Exact WKB Analysis and Cluster Algebras

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Exact WKB Analysis and

Cluster Algebras

Kohei Iwaki (RIMS, Kyoto University)

(joint work with Tomoki Nakanishi)

Winter School on Representation Theory January 21, 2015

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Exact WKB analysis

Schr ¨odinger equation:

(d²

dz² −η²Q(z) )

ψ(z, η)=0

wherezis an complex variable,η=~⁻¹ >0is alarge parameter.

• WKB (Wentzel-Kramers-Brillouin) solutions: ψ±(z, η) = e^±η

∫z √

Q(z⁰)dz⁰ ∑∞ n=0

η⁻ⁿ⁻¹²ψ±,n(z)

In general, WKB solutions aredivergent(i.e., formal solutions).

• Exact WKB analysis=WKB method+Borel resummation. S[ψ_±](z, η)∼ψ_±(z, η) asη→+∞

Monodromy/connection matrices of (Borel resummed) WKB solutions are described by “Voros symbols”.

[Voros 83], [Sato-Aoki-Kawai-Takei 91], [Delabaere-Dillinger-Pham 93], ...

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Exact WKB analysis

(d²

dz² −η²Q(z) )

ψ(z, η)=0

• WKB (Wentzel-Kramers-Brillouin) solutions:

ψ±(z, η) = e^±η

∫z √

Q(z⁰)dz⁰ ∑∞ n=0

η⁻ⁿ⁻¹²ψ±,n(z)

• Exact WKB analysis=WKB method+Borel resummation. S[ψ_±](z, η)∼ψ_±(z, η) asη→+∞

[Voros 83], [Sato-Aoki-Kawai-Takei 91], [Delabaere-Dillinger-Pham 93], ...

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Exact WKB analysis

(d²

dz² −η²Q(z) )

ψ(z, η)=0

• WKB (Wentzel-Kramers-Brillouin) solutions:

ψ±(z, η) = e^±η

∫z √

Q(z⁰)dz⁰ ∑∞ n=0

η⁻ⁿ⁻¹²ψ±,n(z)

• Exact WKB analysis=WKB method+Borel resummation.

S[ψ_±](z, η)∼ψ_±(z, η) asη→+∞

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Cluster algebras (of rank

n≥ 1)

• ^Acluster algebra[Fomin-Zelevinsky 02] is defined in terms ofseeds.

• A seed is a triplet(B,x,y)where

∗ skew-symmetric integer matrixB=(b_{i j})ⁿ_i_,_j₌₁

∗ clusterx-variables x=(x_i)ⁿ_i₌₁

∗ clustery-variables y=(y_i)ⁿ_i₌₁ These two variables satisfyy_i=r_i∏_n

j=1(x_j)^b^ji (r_i : “coefficient”).

• ^A“signed” mutationatk∈ {1, . . . ,n}^{with sign}ε∈ {±}^: µ⁽_k^ε⁾: (B,x,y)7→(B⁰,x⁰,y⁰)defined by

b⁰_{i j}=

−bi j i=kor j=k

bi j+[bik]₊bk j+bik[bk j]₊ otherwise.

x⁰_i =



 xk−1





∏n

j=1

xj[−εb_jk]₊



(1+ykε) i=k

xi i,k.

y⁰_i =

yk−1 i=k

yiyk[εb_ki]₊

(1+ykε)⁻^b^ki i,k.

Here[a]₊=^max(a,0). (The coefficientsr_ialso mutate.)

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Cluster algebras (of rank

n≥ 1)

• ^Acluster algebra[Fomin-Zelevinsky 02] is defined in terms ofseeds.

• A seed is a triplet(B,x,y)where

∗ skew-symmetric integer matrixB=(b_{i j})ⁿ_i_,_j₌₁

∗ clusterx-variables x=(x_i)ⁿ_i₌₁

∗ clustery-variables y=(y_i)ⁿ_i₌₁ These two variables satisfyy_i=r_i∏_n

j=1(x_j)^b^ji (r_i : “coefficient”).

• ^A“signed” mutationatk∈ {1, . . . ,n}^{with sign}ε∈ {±}^: µ⁽_k^ε⁾: (B,x,y)7→(B⁰,x⁰,y⁰)defined by

b⁰_{i j}=

x⁰_i =



 xk−1





∏n

j=1

xj[−εb_jk]₊



(1+ykε) i=k

xi i,k.

y⁰_i =

yk−1 i=k

yiyk[εb_ki]₊

(1+ykε)⁻^b^ki i,k.

