The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers

Bertrand Eynard; Brad Safnuk; Motohico Mulase

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The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers

Bertrand Eynard, Motohico Mulase, Brad Safnuk

Algebraic Geometry · Submitted 29 Jul 2009 · 45 references

Pith's verdict: A genuine proof of Bouchard-Marino, technical but coherent; the only real defect I found is a typo in (2.13).

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CitationBertrand Eynard, Brad Safnuk, Motohico Mulase. "The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers." arXiv:0907.5224 (2009). https://pith.science/paper/LT75TISS

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The thing to know: this is the paper that actually proves Bouchard-Marino, and the proof is not a repackaging of the conjecture. The route is clean in outline — ELSV gives the Laplace transform of Hurwitz numbers, the cut-and-join equation gives the polynomial recursion (3.12) (from Mulase-Zhang, cited), and Sections 5–7 show that the direct image of (3.12) through the Lambert curve is exactly the Eynard-Orantin recursion (1.5). The conceptual payoff is that the Lambert curve, the deck transformation s(t), and the residue kernel all fall out of the Laplace transform rather than being postulated from topological strings. That is the real contribution and it justifies the length. The abstract's claim at [abstract:s2] is accurate as stated: the direct image of the Laplace-transformed equation is the conjectured recursion, not merely structurally similar to it. The parallel to Mirzakhani noted at [abstract:s1] is a genuine observation, not decoration — the topological structure of (1.2) really does match her recursion term by term, and that is what suggested the direct-image route in the first place. What is solid: the residue computations in Theorem 5.2 and Propositions 6.2–6.4 are done explicitly, not waved at. The identification of ξ_n with the Bouchard-Marino ζ_n forms is exact, and the polynomial nature of ξ̂_n(t) is what makes the residues computable — the paper says this and it is true. The ℓ=1 case (Prop 3.4, Cor 7.2) is a clean warm-up that shows the machinery works before the general direct-image step. Soft spots, in proportion: Section 7's 'modulo terms with holomorphic factors in v_k' is the load-bearing step and it is dense. I checked it and the parity argument does hold — F_n(w) is even in v, so under π_* f(v) = f(v)+f(-v) the holomorphic pieces drop out of the principal part, and Lemma 6.5 handles the kernel difference. The paper's phrasing is looser than the actual argument, but the argument is there. Proposition 6.2's proof is compressed at 'starting from a large n and using the recursion backwards'; standard asymptotic analysis, but a reader who wants to check it line by line will have to fill in. The appendix tables have no accompanying code, so they are illustrative rather than reproducible — minor, since they are not load-bearing. One genuine defect: (2.13) writes ξ̂_{-1}(t) = t - 1/t = y, but the y-coordinate is 1 - 1/t. Remark 3.5 and (6.7) both use the correct value, so the slip is isolated to the introduction and (2.13) and does not propagate. Worth fixing in a revision. Who it is for: anyone working on Hurwitz numbers, topological recursion, or the Lambert curve. It deserves a serious referee — the technical density is exactly the kind of thing that needs a careful second reader, but the structure is sound and the result is real. Recommendation: engage with it, cite it, and referee it properly.

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Major objections (2)

[§3, Theorem 3.7, and §1, Eq. (1.2)] The text states that (3.12) 'is equivalent to (1.2) after expanding the generating functions using (2.14)'. As printed, the i<j terms of the two equations have visibly different prefactors: (3.12) carries t_i t_j/(t_i-t_j) multiplying derivatives of Ĥ_{g,l-1}, while (1.2) carries t_i^2 and t_j^2 multiplying ξ̂_{m+1}ξ̂_0. Since (1.2) is stated as Theorem 1.1 and is the equation compared to Mirzakhani's recursion, the equivalence should be checkable by the reader. Please verify the transcription of both equations and, if both are correct as printed, add the intermediate expansion that shows the equivalence. A concrete test is the (g,l)=(1,1) case, which is worked out independently in Proposition 3.4 and should be recovered from both (1.2) and (3.12).
[§7, passage from (7.3) to (7.4)] The reduction discards 'terms with holomorphic factors in v_k' and asserts that 'only those terms containing η_a(v_1)η_b(v_1) survive'. This step is load-bearing: it removes the F_n(w) terms of the decomposition (6.8) and is what makes the direct image equal to (1.5). The argument is correct — F_n(w) is holomorphic in w = v^2/2 and hence even in v, so under π_* f(v_1) = f(v_1)+f(-v_1) any term with an odd number of η's at v_1 vanishes, and any term with F_{n_1}(w_1) times an even product becomes holomorphic in v_1 — but the paper does not spell this out. Please add two or three sentences making the parity argument explicit, since the entire reduction to (1.5) rests on it.

