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Numerical invariants of normed matrix factorizations

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Pith's one-line read For spherical normed Calabi-Yau objects, bounding cochains are classified by one trace value, making the new numerical invariants well defined.

desk verdict New framework and a clean n=1 proof, but Theorem 11 misstates its own target: the bijection lands in F^0(R_s)_{1-n}, not all of (R_s)_{1-n}. read the letter →

arxiv 2412.04437 v1 pith:AUU2OBVH submitted 2024-12-05 math.SG hep-thmath.AG

May Sela , Jake P. Solomon This is my paper · ORCID
classification math.SGhep-thmath.AG MSC 53D3714J33
keywords mirrorsymmetryopenGromov-WitteninvariantsWelschingermatrixfactorizationnon-ArchimedeannormCalabi-Yaustructureboundingcochainstoricgeometry
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 introduces a category of matrix factorizations in which objects and morphism spaces carry non-Archimedean valuations, and uses it to build an algebraic mirror for a real Lagrangian sphere inside a Fano toric manifold. The central claim is that for a spherical normed Calabi-Yau object $M$, the assignment $\varrho([b])=\tau(\operatorname{gexp}(b))$ bijects gauge-equivalence classes of bounding cochains with the degree $1-n$ part of the base Novikov field; this classification makes the numerical invariants $N_{d,k}$ extracted from a point-like bounding cochain independent of all choices. For the $n$-simplex with $n$ odd, the Dirac factorization is shown to be a spherical normed Calabi-Yau object, mirroring $\mathbb{R}P^n \subset \mathbb{C}P^n$. For $n=1$ the invariants are computed exactly and coincide with the open Gromov-Witten invariants of $\mathbb{R}P^1 \subset \mathbb{C}P^1$; for $n=3$ low-degree computations agree with the real enumerative invariants of $\mathbb{R}P^3 \subset \mathbb{C}P^3$. If the construction works, it provides the missing algebraic side of mirror symmetry for real Lagrangians even when the Lagrangian is zero in the ordinary characteristic-zero Lagrangian category.

What carries the argument

The load-bearing object is the normed matrix factorization category $\operatorname{MF}(W,w)$: a DG category whose objects are $\mathbb{Z}$-graded free $S$-modules $M$ with an endomorphism $D_M$ squaring to the superpotential shift $W-w$, together with a family of valuations, and whose morphism complexes carry the induced operator norm. The normed $\infty$-trace $\Theta$, a chain-level Calabi-Yau structure defined by residue and supertrace formulas on the cyclic complex, is what upgrades the category from algebraic to enumerative: it supplies the map $\tau(\operatorname{gexp}(b))$ that classifies bounding cochains and defines the invariants. A point-like bounding cochain is a Maurer-Cartan solution $b\in \operatorname{End}(M)_s$ normalised by $\tau(\operatorname{gexp}(b))=s$; Theorem 11 says its gauge class is unique, so the coefficients $N_{d,k}$ of its potential are canonical. The Dirac matrix factorization $M_{\triangle}$ is the Clifford-algebra quotient $\operatorname{Cl}(\triangle)/\langle e_0\cdots e_n - T^{1/2}\rangle$, constructed from the polytope $\triangle$ together with a combinatorial relative spin structure; it is the proposed algebraic counterpart of the real locus of the associated toric manifold.

What would settle it

Recompute the $n=3$ residues with an independent implementation: the values of Section 1.3.3 must come out as $\Theta_2(b^{\otimes 2}/2)\equiv s^5T/8$, $\Theta_3(b^{\otimes 3}/3)\equiv s^5T/24$, $\Theta_4(b^{\otimes 4}/4)\equiv -3s^5T/10$, $\Theta_5(b^{\otimes 5}/5)\equiv s^5T/6$ modulo $F^6$, and the corresponding three $\Theta$-contributions from $y_{b_{(5)}}$ must be $s^5T/8$, $-11s^5T/30$, $4s^5T/15$; then $\alpha_5=s^5T^{1/2}z_1z_2z_3/120$ follows, and the invariants must be $N_{1/2,2}=-2$, $N_{3/2,6}=2$, with all others zero for $d+k\leq 7$. Any deviation would falsify the normed-trace proof or the advertised matching with real enumerative invariants.