= ,

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Results and Application [I-Nakanishi 14]

• Cluster algebraic structure appears in many contexts:

I representation of quivers

I Teichm ¨uller theory

I hyperbolic geometry

I discrete integrable systems

I Donaldson-Thomas invariants and their wall-crossing

I supersymmetric gauge theory

I · · ·

• Main result: We addExact WKB analysisin the above list: skew-symmetric matrixB ↔ Stokes graph

cluster variables ↔ Voros symbols

cluster mutation ↔ Stokes phenomenon (forη→ ∞)

• Application:Identities of Stokes automorphsimsin the exact WKB analysis (c.f., [Delabaere-Dillinger-Pham 93]) follow from periodicityof corresponding cluster algebras.

For example: S_γ₁S_γ₂=S_γ₂S_γ₂_+γ₁S_γ₁

• Generalized cluster algebras([Chekhov-Shapiro 11]) also appear when Schr ¨odinger equation has a certain type of regular singularity.

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Results and Application [I-Nakanishi 14]

I · · ·

• Main result: We addExact WKB analysisin the above list:

skew-symmetric matrixB ↔ Stokes graph cluster variables ↔ Voros symbols

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Results and Application [I-Nakanishi 14]

I · · ·

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Results and Application [I-Nakanishi 14]

I · · ·

•

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Exact WKB analysis

§2

Main results

Refferences

•

A. Voros, “The return of the quartic oscillator. The complex WKB method”, Ann. Inst. Henri Poincar ´e

39

(1983), 211–338.

•

T. Kawai and Y. Takei, “Algebraic Analysis of Singular Perturbations”, AMS translation, 2005.

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Main results

Refferences

•

A. Voros, “The return of the quartic oscillator. The complex WKB method”, Ann. Inst. Henri Poincar ´e

39

(1983), 211–338.

•

T. Kawai and Y. Takei, “Algebraic Analysis of Singular

Perturbations”, AMS translation, 2005.

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Schr ¨odinger equation and WKB solutions

• Schr ¨odinger equation: (d²

dz² −η²Q(z) )

ψ(z, η)=0

∗ η=~⁻¹: large parameter

∗ Q(z): rational function (“potential”)

∗ Assume that all zeros ofQ(z)are of order 1, and all poles ofQ(z)are of order≥2.

(We may generalizeQ=Q₀(z)+η⁻¹Q₁(z)+η⁻²Q₂(z)+· · ·: finite sum)

• WKB solutions(formal solution ofη⁻¹ with exponential factor): ψ_±(z, η)=e^±η

∫z z0

√Q(z⁰)dz⁰∑∞ n=0

η⁻ⁿ⁻¹²ψ_±,n(z)

• WKB solutions aredivergentin general: (|ψ_±,n(z)| ∼CAⁿn!).

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Schr ¨odinger equation and WKB solutions

dz² −η²Q(z) )

ψ(z, η)=0

• WKB solutions(formal solution ofη⁻¹with exponential factor):

ψ_±(z, η)=e^±η

∫z z0

√Q(z⁰)dz⁰∑∞ n=0

η⁻ⁿ⁻¹²ψ_±,n(z)

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Schr ¨odinger equation and WKB solutions

dz² −η²Q(z) )

ψ(z, η)=0

• WKB solutions(formal solution ofη⁻¹with exponential factor):

ψ_±(z, η)=e^±η

∫z z0

√Q(z⁰)dz⁰∑∞ n=0

η⁻ⁿ⁻¹²ψ_±,n(z)

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Borel resummation method

• Expansion of WKB solution:

ψ_±(z, η)=e^±η

∫z z0

√Q(z⁰)dz⁰∑∞ n=0

η⁻ⁿ⁻¹²ψ_±,n(z) (|ψ_±,n(z)| ∼CAⁿn!).

• ^The^{Borel sum}^ofψ±(as a formal series ofη⁻¹^): S[ψ±]=

∫ _∞

∓a(z)

e⁻^y^ηψ±,B(z,y)dy. Herea(z)=∫_z

z0

√Q(z⁰)dz⁰ and

ψ±,B(z,y)=∑^∞

n=0

ψ_±,n(z) Γ(n+¹₂)

(y±a(z))n−¹2 :Borel transformofψ±

• Borel transform=termwise inverse Laplace transform: (

c.f. η^−α=

∫ _∞

0

e⁻^y^ηy^α−¹

Γ(α)dy ifReα >0. )

• If the Borel sumsS[ψ±]are well-defined, they give analytic solutions of the Sch ¨odinger equation andS[ψ_±]∼ψ_±^whenη→+∞^.

