Skip to main content
SpringerLink
Log in

The extended D-Toda hierarchy

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

In a companion paper (arXiv:1910.03150) to this one, we proved that the Gromov–Witten theory of a Fano orbifold line of type D is governed by a system of Hirota Bilinear Equations. The goal of this paper is to prove that every solution to the Hirota Bilinear Equations determines a solution to a new integrable hierarchy of Lax equations. We suggest the name extended D-Toda hierarchy for this new system of Lax equations, because it should be viewed as the analogue of Carlet’s extended bi-graded Toda hierarchy, which is known to govern the Gromov–Witten theory of Fano orbifold lines of type A.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, M., van Moerbeke, P.: Vertex operator solutions to the discrete KP hierarchy. Commun. Math. Phys. 203, 185–210 (1999)

    Article  MathSciNet  Google Scholar 

  2. Bakalov, B., Milanov, T.: \({\cal{W}}\)-constraints for the total descendant potential of a simple singularity. Compositio Math. 149, 840–888 (2013)

    Article  MathSciNet  Google Scholar 

  3. Carlet, G.: The extended bigraded Toda hierarchy. J. Phys. A: Math. Gen. 39, 9411–9435 (2006)

    Article  MathSciNet  Google Scholar 

  4. Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. Mosc. Math. J. 4, 313–332 (2004)

    Article  MathSciNet  Google Scholar 

  5. Carlet, G., van de Leur, J.: Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of \(CP^1\) orbifolds. J. Phys. A 46, 405205 (2013)

    Article  MathSciNet  Google Scholar 

  6. Cheng,J. P., Milanov, T.: Hirota quadratic equations for the Gromov-Witten invariants of \(\mathbb{P}_{n-2,2,2}^1\). arXiv:1910.03150

  7. Dickey, L.A.: Soliton equations and Hamilonian systems. World Scientific, (2003)

  8. Dubrovin, B.: Geometry of 2D topological field theories. In: “Integrable systems and quantum groups” (Montecatini Terme, 1993), 120–348, Lecture Notes in Math., 1620, Springer, Berlin, (1996)

  9. Dubrovin,B.: Painlevé transcendents in two dimensional topological field theory. The Painlevé property, 287–412, CRM Ser. Math. Phys., Springer, New York, (1999). arXiv: math/9803107

  10. Dubrovin, B., Strachan, Ian A. B., Zhang, Y. J., Zuo, D. F.: Extended affine Weyl groups of BCD-type: their Frobenius manifolds and Landau-Ginzburg superpotentials. Adv. Math. 351, 897-946 (2019)

  11. Dubrovin, B., Zhang, Y. J.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. arXiv:math/0108160

  12. Givental, A.: Semisimple Frobenius structures at higher genus. Internat. Math. Res. Notices 23, 1265–1286 (2001)

    Article  MathSciNet  Google Scholar 

  13. Givental, A.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1, 551–568 (2001)

    Article  MathSciNet  Google Scholar 

  14. Givental, A.: \(A_{n-1}\)-singularities and \(n\)-KdV hierarchies. Mosc. Math. J. 3, 475–505 (2003)

    Article  MathSciNet  Google Scholar 

  15. Iritani, H.: An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. Adv. Math. 222, 1016–1079 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Milanov, T.: The Period map for quantum cohomology of \({\mathbb{P}}^2\). Adv. Math. 351, 804–869 (2019)

    Article  MathSciNet  Google Scholar 

  18. Milanov, T.: The phase factors in singularity theory. arXiv:1502.07444

  19. Milanov, T.: Hirota quadratic equations for the Extended Toda Hierarchy. Duke Math. J. 138, 161–178 (2007)

    Article  MathSciNet  Google Scholar 

  20. Milanov, T., Shen, Y., Tseng, H.-H.: Gromov-Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies. Geometry & Topology 20, 2135–2218 (2016)

