Abstract
AbstractIn this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}$$
S
M
TNN
in the totally non-negative Grassmannian $$Gr^{\text{ TNN }}(k,n)$$
G
r
TNN
(
k
,
n
)
to the spectral data for the relevant class of real regular Kadomtsev–Petviashvili II (KP-II) solutions, thus completing the search of real algebraic-geometric data for the KP-II equation started in Abenda and Grinevich (Commun Math Phys 361(3):1029–1081, 2018; Sel Math New Ser 25(3):43, 2019). The spectral curve is modeled on the Krichever construction for degenerate finite-gap solutions and is a rationally degenerate $${\texttt {M}}$$
M
-curve, $$\Gamma $$
Γ
, dual to the graph. The divisors are real regular KP-II divisors in the ovals of $$\Gamma $$
Γ
, i.e. they fulfill the conditions for selecting real regular finite-gap KP-II solutions in Dubrovin and Natanzon (Izv Akad Nauk SSSR Ser Mat 52:267–286, 1988). Since the soliton data are described by points in $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}$$
S
M
TNN
, we establish a bridge between real regular finite-gap KP-II solutions (Dubrovin and Natanzon, 1988) and real regular multi-line KP-II solitons which are known to be parameterized by points in $$Gr^{\text{ TNN }}(k,n)$$
G
r
TNN
(
k
,
n
)
(Chakravarty and Kodama in Stud Appl Math 123:83–151, 2009; Kodama and Williams in Invent Math 198:637–699, 2014). We use the geometric characterization of spaces of relations on plabic networks introduced in Abenda and Grinevich (Adv Math 406:108523, 2022; Int Math Res Not 2022:rnac162, 2022. https://doi.org/10.1093/imrn/rnac162) to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in Lam (in: Current developments in mathematics, International Press, Somerville, 2014) for the computation of scattering amplitudes for on-shell diagrams $$N=4$$
N
=
4
SYM (Arkani-Hamed et al. in Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, 2016) and govern the totally non-negative amalgamation of the little positive Grassmannians, $$Gr^{\text{ TP }}(1,3)$$
G
r
TP
(
1
,
3
)
and $$Gr^{\text{ TP }}(2,3)$$
G
r
TP
(
2
,
3
)
, into any given positroid cell $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}\subset {Gr^{\text{ TNN }} (k,n)}$$
S
M
TNN
⊂
G
r
TNN
(
k
,
n
)
. In our setting they control the reality and regularity properties of the KP-II divisor. Finally, we explain the transformation of both the curve and the divisor both under Postnikov’s moves and reductions and under amalgamation of positroid cells, and apply our construction to some examples.
Funder
Russian Foundation of Basic Research
RFO - University of Bologna
GNFM-INDAM
Istituto Nazionale di Fisica Nucleare
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Cited by
1 articles.
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