Minor objections (8)

[(2.13) and the introduction] The paper writes ξ̂_{-1}(t) = t - 1/t = y. The y-coordinate of the Lambert curve with x = (1-1/t)e^{-(1-1/t)} is y = 1 - 1/t, and Remark 3.5 and (6.7) both use the correct value. The typo appears twice (introduction and (2.13)) and should be corrected.
[§6, Proposition 6.2] The proof of (6.5) uses the phrase 'starting from a large n and using the recursion backwards'. The inverse induction is standard but compressed; please expand it slightly, especially the passage from the asymptotic expansion (6.3) to the Laurent series (6.4).
[§6, Lemma 6.5] The argument that 'each coefficient does not contain a constant term because it is cancelled by taking the difference' is terse. Please spell out the cancellation, since the lemma is used in the proof of Proposition 6.4.
[Abstract] The abstract says 'the direct image of this Laplace transformed equation via the inverse of the Lambert W-function'. The direct image is taken via the projection π: C → C, not via the inverse Lambert W-function, which is the coordinate change w ↦ t. Please reword.
[§1] The narrative says 'the direct image of (1.2) then becomes identical to the conjectured formula (1.5)', but §7 starts from (3.12), not (1.2). Since the two are claimed equivalent, this is not an error, but the narrative should be made consistent.
[Appendix] The tables of linear Hodge integrals and Hurwitz numbers are presented without a statement of what checks were performed. Please state whether they were cross-checked against known values (e.g., the λ_g formula, or the Witten-Kontsevich values).
[§2, around (2.10)-(2.13)] The paper uses ξ_n, ξ̂_n, and η_n, related by dξ̂_n = ξ_n and (6.8). A short table of these relations near (2.10)-(2.13) would help the reader.
[§4, (4.15)] The notation 'dt⊗1/dz' in the expression for K is unclear; please write the kernel as a 1-form in dt tensored with a 1-form in dz.

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The paper

The paper claims

The paper proves that Hurwitz numbers — counts of ramified coverings of the sphere by a genus-$g$ curve with prescribed ramification — are generated by the Eynard–Orantin topological recursion on the Lambert curve $x = y e^{-y}$, confirming

The reading

The paper sets out to prove a conjecture of Bouchard and Mariño: that the generating functions of Hurwitz numbers are produced by the topological recursion of Eynard and Orantin applied to one specific plane curve, the Lambert curve $x = y e^{-y}$. The route is indirect. The authors take the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil — an identity relating the Hurwitz number of a given profile to those obtained by splitting one part or merging two — and show the transform is a polynomial recursion on primitives whose topological shape matches Mirzakhani's recursion for Weil–Petersson volumes of bordered hyperbolic surfaces. Passing to differential forms and pushing forward along the two-sheeted projection of the Lambert curve, an averaging over the deck involution $t$ sent to $s(t)$, turns that equation into exactly the recursion Bouchard and Mariño had written down from topological string theory. If the argument holds, a conjecture imported from the B-model is settled by combinatorics, and the Lambert curve's appearance is explained rather than postulated.

What carries the argument

The load-bearing objects are the Lambert curve $C = {(x,t) : x = frac{1}{e}(1 - frac{1}{t}) e^{1/t}}$ with its global coordinate $t$, the deck transformation $s(t)$ of the two-sheeted projection near the unique critical point $t = infty$, and the polynomials $hat xi_n(t)$ that arise as Laplace transforms of the terms $k^{k+n}/k!$. The mechanism that carries the proof is a direct image: because the Lambert curve is a double cover in the $x$-variable, the natural coordinate near the branch point is $v$ with $frac{1}{2}v^2 = w$, and the involution becomes $v$ sent to $-v$, so pushing a function forward along the projection is just keeping its even part, $f(v) + f(-v)$. Proposition 6.2 supplies the bridge, $eta_n(v) = frac{1}{2}(hat xi_n(t) - hat xi_n(s(t)))$ up to a holomorphic function of $w$, with $eta_{-1}(v) = frac{1}{2}(t - s(t))/(t s(t))$ playing the role of the recursion kernel denominator. Theorem 5.2 and Corollary 6.3 then convert the residue integrals of the topological recursion into these truncated polynomial expressions, which is what makes the comparison with the transformed cut-and-join equation possible at all.