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Borel resummation method

ψ_±(z, η)=e^±η

∫z z0

√Q(z⁰)dz⁰∑∞ n=0

η⁻ⁿ⁻¹²ψ_±,n(z) (|ψ_±,n(z)| ∼CAⁿn!).

• ^The^{Borel sum}^ofψ±(as a formal series ofη⁻¹^):

S[ψ±]=

∫ _∞

∓a(z)

z0

ψ±,B(z,y)=∑^∞

n=0

ψ_±,n(z) Γ(n+¹₂)

• Borel transform=termwise inverse Laplace transform: (

c.f. η^−α=

∫ _∞

0

e⁻^y^ηy^α−¹

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Borel resummation method

ψ_±(z, η)=e^±η

∫z z0

√Q(z⁰)dz⁰∑∞ n=0

η⁻ⁿ⁻¹²ψ_±,n(z) (|ψ_±,n(z)| ∼CAⁿn!).

S[ψ±]=

∫ _∞

∓a(z)

z0

ψ±,B(z,y)=∑^∞

n=0

ψ_±,n(z) Γ(n+¹₂)

• Borel transform=termwise inverse Laplace transform:

(

c.f. η^−α=

∫ _∞

0

e⁻^y^ηy^α−¹

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Borel resummation method

ψ_±(z, η)=e^±η

∫z z0

√Q(z⁰)dz⁰∑∞ n=0

η⁻ⁿ⁻¹²ψ_±,n(z) (|ψ_±,n(z)| ∼CAⁿn!).

S[ψ±]=

∫ _∞

∓a(z)

z0

ψ±,B(z,y)=∑^∞

n=0

ψ_±,n(z) Γ(n+¹₂)

• Borel transform=termwise inverse Laplace transform:

(

c.f. η^−α=

∫ _∞

0

e⁻^y^ηy^α−¹

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Stokes graph and Stokes segent

• Stokes graph:

∗ Vertices:turning points(i.e., zeros ofQ(z)) and singular points.

∗ Edges:Stokes curvesemanating from turning points.

(real one-dimensional curves defined byIm∫z √

Q(z⁰)dz⁰=const.) Stokes curves aretrajectoriesof the quadratic differentialQ(z)dz^⊗².

Q(z)=z. Q(z)=z(z+1)(z+i). Q(z)=(z−2)(z−3) z²(z−1)² .

Q(z)=1−z².

• Stokes segmentis a Stokes curve connecting turning points (= saddle trajectory ofQ(z)dz^⊗2).

• Stokes graph is said to besaddle-freeif it doesn’t contain Stokes segments.

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Stokes graph and Stokes segent

Q(z)=1−z².

• Stokes graph is said to besaddle-freeif it doesn’t contain Stokes segments.

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Stokes graph and Stokes segent

• Stokes graph is said to besaddle-freeif it doesn’t

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Stokes graph and Borel summability

.

Theorem (Koike-Sch ¨afke)

.

.. .

. .

Suppose that the Stokes graph issaddle-free. Then,

• ψ±(z, η)areBorel summable(as a formal series ofη⁻¹^{) on each} Stokes region(= a face of the Stokes graph).

• The Borel sumsS[ψ±](z, η)giveanalytic(in bothzandη) solutions of the Schr ¨odinger equation on each Stokes region satisfying

S[ψ±](z, η)∼ψ±(z, η) as η→+∞.

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Stokes graph and Borel summability

.

Theorem (Koike-Sch ¨afke)

.

. .

S[ψ±](z, η)∼ψ±(z, η) as η→+∞.

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Stokes graph and Borel summability

.

Theorem (Koike-Sch ¨afke)

.

.. .

. .

S[ψ±](z, η)∼ψ±(z, η) as η→+∞.

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Voros symbols

Again suppose that the Stokes graph issaddle-free. Then,

• An explicit connection formula for (Borel resummed) WKB solutions on Stokes curves emanating from a turning point of order 1 ([Voros 83], [Aoki-Kawai-Takei 91]).