Watch

Extended reading notes

Core claim

The paper's main result is Theorem 11: if $M$ is a spherical object that is normed Calabi-Yau, the map $\varrho: x_M(\operatorname{End}(M)_s)/{\sim} \to (R_s)_{1-n}$, $[b] \mapsto \tau(\operatorname{gexp}(b))$, is bijective. In words, the only invariant of a bounding cochain up to gauge equivalence is the single element $\tau(\operatorname{gexp}(b))$, obtained by applying the normed $\infty$-trace to the cyclic-chain element $\operatorname{gexp}(b)$ formed from the exponential of $b$ after subtracting a canonical correction. Because a point-like bounding cochain is defined by the normalisation $\tau(\operatorname{gexp}(b))=s$, the potential $c=\sum_{k,d} N_{d,k+1} \frac{s^k}{k!}T^d$ of that cochain yields well-defined numerical invariants $N_{d,k}$ (Definition 13). On the toric side, the paper constructs a valued Landau-Ginzburg model for every Delzant polytope and, under a combinatorial relative spin condition, a Dirac factorization; for the odd $n$-simplex this object is spherical, with $H^*(\operatorname{End}(M_{\triangle_n})_0) \cong H^*(\mathbb{R}P^n;(R_{\triangle_n})_0)$, and the $\infty$-trace $\Theta$ is normed. For $\triangle_1$, the invariants are $N_{1/2,2}=2$ and all others zero, matching the open Gromov-Witten invariants of $\mathbb{R}P^1 \subset \mathbb{C}P^1$; for $\triangle_3$, low-degree computer calculations give $N_{1/2,2}=-2$, $N_{3/2,6}=2$ with all other $N_{d,k}$ vanishing for $d+k \leq 7$, matching the real enumerative invariants of $\mathbb{R}P^3 \subset \mathbb{C}P^3$ up to a factor of 2.

Load-bearing premise

The construction of point-like bounding cochains and of the invariants $N_{d,k}$ depends on the intricate residue estimates (Lemmas 6.9-6.12) proving that the $\infty$-trace $\Theta$ on the projective-space Landau-Ginzburg model is normed; if any sign or convergence check is wrong, the normed Calabi-Yau structure, the sphere-cohomology computation, and the numerical invariants all collapse, and the $n=3$ computer verification is a further load-bearing premise with no code or raw data supplied.

Editorial extensions

If this is right

  • For every spherical normed Calabi-Yau object, bounding cochains up to gauge equivalence are parameterised by the base field: the map $[b] \mapsto \tau(\operatorname{gexp}(b))$ is a bijection, so the numerical invariants of Definition 13 do not depend on any choice.
  • For each odd $n$, the Dirac factorization of the $n$-simplex is a spherical normed Calabi-Yau object with $H^*(\operatorname{End}(M_{\triangle_n})_0) \cong H^*(\mathbb{R}P^n;(R_{\triangle_n})_0)$, realising the mirror of $\mathbb{R}P^n \subset \mathbb{C}P^n$ at the cohomological level.
  • For $n=1$, the invariants are $N_{1/2,2}=2$ and all others zero, exactly the open Gromov-Witten invariants of $\mathbb{R}P^1 \subset \mathbb{C}P^1$.
  • For $n=3$, the low-degree invariants $N_{1/2,2}=-2$, $N_{3/2,6}=2$, and $N_{d,k}=0$ for $d+k\leq 7$ coincide with the real enumerative invariants of $\mathbb{R}P^3\subset\mathbb{C}P^3$ up to a factor of 2.
  • These statements hold over a Novikov ring, so the invariants remain non-trivial even though $\mathbb{R}P^n$ is the zero object in the ordinary characteristic-zero Lagrangian category.