    Article  MathSciNet  Google Scholar 

  21. Milanov, T., Tseng, H.-H.: The spaces of Laurent polynomials, Gromov-Witten theory of \({\mathbb{P}}^1\)-orbifolds, and integrable hierarchies. J. Reine Angew. Math. 622, 189–235 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Shiota, T.: Prym varieties and soliton equations. In: “Infinite dimensional Lie algebras and groups”, Proc. of the Conference held at CIRM (Luminy, Marseille, 1988), Victor Kac (editor), 407-448, Singapore: World Scientific, (1989)

  23. Shiraishi, Y.: On Frobenius manifolds from Gromov-Witten theory of orbifold projective lines with r orbifold points. Tohoku Math. J. 70, 17–37 (2018)

    Article  MathSciNet  Google Scholar 

  24. Teleman, C.: The structure of 2D semi-simple field theories. Invent. Math. 188, 525–588 (2012)

    Article  MathSciNet  Google Scholar 

  25. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv. Diff. Geom. 1, 243–310 (1991)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

T.M. would like to thank Mikhail Kapranov for a very useful discussion on localization of difference operators, which helped us to define the ring of rational difference operators in Section 2.2. We would like to thank also Atsushi Takahashi for letting us know the reference [23] about quantum cohomology of orbifold lines. The work of T.M. is partially supported by JSPS Grant-In-Aid (Kiban C) 17K05193 and by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

Author information

Authors and Affiliations

  1. School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, P.R. China

    Jipeng Cheng

  2. Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Todor Milanov

Authors
  1. Jipeng Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Todor Milanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jipeng Cheng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

In this section, the examples of the Lax operator and flow equations are given.

1.1 Examples of Lax operator

Take \(n=4\) as an example. The Lax operator will be

$$\begin{aligned}&{\mathcal {L}}=a\Lambda ^{2}+\frac{1}{4}(c_2-c_3)\Lambda +\frac{1}{2}(c_2+c_3)-a-a[-1]\\&\quad +\frac{1}{4}(c_2-c_3)\Lambda ^{-1}+a[-1]\Lambda ^{-2}+\frac{1}{2}\partial _2^2+\frac{1}{2}\partial _3^2, \end{aligned}$$

where \(a=\frac{1}{2}e^{2\alpha }\). Then we will obtain

$$\begin{aligned} \pi _+({\mathcal {L}})&=a\Lambda ^{2}+\frac{1}{4}(c_2-c_3)\Lambda +v_{1,0}+v_{1,1}\Lambda ^{-1}+v_{1,2}\Lambda ^{-2}+O(\Lambda ^{-3}),\\ \pi _i({\mathcal {L}})&=\frac{1}{2}\partial _i^2+c_i+v_{i,1}\partial _i^{-1}+v_{i,2}\partial _{i}^{-2}+O(\partial _i^{-3}), \end{aligned}$$

where

$$\begin{aligned} v_{1,0}&=\frac{1}{4}(c_2+c_3)+\frac{1}{4}(c_2+c_3)[1]-a-a[-1]+\frac{1}{2}\Big (q_2^{2}+q_3^2\Big )[-1],\\ v_{1,1}&=\frac{1}{4}(c_2-c_3)[-2]+\frac{1}{2}\sum _{l=2}^3(-1)^l(q_l[-2]+q_l[-1])^2,\\ v_{1,2}&=a[-1]+\frac{1}{4}(c_2+c_3)[-3]-\frac{1}{4}(c_2+c_3)[-1]\\&+\frac{1}{2}\sum _{l=2}^3(q_l[-3]+q_l[-2])(2q_l[-1]+q_l[-2]+q_l[-3]),\\ v_{i,1}&=2a(q_i+q_i[1])+\frac{(-1)^i}{2}(c_2-c_3)(q_i+q_i[-1])-2a[-1](q_i[-1]+q_i[-2]),\\ v_{i,2}&=2a((q_i+q_i[1])^2-\partial _i(q_i+q_i[1]))+\frac{(-1)^i}{2}(c_2-c_3)(q_i^2-q_i[-1]^2\\&-\partial _i(q_i-q_i[-1]))+2a[-1]((q_i[-1]+q_i[-2])^2-\partial _i(q_i[-1]+q_i[-2])). \end{aligned}$$