• Connection formulas and monodromy matrices of WKB solutions are written by (the Borel sum of)Voros symbolse^W^β⁽^η⁾ande^V^γ⁽^η⁾, where

W_β(η)=

∫

β

(S_odd(z, η)−η√ Q(z))

dz, V_γ(η)= I

γS_odd(z, η)dz. (c.f., [Kawai-Takei 05,§^3]). ^Here

I S_±(z, η)= d

dzlogψ_±(z, η)=±η√

Q(z)+· · ·, and

Sodd(z, η)=1

2(S₊(z, η)−S₋(z, η))=η√

Q(z)+· · ·

I β∈H1(R,P;Z)(“path”), γ∈H1(R;Z)(“cycle”). R=Riemann surface of √

Q(z), P=the set of poles ofQ(z).

• Voros symbolse^W^β⁽^η⁾ande^V^γ⁽^η⁾(for any pathβand any cycleγ^{) are} Borel summableif the Stokes graph is saddle-free.

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Voros symbols

W_β(η)=

∫

β

(S_odd(z, η)−η√ Q(z))

dz, V_γ(η)= I

I S_±(z, η)= d

Q(z)+· · ·, and

Sodd(z, η)=1

2(S₊(z, η)−S₋(z, η))=η√

Q(z)+· · ·

I β∈H1(R,P;Z)(“path”), γ∈H1(R;Z)(“cycle”). R=Riemann surface of √

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Voros symbols

W_β(η)=

∫

β

(S_odd(z, η)−η√ Q(z))

dz, V_γ(η)= I

I S_±(z, η)= d

Q(z)+· · ·, and

Sodd(z, η)=1

2(S₊(z, η)−S₋(z, η))=η√

Q(z)+· · ·

I β∈H1(R,P;Z)(“path”), γ∈H1(R;Z)(“cycle”).

R=Riemann surface of √

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Voros symbols

W_β(η)=

∫

β

(S_odd(z, η)−η√ Q(z))

dz, V_γ(η)= I

I S_±(z, η)= d

Q(z)+· · ·, and

Sodd(z, η)=1

2(S₊(z, η)−S₋(z, η))=η√

Q(z)+· · ·

I β∈H1(R,P;Z)(“path”), γ∈H1(R;Z)(“cycle”).

R=Riemann surface of √

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Mutation of Stokes graphs

G_+δ

G0

G_−δ

(The figure describes a part of Stokes graph.)

• Suppose that the Stokes graphG0has aStokes segment.

• Consider theS¹-family of the potential: Q⁽^θ⁾(z)=e²ⁱ^θQ(z) (θ∈R). G_θ: Stokes graph forQ⁽^θ⁾(z).

• For any sufficiently smallδ >0,G_±δaresaddle-freesince the existence of the Stokes segment implies

∫

along Stokes segment

√Q(z)dz∈R,0

• S¹-action causes a “mutation of Stokes graphs” (= a discontinuous change of topology of Stokes graphs caused by a Stokes segment).

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Mutation of Stokes graphs

G_+δ

G0

G_−δ

∫

√Q(z)dz∈R,0

• S¹-action causes a “mutation of Stokes graphs” (= a discontinuous change of topology of Stokes graphs caused by a Stokes segment).

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Mutation of Stokes graphs

G_+δ G0 G_−δ

∫

√Q(z)dz∈R,0

• S¹-action causes a “mutation of Stokes graphs” (= a discontinuous

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DDP’s jump formula of Voros symbols

G_+δ G0 G_−δ

γ0

• Suppose thatG0has a Stokes segment connecting two distinct turning points, and no other Stokes segments.

• ^LetS[e^W

(θ)

β ],S[e^V^γ⁽^θ⁾]be the Borel sum of Voros symbols for Q⁽^θ⁾(z)and S±[e^W^β] := lim

θ→±0S[e^W

(θ)

β ], S±[e^V^γ] := lim

θ→±0S[e^V^γ^(θ)].

.

Theorem (Delabaere-Dillinger-Pham 93)

.

.. .

.

S₋[e^W^β]=S₊[e^W^β](1+S₊[e^V^γ0])^−hγ⁰^{, βi}, S₋[e^V^γ]=S₊[e^V^γ](1+S₊[e^V^γ0])^−hγ⁰^,γi.

Hereh, iis the intersection form (normalized ashx-axis,y-axisi= +1), and γ0is the cycle around the Stokes segment oriented asH

γ0

√Q(z)dz∈R_<0.

