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Fredholm Pfaffian \(\tau \)-Functions for Orthogonal Isospectral and Isomonodromic Systems

Abstract

We extend the approach to \(\tau \)-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld–Sokolov reductions and isomonodromic deformations systems. The combinatorial expansion of the \(\tau \)-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and dependence on the time parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal \(\tau \)-functions, both in the Drinfeld–Sokolov and isomonodromic case, become squares of \(\tau \)-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial \(\tau \)-functions of orthogonal Drinfeld–Sokolov type and isomonodromic ones with four regular singular points.

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Notes

  1. \(\tau _O[J]\) can also be identified with the relative Pfaffian of the operators \(S\mathsf{{a}}\) and \(\mathsf{{d}}S\), which was introduced in [20]. The definition of the relative Pfaffian is essentially the expansion (4.44).

  2. More generally, one could set \(h_i:= H_i+\delta _{i,0}c\), where c is the central extension element. But since the central extension will play no part, we omit it, and just deal with the loop algebra \(L\mathfrak {g}\).

  3. In general, the \(D_\ell ^{(1)}\) case admits two sets of abelian flows, labeled by times \(t_{2k+1}\) and \(t'_{2k+1}\), as in [7]. For the sake of clarity of exposition, we restrict ourselves to the first only, and set \(t'_{2k+1}=0\).

  4. See [17], Appendix E, or [3, 18, 19] for the definition of Cartan coordinates in the context of the BKP hierarchy. In our context, by “Cartan coordinate”, we simply mean the Pfaffian expression \(\mathrm {Pf}(\mathsf{{d}}_{\vec {\lambda }})\), specifying the initial condition in the isotropic Grassmannian.

  5. See [23,24,25, 33] for a thorough account of polynomial \(\tau \)-functions of KP, BKP, DKP, mKP and multicomponent KP type.

  6. Note that in the previous sections we dealt with Riemann–Hilbert problems on the unit circle, while here the circle has radius R. Everything is easily generalized to this case by reinserting the radius R in the expressions where needed. All our quantities depend only on the splitting of the space \(L^2(S^1)\) into the subspace \({\mathcal {H}}^N_+\) of functions admitting analytic continuation inside the circle and those \({\mathcal {H}}^N_-\) admitting analytic continuation outside.

References

  1. Bershtein, M., Gavrylenko, P., Marshakov, A.: Twist-field representations of \(W\)-algebras, exact conformal blocks and character identities. JHEP 08, 108 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  2. Bonelli, G., Globlek, F., Tanzini, A.: Counting Yang–Mills instantons by surface operator renormalization group flow. Phys. Rev. Lett. 126(23), 231602 (2021)

    ADS  MathSciNet  Article  Google Scholar 

  3. Balogh, F., Harnad, J., Hurtubise, J.: Isotropic Grassmannians, Plücker and Cartan maps. J. Math. Phys. 62(2), 021701 (2021)

    ADS  Article  Google Scholar 

  4. Cafasso, M., Gavrylenko, P., Lisovyy, O.: Tau functions as Widom constants. Commun. Math. Phys. 365(2), 741–772 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  5. Cafasso, M., du CrestdeVilleneuve, A., Yang, D.: Drinfeld–Sokolov hierarchies, tau functions, and generalized schur polynomials. SIGMA 14, 104 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  6. Cafasso, M., Chao-Zhong, W.: Tau functions and the limit of block Toeplitz determinants. Int. Math. Res. Not. 2015(20), 10339–10366 (2015)

    MathSciNet  Article  Google Scholar 

  7. Cafasso, M., Chao-Zhong, W.: Borodin–Okounkov formula, string equation and topological solutions of Drinfeld–Sokolov hierarchies. Lett. Math. Phys. 109(12), 2681–2722 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  8. Del Monte, F., Desiraju, H., Gavrylenko, P.: Isomonodromic tau functions on a torus as fredholm determinants, and charged partitions. arXiv preprint arXiv:2011.06292 (2020)

  9. Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. J. Math. Sci. 30, 1975–2036 (1985)