Reading between the lines

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

  • Editorial inference: the classification mechanism should produce point-like bounding cochains and numerical invariants for other Fano toric pairs with spherical real loci, since the proof of Theorem 11 uses only sphericity and the normed Calabi-Yau property; testing Conjecture 19 on the next odd simplex $n=5$ would separate the general method from the specific residue computations.
  • Editorial inference: the six residue values listed for the $n=3$ computation (for example $\Theta_4(b^{\otimes 4}/4)\equiv -3s^5T/10$ modulo $F^6$) are checkable by an independent implementation, and a mismatch would localise the error to the canonical choice of $y_b$ or to the residue bounds, not to the classification theorem.
  • Editorial inference: because the normed structure is valued in $\frac{1}{n+1}\mathbb{Z}$, the invariants may admit a combinatorial reading in terms of polytope data that the paper does not pursue.
  • Editorial inference: the exceptional nontrivial background class chosen for $\triangle_1$, whereas higher simplices use the tautological class, suggests that the sign conventions of the invariants are tied to the combinatorial relative spin structure; varying that structure is a concrete test of the factor of 2 appearing in the $\mathbb{R}P^3$ comparison.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces a normed DG category of matrix factorizations over a Novikov field, defines a notion of bounding cochain for spherical normed Calabi-Yau objects, and proves a classification theorem (Theorem 11) asserting that gauge-equivalence classes of bounding cochains are in bijection with the degree-(1-n) part of R_s. The numerical invariants extracted from a point-like bounding cochain are then matched, for the simplex Delta_n with n odd, against open Gromov-Witten invariants: exactly for n=1 (Theorem 20) and by computer calculation for n=3 in low degrees (Section 1.3.3). The construction passes through a toric valued Landau-Ginzburg model, the Dirac factorization M_n, proofs of sphericity and of the normed property of Shklyarov's infinity-trace, and a detailed inductive construction of bounding cochains.

Significance. If the classification theorem is corrected as described below, this is a substantial piece of mirror-symmetry infrastructure: it gives an algebraic setting in which one can talk about point-like bounding cochains and numerical invariants for objects that are invisible in the characteristic-zero Fukaya category, and the n=1 computation is a complete, checkable match with the open Gromov-Witten invariants of RP^1 in CP^1. The toric construction of valued Landau-Ginzburg models and the Dirac factorization are natural and likely to be reused. However, the paper's central theorem as stated has a false codomain, and the advertised n=3 evidence is not reproducible from the text.

