Phases of flux vacua and their quantum symmetries

Dan Israel; Ilarion V. Melnikov; Yann Proto

REVIEW

Phases of flux vacua and their quantum symmetries

Dan Israel, Ilarion V. Melnikov, Yann Proto

High Energy Physics - Theory · Submitted 22 Oct 2026 · 65 references

Pith's verdict: This paper delivers the first explicit TLSM construction of a heterotic flux vacuum with a Gepner point, backed by multiple consistency checks and a detailed analysis of the field redefinition structure — a genuine advance that deserves a serious referee despite one load-bearing assumption that the field cannot yet fully verify.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 23, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 3 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2610.30865 (version 1)

Pith IDZYXEXXEL, Pith's permanent record of this paper and its review

CitationDan Israel, Ilarion V. Melnikov, Yann Proto. "Phases of flux vacua and their quantum symmetries." arXiv:2610.30865 (2026). https://pith.science/paper/ZYXEXXEL

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This is a strong paper. The central construction — a (0,2) torsional linear sigma model whose large-radius phase describes a five-dimensional heterotic flux vacuum and whose strongly coupled regime connects to an asymmetric orbifold of a Gepner model and a circle — is new and non-trivial. The authors do not stop at presenting the model; they subject it to a battery of independent checks: anomaly cancellation in the required factorized form, a detailed classical phase analysis, field redefinition arguments showing that the FI parameter space collapses to a one-dimensional marginal deformation connecting the large-radius and LGO phases, a dressed elliptic genus computation that matches the geometric formula via localization, and an explicit LGO partition function that reveals symmetry enhancement and free fermion decoupling. The paper is transparent about its limitations, including the unresolved central charge discrepancy in the hybrid phase and the conjectural uniqueness of the Gepner fixed point.

The stress-test concern about the ‘protected class’ assumption is fair as a matter of principle, but the paper handles it honestly: it states the reliance explicitly (Section 1.1) and acknowledges that sufficient conditions for this specific model have not been verified. The paper does not claim to have proved instanton stability; it provides the circumstantial evidence that is currently standard in the field. The multiple consistency checks — anomaly cancellation, field redefinition equivalences, elliptic genus matching — collectively make the case that the construction is physically meaningful. The protected class assumption is a known open problem in (0,2) GLSMs, not a loophole unique to this paper. The authors’ discussion of it is proportionate.

What this paper does well is to show that a heterotic flux vacuum can be realized in a linear sigma model and connected to a solvable CFT point, with enough technical machinery (field redefinitions for line bundle stability, new methods for constructing anomaly-free models via trivial bundle factors) to be useful for future constructions. The soft spots are genuine but narrow: the hybrid phase central charge discrepancy, the conjectural nature of the Gepner fixed point uniqueness, and the reliance on the protected class assumption. None of these undermine the central argument, which holds up on its own terms.

Who is this for? Specialists in heterotic compactifications, (0,2) linear sigma models, and flux vacua. It is a substantial advance that should be published. Yes, it deserves a serious referee.

Referee report

Major objections (2)

[Section 1.1, paragraph starting 'Remarkably, there are arguments...'] The paper explicitly assumes (citing Refs. [16–18]) that the TLSM belongs to the 'protected class' of (0,2) GLSMs for which gauge instanton corrections do not destabilize the vacuum. This assumption is load-bearing for the interpretation of the large-radius phase as a consistent heterotic flux vacuum. While the necessary checks (anomaly cancellation, dressed elliptic genus matching, phase analysis) are performed, no argument is given that the specific model (with its additional chiral and fermi multiplets and superpotential couplings) satisfies the sufficient conditions of [16–18]. A brief discussion of why this model is expected to fall into that class would significantly strengthen the claim.
[Section 3.1, around Eq. (3.12)] In the hybrid phase, c-extremization gives c_R = 15/2 + 3/5, exceeding the expected value. The paper argues that this phase is not connected to the geometric phase by a marginal deformation but may be connected via relevant RG flow. The analysis here is plausible but somewhat speculative; a more quantitative discussion (e.g., the nature of the putative relevant operator) would be helpful.

Minor objections (3)

[Eq. (3.1)] The superpotential includes the term Φ7 Γ1 Φ5^1, but the charge table in Table 1 shows Γ1 with charges (1,0) and Φ5 with charges (1,1); it might be helpful to explicitly verify gauge invariance for this term.
[Table 1] The table lists charges for fields including the shift multiplet Ω, but Ω transforms inhomogeneously; the charge listed (1,-1) is the shift charge, not a standard gauge charge. Clarifying this in the caption would avoid confusion.
[Eq. (4.5) and surrounding discussion] The geometric formula for the dressed elliptic genus is given for the factorized geometry X×S^1. It would be useful to note explicitly that Z_Γ(τ,τ̄) is the partition function of the free circle at a generic radius, and that the radius r=√3 is used later.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

First TLSM connects heterotic flux vacuum to a stringy Gepner orbifold.