    Article  Google Scholar 

  10. Dubrovin, B.: Geometry of 2-D topological field theories. Lect. Notes Math. 1620, 120–348 (1996)

    MathSciNet  Article  Google Scholar 

  11. Fan, H., Francis, A., Jarvis, T., Merrell, E., Ruan, Y.: Witten’s \(D_4\) integrable hierarchies conjecture. Chin. Ann. Math. Ser. B 37(2), 175–192 (2016)

    Article  Google Scholar 

  12. Feigin, E., van de Leur, J., Shadrin, S.: Givental symmetries of Frobenius manifolds and multi-component KP tau-functions. Adv. Math. 224(3), 1031–1056 (2010)

    MathSciNet  Article  Google Scholar 

  13. Gavrylenko, P., Iorgov, N., Lisovyy, O.: Higher rank isomonodromic deformations and \(W\)-algebras. Lett. Math. Phys. 110(2), 327–364 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  14. Gavrylenko, P., Lisovyy, O.: Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions. Commun. Math. Phys. 363, 1–58 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  15. Gavrylenko, P., Lisovyy, O.: Pure \(SU(2)\) gauge theory partition function and generalized Bessel kernel. Proc. Symp. Pure Math. 18, 181–208 (2018)

    MathSciNet  Article  Google Scholar 

  16. Gavrylenko, P.G., Marshakov, A.V.: Free fermions, \(W\)-algebras and isomonodromic deformations. Theor. Math. Phys. 187(2), 649–677 (2016)

    MathSciNet  Article  Google Scholar 

  17. Harnad, J., Balogh, F.: Tau Functions and Their Applications. Cambridge University Press (2021)

  18. Harnad, J., Orlov, A.Y.: Bilinear expansions of lattices of KP \(\tau \)-functions in BKP \(\tau \)-functions: a fermionic approach. J. Math. Phys. 62(1), 013508 (2021)

    ADS  MathSciNet  Article  Google Scholar 

  19. Harnad, J., Orlov, A.Y.: Polynomial KP and BKP \(\tau \)-functions and correlators. Annales Henri Poincaré (2021)

  20. Jaffe, A., Lesniewski, A., Weitsman, J.: Pfaffians on Hilbert space. J. Fun. Anal. 83(2), 348–363 (1989)

    MathSciNet  Article  Google Scholar 

  21. Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS Kyoto Univ. 19(2), 943–1001 (1983)

    MathSciNet  Article  Google Scholar 

  22. Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge University Press (1990)

  23. Kac, V.G., Rozhkovskaya, N., van de Leur, J.: Polynomial tau-functions of the KP, BKP, and the s-component KP hierarchies. J. Math. Phys. 62(2), 021702 (2021)

    ADS  MathSciNet  Article  Google Scholar 

  24. Kac, V.G., van de Leur, J.W.: Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions. Jpn. J. Math. 13(2), 235–271 (2018)

    MathSciNet  Article  Google Scholar 

  25. Kac, V., van de Leur, J.: Polynomial tau-functions of bkp and dkp hierarchies. J. Math. Phys. 60(7), 071702 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  26. Liu, S.-Q., Ruan, Y., Zhang, Y.: BCFG Drinfeld–Sokolov hierarchies and FJRW-theory. Invent. Math. 201(2), 711–772 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  27. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press (1998)

  28. Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite-Dimensional Algebras, Volume 135 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)

  29. Rains, E.M.: Correlation functions for symmetrized increasing subsequences. arXiv preprint math/0006097 (2000)

  30. Sato, M.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. Publ. RIMS Kyoto Univ. 2, 30–46 (1981)

    Google Scholar 

  31. Sato, M., Jimbo, M., Miwa, K.: Studies on holonomic quantum fields I–V. Proc. Japan Acad. 53A, 219–224 (1977)

    MathSciNet  Google Scholar 

  32. Sato, M., Jimbo, M., Miwa, K.: Studies on holonomic quantum fields VI–VII. Proc. Japan Acad. 54A(1–5), 136–141 (1978)