major comments (3)
  1. [Section 1.2.4, Theorem 11; Section 5.2, Proposition 5.4; Section 5.4, Proposition 5.19] Theorem 11 states that the map from gauge-equivalence classes of bounding cochains to the full degree-(1-n) part of R_s is bijective, but the proof only treats values in the positive-valuation subspace F^0 R_s. Proposition 5.4 constructs bounding cochains only for r in F^0 R_s with |r| = 1-n, and Proposition 5.19 only compares bounding cochains with a common value r in F^0 R_s. The stated codomain is strictly larger: for n=3, the element T^{-1/4} has degree 2(n+1)(-1/4) = -2 = 1-n and valuation -1/4 < 0. Meanwhile every bounding cochain b satisfies nu_{M,s}(b) > 0, and since tau is normed, nu_R(tau(gexp(b))) >= nu(gexp(b)) > 0. Hence T^{-1/4} lies in the stated codomain but not in the image. The correct target is F^0 (R_s)_{1-n} (indeed the image has positive valuation). The classification theorem, the surrounding statements such as Definition 13's point-like bounding cochain, and Propositions 5.4 and 5.19 need to be restated consistently with this corrected target. The point-like value s lies in F^0 R_s, so the numerical invariants themselves are not destroyed, but Theorem 11 as written is false.
  2. [Section 1.3.3 and Section 1.2.7, Theorem 20] The n=3 agreement with open Gromov-Witten-Welschinger invariants is a central advertised claim, but it rests entirely on computer calculations for which no code, scripts, or raw residue data are provided. The text reports only final assembled values, e.g. N_{1/2,2} = -2 and N_{3/2,6} = 2 in Equation (9), after mentioning 345,912 residues appearing at m=7. This is not independently verifiable. In addition, the sign of the first value is not discussed in relation to the claimed coincidence with Welschinger invariants, and it differs from the n=1 value N_{1/2,2} = 2 of Corollary 23 without explanation. Please supply a reproducible computation, including code or detailed residue tables, and state explicitly, with signs, the OGW/Welschinger values being matched.
  3. [Section 6.3.3, Lemmas 6.9-6.12 and Corollary 6.10] The proof that Theta is normed reduces to residue estimates whose boundary cases need justification. In Corollary 6.10, the assertion that the assumption sum_j m_j >= (n+1)(n+l-1)-n implies 'not all m_j are negative' is not immediate from the inequality alone, and the subsequent identification of the sum over critical points of Res_x(g_vec m) with -Res_infinity(g_vec m) requires ruling out residues at coordinate hyperplanes z_j=0, which can appear when some m_j<0. Since Theorem 15 and hence Theorem 16 rely on this normedness, please either prove these two facts explicitly or reformulate the argument so that the pole structure at the coordinate hyperplanes is controlled.
minor comments (4)
  1. [Section 1.4, Outline] The sentence 'In Section 6.5 we prove Proposition 5.4 which yields the numerical invariants of M_n for n=1' refers to the wrong statement; it should refer to Proposition 22 (or Corollary 23), not Proposition 5.4.
  2. [Section 5.2, proof of Proposition 5.4] Several congruences in the inductive step are written modulo F^{E_l} End(M)_s when the quantity is a scalar in R_s; they should be modulo F^{E_l} R_s, e.g. the line containing tau(gexp(b_(l))) = r.
  3. [Section 6.3, first paragraph] The reference to 'Lemma 4.2' for the statement that Theta is an infinity-trace should be to Theorem 4.2.
  4. [Section 1.3.3, Equation (9)] Equation (9) reads N_{1/2,2} = -2, N_{3/2,6} = 2, N_{d,k} = 0, d+k <= 7; since 1/2+2 and 3/2+6 are not greater than 7, the last clause should be 'for all other (d,k) with d+k <= 7'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical invariants are computed from an internal point-like cochain construction and then matched to independently defined OGW/Welschinger invariants.

full rationale

The paper's central invariants are defined internally: a point-like bounding cochain is characterized by tau(gexp(b)) = s, existence is proved in Proposition 5.4 using the normed Calabi-Yau structure, and the invariants N_{d,k} are read off from the associated potential c in Definition 13. The normed infinity-trace Theta is Shklyarov's external construction, and the property that Theta is normed is proved by residue estimates in Section 6.3.3 rather than assumed. The n = 1 match with open Gromov-Witten invariants is a genuine comparison: Proposition 22 computes an explicit bounding cochain b and potential c for the matrix factorization, Corollary 23 extracts N_{1/2,2} = 2, and Theorem 20 independently computes the OGW invariants of RP^1 from the top-degree axiom and the superpotential. No parameter is fitted to the OGW numbers. The n = 3 computer verification is presented as evidence, not as a derivation from the target invariants; the absence of code or raw residue data is a reproducibility concern, not a circularity. Self-citations to Solomon-Tukachinsky motivate terminology and the mirror expectation, but Theorem 11 is proved in Section 5 rather than imported from [22], so the classification does not reduce to a self-citation. The gap noted by a skeptic, namely that bounding cochains constructed in Section 5 land in F^0(R_s) rather than all of (R_s)_{1-n}, would be a correctness issue in the stated codomain of Theorem 11, not a circularity: the point-like construction with r = s uses s in F^0(R_s), so Definition 13 and the numerical comparisons are unaffected unless the proof itself fails. Overall, the derivation chain does not identify any equation in which an input is re-labeled as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central claim rests on standard results in matrix factorization and symplectic geometry, plus domain assumptions in the definition of the normed category. No free parameters are fitted to data: the invariants are computed from a point-like bounding cochain fixed by τ(gexp(b)) = s, so no constant is adjusted to force agreement with OGW invariants. The only ad hoc input is the correctness of the unshipped n=3 computer calculations.