The reading

This paper presents the first torsional linear sigma model (TLSM) with (0,2) supersymmetry that realizes, in its large-radius phase, a consistent five-dimensional heterotic flux vacuum: a U(1) bundle over a K3 surface with NS-NS flux. In the strongly coupled regime, the same TLSM flows to an asymmetric orbifold of a Gepner model and a circle CFT, providing an exactly solvable description. The authors show that the two phases are connected by a marginal deformation and describe the same infrared SCFT, as evidenced by matching dressed elliptic genera and massless spectra. If correct, this is the first example of a linear sigma model that achieves such a flux vacuum and can be linked to a rational CFT.

What carries the argument

The key machinery is the shift multiplet Ω, a chiral superfield that transforms inhomogeneously under gauge symmetries and couples axially to the gauge field strengths. Its classical gauge variation cancels the one-loop gauge anomaly provided the anomaly matrix factorizes as Aαβ = −2r² wα wβ, where wα are integer shift charges. This anomaly condition encodes the heterotic Bianchi identity. A field redefinition that rescales the matter multiplets induces an equivalence relation on the complexified FI parameters, τα ~ τα + ξ wα, which collapses the two-dimensional FI phase diagram to a one-dimensional line connecting the large-radius phase and the Landau-Ginzburg orbifold phase. In the LG phase, the superpotential at the Fermat point yields a (2,2) Gepner model, and the Z6×Z6 orbifold acts chirally on both the LG fields and the circle, preserving only (0,2) supersymmetry.

Extended reading notes

Core claim

The central discovery is a class of (0,2) torsional linear sigma models whose large-radius phase describes a five-dimensional heterotic flux vacuum (a principal U(1) bundle over a K3 surface with NS-NS flux, satisfying the Bianchi identity with a trivial rank-two gauge bundle). The same TLSM has a Landau-Ginzburg orbifold phase where the gauge symmetry is broken to Z6×Z6, and the low-energy theory becomes an asymmetric orbifold of the product of a Gepner model (built from four minimal models) and a compact boson at radius squared 3. The two phases are shown to be connected by a marginal deformation, and the equality of the dressed elliptic genus computed in each phase, together with the matching of the spacetime massless spectrum, confirms that they describe the same SCFT. At the Gepner point, the left-moving current algebra enhances to include two free Weyl fermions, which reconstruct the second E8 gauge factor in the heterotic string.

Load-bearing premise

The construction assumes the TLSM belongs to the 'protected class' of (0,2) GLSMs for which gauge instanton corrections do not destabilize the vacuum, as argued in references [16-18].

Editorial extensions

If this is right

This construction provides the first explicit worldsheet realization of a heterotic flux vacuum with eight supercharges, allowing precise CFT calculations in a regime previously accessible only through supergravity approximations.
The equivalence between the large-radius and Gepner orbifold phases demonstrates that the torsional sigma model SCFT is well-defined and exhibits expected features such as free left-moving fermions from a trivial bundle.
The analysis of field redefinitions in (0,2) GLSMs shows that FI parameters can be redundant, collapsing phase diagrams and revealing that the LG phase consists of a single Gepner point.
The enhanced gauge symmetry at the Gepner point (E8×E8×SO(4)×U(1)) and its Higgsing when moving to the large-radius phase match expectations from supergravity and support the consistency of the flux vacuum.
The techniques developed, including the GLSM realization of line bundle stability and the use of spectator fields, can be applied to construct more general (0,2) GLSMs for heterotic vacua with non-trivial bundles.

Reading between the lines

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

If the construction is robust, it suggests that many heterotic flux vacua may admit similar TLSM descriptions, potentially enabling a systematic exploration of their moduli spaces and dualities.
The appearance of free fermions in the twisted sectors of the orbifold hints at a general mechanism: asymmetric orbifolds of Gepner models can produce decoupled fermionic degrees of freedom, which may be useful for realizing realistic gauge groups in heterotic string compactifications.
The field-redefinition equivalence between FI parameters and superpotential couplings implies that the usual phase diagram of (0,2) GLSMs should be interpreted with caution; the actual space of SCFTs may be smaller than suggested by the classical FI parameters.
This work paves the way for constructing (0,2) linear sigma models for four-dimensional heterotic flux vacua, possibly by orbifolding the present TLSM by an Enriques involution.

Original abstract

We present a torsional linear sigma model with (0,2) supersymmetry that realizes, in its large radius phase, a five-dimensional heterotic flux vacuum, and we use the linear sigma model to connect this geometry to a stringy regime realized as an asymmetric orbifold of a Gepner model and a circle. Along the way we develop a number of tools relevant for more general (0,2) gauged linear sigma model constructions of heterotic vacua.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-23: 1 request, 2,385 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.11 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a torsional linear sigma model with (0,2) supersymmetry that realizes, in its large radius phase, a five-dimensional heterotic flux vacuum, and we use the linear sigma model to connect this geometry to a stringy regime realized as an asymmetric orbifold of a Gepner model and a circle. Along the way we develop a number of tools relevant for more general (0,2) gauged linear sigma model constructions of heterotic vacua.