Extended reading notes

Core claim

The central claim is Theorem 1.3, which is Conjecture 4.8 of Bouchard and Mariño: the symmetric differential forms $d^{tensor l} H_{g,l}(t_1,...,t_l) = sum over $n_1+...+n_l$ of $<tau_{n_1}...tau_{n_l} Lambda^vee_g(1)>$ times the tensor product of $xi_{n_i}(t_i)$ satisfy the Eynard–Orantin recursion (4.16) on the Lambert curve, with initial data $W_{0,3} = dt_1 dt_2 dt_3$ and $W_{1,1} = frac{1}{24}(t-1)(3t+1)dt$. The proof factors into two steps. First, the Laplace transform of the cut-and-join equation is computed explicitly and becomes a polynomial equation on primitives, Theorem 3.7, in which the differential operator $$t^{2}$(t-1) d/dt$ acts on polynomials $hat xi_n(t)$ defined by $hat xi_{n+1} = $t^{2}$(t-1) d hat xi_n/dt$ from $hat xi_0 = t-1$. Second, the exterior differential of that equation is pushed forward through the projection of the Lambert curve, and the residues in the recursion are replaced by an algebraic truncation to the polynomial part at the point $t$ and its Galois conjugate $s(t)$. The direct image is then identical, coefficient by coefficient, to the conjectured recursion.

Load-bearing premise

The reduction hinges on the principal-part extraction in Section 7, where terms carrying holomorphic factors in the $v$-variables are discarded on the grounds that only the $eta_{a+1}(v_1) eta_{b+1}(v_1)$ products survive; this rests on the decomposition of Proposition 6.2 and on Lemma 6.5, and if the holomorphic remainders could contribute after multiplication by $eta_{-1}(v)$ the passage to the recursion would not go through.

Editorial extensions

If this is right

The recursion gives an effective algorithm: linear Hodge integrals of complexity $2g-2+l$ are computed from those of complexity one less, and Hurwitz numbers follow through the ELSV formula.
Restricting the polynomial equation to its top-degree homogeneous terms recovers the Witten–Kontsevich theorem, that is, the Virasoro constraints on $psi$-class intersection numbers.
The lowest-degree homogeneous terms yield the coefficients of the $lambda_g$-formula, giving a combinatorial derivation of that Faber–Pandharipande statement.
The ingredients of topological recursion — the spectral curve, its critical point, the Cauchy differentiation kernel, and the residue calculus — acquire a combinatorial reading as Laplace-transformed cut-and-join operations.
The unstable geometries $(0,1)$ and $(0,2)$ are absorbed as initial data rather than exceptions: $W_{0,1} = 0$ and $W_{0,2}$ equal to the Bergman kernel, matching the logarithm of $(hat xi_{-1}(t_1) - hat xi_{-1}(t_2))/(x_1 - x_2)$ computed in Proposition 3.6.

Reading between the lines

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

The strategy — Laplace transform a combinatorial cut-and-join equation, then take a direct image along the spectral curve — looks reusable for other enumerative problems whose recursion comes from a join-and-cut operation, and the authors themselves ask what the inverse Laplace transform of a general topological recursion should be.
Because a matrix-model derivation of the same generating functions on the Lambert curve exists, the proof suggests the combinatorial and matrix-integral routes are two descriptions of one mechanism rather than independent coincidences.
A reader wanting a quick sanity test can compare the recursion's output against the appendix tables, for instance $langle tau_3 lambda_1 rangle_{2,1} = 1/480$ and $d hat H_{1,1} = frac{1}{24}(t-1)(3t+1)dt$, which are small enough to check by hand.
The recursion's right-hand side is not visibly symmetric in its arguments, and the paper notes symmetry follows from the geometry of the Hurwitz differential rather than from the recursion itself; this is the same situation as Mirzakhani's volume recursion, which suggests symmetry is a general structural feature of such recursions worth isolating.

Original abstract

We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Marino using topological string theory.

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The abstract, sentence by sentence

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

We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Marino using topological string theory.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil.	contribution 0.92	0.96	0.07	asserted 1.00
s1	The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces.	result 0.93	0.84	0.25	asserted 1.00
s2	We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Marino using topological string theory.	result 0.97	0.93	0.70	asserted 1.00

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Figures

**Figure 1.** 1: The Lambert curve C ⊂ C ∗ × C ∗ defined by x = ye−y . The coordinate w is the parameter of the Laplace transformation, which changes a function in positive integers to a complex analytic function in w. Recall the Stirling expansion e −k k k+n k! ∼ 1 √ 2π k n− 1 2 , which makes its Laplace transform a function of √ w. Note that the x-projection π of the Lambert curve (1.3) is locally a double-sheeted covering around its unique critical point (x = e −1 , y = 1). Therefore, the Laplace transform of the ELSV formula (1.1) naturally lives on the Lambert curve C rather than