assumptions (5)
  • standard math Orlov's equivalence D^b_sing ≅ H^0 MF(W) and Shklyarov's Θ is a chain-level Calabi-Yau structure (Theorem 4.2).
    Invoked in Sections 1.1 and 4.1; the paper extends Θ to the normed setting.
  • standard math Solomon-Tukachinsky classification of bounding cochains for Lagrangian rational cohomology spheres (Theorem 1 of [22]).
    Used as the symplectic-side mirror that Theorem 11 mirrors; defines the OGW invariants to which the paper compares.
  • domain assumption Lemma 3.12: Im νM1,M2 = Im νR for objects of MF(W,w), which requires Schauder basis valuations to limit at polytope vertices to values in Im νR.
    This equality is used in Theorem 15 and Proposition 5.4; it is part of the definition of the valued Landau-Ginzburg model.
  • standard math Clifford algebra CSGA facts (Lam) and the identification Cl^+_even ⊗ k ≅ Cl(k^{n-1}) (Lemma 2.15).
    Used in Lemma 6.6 to prove the Dirac factorization is spherical.
  • ad hoc to paper The computer calculations in Section 1.3.3 are correct.
    The n=3 invariants N_{1/2,2}=-2 and N_{3/2,6}=2 are presented as verified by computation, but no code is shipped.
invented entities (3)
  • Normed matrix factorization category MF(W,w) independent evidence
    purpose: An algebraic category with non-Archimedean norms mirroring the Fukaya category over a Novikov ring.
    Supported by the n=1 theorem match and by the normed ∞-trace Theorems 15-16.
  • Dirac factorization M_Δ independent evidence
    purpose: Object mirroring the real locus X^R_Δ; shown spherical for Δ_n, n odd.
    Theorem 18 gives H^*(End(M_Δn)_0) ≅ H^*(RP^n; R_0), matching a known symplectic fact; n=1 invariants match OGW.
  • Normed ∞-trace / normed Calabi-Yau structure independent evidence
    purpose: Refined trace playing the role of the integral over the Lagrangian in the classification of bounding cochains.
    Proved normed for CP^n in Theorem 15; determines the numerical invariants and mirrors the symplectic top-degree axiom.

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Pith. "Pith review of Numerical invariants of normed matrix factorizations." pith.science (2026). https://pith.science/paper/AUU2OBVH

@misc{pith2026241204437,
  author       = {Pith},
  title        = {Pith review of: Numerical invariants of normed matrix factorizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUU2OBVH}},
  note         = {Machine review of arXiv:2412.04437}
}
abstract

We define a normed matrix factorization category and a notion of bounding cochains for objects of this category. We classify bounding cochains up to gauge equivalence for spherical objects and use this classification to define numerical invariants. These invariants are expected to correspond under mirror symmetry to the open Gromov-Witten invariants with only boundary constraints of Lagrangian rational cohomology spheres defined by the second author and Tukachinsky. For each Delzant polytope, we construct a normed matrix factorization category. For Delzant polytopes satisfying a combinatorial relative spin condition, we construct an object of this category called the Dirac factorization. The Dirac factorization is expected to correspond under mirror symmetry to the Lagrangian submanifold given by the real locus of the toric symplectic manifold associated to the Delzant polytope. In the case of the $n$-simplex for $n$ odd, we show that the Dirac factorization is spherical, mirroring the fact that $\mathbb{R} P^n$ is a rational cohomology sphere. For $n = 1,$ we show the numerical invariants of the Dirac factorization coincide with the open Gromov-Witten invariants of $\mathbb{R} P^1 \subset \mathbb{C} P^1.$ For $n = 3$ in low degrees, computer calculations verify that the numerical invariants of the Dirac factorization coincide with the open Gromov-Witten-Welschinger invariants of $\mathbb{R}P^3 \subset \mathbb{C} P^3.$ Although $\mathbb{R} P^n$ is trivial in the Fukaya category of $\mathbb{C} P^n$ over any field of characteristic zero, the above results can be seen as a manifestation of mirror symmetry over a Novikov ring.