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 present a torsional linear sigma model with (0,2) supersymmetry that realizes, in its large radius phase, a five-dimensional heterotic flux vacuum, and we use the linear sigma model to connect this geometry to a stringy regime realized as an asymmetric orbifold of a Gepner model and a circle.	contribution 0.99	0.94	0.70	asserted 1.00
s1	Along the way we develop a number of tools relevant for more general (0,2) gauged linear sigma model constructions of heterotic vacua.	contribution 0.98	0.92	0.06	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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

levels run

P abstract

not run

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

References

Works cited (65)

[1]

K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror symmetry, vol. 1 ofClay Mathematics Monographs. American Mathematical Society, Providence, RI, 2003. With a preface by Vafa

View on Pith 2003
[2]

P. S. Aspinwall, T. Bridgeland, A. Craw, M. R. Douglas, A. Kapustin, G. W. Moore, M. Gross, G. Segal, B. Szendroi, and P. M. H. Wilson,Dirichlet branes and mirror symmetry, vol. 4 of – 63 – Clay Mathematics Monographs. AMS, Providence, RI, 2009. http://people.maths.ox.ac.uk/cmi/library/monographs/cmim04c.pdf

View on Pith 2009
[3]

E. Witten, “Phases of N = 2 theories in two dimensions,”Nucl. Phys.B403(1993) 159–222, arXiv:hep-th/9301042

View on Pith Pith review arXiv 1993
[4]

K. Dasgupta, G. Rajesh, and S. Sethi, “M theory, orientifolds and G-flux,”JHEP08(1999) 023,arXiv:hep-th/9908088

View on Pith Pith review arXiv 1999
[5]

E. Goldstein and S. Prokushkin, “Geometric model for complex non-Kaehler manifolds with SU(3) structure,”Commun. Math. Phys.251(2004) 65–78,arXiv:hep-th/0212307

View on Pith Pith review arXiv 2004
[6]

K. Becker, M. Becker, J.-X. Fu, L.-S. Tseng, and S.-T. Yau, “Anomaly cancellation and smooth non-Kaehler solutions in heterotic string theory,”Nucl. Phys.B751(2006) 108–128, arXiv:hep-th/0604137

View on Pith Pith review arXiv 2006
[7]

J.-X. Fu and S.-T. Yau, “The theory of superstring with flux on non-Kaehler manifolds and the complex Monge-Ampere equation,”J. Diff. Geom.78(2009) 369–428,arXiv:hep-th/0604063

View on Pith Pith review arXiv 2009
[8]

A. Adams, M. Ernebjerg, and J. M. Lapan, “Linear models for flux vacua,” arXiv:hep-th/0611084

View on Pith Pith review arXiv
[9]

L. B. Anderson and M. Karkheiran, “TASI Lectures on Geometric Tools for String Compactifications,”PoST ASI2017(2018) 013,arXiv:1804.08792 [hep-th]

View on Pith Pith review arXiv 2018
[10]

J. Distler and S. Kachru, “(0,2) Landau-Ginzburg theory,”Nucl. Phys.B413(1994) 213–243, hep-th/9309110

View on Pith Pith review arXiv 1994
[11]

J. Distler, “Notes on (0,2) superconformal field theories,”hep-th/9502012

View on Pith Pith review arXiv
[12]

F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic genera of 2d N=2 gauge theories,” arXiv:1308.4896 [hep-th]

View on Pith Pith review arXiv
[13]

T. Kawai and K. Mohri, “Geometry of (0,2) Landau-Ginzburg orbifolds,”Nucl. Phys.B425 (1994) 191–216,hep-th/9402148

View on Pith Pith review arXiv 1994
[14]

M. Dine, N. Seiberg, X. G. Wen, and E. Witten, “Nonperturbative Effects on the String World Sheet,”Nucl. Phys.B278(1986) 769

View on Pith 1986
[15]

M. Dine, N. Seiberg, X. G. Wen, and E. Witten, “Nonperturbative Effects on the String World Sheet. 2,”Nucl. Phys.B289(1987) 319

View on Pith 1987
[16]

E. Silverstein and E. Witten, “Criteria for conformal invariance of (0,2) models,”Nucl. Phys. B444(1995) 161–190,arXiv:hep-th/9503212

View on Pith Pith review arXiv 1995
[17]

C. Beasley and E. Witten, “Residues and world-sheet instantons,”JHEP10(2003) 065, arXiv:hep-th/0304115

View on Pith Pith review arXiv 2003
[18]

M. Bertolini and M. R. Plesser, “Worldsheet instantons and (0,2) linear models,”JHEP08 (2015) 081,arXiv:1410.4541 [hep-th]

View on Pith Pith review arXiv 2015
[19]

M. Bertolini, I. V. Melnikov, and M. R. Plesser, “Fixed points of (0,2) Landau-Ginzburg renormalization group flows and the chiral algebra,”JHEP09(2022) 230,arXiv:2106.00105 [hep-th]

View on Pith Pith review arXiv 2022
[20]

I. V. Melnikov,An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry, vol. 951 ofLect. Notes Phys.Springer, 2019. – 64 –

View on Pith 2019
[21]

E. J. Martinec, “Algebraic Geometry and Effective Lagrangians,”Phys. Lett.B217(1989) 431

View on Pith 1989
[22]

