, we will explain how to construct mirror pairs of (0,2) models using worldsheet duality [ref] , generalizing the approach used in the (2,2) setting by [ref] .

Recent Developments in (0,2) Mirror Symmetry · Worldsheet duality · 0.96
More recently, mirror descriptions of $(0,2)$ gauged linear sigma models were proposed [ref] using a generalization of [ref] .

Half-Twisted (0,2) Landau-Ginzburg Models · Introduction · 0.81

Uses its method or tool 33 papers

In section 3 we use the results of [ref] to derive the Landau-Ginzburg model that is dual to the gauged sigma-model with target $X$.

The quantum equivariant cohomology of toric manifolds through mirror symmetry · Introduction · 0.81
Using the explicit dualisation of [ref] allows us to show that appropriate anti-holomorphic involutions are identified under the mirror map.

$G_2$ Mirrors from Calabi-Yau Mirrors · Introduction · 0.92
To this end we will rely on the mirror formulation [ref] of the weighted CP model ([ref]).

Non-Abelian Confinement in N=2 Supersymmetric QCD: Duality and Kinks on Confining Strings · Mirror superpotential · 1.00

Compares with it 17 papers

In [ref] all the gauge terms were gotten rid of, because there one is only interested in the non-gauged sigma-model with target $X$.

The quantum equivariant cohomology of toric manifolds through mirror symmetry · The dual gauged lagrangian · 0.92
We implement T-dualities in GLSM [ref] by a different method than Hori and Vafa [ref] , that for a particular Abelian case leads to the same results, i.e. MS between dual models.

T-dualities in gauged linear sigma models · T-duality in the GLSM · 0.98
The GW invariants do not depend on the choice of coordinates, hence remain the same for any non-zero value of $$. * In [ref] we derived the tropical mirror superpotential for toric $X$ and it matches with exponential mirror superpotential derived in [ref] , [ref] for the same toric space $X$.

Tropical Mirror Symmetry: Correlation functions · Tropical GW invariants · 0.88

Cites it as support 1 paper

In any case, the mirror symmetry result established in [ref] justifies our assumption a posteriori.

Refined instanton analysis of the 2D $\mathbb{C}P^{N-1}$ model: mass gap, theta dependence, and mirror symmetry · Vortex operators for the CPN-1 model · 1.00

Cites it as background 161 papers

Mirror symmetry was described by Hori and Vafa in [ref] as a duality between sigma-models and Landau-Ginzburg models.

The quantum equivariant cohomology of toric manifolds through mirror symmetry · Introduction · 0.90
In [ref] , Teleman, based on toric mirror constructions by Givental [ref] and Hori-Vafa [ref] , proposed the following conjecture when $G=T$ is abelian:

Equivariant Lagrangian correspondence and a conjecture of Teleman · Introduction · 0.81
In [ref] Abouzaid established an equivalence between $𝒟^b(X)$, the derived category of coherent sheaves of $X$, and a certain full subcategory of $𝒟^π (Fuk( (ℂ^∗)^n, W))$, the derived Fukaya category of a specific mirror model $((ℂ^∗)^n, W)$, affirming Kontsevich's homological mirror symmetry conjecture for toric Fano manifolds, see [ref] .

On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces · Introduction and Summary of Main Results · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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