    Google Scholar 

  33. van de Leur J.: BKP tau-functions as square roots of KP tau-functions. arXiv:2103.16290 (2021)

  34. You, Y.: Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups. Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988). Adv. Ser. Math. Phys. 7, 449–464 (1989)

    Google Scholar 

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Acknowledgements

The authors would like to thank M. Caffasso, P. Gavrylenko, O. Lisovyy and D. Yang for helpful discussions that contributed much to clarifying the results presented here. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Appendices

A Appendix. A Polynomial \(\tau \)-Function of \(B^{(1)}_2\) Type

Recall equation (4.28) in the main text, representing the map between decreasing sequences of positive half-integers \(\{p_{\alpha ,i}\}_{i=1}^{n_\alpha }\) and strict partitions

$$\begin{aligned} \lambda ^{(\alpha )}=\left( p_{\alpha ,1}-\frac{1}{2},\dots ,p_{\alpha ,n_{\alpha }}-\frac{1}{2}\right) . \end{aligned}$$
(A.1)

In the tables below we group the coefficients in the Pfaffian minor expansions by the total weight of the N-tuple of (positive) strict partitions obtained by adding 1 to each part:

$$\begin{aligned} |\vec {\lambda }|:=\sum _{\alpha =1}^N\sum _{i=1}^{n_\alpha }\left( p_{\alpha ,i}+\frac{1}{2}\right) , \end{aligned}$$
(A.2)

counting the overall power of z (resp. \(z^{-1}\)) in the Fourier expansion of \(\mathsf{{a}}\) (respectively \(\mathsf{{d}}\)) that contribute to the Pfaffian minor. For example

$$\begin{aligned} \mathrm {Pf}\left( \begin{array}{cc} 0 &{} \quad \left( S\mathsf{{a}}^{\frac{1}{2}}_{-\frac{1}{2}} \right) _{12} \\ \left( S\mathsf{{a}}^{\frac{1}{2}}_{-\frac{1}{2}} \right) _{12} &{}\quad 0 \end{array} \right) \end{aligned}$$
(A.3)

has weight \(|\vec {\lambda }|=2\), since it contains two Fourier coefficients of \(\mathsf{{a}}\) of weight 0 that multiply \(z^0w^0\), while

$$\begin{aligned} \mathrm {Pf}\left( \begin{array}{cc} 0 &{} \left( S\mathsf{{a}}^{\frac{3}{2}}_{-\frac{3}{2}} \right) _{12} \\ \left( S\mathsf{{a}}^{\frac{3}{2}}_{-\frac{3}{2}} \right) _{12} &{} 0 \end{array} \right) \end{aligned}$$
(A.4)

has weight \(|\vec {\lambda }|=4\), since it contains two Fourier coefficients of \(\mathsf{{a}}\) of weight 2, that multiply zw.