C. Vafa and N. P. Warner, “Catastrophes and the Classification of Conformal Theories,”Phys. Lett.B218(1989) 51

View on Pith 1989
[23]

P. S. Aspinwall, B. R. Greene, and D. R. Morrison, “The monomial divisor mirror map,” Internat. Math. Res. Noticesno. 12, (1993) 319–337,arXiv:alg-geom/9309007

View on Pith Pith review arXiv 1993
[24]

D. A. Cox and S. Katz,Mirror symmetry and algebraic geometry. Amer. Math. Soc., 1999. Providence, USA: AMS (2000) 469 p

View on Pith 2000
[25]

M. Kreuzer, J. McOrist, I. V. Melnikov, and M. Plesser, “(0,2) deformations of linear sigma models,”JHEP1107(2011) 044,arXiv:1001.2104 [hep-th]

View on Pith Pith review arXiv 2011
[26]

H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison, and M. Romo, “Two-Sphere Partition Functions and Gromov-Witten Invariants,”arXiv:1208.6244 [hep-th]

View on Pith Pith review arXiv
[27]

D. Israel, I. V. Melnikov, R. Minasian, and Y. Proto, “Topology change and heterotic flux vacua,”JHEP06(2024) 204,arXiv:2312.08923 [hep-th]

View on Pith Pith review arXiv 2024
[28]

J. Evslin and R. Minasian, “Topology change from (heterotic) Narain T-duality,”Nucl. Phys. B820(2009) 213–236,arXiv:0811.3866 [hep-th]

View on Pith Pith review arXiv 2009
[29]

P. Griffiths and J. Harris,Principles of algebraic geometry. Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics

View on Pith 1978
[30]

I. V. Melnikov, R. Minasian, and S. Sethi, “Heterotic fluxes and supersymmetry,”JHEP06 (2014) 174,arXiv:1403.4298 [hep-th]

View on Pith Pith review arXiv 2014
[31]

K. Uhlenbeck and S. Yau, “On the existence of Hermitian-Yang-Mills connections in stable vector bundles,”Comm. Pure and Applied Mathematics39(1986) 257–293

View on Pith 1986
[32]

H. J. Hoppe, “Generischer Spaltungstyp und zweite Chernklasse stabiler Vektorraumb¨ undel vom Rang 4 aufP 4,”Mathematische Zeitschrift187no. 3, (Sept., 1984) 345–360

View on Pith 1984
[33]

R. Pardini, “Abelian covers of algebraic varieties,”Journal f¨ ur die reine und angewandte Mathematik1991no. 417, (1991) 191–214

View on Pith 1991
[34]

D. Festi, D. Platt, R. Singhal, and Y. Tanaka, “Examples of real stable bundles on K3 surfaces,” 2025.https://arxiv.org/abs/2503.02937

View on Pith Pith review arXiv 2025
[35]

T. Hayashi, “Finite Abelian Groups of K3 Surfaces with Smooth Quotient,”Chinese Annals of Mathematics, Series B44no. 1, (Jan., 2023) 99–162

View on Pith 2023
[36]

F. Benini and N. Bobev, “Exact two-dimensional superconformal R-symmetry and c-extremization,”Phys.Rev.Lett.110(2013) 061601,arXiv:1211.4030 [hep-th]

View on Pith Pith review arXiv 2013
[37]

M. Bertolini, I. V. Melnikov, and M. R. Plesser, “Hybrid conformal field theories,” arXiv:1307.7063 [hep-th]

View on Pith Pith review arXiv
[38]

M. Bertolini and M. R. Plesser, “(0,2) hybrid models,”JHEP09(2018) 067, arXiv:1712.04976 [hep-th]

View on Pith Pith review arXiv 2018
[39]

I. V. Melnikov, R. Minasian, and S. Theisen, “Heterotic flux backgrounds and their IIA duals,” JHEP07(2014) 023,arXiv:1206.1417 [hep-th]

View on Pith Pith review arXiv 2014
[40]

J. Gomis, Z. Komargodski, H. Ooguri, N. Seiberg, and Y. Wang, “Shortening Anomalies in Supersymmetric Theories,”JHEP01(2017) 067,arXiv:1611.03101 [hep-th]. – 65 –

View on Pith Pith review arXiv 2017
[41]

P. S. Aspinwall, “K3 surfaces and string duality,”arXiv:hep-th/9611137

View on Pith Pith review arXiv
[42]

J. McOrist and I. V. Melnikov, “Old issues and linear sigma models,”Adv.Theor.Math.Phys.16 (2012) 251–288,arXiv:1103.1322 [hep-th]

View on Pith Pith review arXiv 2012
[43]

E. Witten, “Elliptic genera and quantum field theory,”Commun.Math.Phys.109(1987) 525

View on Pith 1987
[44]

T. Kawai, Y. Yamada, and S.-K. Yang, “Elliptic genera and N=2 superconformal field theory,” Nucl.Phys.B414(1994) 191–212,arXiv:hep-th/9306096 [hep-th]