The expressions of \(B_{i,k}\) are listed as follows

$$\begin{aligned}&B_{0,1}={\mathcal {L}}\cdot \epsilon \partial _x-\frac{1}{2}(a_{2,1}\partial _2+a_{3,1}\partial _3)+\sum _{l=-2}^2b_{0,l}\Lambda ^{l}\\&B_{1,1}=e^\beta (\Lambda -\Lambda ^{-1}),\quad B_{2,1}=\partial _2,\quad B_{3,1}=\partial _3\\&B_{1,2}=\Big (2a\Lambda +\frac{1}{2}(c_2-c_3)-2a[-1]\Lambda ^{-1}\Big )\cdot (\Lambda -\Lambda ^{-1})\\&B_{2,3}=\partial _2^3+3c_2\partial _2,\quad B_{3,3}=\partial _3^2+3c_3\partial _3, \end{aligned}$$

where

$$\begin{aligned}&\beta =\frac{2}{1+e^{\epsilon \partial _x}}(\alpha )\quad b_{0,2}=-a\Big (\frac{1}{2}+a_{1,0}[2]\Big ),\\&b_{0,1}=-\frac{1}{4}(c_2-c_3)(\frac{1}{2}+a_{1,0}[1])-a a_{1,1}[2],\\&b_{0,0}=-(b_{0,2}+b_{0,-2})+\epsilon \partial _x a[-1]-a_{1,0}[1]\cdot a[-1]-a_{2,1}-\frac{1}{2}a_{2,2},\\&b_{0,-1}=-b_{0,1}+\frac{1}{4}\epsilon \partial _x(c_2-c_3),\\&b_{0,-2}=\Big (\frac{1}{2}+a_{1,0}[2]\Big )\cdot a[-1]+\epsilon \partial _x a[-1]. \end{aligned}$$

Here \(a_{k,l}\) comes from \(\ell _i=\epsilon \partial _x(S_i)\cdot S_i^{-1}\) with \(i=1,2,3\),

$$\begin{aligned}&\ell _1=a_{1,0}+a_{1,1}\Lambda ^{-1}+O(\Lambda ^{-2})\\&\ell _2=a_{2,1}\partial _2^{-1}+a_{2,2}\partial _2^{-2}+O(\partial _2^{-3}),\quad \ell _3=a_{3,1}\partial _3^{-1}+a_{3,2}\partial _3^{-2}+O(\partial _3^{-3}), \end{aligned}$$

where

$$\begin{aligned}&a_{1,0}=\frac{\epsilon \partial _x}{1-e^{\epsilon \partial _x}}(\beta ),\quad a_{1,1}=\frac{1}{2}e^{\beta }\frac{\epsilon \partial _x}{1-e^{2\epsilon \partial _x}}\Big ((c_2-c_3)e^{-\beta }\Big ),\quad a_{i,1}=\frac{2\epsilon \partial _x}{e^{\epsilon \partial _x}-1}(q_i),\\&a_{i,2}=\frac{2\epsilon \partial _x}{e^{\epsilon \partial _x}-1}\Big (-\partial _i(q_i)-q_i^2+\sum _{m=0}^{+\infty }\sum _{i=1}^{m+1}\sum _{j=0}^{i-1}\left( {\begin{array}{c}i-1\\ j\end{array}}\right) (\epsilon \partial _x)^j(a_{i,1})(\epsilon \partial _x)^{m-j}(a_{i,1})\Big ),\quad \\&\qquad i=2,3. \end{aligned}$$

1.2 Examples of flows of \(t_{i,1}\) for \(i=1,2,3\)

Flows of \(t_{1,1}\) are

$$\begin{aligned}&\partial _{1,1}(a)=\frac{1}{4}\Big (e^{\beta }\cdot (c_2-c_3)[1]-e^{\beta [1]}\cdot (c_2-c_3)\Big ),\\&\partial _{1,1}(q_2)=e^{\beta [1]}(q_2+q_2[1])-e^{\beta }(q_2+q_2[-1]),\\&\partial _{1,1}(q_3)=e^{\beta }(q_3+q_3[-1])-e^{\beta [1]}(q_3+q_3[1]),\\&\partial _{1,1}(c_2)=c_2[1]-c_2[-1],\quad \partial _{1,1}(c_3)=c_3[-1]-c[1]. \end{aligned}$$