Figures

Figures reproduced from arXiv: 2412.04437 by the authors.

Figure 1
Figure 1. The constants of level 1, 2, 3, 4. Let E, E′ ∈ R>0 such that E ′ > E. Let b ∈ Mx(A, c)E and q ∈ (A) 2 satisfying ν(q) ≥ E ′ and dA(b) − b 2 ≡ c · 1A + q (mod F E′ A). Assume there is a projection π : A → R such that π(b) = 0. Lemma 4.21. dA(q) ≡ 0 (mod F E′ A). Proof. 0 = −dA(b 2 ) − bdA(b) + dA(b)b = dA(dA(b) − b 2 ) − b(dA(b) − b 2 ) + (dA(b) − b 2 )b ≡ dA(c · 1A + q) − b(c · 1A + q) + (c · 1A + q)b ≡ dA(q) − bq +… view at source ↗

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Works this paper leans on

26 extracted references · 15 canonical work pages

  1. [1]

    Auroux, Mirror symmetry and T -duality in the complement of an anticanonical divisor , J

    D. Auroux, Mirror symmetry and T -duality in the complement of an anticanonical divisor , J. G¨ okova Geom. Topol. GGT 1 (2007), 51–91

    work page 2007

  2. [2]

    Cannas da Silva, Symplectic toric manifolds, Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001), Adv

    A. Cannas da Silva, Symplectic toric manifolds, Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001), Adv. Courses Math. CRM Barcelona, Birkh¨ auser, Basel, 2003, pp. 85–173

    work page 2001

  3. [3]

    Cho, Counting real J-holomorphic discs and spheres in dimension four and six , J

    C.-H. Cho, Counting real J-holomorphic discs and spheres in dimension four and six , J. Korean Math. Soc. 45 (2008), no. 5, 1427–1442, doi:10.4134/JKMS.2008.45.5.1427

    work page doi:10.4134/jkms.2008.45.5.1427 2008

  4. [4]

    Delzant, Hamiltoniens p´ eriodiques et images convexes de l’application moment , Bull

    T. Delzant, Hamiltoniens p´ eriodiques et images convexes de l’application moment , Bull. Soc. Math. France 116 (1988), no. 3, 315–339

    work page 1988

  5. [5]

    Eisenbud, Homological algebra on a complete intersection, with an application to group representa- tions, Trans

    D. Eisenbud, Homological algebra on a complete intersection, with an application to group representa- tions, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64, doi:10.2307/1999875

    doi:10.2307/1999875 1980

  6. [6]

    Fukaya, Y.-G

    K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and ob- struction. Parts I and II , AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009, doi:10.1090/crmp/049/07. 81

    doi:10.1090/crmp/049/07 2009

  7. [7]

    Georgieva, Open Gromov-Witten disk invariants in the presence of an anti-symplectic involution , Adv

    P. Georgieva, Open Gromov-Witten disk invariants in the presence of an anti-symplectic involution , Adv. Math. 301 (2016), 116–160, doi:10.1016/j.aim.2016.06.009

    work page doi:10.1016/j.aim.2016.06.009 2016

  8. [8]

    Giterman, Generalized superpotential, to appear

    P. Giterman, Generalized superpotential, to appear

    work page

Show all 26 references
  1. [9]

    Hori and C

    K. Hori and C. Vafa, Mirror symmetry, (2000), arXiv:hep-th/0002222

    2000 arXiv

  2. [10]

    Kapustin and Y

    A. Kapustin and Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry , Journal of High Energy Physics 2003 (2003), no. 12, 005–005, doi:10.1088/1126-6708/2003/12/005