• This formula describes theStokes phenomenonfor Voros symbols.

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DDP’s jump formula of Voros symbols

G_+δ G0 G_−δ

γ0

• ^LetS[e^W

(θ)

β ],S[e^V^γ⁽^θ⁾]be the Borel sum of Voros symbols forQ⁽^θ⁾(z)and S±[e^W^β] := lim

θ→±0S[e^W

(θ)

β ], S±[e^V^γ] := lim

θ→±0S[e^V^γ^(θ)].

.

Theorem (Delabaere-Dillinger-Pham 93)

.

.. .

.

S₋[e^W^β]=S₊[e^W^β](1+S₊[e^V^γ0])^−hγ⁰^{, βi}, S₋[e^V^γ]=S₊[e^V^γ](1+S₊[e^V^γ0])^−hγ⁰^,γi.

γ0

√Q(z)dz∈R_<0.

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DDP’s jump formula of Voros symbols

G_+δ G0 G_−δ

γ0

• ^LetS[e^W

(θ)

θ→±0S[e^W

(θ)

β ], S±[e^V^γ] := lim

θ→±0S[e^V^γ^(θ)].

.

Theorem (Delabaere-Dillinger-Pham 93)

.

.. .

.

S₋[e^W^β]=S₊[e^W^β](1+S₊[e^V^γ0])^−hγ⁰^{, βi}, S₋[e^V^γ]=S₊[e^V^γ](1+S₊[e^V^γ⁰])^−hγ⁰^,γi.

γ0

√Q(z)dz∈R_<0.

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DDP’s jump formula of Voros symbols

G_+δ G0 G_−δ

γ0

• ^LetS[e^W

(θ)

θ→±0S[e^W

(θ)

β ], S±[e^V^γ] := lim

θ→±0S[e^V^γ^(θ)].

.

Theorem (Delabaere-Dillinger-Pham 93)

.

S₋[e^W^β]=S₊[e^W^β](1+S₊[e^V^γ0])^−hγ⁰^{, βi}, S₋[e^V^γ]=S₊[e^V^γ](1+S₊[e^V^γ⁰])^−hγ⁰^,γi.

γ0

√Q(z)dz∈R_<0.

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Exact WKB analysis

§2 Main results

Refferences

•

K. I and T. Nakanishi, “Exact WKB analysis and cluster algebras”, J. Phys. A: Math. Theor. 47 (2014) 474009.

•

K. I and T. Nakanishi, “Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras”, arXiv:1401.7094.

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Dictionary

Exact WKB analysis Cluster algebras

saddle-free Stokes graph skew-symmetric matirxB mutation of Stokes graphs mutation ofB (Borel sum of) Voros symbole^W^βi clusterx-variablexi

(Borel sum of) Voros symbole^V^γⁱ clustery-variabley_i e^η

H

γi

√Q(z)dz

coefficientri

Stokes phenomenon for Voros symbols mutation of cluster variables

W_β(η)=

∫

β

(Sodd(z, η)−η√ Q(z))

dz, V_γ(η)= I

γSodd(z, η)dz.

b⁰_{i j}=

−bi j i=korj=k

x⁰_i=



 xk−1





∏n

j=1

xj[−εb_jk]₊



(1+ykε) i=k

xi i,k.

y⁰_i=

yk−1 i=k

yiyk[εb_ki]₊(1+ykε)⁻^b^ki i,k.

∏

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Stokes graph

{

Skew-symmetric matrix

• A saddle-free Stokes graph

{^Atriangulated surface: (Three Stokes curve emanate from an order 1 turning point.) [Gaiotto-Moore-Neitzke 09]

• A triangulated surface{^A^quiver [Fomin-Shapiro-Thurston 08]:

∗ Put vertices on edges of triangulation.

∗ Draw arrows on each triangle in clockwise direction.

∗ Remove vertices on “boundary edges” together with attached arrows. (boundary / internal edge↔digon-type / rectangular Stokes region)

Stokes graph

{

Triangulated surface {

Quiver

• A quiver{Askew-symmetric matrixB=(bi j)ⁿ_i_,_j₌₁by b_{i j}=(]^{of arrows}◦i→ ◦j)−(]^{of arrows}◦j → ◦i) (Assign labelsi∈ {1, . . . ,n}to rectangular Stokes regions.)