View on Pith Pith review arXiv 1994
[45]

D. Israel and M. Sarkis, “New supersymmetric index of heterotic compactifications with torsion,”JHEP12(2015) 069,arXiv:1509.05704 [hep-th]

View on Pith Pith review arXiv 2015
[46]

D. Israel and M. Sarkis, “Dressed elliptic genus of heterotic compactifications with torsion and general bundles,”JHEP08(2016) 176,arXiv:1606.08982 [hep-th]

View on Pith Pith review arXiv 2016
[47]

I. V. Melnikov and R. Minasian, “Heterotic sigma models with N=2 space-time supersymmetry,”JHEP1109(2011) 065,arXiv:1010.5365 [hep-th]

View on Pith Pith review arXiv 2011
[48]

T. Banks and L. J. Dixon, “Constraints on string vacua with space-time supersymmetry,”Nucl. Phys.B307(1988) 93–108

View on Pith 1988
[49]

K. S. Narain, M. H. Sarmadi, and C. Vafa, “Asymmetric Orbifolds,”Nucl. Phys. B288(1987) 551

View on Pith 1987
[50]

D. Israel, I. V. Melnikov, and Y. Proto, “Shift orbifolds, decompactification limits, and lattices,”SciPost Phys.20no. 2, (2026) 039,arXiv:2502.18453 [hep-th]

View on Pith Pith review arXiv 2026
[51]

J. Fuchs, “Simple WZW currents,”Commun. Math. Phys.136(1991) 345–356

View on Pith 1991
[52]

G. Lockhart, L. Novelli, and Y. Proto, “String probes, simple currents, and the no global symmetries conjecture,”arXiv:2605.12594 [hep-th]

View on Pith Pith review arXiv
[53]

V. A. Fateev and A. B. Zamolodchikov, “Parafermionic Currents in the Two-Dimensional Conformal Quantum Field Theory and Selfdual Critical Points in Z(n) Invariant Statistical Systems,”Sov. Phys. JETP62(1985) 215–225

View on Pith 1985
[54]

Z.-a. Qiu, “Nonlocal Current Algebra andN= 2 Superconformal Field Theory in Two-dimensions,”Phys. Lett. B188(1987) 207–213

View on Pith 1987
[55]

D. Gepner, “Space-Time Supersymmetry in Compactified String Theory and Superconformal Models,”Nucl. Phys.B296(1988) 757

View on Pith 1988
[56]

M. Bertolini, I. V. Melnikov, and M. R. Plesser, “Accidents in (0,2) Landau-Ginzburg theories,” JHEP12(2014) 157,arXiv:1405.4266 [hep-th]

View on Pith Pith review arXiv 2014
[57]

P. C. Argyres, M. R. Plesser, and N. Seiberg, “The Moduli space of vacua of N=2 SUSY QCD and duality in N=1 SUSY QCD,”Nucl.Phys.B471(1996) 159–194,arXiv:hep-th/9603042 [hep-th]

View on Pith Pith review arXiv 1996
[58]

P. C. Argyres, M. R. Plesser, and A. D. Shapere, “N=2 moduli spaces and N=1 dualities for SO(n(c)) and USp(2n(c)) superQCD,”Nucl.Phys.B483(1997) 172–186, arXiv:hep-th/9608129 [hep-th]

View on Pith Pith review arXiv 1997
[59]

M. Jardim, G. Menet, D. M. Prata, and H. N. Sa Earp, “Holomorphic bundles for higher dimensional gauge theory,”Bull. London Math. Soc.49no. 1, (2017) 117–132, arXiv:1109.2750 [math.AG]. – 66 –

View on Pith Pith review arXiv 2017
[60]

S. Kachru and E. Witten, “Computing the complete massless spectrum of a Landau- Ginzburg orbifold,”Nucl. Phys.B407(1993) 637–666,arXiv:hep-th/9307038

View on Pith Pith review arXiv 1993
[61]

M. Becker, L.-S. Tseng, and S.-T. Yau, “New Heterotic Non-Kahler Geometries,” arXiv:0807.0827 [hep-th]

View on Pith Pith review arXiv
[62]

S. Ferrara, J. A. Harvey, A. Strominger, and C. Vafa, “Second quantized mirror symmetry,” Phys.Lett.B361(1995) 59–65,arXiv:hep-th/9505162 [hep-th]

View on Pith Pith review arXiv 1995
[63]

D. Israel and Y. Proto, “A Worldsheet Approach toN= 1 Heterotic Flux Backgrounds,” arXiv:2302.01889 [hep-th]

View on Pith Pith review arXiv
[64]

M. Maio and A. N. Schellekens, “Permutation Orbifold of N=2 Supersymmetric Minimal Models,”Nucl. Phys. B845(2011) 212–245,arXiv:1011.0934 [hep-th]

View on Pith Pith review arXiv 2011
[65]

D. Isra¨ el, “T-Duality in Gauged Linear Sigma-Models with Torsion,”JHEP11(2013) 093, arXiv:1306.6609 [hep-th]. – 67 –

View on Pith Pith review arXiv 2013

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.