Flows of \(t_{2,1}\) are

$$\begin{aligned} \partial _2(a)&=a(q_2[-1]-q_2[1]), \quad \partial _2(q_2)=\frac{1}{2}(c_2-c_2[1]), \quad \partial _2(q_3)=\frac{e^{\epsilon \partial _x}-1}{e^{\epsilon \partial _x}+1}(q_2q_3),\\ \partial _2(c_2)&=\frac{1}{1+e^{-\epsilon \partial _x}}\Big (-(c_2-c_3)q_2+(c_2-c_3)[-1]\cdot q_2[-2]-4a(q_2+q_2[1])\\&+4a[-1](q_2[-2]-q_2)+4a[-2](q_2[-2]+q_2[-3])-q_2[-1](c_2[-1]-c_2)\\&-2q_3[-1]\frac{1-e^{-\epsilon \partial _x}}{1+e^{\epsilon \partial _x}}(q_2q_3)\Big ),\\ \partial _2(c_3)&=\frac{1}{1+e^{-\epsilon \partial _x}}\Big (q_2[-1](c_3-c_3[-1])-2q_3[-1]\frac{1-e^{-\epsilon \partial _x}}{1+e^{\epsilon \partial _x}}(q_2q_3)\Big ). \end{aligned}$$

And flows of \(t_{3,1}\) are given by

$$\begin{aligned} \partial _3(a)&=a(q_3[-1]-q_3[1]), \quad \partial _3(q_3)=\frac{1}{2}(c_3-c_3[1]), \quad \partial _3(q_3)=\frac{e^{\epsilon \partial _x}-1}{e^{\epsilon \partial _x}+1}(q_2q_3),\\ \partial _3(c_2)&=\frac{1}{1+e^{-\epsilon \partial _x}}\Big (q_3[-1](c_2-c_2[-1])-2q_2[-1]\frac{1-e^{-\epsilon \partial _x}}{1+e^{\epsilon \partial _x}}(q_2q_3)\Big ),\\ \partial _3(c_3)&=\frac{1}{1+e^{-\epsilon \partial _x}}\Big ((c_2-c_3)q_3-(c_2-c_3)[-1]\cdot q_3[-2]-4a(q_3+q_3[1])\\&+4a[-1](q_3[-2]-q_3)+4a[-2](q_3[-2]+q_3[-3])-q_3[-1](c_3[-1]-c_3)\\&-2q_2[-1]\frac{1-e^{-\epsilon \partial _x}}{1+e^{\epsilon \partial _x}}(q_2q_3)\Big ). \end{aligned}$$