**Figure 5.** 1: The contours of integration. γ∞ is the circle of a large radius, and γ[0,1] is the thin loop surrounding the closed interval [0, 1]. Definition 5.1. For a Laurent series P n∈Z ant n , we denote "X n∈Z ant n # + = X n≥0 ant n . Theorem 5.2. In terms of the primitives ˆξn(t), we have (5.1) Ra,b(t) = − 1 2πi I γ∞ K(t 0 , t)ξa(t 0 )ξb

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References

Works cited (45)

[38]

M. Mulase and N. Zhang, Polynomial recursion formula for linear Hodge integrals , Communications in Number Theory and Physics 4 No.2, 267–294 (2010)

View on Pith 2010
[1]

G. Borot, B. Eynard, M. Mulase and B. Safnuk, Hurwitz numbers, matrix models and topological recursion, to appear in Journal of Geometry and Physics (2011)

View on Pith 2011
[2]

V. Bouchard, A. Klemm, M. Mari˜ no, and S. Pasquetti, Remodeling the B-model , Commun. Math. Phys. 287, 117–178 (2008)

View on Pith 2008
[3]

V. Bouchard and M. Mari˜ no,Hurwitz numbers, matrix models and enumerative geometry , Proc. Symposia Pure Math. 78, 263–283 (2008). LAPLACE TRANSFORM OF HUR WITZ NUMBERS 33

View on Pith 2008
[4]

L. Chen, Y. Li, and K. Liu, Localization, Hurwitz numbers and the Witten conjecture , arXiv:math.AG/0609263 [math.AG] (2006)

View on Pith arXiv 2006
[5]

R. Dijkgraaf, Intersection Theory, Integrable Hierarchies and Topological Field Theory, New symmetry principles in quantum ﬁeld theory (Carg´ ese, 1991), 95–158, NATO Adv. Sci. Inst. Ser. B Phys., 295, Plenum, New York, 1992

View on Pith 1991
[6]

R. Dijkgraaf and C. Vafa, Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity, arXiv:0711.1932 [hep-th]

View on Pith Pith review arXiv 1932
[7]

R. Dijkgraaf, E. Verlinde, and H. Verlinde, Loop equations and Virasoro constraints in non-perturbative two- dimensional quantum gravity , Nucl. Phys. B348, 435–456 (1991)

View on Pith 1991
[8]

T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves , Invent. Math. 146, 297–327 (2001)

View on Pith 2001
[9]

B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, arXiv:hep-th/0407261

View on Pith Pith review arXiv
[10]

B. Eynard, Recursion between volumes of moduli spaces , arXiv:0706.4403 [math-ph]

View on Pith Pith review arXiv
[11]

B. Eynard, All order asymptotic expansion of large partitions , arXiv:0804.0381 [math-ph]

View on Pith Pith review arXiv
[12]

B. Eynard, M. Mari˜ no and N. Orantin,Holomorphic anomaly and matrix models, Journal of High Energy Physics 06 058, (2007) [arXiv:hep-th/0702110]

View on Pith Pith review arXiv 2007
[13]

B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Communications in Number Theory and Physics 1, 347–452 (2007)

View on Pith 2007
[14]

B. Eynard and N. Orantin, Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models , arXiv:0705.3600 [math-ph]

View on Pith Pith review arXiv
[15]

C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139, 173–199 (2000)

View on Pith 2000
[16]

C. Faber and R. Pandharipande, Hodge integrals, partition matrices, and the λg conjecture, Ann. of Math. 157, 97–124 (2003)

View on Pith 2003
[17]

I.P. Goulden and D.M. Jackson, Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. A.M.S., 125, 51–60 (1997)

View on Pith 1997
[18]

I.P. Goulden, D.M. Jackson and A. Vainshtein, The number of ramiﬁed coverings of the sphere by the torus and surfaces of higher genera , Ann. of Comb. 4, 27–46 (2000)

View on Pith 2000
[19]

I.P. Goulden, D.M. Jackson and R. Vakil, The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. London Math. Soc. 83:3, 563–581 (2001)

View on Pith 2001
[20]

I.P. Goulden, D.M. Jackson and R. Vakil, A short proof of the λg-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves , arXiv:math/0604297v1 [math.AG] (2006)

View on Pith Pith review arXiv 2006
[21]

T. Graber, R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization , Compositio Math. 135, 25–36 (2003)

View on Pith 2003
[22]