A.1 The Pfaffian \(\mathsf{{d}}\) Coefficients

A.2 The Pfaffian \(\mathsf{{a}}\) Coefficients

$$\begin{aligned} \begin{array}{c|c} \mathrm {Pf}(\mathsf{{a}}_{\vec {\lambda }}) &{} \vec {\lambda },\,|\vec {\lambda }|=2 \\ \hline -\frac{1^{}}{12} \left( t_1^3+6 t_3\right) =-\frac{1}{2}Q_{(3,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad (0) , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) \\ \end{array} \right) \\ t_3-\frac{t_1^{3^{}}}{12}=-\frac{1}{2}Q_{(2,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (0) , &{}\quad \emptyset \\ \end{array} \right) \\ -\frac{t_1^{2^{}}}{4}=-\frac{1}{2}Q_{(2,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad \emptyset , &{}\quad (0) \\ \end{array} \right) \\ -\frac{t_1^{}}{2}=-\frac{1}{2}Q_{(1,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (0) \\ \end{array} \right) \\ \hline &{} \vec {\lambda },\,|\vec {\lambda }|=3 \\ \hline \frac{1}{360} \left( -t_1^6-30 t_3 t_1^3+180 t_5 t_1+180 t_3^2\right) ^{}=-\frac{1}{2}Q_{(5,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1,0) , &{}\quad \emptyset \\ \end{array} \right) \\ -\frac{1}{96} t_1 \left( t_1^3+24 t_3\right) =-\frac{1}{4}Q_{(4,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1,0) \\ \end{array} \right) \\ -\frac{t_1^6}{1440}-\frac{1}{12} t_3 t_1^3-\frac{t_5 t_1}{2}-\frac{t_3^2}{4}=-\frac{1}{2}Q_{(6,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (1) , &{}\quad \emptyset , &{}\quad (0) \\ \end{array} \right) \\ \frac{1}{576} \left( t_1^3-12 t_3\right) ^2=\frac{1}{4}Q_{(4,2)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad \emptyset , &{}\quad (1) \\ \end{array} \right) \\ -\frac{t_1^5}{240}-\frac{1}{4} t_3 t_1^2-\frac{t_5}{2}=-\frac{1}{2}Q_{(5,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1) , &{}\quad (0) \\ \end{array} \right) \\ \frac{1}{160} \left( t_1^5-80 t_5\right) =\frac{1}{4}Q_{(4,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (1) \\ \end{array} \right) \\ \hline &{} \vec {\lambda },\,|\vec {\lambda }|=4 \\ \hline -\frac{t_1^9}{725760}-\frac{t_3 t_1^6}{1440}-\frac{1}{48} t_5 t_1^4-\frac{1}{24} t_3^2 t_1^3-\frac{1}{2} t_3 t_5 t_1-\frac{t_3^3}{12}=-\frac{1}{2}Q_{(9,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (2) , &{}\quad (0) \\ \end{array} \right) \\ \frac{t_1^9}{207360}+\frac{1}{720} t_3 t_1^6+\frac{1}{48} t_5 t_1^4+\frac{1}{48} t_3^2 t_1^3-\frac{1}{4} t_3 t_5 t_1-\frac{t_3^3}{12}=\frac{1}{4}Q_{(8,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (2) \\ \end{array} \right) \\ -\frac{t_1^9}{103680}+\frac{t_3 t_1^6}{2880}+\frac{1}{96} t_5 t_1^4-\frac{1}{24} t_3^2 t_1^3+\frac{1}{4} t_3 t_5 t_1+\frac{t_3^3}{6}=-\frac{1}{4}Q_{(5,4)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1) , &{}\quad (1) \\ \end{array} \right) \\ \frac{t_1^6-60 t_3 t_1^3+720 t_5 t_1-720 t_3^2}{1440}=\frac{1}{4}Q_{(3,2,1,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad (0) , &{}\quad (0) , &{}\quad (0) , &{}\quad (0) \\ \end{array} \right) \\ \end{array} \end{aligned}$$

B Appendix. Matrix Representation of Orthogonal Affine Lie Algebras

In the following, \(e_{ij}\) is the elementary matrix \((e_{ij})_{\alpha \beta }:=\delta _{\alpha i}\delta _{\beta j}\).

B.1 Matrix Realization of \(B_\ell ^{(1)}\)

  • Weyl generators:

    $$\begin{aligned}&E_0=\frac{1}{2}(e_{1,2\ell }+e_{2,2\ell +1}), \quad F_0=2\left( e_{2\ell ,1}+e_{2\ell +1,2} \right) , \nonumber \\&H_0 =e_{1,1}+e_{2,2}-e_{2l,2l}-e_{2l+1,2l+1}, \nonumber \\&E_i=e_{i+1,i}+e_{2l+2-i,2l+1-i}, \quad F_i=e_{i,i+1}+e_{2\ell +1-i,2\ell +2-i} \nonumber \\&H_i=-e_{i,i}+e_{i+1,i+1}-e_{2\ell +1-i,2\ell +1-i}+e_{2\ell +2-i,2\ell +2-i}, \quad i=1,\dots ,\ell -1, \nonumber \\&E_\ell =e_{\ell +1,\ell }+e_{\ell +2,\ell +1}, \quad F_{\ell }=e_{\ell ,\ell +1}+e_{\ell +1,\ell +2}, \quad H_\ell =-e_{\ell ,\ell }+e_{\ell +2,\ell +2}. \nonumber \\ \end{aligned}$$
    (B.1)
  • Cartan matrix:

    $$\begin{aligned} A=\left( \begin{array}{cccccc} 2 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 2 &{}\quad -1 &{}\quad &{}\quad &{}\quad \vdots \\ -1 &{}\quad -1 &{}\quad 2 &{}\quad \ddots &{}\quad &{}\quad 0 \\ 0 &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad -1 &{}\quad 0 \\ \vdots &{}\quad &{}\quad &{}\quad -1 &{}\quad 2 &{}\quad -1 \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad -2 &{}\quad 2 \end{array} \right) _{(\ell +1)\times (\ell +1)}, \end{aligned}$$
    (B.2)
  • Chevalley involution:

    $$\begin{aligned} S= & {} \mathrm {antidiag}\left( 1,-1,1,\dots ,-1,1 \right) _{2\ell +1\times 2\ell +1} \nonumber \\= & {} \left( \begin{array}{cccccc} 0 &{}\quad 0 &{}\quad \cdots &{}\quad &{}\quad &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \ddots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 0 &{}\quad &{}\quad \cdots &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad &{}\quad \cdots &{}\quad 0 \end{array} \right) _{2\ell +1\times 2\ell +1}. \end{aligned}$$
    (B.3)

B.2 Matrix Realization of \(D_\ell ^{(1)}\)

  • Weyl generators:

    $$\begin{aligned}&E_0=\frac{1}{2}(e_{1,2\ell -1}+e_{2,2\ell }), \quad F_0=2\left( e_{2\ell -1,1}+e_{2\ell ,2}\right) , \nonumber \\&H_0 =e_{1,1}+e_{2,2}-e_{2l-1,2l-1}-e_{2l,2l}, \nonumber \\&E_i=e_{i+1,i}+e_{2l+1-i,2l-i}, \quad F_i=e_{i,i+1}+e_{2\ell -i,2\ell +1-i} \nonumber \\&H_i=-e_{i,i}+e_{i+1,i+1}-e_{2\ell -i,2\ell -i}+e_{2\ell +1-i,2\ell +1-i}, \quad i=1,\dots ,\ell -1, \nonumber \\&E_\ell =\frac{1}{2}\left( e_{\ell +1,\ell -1}+e_{\ell +2,\ell }\right) , \quad F_{\ell }=2\left( e_{\ell -1,\ell +1}+e_{\ell ,\ell +2}\right) , \nonumber \\&H_\ell =-e_{\ell -1,\ell -1}-e_{\ell ,\ell }+e_{\ell +1,\ell +1} +e_{\ell +2,\ell +2}. \end{aligned}$$
    (B.4)
  • Cartan matrix:

    $$\begin{aligned} A=\left( \begin{array}{cccccc} 2 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 2 &{}\quad -1 &{}\quad &{}\quad &{}\quad \vdots \\ -1 &{}\quad -1 &{}\quad 2 &{}\quad \ddots &{}\quad &{}\quad 0 \\ 0 &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad -1 &{}\quad -1 \\ \vdots &{}\quad &{}\quad &{}\quad -1 &{}\quad 2 &{}\quad 0 \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 2 \end{array} \right) _{(\ell +1)\times (\ell +1)}. \end{aligned}$$
    (B.5)
  • Chevalley involution:

    $$\begin{aligned} S=\mathrm {antidiag}\left( 1,-1,1,\dots ,(-1)^{\ell },(-1)^{\ell },(-1)^{\ell +1},\dots ,-1,1\right) _{2\ell \times 2\ell }. \end{aligned}$$
    (B.6)

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Bertola, M., Del Monte, F. & Harnad, J. Fredholm Pfaffian \(\tau \)-Functions for Orthogonal Isospectral and Isomonodromic Systems. Ann. Henri Poincaré (2022). https://doi.org/10.1007/s00023-022-01204-x

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