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Stokes graph

{

Skew-symmetric matrix

• A saddle-free Stokes graph {^Atriangulated surface:

(Three Stokes curve emanate from an order 1 turning point.) [Gaiotto-Moore-Neitzke 09]

∗ Remove vertices on “boundary edges” together with attached arrows. (boundary / internal edge↔digon-type / rectangular Stokes region)

Stokes graph

{

Triangulated surface

{

Quiver

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Stokes graph

{

Skew-symmetric matrix

∗ Remove vertices on “boundary edges” together with attached arrows.

(boundary / internal edge↔digon-type / rectangular Stokes region)

Stokes graph

{

Quiver

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Stokes graph

{

Skew-symmetric matrix

∗ Remove vertices on “boundary edges” together with attached arrows.

(boundary / internal edge↔digon-type / rectangular Stokes region)

Stokes graph

{

Quiver

• A quiver{Askew-symmetric matrixB=(bi j)ⁿ_i,_j=1by b_{i j}=(]^{of arrows}◦i→ ◦j)−(]^{of arrows}◦j → ◦i)

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Muation of Stokes graph and quiver mutation

• S¹-family of potentials:Q⁽^θ⁾(z)=e²ⁱ^θQ(z).

• Mutation of Stokes graph{Quiver mutationatk-th vertex:

(k= label of Stokes region which “degenerates” to a Stokes segment under the mutation of Stokes graph)

G_+δ G_−δ

←→

µk

(Figures describes a part of Stokes graphs.)

• Quiver muation is compatible withmutationofB-matix: b⁰_{i j}=

b_{i j}+[b_ik]₊b_{k j}+b_ik[b_{k j}]₊ otherwise. ([a]₊=max(a,0))

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Exact WKB Analysis and Cluster Algebras

Exact WKB Analysis and

Cluster Algebras

Exact WKB analysis

Exact WKB analysis

Exact WKB analysis

Cluster algebras (of rank

Cluster algebras (of rank

Results and Application [I-Nakanishi 14]

Results and Application [I-Nakanishi 14]

Results and Application [I-Nakanishi 14]

Results and Application [I-Nakanishi 14]

Contents

Exact WKB analysis

Main results

A. Voros, “The return of the quartic oscillator. The complex WKB method”, Ann. Inst. Henri Poincar ´e

(1983), 211–338.

T. Kawai and Y. Takei, “Algebraic Analysis of Singular Perturbations”, AMS translation, 2005.

Contents

Main results

A. Voros, “The return of the quartic oscillator. The complex WKB method”, Ann. Inst. Henri Poincar ´e

(1983), 211–338.

T. Kawai and Y. Takei, “Algebraic Analysis of Singular

Perturbations”, AMS translation, 2005.

Schr ¨odinger equation and WKB solutions

Schr ¨odinger equation and WKB solutions

Schr ¨odinger equation and WKB solutions

Borel resummation method

Borel resummation method

Borel resummation method

Borel resummation method

Stokes graph and Stokes segent

Stokes graph and Stokes segent

Stokes graph and Stokes segent

Stokes graph and Borel summability

Theorem (Koike-Sch ¨afke)

Stokes graph and Borel summability

Theorem (Koike-Sch ¨afke)

Stokes graph and Borel summability

Theorem (Koike-Sch ¨afke)

Voros symbols

Voros symbols

Voros symbols

Voros symbols

Mutation of Stokes graphs

Mutation of Stokes graphs

Mutation of Stokes graphs

DDP’s jump formula of Voros symbols

Theorem (Delabaere-Dillinger-Pham 93)

DDP’s jump formula of Voros symbols

Theorem (Delabaere-Dillinger-Pham 93)

DDP’s jump formula of Voros symbols

Theorem (Delabaere-Dillinger-Pham 93)

DDP’s jump formula of Voros symbols

Theorem (Delabaere-Dillinger-Pham 93)

Contents

Exact WKB analysis

K. I and T. Nakanishi, “Exact WKB analysis and cluster algebras”, J. Phys. A: Math. Theor. 47 (2014) 474009.

K. I and T. Nakanishi, “Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras”, arXiv:1401.7094.

Dictionary

Stokes graph

Skew-symmetric matrix

Stokes graph

Skew-symmetric matrix

Stokes graph

Skew-symmetric matrix

Stokes graph

Skew-symmetric matrix

Muation of Stokes graph and quiver mutation