T. Graber, R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves , Duke Math. J. 130, 1–37 (2005)

View on Pith 2005
[23]

A. Hurwitz, ¨Uber Riemann’sche Fl¨ achen mit gegebene Verzweigungspunkten, Mathematische Annalen 39, 1–66 (1891)

View on Pith
[24]

M. Kazarian, KP hierarchy for Hodge integrals , arXiv:0809.3263

View on Pith Pith review arXiv
[25]

M. Kazarian, S. Lando, An algebro-geometric proof of Witten’s conjecture , J. Amer. Math. Soc. 20, 1079–1089 (2007)

View on Pith 2007
[26]

Y.S. Kim and K. Liu, A simple proof of Witten conjecture through localization , preprint arXiv:math/0508384 [math.AG] (2005)

View on Pith Pith review arXiv 2005
[27]

M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Commun. Math. Phys. 147, 1–23 (1992)

View on Pith 1992
[28]

A.M. Li, G. Zhao, and Q. Zheng, The number of ramiﬁed coverings of a Riemann surface by Riemann surface , Commun. Math. Phys. 213, 685–696 (2000)

View on Pith 2000
[29]

C.-C. M. Liu, Formulae of one-partition and two-partition Hodge Integrals, Geom. & Top. Monographs8, 105–128 (2006)

View on Pith 2006
[30]

C.-C. M. Liu, K. Liu, J. Zhou, A proof of a conjecture of Mari˜ no-Vafa on Hodge Integrals, J. Diﬀerential Geom. 65, no. 2, 289–340 (2003)

View on Pith 2003
[31]

K. Liu and H. Xu, A simple proof of Mirzakhani’s recursion formula of Weil–Petersson volumes, arXiv:0705.2086 [math.AG]

View on Pith Pith review arXiv
[32]

M. Mari˜ no,Chern-Simons theory, matrix models, and topological strings , Oxford University Press, 2005

View on Pith 2005
[33]

M. Mari˜ no,Open string amplitudes and large order behavior in topological string theory , Journal of High Energy Physics 03 060, (2008)

View on Pith 2008
[34]

A. Mironov and A. Morozov, Virasoro constraints for Kontsevich-Hurwitz partition function , arXiv:0807.2843

View on Pith Pith review arXiv
[35]

M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces , Invent. Math. 167, 179–222 (2007)

View on Pith 2007
[36]

M. Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves , J. Amer. Math. Soc. 20, 1–23 (2007)

View on Pith 2007
[37]

M. Mulase and B. Safnuk, Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy , Indian J. Math. 50, 189–228 (2008). 34 B. EYNARD, M. MULASE, AND B. SAFNUK

View on Pith 2008
[39]

A. Okounkov, Random matrices and random perputations , International Mathematics Research Notices 2000, 1043–1095 (2000)

View on Pith 2000
[40]

A. Okounkov, Toda equations for Hurwitz numbers , Math. Res. Lett. 7, 447 (2000) [arXiv:math.AG/0004128]

View on Pith arXiv 2000
[41]

A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, I , Proc. Symposia Pure Math. 80, 325–414 (2009)

View on Pith 2009
[42]

A. Okounkov and R. Pandharipande, The equivariant Gromov-Witten theory of P1, math.AG/0207233 (2002)

View on Pith arXiv 2002
[43]

R. Vakil, Harvard Thesis 1997

View on Pith 1997
[44]

E. Witten, Two dimensional gravity and intersection theory on moduli space , Surveys in Diﬀerential Geometry 1, 243–310 (1991)

View on Pith 1991
[45]

J. Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups , preprint, math.AG/0308024. Service de Physique Th´eorique de Saclay, Gif-sur-Yvette, F-91191 Cedex, France E-mail address: bertrand.eynard@cea.fr Department of Mathematics, University of California, Davis, CA 95616–8633 E-mail address: mulase@math.ucdavis.edu Department of Mathematics, Cent...

View on Pith arXiv

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Abstract loop equations, topological recursion, and applications
math-ph 2013-03 conditional novelty 7.0 of 10

A single abstract set of loop equations, solved by one universal topological recursion, is shown to govern hermitian matrix models, repulsive particle systems, loop-decorated random maps and large-N torus knot invariants.
The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
hep-th 2010-10 conditional novelty 5.0 of 10

Perturbative knot invariants and topological-recursion free energies agree to fourth order for two hyperbolic knot complements, but only after ad-hoc tuning of a constant in the Bergman kernel.
Local Mirror Symmetry for One-Legged Topological Vertex
math.AG 2009-10 accept novelty 5.0 of 10

The Bouchard-Marino conjecture for the framed one-legged topological vertex is proved by deriving the Eynard-Orantin recursion from the cut-and-join equation for triple Hodge integrals.