[1] [1]

K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror symmetry, vol. 1 ofClay Mathematics Monographs. American Mathematical Society, Providence, RI, 2003. With a preface by Vafa

View on Pith 2003

[2] [2]

P. S. Aspinwall, T. Bridgeland, A. Craw, M. R. Douglas, A. Kapustin, G. W. Moore, M. Gross, G. Segal, B. Szendroi, and P. M. H. Wilson,Dirichlet branes and mirror symmetry, vol. 4 of – 63 – Clay Mathematics Monographs. AMS, Providence, RI, 2009. http://people.maths.ox.ac.uk/cmi/library/monographs/cmim04c.pdf

View on Pith 2009

[3] [3]

E. Witten, “Phases of N = 2 theories in two dimensions,”Nucl. Phys.B403(1993) 159–222, arXiv:hep-th/9301042

View on Pith Pith review arXiv 1993

[4] [4]

K. Dasgupta, G. Rajesh, and S. Sethi, “M theory, orientifolds and G-flux,”JHEP08(1999) 023,arXiv:hep-th/9908088

View on Pith Pith review arXiv 1999

[5] [5]

E. Goldstein and S. Prokushkin, “Geometric model for complex non-Kaehler manifolds with SU(3) structure,”Commun. Math. Phys.251(2004) 65–78,arXiv:hep-th/0212307

View on Pith Pith review arXiv 2004

[6] [6]

K. Becker, M. Becker, J.-X. Fu, L.-S. Tseng, and S.-T. Yau, “Anomaly cancellation and smooth non-Kaehler solutions in heterotic string theory,”Nucl. Phys.B751(2006) 108–128, arXiv:hep-th/0604137

View on Pith Pith review arXiv 2006

[7] [7]

J.-X. Fu and S.-T. Yau, “The theory of superstring with flux on non-Kaehler manifolds and the complex Monge-Ampere equation,”J. Diff. Geom.78(2009) 369–428,arXiv:hep-th/0604063

View on Pith Pith review arXiv 2009

[8] [8]

A. Adams, M. Ernebjerg, and J. M. Lapan, “Linear models for flux vacua,” arXiv:hep-th/0611084

View on Pith Pith review arXiv

[9] [9]

L. B. Anderson and M. Karkheiran, “TASI Lectures on Geometric Tools for String Compactifications,”PoST ASI2017(2018) 013,arXiv:1804.08792 [hep-th]

View on Pith Pith review arXiv 2018

[10] [10]

J. Distler and S. Kachru, “(0,2) Landau-Ginzburg theory,”Nucl. Phys.B413(1994) 213–243, hep-th/9309110

View on Pith Pith review arXiv 1994

[11] [11]

J. Distler, “Notes on (0,2) superconformal field theories,”hep-th/9502012

View on Pith Pith review arXiv

[12] [12]

F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic genera of 2d N=2 gauge theories,” arXiv:1308.4896 [hep-th]

View on Pith Pith review arXiv

[13] [13]

T. Kawai and K. Mohri, “Geometry of (0,2) Landau-Ginzburg orbifolds,”Nucl. Phys.B425 (1994) 191–216,hep-th/9402148

View on Pith Pith review arXiv 1994

[14] [14]

M. Dine, N. Seiberg, X. G. Wen, and E. Witten, “Nonperturbative Effects on the String World Sheet,”Nucl. Phys.B278(1986) 769

View on Pith 1986

[15] [15]

M. Dine, N. Seiberg, X. G. Wen, and E. Witten, “Nonperturbative Effects on the String World Sheet. 2,”Nucl. Phys.B289(1987) 319

View on Pith 1987

[16] [16]

E. Silverstein and E. Witten, “Criteria for conformal invariance of (0,2) models,”Nucl. Phys. B444(1995) 161–190,arXiv:hep-th/9503212

View on Pith Pith review arXiv 1995

[17] [17]

C. Beasley and E. Witten, “Residues and world-sheet instantons,”JHEP10(2003) 065, arXiv:hep-th/0304115

View on Pith Pith review arXiv 2003

[18] [18]

M. Bertolini and M. R. Plesser, “Worldsheet instantons and (0,2) linear models,”JHEP08 (2015) 081,arXiv:1410.4541 [hep-th]

View on Pith Pith review arXiv 2015

[19] [19]

M. Bertolini, I. V. Melnikov, and M. R. Plesser, “Fixed points of (0,2) Landau-Ginzburg renormalization group flows and the chiral algebra,”JHEP09(2022) 230,arXiv:2106.00105 [hep-th]

View on Pith Pith review arXiv 2022

[20] [20]

I. V. Melnikov,An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry, vol. 951 ofLect. Notes Phys.Springer, 2019. – 64 –

View on Pith 2019

[21] [21]

E. J. Martinec, “Algebraic Geometry and Effective Lagrangians,”Phys. Lett.B217(1989) 431

View on Pith 1989

[22] [22]

C. Vafa and N. P. Warner, “Catastrophes and the Classification of Conformal Theories,”Phys. Lett.B218(1989) 51

View on Pith 1989

[23] [23]