1.3 Examples of flows of \(t_{0,1}\)

Flows of \(t_{0,1}\) are

$$\begin{aligned} \partial _{0,1}(a)&=a\cdot \epsilon \partial _x v_{1,0}[2] +\frac{1}{16}(c_2-c_3)\cdot \epsilon \partial _x(c_2-c_3)[1]\\&\quad +v_{1,0}\cdot \epsilon \partial _x(a)+b_{0,2}(v_{1,0}[2]-v_{1,0})\\&\quad +\frac{1}{4}\Big (b_{0,1}\cdot (c_2-c_3)[1]-(c_2-c_3)\cdot b_{1,0}[1]\Big )\\&\quad +a\Big (b_{0,0}-b_{0,0}[2]-\frac{1}{2}(a_{2,1}\cdot q_2[-1]\\&\quad +a_{3,1}\cdot q_3[-1])+\frac{1}{2}(a_{2,1}[2]q_2[1]+a_{3,1}[2]\cdot q_3[1])\Big ),\\ \partial _{0,1}(c_2)&=\frac{1}{2}\epsilon \partial _x\Big (\partial _2^2(c_2)+c_2^2 +2\partial _2(v_{2,1})+v_{2,2}\Big )\\&\quad -\frac{1}{2}a_{2,1}\partial _2(c_2)-\frac{1}{2}\partial _2^2(b_{0,0,2})-\partial _2(b_{0,1,2}),\\ \partial _{0,1}(c_3)&=\frac{1}{2}\epsilon \partial _x\Big (\partial _3^2(c_3)+c_2^2 +2\partial _3(v_{3,1})+v_{3,2}\Big )\\&\quad -\frac{1}{2}a_{3,1}\partial _3(c_3)-\frac{1}{2}\partial _3^2(b_{0,0,3})-\partial _3(b_{0,1,3}),\\ \partial _{0,1}(q_2)&=\partial _2\Big (b_{0,0}[1]-b_{0,1}-\frac{1}{2}a_{2,1}[1]q_2-\frac{1}{2}a_{3,1}[1]q_3 \Big )\\&\quad +q_2\Big (b_{0,2}[1]-b_{0,2}+b_{0,1}[1]-2b_{0,1}\Big )\\&\quad +q_2[1]\Big (b_{0,2}[1]-b_{0,2}+b_{0,1}[1]\Big )-\epsilon \partial _xA_2,\\ \partial _{0,1}(q_3)&=\partial _3\Big (b_{0,0}[1]+b_{0,1}-\frac{1}{2}a_{2,1}[1]q_2-\frac{1}{2}a_{3,1}[1]q_3 \Big )\\&\quad +q_2\Big (b_{0,2}[1]+b_{0,2}+b_{0,1}[1]+2b_{0,1}\Big )\\&\quad +q_2[1]\Big (b_{0,2}[1]+b_{0,2}+b_{0,1}[1]\Big )-\epsilon \partial _xA_3. \end{aligned}$$

Here

$$\begin{aligned} b_{0,0,2}&=\epsilon \partial _x\Big (a[-1]+\frac{1}{4}(c_2-c_3)\Big )-a_{1,0}[1]\cdot a[-1]-a_{2,1}-\frac{1}{2}a_{2,2},\\ b_{0,0,3}&=\epsilon \partial _x\Big (a[-1]-\frac{1}{4}(c_2-c_3)\Big )-a_{1,0}[1]\cdot a[-1]-a_{2,1}-\frac{1}{2}a_{2,2},\\ b_{0,1,2}&=2b_{0,2}\Big (q_2+q_2[1]\Big )+2b_{0,1} q_2+2b_{0,-1}q_2[-1]\\&\quad -2b_{0,-2}\Big (q_2[-1]+q_2[-2]\Big )+\frac{1}{2}a_{3,1}q_1,\\ b_{0,1,3}&=2b_{0,2}\Big (q_3+q_3[1]\Big )-2b_{0,1} q_3-2b_{0,-1}q_3[-1]\\&\quad -2b_{0,-2}\Big (q_3[-1]+q_3[-2]\Big )+\frac{1}{2}a_{2,1}q_1,\\ A_2&=\partial _2\Big (\frac{1}{2}(c_2+c_3)[1]-a-a[1]-\frac{1}{4}(c_2-c_3)\Big )\\&\quad +q_2\Big (a[1]-a+\frac{1}{4}(c_2-c_3)[1]-\frac{1}{2}(c_2-c_3)\\&\quad -\frac{1}{2}(\partial _2(q_2)+\partial _3(q_3)+q_2^2+q_3^2)\Big )+ q_2[1]\\&\quad \cdot \Big (a[1]-a+\frac{1}{4}(c_2-c_3)[1]\Big ),\\ A_3&=\partial _3\Big (\frac{1}{2}(c_2+c_3)[1]-a-a[1]-\frac{1}{4}(c_2-c_3)\Big )\\&\quad +q_3\Big (a[1]-a -\frac{1}{4}(c_2-c_3)[1]+\frac{1}{2}(c_2-c_3)\\&\quad +\frac{1}{2}(\partial _2(q_2)+\partial _3(q_3)+q_2^2+q_3^2)\Big )\\&\quad + q_3[1]\cdot \Big (a[1]-a-\frac{1}{4}(c_2-c_3)[1]\Big ). \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, J., Milanov, T. The extended D-Toda hierarchy. Sel. Math. New Ser. 27, 24 (2021). https://doi.org/10.1007/s00029-021-00646-1

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00029-021-00646-1

Mathematics Subject Classification

Search

Navigation