How the literature uses it

43 arXiv papers cite this in 104 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 · 69 sentences without a confident label

Builds on it 3 papers

We then give a rigorous proof of the recursion formula, generalizing the result of [ref] to the orbifold case.

Mirror symmetry for orbifold Hurwitz numbers · Overview · 1.00
The second is a generalization of the (now proved) Bouchard-Mariño conjecture [ref] relating Hurwitz numbers and the topological recusion procedure of Chekhov, Eynard, and Orantin [ref] , that we call the $r$-BM conjecture.

Equivalence of ELSV and Bouchard-Mari\~no conjectures for $r$-spin Hurwitz numbers · Introduction · 0.82
Généralisant [ref] , Zhou a montré dans [ref] qu'elles sont équivalentes aux équations de boucles pour la courbe spectrale miroir de $ℂ^3$, dont l'équation est $e^fy + e^x + e^y(f + 1) = 0$ (cf. [ref]).

Quelques probl\`emes de g\'eom\'etrie \'enum\'erative, de matrices al\'eatoires, d'int\'egrabilit\'e, \'etudi\'es via la g\'eometrie des surfaces de Riemann · Relations combinatoires et équations de boucles · 0.90

Uses its method or tool 7 papers

The starting point of our proof, as in [ref] , is the cut-and-join equation as suggested in [ref] .

Local Mirror Symmetry for One-Legged Topological Vertex · Introduction · 0.96
The proof is based on the idea of [ref] .

Mirror symmetry for orbifold Hurwitz numbers · Main results · 0.98
Based on the method in [ref] , two proofs, one by Chen [ref] and one by the author [ref] , have appeared for Bouchard-Mariño Conjecture for the case of $^3$ with one D-brane.

Open String Invariants and Mirror Curve of the Resolved Conifold · Introduction · 0.92

Compares with it 5 papers

After the completion of this work, there appears a paper [ref] which gets the same result by the method of [ref] .

Local Mirror Symmetry for One-Legged Topological Vertex · Introduction · 0.98
Let us stress that in [ref] this polynomiality was derived from the ELSV formula, while in the present paper we have it independently due to the results of Section [ref], as noted above. * The principal part of the pole of $H_g,n$ is odd with respect to the involution $$ on each $t_i$-line (as it follows from Lemma [ref]).

Polynomiality of Hurwitz numbers, Bouchard-Mari\~no conjecture, and a new proof of the ELSV formula · Reduction by symmetrization · 0.98
Unlike the Hurwitz number case studied in [ref] , the LHS of our Symmetrized Cut-Join Equation involves taking derivative with respect to the parameter $f$ that need more special treatment.

Bouchard-Klemm-Marino-Pasquetti Conjecture for $\mathbb{C}^3$ · The left hand side · 0.96

Cites it as background 12 papers

Eynard and Orantin have defined it axiomatically in the context of algebraic geometry [ref] , and in this form, it enjoys many interesting properties (symplectic invariance, special geometry, WDVV equations, …), and has appeared provably or experimentally in many problems of 2D enumerative geometry: the two hermitian matrix model [ref] and the chain of hermitian matrices [ref] , topological string theory and Gromov-Witten invariants [ref] , integrable systems [ref] , intersection numbers on the moduli space of curves [ref] , asymptotic of knot invariants [ref] , …

Formal multidimensional integrals, stuffed maps, and topological recursion · Problem and main results · 0.91
Subsequently, Bouchard and Mariño conjectured [ref] that single Hurwitz numbers are governed by the topological recursion of Chekhov, Eynard and Orantin [ref] ; the first complete proof of this conjecture was given by Eynard, Mulase and Safnuk [ref] .

Double Hurwitz numbers: polynomiality, topological recursion and intersection theory · Review of Hurwitz numbers · 0.93
Since 1990, other correspondences between the four sides A, B, C and D have been found, among them the simple Hurwitz numbers (A) are computed by topological recursion (conjectured in [ref] and proved in [ref] ) with the Lambert curve (B), related by the ELSV formula to the Hodge class [ref] (C) and determined by the Toda hierarchy [ref] (D).

Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers · Introduction · 0.89

What it cites · 6 sentences without a confident label

Compares with 2 papers

This derivation is in the same spirit as those found in [ref] , though the argument is much clearer due to the polynomial nature of our equation.