P. S. Aspinwall, B. R. Greene, and D. R. Morrison, “The monomial divisor mirror map,” Internat. Math. Res. Noticesno. 12, (1993) 319–337,arXiv:alg-geom/9309007

View on Pith Pith review arXiv 1993

[24] [24]

D. A. Cox and S. Katz,Mirror symmetry and algebraic geometry. Amer. Math. Soc., 1999. Providence, USA: AMS (2000) 469 p

View on Pith 2000

[25] [25]

M. Kreuzer, J. McOrist, I. V. Melnikov, and M. Plesser, “(0,2) deformations of linear sigma models,”JHEP1107(2011) 044,arXiv:1001.2104 [hep-th]

View on Pith Pith review arXiv 2011

[26] [26]

H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison, and M. Romo, “Two-Sphere Partition Functions and Gromov-Witten Invariants,”arXiv:1208.6244 [hep-th]

View on Pith Pith review arXiv

[27] [27]

D. Israel, I. V. Melnikov, R. Minasian, and Y. Proto, “Topology change and heterotic flux vacua,”JHEP06(2024) 204,arXiv:2312.08923 [hep-th]

View on Pith Pith review arXiv 2024

[28] [28]

J. Evslin and R. Minasian, “Topology change from (heterotic) Narain T-duality,”Nucl. Phys. B820(2009) 213–236,arXiv:0811.3866 [hep-th]

View on Pith Pith review arXiv 2009

[29] [29]

P. Griffiths and J. Harris,Principles of algebraic geometry. Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics

View on Pith 1978

[30] [30]

I. V. Melnikov, R. Minasian, and S. Sethi, “Heterotic fluxes and supersymmetry,”JHEP06 (2014) 174,arXiv:1403.4298 [hep-th]

View on Pith Pith review arXiv 2014

[31] [31]

K. Uhlenbeck and S. Yau, “On the existence of Hermitian-Yang-Mills connections in stable vector bundles,”Comm. Pure and Applied Mathematics39(1986) 257–293

View on Pith 1986

[32] [32]

H. J. Hoppe, “Generischer Spaltungstyp und zweite Chernklasse stabiler Vektorraumb¨ undel vom Rang 4 aufP 4,”Mathematische Zeitschrift187no. 3, (Sept., 1984) 345–360

View on Pith 1984

[33] [33]

R. Pardini, “Abelian covers of algebraic varieties,”Journal f¨ ur die reine und angewandte Mathematik1991no. 417, (1991) 191–214

View on Pith 1991

[34] [34]

D. Festi, D. Platt, R. Singhal, and Y. Tanaka, “Examples of real stable bundles on K3 surfaces,” 2025.https://arxiv.org/abs/2503.02937

View on Pith Pith review arXiv 2025

[35] [35]

T. Hayashi, “Finite Abelian Groups of K3 Surfaces with Smooth Quotient,”Chinese Annals of Mathematics, Series B44no. 1, (Jan., 2023) 99–162

View on Pith 2023

[36] [36]

F. Benini and N. Bobev, “Exact two-dimensional superconformal R-symmetry and c-extremization,”Phys.Rev.Lett.110(2013) 061601,arXiv:1211.4030 [hep-th]

View on Pith Pith review arXiv 2013

[37] [37]

M. Bertolini, I. V. Melnikov, and M. R. Plesser, “Hybrid conformal field theories,” arXiv:1307.7063 [hep-th]

View on Pith Pith review arXiv

[38] [38]

M. Bertolini and M. R. Plesser, “(0,2) hybrid models,”JHEP09(2018) 067, arXiv:1712.04976 [hep-th]

View on Pith Pith review arXiv 2018

[39] [39]

I. V. Melnikov, R. Minasian, and S. Theisen, “Heterotic flux backgrounds and their IIA duals,” JHEP07(2014) 023,arXiv:1206.1417 [hep-th]

View on Pith Pith review arXiv 2014

[40] [40]

J. Gomis, Z. Komargodski, H. Ooguri, N. Seiberg, and Y. Wang, “Shortening Anomalies in Supersymmetric Theories,”JHEP01(2017) 067,arXiv:1611.03101 [hep-th]. – 65 –

View on Pith Pith review arXiv 2017

[41] [41]

P. S. Aspinwall, “K3 surfaces and string duality,”arXiv:hep-th/9611137

View on Pith Pith review arXiv

[42] [42]

J. McOrist and I. V. Melnikov, “Old issues and linear sigma models,”Adv.Theor.Math.Phys.16 (2012) 251–288,arXiv:1103.1322 [hep-th]

View on Pith Pith review arXiv 2012

[43] [43]

E. Witten, “Elliptic genera and quantum field theory,”Commun.Math.Phys.109(1987) 525

View on Pith 1987

[44] [44]

T. Kawai, Y. Yamada, and S.-K. Yang, “Elliptic genera and N=2 superconformal field theory,” Nucl.Phys.B414(1994) 191–212,arXiv:hep-th/9306096 [hep-th]

View on Pith Pith review arXiv 1994

[45] [45]