    2003 doi

  3. [11]

    Kapustin and Y

    A. Kapustin and Y. Li, Topological correlators in Landau-Ginzburg models with boundaries, Adv. Theor. Math. Phys. 7 (2003), no. 4, 727–749, arXiv:hep-th/0305136, doi:10.4310/ATMP.2003.v7.n4.a5

    2003 arXiv

  4. [12]

    Kontsevich and Y

    M. Kontsevich and Y. Soibelman, Notes on A∞-algebras, A∞-categories and non-commutative geometry, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 153–219

    2009

  5. [13]

    Kontsevich, Homological algebra of mirror symmetry , Proceedings of the International Congress of Mathematicians, Vol

    M. Kontsevich, Homological algebra of mirror symmetry , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), Birkh¨ auser, Basel, 1995, pp. 120–139

    1994

  6. [14]

    T. Y. Lam, Introduction to quadratic forms over fields , Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005, doi:10.1090/gsm/067

    2005 doi

  7. [15]

    Loday, Cyclic homology , second ed., Grundlehren der mathematischen Wissenschaften [Fun- damental Principles of Mathematical Sciences], vol

    J.-L. Loday, Cyclic homology , second ed., Grundlehren der mathematischen Wissenschaften [Fun- damental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by Mar ´ ıa O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Piras...

    1998 doi

  8. [16]

    J. W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds , Mathematical Notes, vol. 44, Princeton University Press, Princeton, NJ, 1996

    1996

  9. [17]

    D. R. Morrison and J. Walcher, D-branes and normal functions , arXiv: High Energy Physics - Theory (2007)

    2007

  10. [18]

    D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), 240–262

    2004

  11. [19]

    Pandharipande, J

    R. Pandharipande, J. Solomon, and J. Walcher, Disk enumeration on the quintic 3-fold , J. Amer. Math. Soc. 21 (2008), no. 4, 1169–1209, doi:10.1090/S0894-0347-08-00597-3

    2008 doi

  12. [20]

    Shklyarov, Calabi-Yau structures on categories of matrix factorizations , J

    D. Shklyarov, Calabi-Yau structures on categories of matrix factorizations , J. Geom. Phys. 119 (2017), 193–207, doi:10.1016/j.geomphys.2017.05.006

    2017 doi

  13. [21]

    J. P. Solomon, Intersection theory on the moduli space of holomorphic curves with lagrangian boundary conditions, 2006, arXiv:math/0606429

    2006 arXiv

  14. [22]

    J. P. Solomon and S. B. Tukachinsky, Point-like bounding chains in open Gromov-Witten theory , Geom. Funct. Anal. 31 (2021), no. 5, 1245–1320, doi:10.1007/s00039-021-00583-3

    2021 doi

  15. [23]

    J. P. Solomon and S. B. Tukachinsky, Relative quantum cohomology , Journal of the European Mathe- matical Society (2023), doi:10.4171/jems/1337

    2023 doi

  16. [24]

    Walcher, Opening mirror symmetry on the quintic , Comm

    J. Walcher, Opening mirror symmetry on the quintic , Comm. Math. Phys. 276 (2007), no. 3, 671–689, doi:10.1007/s00220-007-0354-8

    2007 doi

  17. [25]

    Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative ge- ometry, Invent

    J.-Y. Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative ge- ometry, Invent. Math. 162 (2005), no. 1, 195–234, doi:10.1007/s00222-005-0445-0

    2005 doi

  18. [26]

    Welschinger, Spinor states of real rational curves in real algebraic convex 3-manifolds and enu- merative invariants, Duke Math

    J.-Y. Welschinger, Spinor states of real rational curves in real algebraic convex 3-manifolds and enu- merative invariants, Duke Math. J. 127 (2005), no. 1, 89–121, doi:10.1215/S0012-7094-04-12713-7 . Institute of Mathematics, Hebrew University Email address : mayysela@gmail.c...

    2005 doi

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