A simple proof of Witten conjecture through localization · The cut-and-join equation and its Laplace transform · 0.94
This derivation is in the same spirit as those found in [ref] , though the argument is much clearer due to the polynomial nature of our equation.

math.AG/0609263 · The cut-and-join equation and its Laplace transform · 0.94

Cites as background 11 papers

It has been noted in [ref] that the Airy curve $w = v^2$ is a universal object of the topological recursion for the case of a genus $0$ spectral curve with only one critical point.

All orders asymptotic expansion of large partitions · The Laplace transform of the ELSV formula · 0.89
It has been noted in [ref] that the Airy curve $w = v^2$ is a universal object of the topological recursion for the case of a genus $0$ spectral curve with only one critical point.

Recursion between Mumford volumes of moduli spaces · The Laplace transform of the ELSV formula · 0.89
It has been noted in [ref] that the Airy curve $w = v^2$ is a universal object of the topological recursion for the case of a genus $0$ spectral curve with only one critical point.

Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models · The Laplace transform of the ELSV formula · 0.89

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] [38]

M. Mulase and N. Zhang, Polynomial recursion formula for linear Hodge integrals , Communications in Number Theory and Physics 4 No.2, 267–294 (2010)

View on Pith 2010

[2] [1]

G. Borot, B. Eynard, M. Mulase and B. Safnuk, Hurwitz numbers, matrix models and topological recursion, to appear in Journal of Geometry and Physics (2011)

View on Pith 2011

[3] [2]

V. Bouchard, A. Klemm, M. Mari˜ no, and S. Pasquetti, Remodeling the B-model , Commun. Math. Phys. 287, 117–178 (2008)

View on Pith 2008

[4] [3]

V. Bouchard and M. Mari˜ no,Hurwitz numbers, matrix models and enumerative geometry , Proc. Symposia Pure Math. 78, 263–283 (2008). LAPLACE TRANSFORM OF HUR WITZ NUMBERS 33

View on Pith 2008

[5] [4]

L. Chen, Y. Li, and K. Liu, Localization, Hurwitz numbers and the Witten conjecture , arXiv:math.AG/0609263 [math.AG] (2006)

View on Pith arXiv 2006

[6] [5]

R. Dijkgraaf, Intersection Theory, Integrable Hierarchies and Topological Field Theory, New symmetry principles in quantum ﬁeld theory (Carg´ ese, 1991), 95–158, NATO Adv. Sci. Inst. Ser. B Phys., 295, Plenum, New York, 1992

View on Pith 1991

[7] [6]

R. Dijkgraaf and C. Vafa, Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity, arXiv:0711.1932 [hep-th]

View on Pith Pith review arXiv 1932

[8] [7]

R. Dijkgraaf, E. Verlinde, and H. Verlinde, Loop equations and Virasoro constraints in non-perturbative two- dimensional quantum gravity , Nucl. Phys. B348, 435–456 (1991)

View on Pith 1991

[9] [8]

T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves , Invent. Math. 146, 297–327 (2001)

View on Pith 2001

[10] [9]

B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, arXiv:hep-th/0407261

View on Pith Pith review arXiv

[11] [10]

B. Eynard, Recursion between volumes of moduli spaces , arXiv:0706.4403 [math-ph]

View on Pith Pith review arXiv

[12] [11]

B. Eynard, All order asymptotic expansion of large partitions , arXiv:0804.0381 [math-ph]

View on Pith Pith review arXiv

[13] [12]

B. Eynard, M. Mari˜ no and N. Orantin,Holomorphic anomaly and matrix models, Journal of High Energy Physics 06 058, (2007) [arXiv:hep-th/0702110]

View on Pith Pith review arXiv 2007

[14] [13]

B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Communications in Number Theory and Physics 1, 347–452 (2007)

View on Pith 2007

[15] [14]

B. Eynard and N. Orantin, Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models , arXiv:0705.3600 [math-ph]

View on Pith Pith review arXiv

[16] [15]

C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139, 173–199 (2000)

View on Pith 2000

[17] [16]

C. Faber and R. Pandharipande, Hodge integrals, partition matrices, and the λg conjecture, Ann. of Math. 157, 97–124 (2003)

View on Pith 2003

[18] [17]

I.P. Goulden and D.M. Jackson, Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. A.M.S., 125, 51–60 (1997)

View on Pith 1997

[19] [18]

I.P. Goulden, D.M. Jackson and A. Vainshtein, The number of ramiﬁed coverings of the sphere by the torus and surfaces of higher genera , Ann. of Comb. 4, 27–46 (2000)

View on Pith 2000

[20] [19]