D. Israel and M. Sarkis, “New supersymmetric index of heterotic compactifications with torsion,”JHEP12(2015) 069,arXiv:1509.05704 [hep-th]

View on Pith Pith review arXiv 2015

[46] [46]

D. Israel and M. Sarkis, “Dressed elliptic genus of heterotic compactifications with torsion and general bundles,”JHEP08(2016) 176,arXiv:1606.08982 [hep-th]

View on Pith Pith review arXiv 2016

[47] [47]

I. V. Melnikov and R. Minasian, “Heterotic sigma models with N=2 space-time supersymmetry,”JHEP1109(2011) 065,arXiv:1010.5365 [hep-th]

View on Pith Pith review arXiv 2011

[48] [48]

T. Banks and L. J. Dixon, “Constraints on string vacua with space-time supersymmetry,”Nucl. Phys.B307(1988) 93–108

View on Pith 1988

[49] [49]

K. S. Narain, M. H. Sarmadi, and C. Vafa, “Asymmetric Orbifolds,”Nucl. Phys. B288(1987) 551

View on Pith 1987

[50] [50]

D. Israel, I. V. Melnikov, and Y. Proto, “Shift orbifolds, decompactification limits, and lattices,”SciPost Phys.20no. 2, (2026) 039,arXiv:2502.18453 [hep-th]

View on Pith Pith review arXiv 2026

[51] [51]

J. Fuchs, “Simple WZW currents,”Commun. Math. Phys.136(1991) 345–356

View on Pith 1991

[52] [52]

G. Lockhart, L. Novelli, and Y. Proto, “String probes, simple currents, and the no global symmetries conjecture,”arXiv:2605.12594 [hep-th]

View on Pith Pith review arXiv

[53] [53]

V. A. Fateev and A. B. Zamolodchikov, “Parafermionic Currents in the Two-Dimensional Conformal Quantum Field Theory and Selfdual Critical Points in Z(n) Invariant Statistical Systems,”Sov. Phys. JETP62(1985) 215–225

View on Pith 1985

[54] [54]

Z.-a. Qiu, “Nonlocal Current Algebra andN= 2 Superconformal Field Theory in Two-dimensions,”Phys. Lett. B188(1987) 207–213

View on Pith 1987

[55] [55]

D. Gepner, “Space-Time Supersymmetry in Compactified String Theory and Superconformal Models,”Nucl. Phys.B296(1988) 757

View on Pith 1988

[56] [56]

M. Bertolini, I. V. Melnikov, and M. R. Plesser, “Accidents in (0,2) Landau-Ginzburg theories,” JHEP12(2014) 157,arXiv:1405.4266 [hep-th]

View on Pith Pith review arXiv 2014

[57] [57]

P. C. Argyres, M. R. Plesser, and N. Seiberg, “The Moduli space of vacua of N=2 SUSY QCD and duality in N=1 SUSY QCD,”Nucl.Phys.B471(1996) 159–194,arXiv:hep-th/9603042 [hep-th]

View on Pith Pith review arXiv 1996

[58] [58]

P. C. Argyres, M. R. Plesser, and A. D. Shapere, “N=2 moduli spaces and N=1 dualities for SO(n(c)) and USp(2n(c)) superQCD,”Nucl.Phys.B483(1997) 172–186, arXiv:hep-th/9608129 [hep-th]

View on Pith Pith review arXiv 1997

[59] [59]

M. Jardim, G. Menet, D. M. Prata, and H. N. Sa Earp, “Holomorphic bundles for higher dimensional gauge theory,”Bull. London Math. Soc.49no. 1, (2017) 117–132, arXiv:1109.2750 [math.AG]. – 66 –

View on Pith Pith review arXiv 2017

[60] [60]

S. Kachru and E. Witten, “Computing the complete massless spectrum of a Landau- Ginzburg orbifold,”Nucl. Phys.B407(1993) 637–666,arXiv:hep-th/9307038

View on Pith Pith review arXiv 1993

[61] [61]

M. Becker, L.-S. Tseng, and S.-T. Yau, “New Heterotic Non-Kahler Geometries,” arXiv:0807.0827 [hep-th]

View on Pith Pith review arXiv

[62] [62]

S. Ferrara, J. A. Harvey, A. Strominger, and C. Vafa, “Second quantized mirror symmetry,” Phys.Lett.B361(1995) 59–65,arXiv:hep-th/9505162 [hep-th]

View on Pith Pith review arXiv 1995

[63] [63]

D. Israel and Y. Proto, “A Worldsheet Approach toN= 1 Heterotic Flux Backgrounds,” arXiv:2302.01889 [hep-th]

View on Pith Pith review arXiv

[64] [64]

M. Maio and A. N. Schellekens, “Permutation Orbifold of N=2 Supersymmetric Minimal Models,”Nucl. Phys. B845(2011) 212–245,arXiv:1011.0934 [hep-th]

View on Pith Pith review arXiv 2011

[65] [65]

D. Isra¨ el, “T-Duality in Gauged Linear Sigma-Models with Torsion,”JHEP11(2013) 093, arXiv:1306.6609 [hep-th]. – 67 –

View on Pith Pith review arXiv 2013