I.P. Goulden, D.M. Jackson and R. Vakil, The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. London Math. Soc. 83:3, 563–581 (2001)

View on Pith 2001

[21] [20]

I.P. Goulden, D.M. Jackson and R. Vakil, A short proof of the λg-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves , arXiv:math/0604297v1 [math.AG] (2006)

View on Pith Pith review arXiv 2006

[22] [21]

T. Graber, R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization , Compositio Math. 135, 25–36 (2003)

View on Pith 2003

[23] [22]

T. Graber, R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves , Duke Math. J. 130, 1–37 (2005)

View on Pith 2005

[24] [23]

A. Hurwitz, ¨Uber Riemann’sche Fl¨ achen mit gegebene Verzweigungspunkten, Mathematische Annalen 39, 1–66 (1891)

View on Pith

[25] [24]

M. Kazarian, KP hierarchy for Hodge integrals , arXiv:0809.3263

View on Pith Pith review arXiv

[26] [25]

M. Kazarian, S. Lando, An algebro-geometric proof of Witten’s conjecture , J. Amer. Math. Soc. 20, 1079–1089 (2007)

View on Pith 2007

[27] [26]

Y.S. Kim and K. Liu, A simple proof of Witten conjecture through localization , preprint arXiv:math/0508384 [math.AG] (2005)

View on Pith Pith review arXiv 2005

[28] [27]

M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Commun. Math. Phys. 147, 1–23 (1992)

View on Pith 1992

[29] [28]

A.M. Li, G. Zhao, and Q. Zheng, The number of ramiﬁed coverings of a Riemann surface by Riemann surface , Commun. Math. Phys. 213, 685–696 (2000)

View on Pith 2000

[30] [29]

C.-C. M. Liu, Formulae of one-partition and two-partition Hodge Integrals, Geom. & Top. Monographs8, 105–128 (2006)

View on Pith 2006

[31] [30]

C.-C. M. Liu, K. Liu, J. Zhou, A proof of a conjecture of Mari˜ no-Vafa on Hodge Integrals, J. Diﬀerential Geom. 65, no. 2, 289–340 (2003)

View on Pith 2003

[32] [31]

K. Liu and H. Xu, A simple proof of Mirzakhani’s recursion formula of Weil–Petersson volumes, arXiv:0705.2086 [math.AG]

View on Pith Pith review arXiv

[33] [32]

M. Mari˜ no,Chern-Simons theory, matrix models, and topological strings , Oxford University Press, 2005

View on Pith 2005

[34] [33]

M. Mari˜ no,Open string amplitudes and large order behavior in topological string theory , Journal of High Energy Physics 03 060, (2008)

View on Pith 2008

[35] [34]

A. Mironov and A. Morozov, Virasoro constraints for Kontsevich-Hurwitz partition function , arXiv:0807.2843

View on Pith Pith review arXiv

[36] [35]

M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces , Invent. Math. 167, 179–222 (2007)

View on Pith 2007

[37] [36]

M. Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves , J. Amer. Math. Soc. 20, 1–23 (2007)

View on Pith 2007

[38] [37]

M. Mulase and B. Safnuk, Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy , Indian J. Math. 50, 189–228 (2008). 34 B. EYNARD, M. MULASE, AND B. SAFNUK

View on Pith 2008

[39] [39]

A. Okounkov, Random matrices and random perputations , International Mathematics Research Notices 2000, 1043–1095 (2000)

View on Pith 2000

[40] [40]

A. Okounkov, Toda equations for Hurwitz numbers , Math. Res. Lett. 7, 447 (2000) [arXiv:math.AG/0004128]

View on Pith arXiv 2000

[41] [41]

A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, I , Proc. Symposia Pure Math. 80, 325–414 (2009)

View on Pith 2009

[42] [42]

A. Okounkov and R. Pandharipande, The equivariant Gromov-Witten theory of P1, math.AG/0207233 (2002)

View on Pith arXiv 2002

[43] [43]

R. Vakil, Harvard Thesis 1997

View on Pith 1997

[44] [44]

E. Witten, Two dimensional gravity and intersection theory on moduli space , Surveys in Diﬀerential Geometry 1, 243–310 (1991)

View on Pith 1991

[45] [45]

J. Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups , preprint, math.AG/0308024. Service de Physique Th´eorique de Saclay, Gif-sur-Yvette, F-91191 Cedex, France E-mail address: bertrand.eynard@cea.fr Department of Mathematics, University of California, Davis, CA 95616–8633 E-mail address: mulase@math.ucdavis.edu Department of Mathematics, Cent...

View on Pith arXiv