Strebel differentials and string field theory

JPS meeting at Hokkaido University
Nobuyuki Ishibashi (University of Tsukuba)
Sep. 17, 2024

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Quadratic differentials• On a punctured Riemann surface (∼Feynman diagram of string theory), let
us consider a quadratic differential ϕ(z)dz 2 such that• near punctures (z ∼ za (a = 1, ⋯n))ϕ(z)dz 2 ∼ − (La 2dz 2)2π(z − za )2

with La > 0 and holomorphic for z ≠ za

• One can define a locally flat metricds2=∣ϕ(z)∣ dzdz̄ = dwdw̄√zw = ∫ dz ′ ϕ(z ′ )2 / 12

Strebel’s theorem• Given a punctured Riemann surface and positive numbers L1 , ⋯, Ln , thereexists the unique quadratic differential ϕ(z)dz 2 (Strebel differential) suchthat2a• for z ∼ za , ϕ(z)dz 2 ∼ − ( L)
2π
• holomorphic for z ≠ za
• with the metricds2=dz 2(z−za )2∣ϕ(z)∣ dzdz̄ = dwdw̄
√zw = ∫ dz ′ ϕ(z ′ )the surface looks like3 / 12

Strebel differentials• To any punctured Riemann surface, there corresponds a graph called
critical graph.
• Moduli spaces of punctured Riemann surfaces can be perametrized by thelengths of the edges of the critical graphs (combinatorial moduli space).
• Strebel differentials play important roles in• Kontsevich’s proof of Witten conjecture• studying the free field limit of AdS/CFT (Gopakumar, ...)

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Strebel differentials and string field theory• Strebel differentials may be useful in constructing a string field theory.
• They were used to construct the interaction vertices of a closed bosonicstring field theory in Saadi-Zwiebach, Kugo-Kunitomo-Suehiro,Kugo-Suehiro.
• The nonpolynomial SFT reproduces the tree amplitudes.
• Strebel’s theorem and combinatorial moduli space are not relevant.

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Strebel differentials and string field theory• If Strebel differentials are really important in describing the free field limit
in AdS/CFT, it is worthwhile to construct an SFT using the combinatorial
moduli space as basic tool.
• We need to give up the conventional form of propagators.

• It is possible to construct such an SFT. (PTEP 2024 (2024) 7, 073B02)

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1. Combinatorial pants decomposition

1. Combinatorial pants decomposition• The critical graphs can be decomposed into three string vertices.
(combinatorial pants decomposition)• We may be able to construct a theory with

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Pants decomposition• A Riemann surface with boundaries (2g − 2 + n > 0) admits a hyperbolic
metric (R = −2) such that the boundaries are geodesics.
• It can be decomposed into pairs of pants whose boundaries are geodesics.

• The critical graphs in the combinatorial moduli space can be regarded asthe long string limit of the hyperbolic surfaces. (Mondello, Do)

8 / 12SFT• The combinatorial pants decomposition can be considered as the longstring limit of the hyperbolic pants decomposition.

• An SFT based on the hyperbolic pants decomposition was constructed(N.I.).
• The construction is based on the Fenchel-Nielsen coordinates, Mirzkhani’srecursion relation ...

• The combinatorial version of all these were defined in Andersen et al..
• It is possible to construct an SFT using these results.

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2. The Fokker-Planck formalism

2. The Fokker-Planck formalism
In order to construct the theory, we need to employ the Fokker-Planckformalism• Euclidean field theory⟨ϕ(x1 )⋯ϕ(xn )⟩ =−S[ϕ]ϕ(x1 )⋯ϕ(xn )∫ [dϕ]e∫ [dϕ]e−S[ϕ]• Fokker-Planck formalism⟨ϕ(x1 )⋯ϕ(xn )⟩ = lim ⟨0∣e−τ ĤFP

τ →∞ϕ̂(x1 )⋯ϕ̂(xn )∣0⟩[π̂(x), ϕ̂(y)] = δ(x − y) , [π̂, π̂] = [ϕ̂, ϕ̂] = 0⟨0∣ϕ̂(x) = π̂(x)∣0⟩ = 0ĤFP = − ∫ dx (π̂(x) +δS[ϕ̂]) π̂(x)δϕ(x)• path integral⟨ϕ(x1 )⋯ϕ(xn )⟩ =IFP = ∫0−I
∫ [dπdϕ]e FP ϕ(0, x1 )⋯ϕ(0, xn )−I∫ [dπdϕ]e FP∞dτ [− ∫ dxπ∂τ ϕ + HFP ]10 / 12SFTIFP=∞∫0∞dτ [− ∫0+∫0H=∫−∞
012dL⟨R∣πα (τ, L)⟩∞∂ α∣ϕ (τ, L)⟩ + H(τ )∂τQααTdL (⟨R∣Q (τ, L)⟩∣λα (τ, L)⟩ + ⟨R∣T (τ, L)⟩∣λα (τ, L)⟩) ]αdLL [⟨R12 ∣ϕ (L)⟩1 ∣πα (L)⟩2 − ⟨R12 ∣πα (L)⟩1 ∣π−α (L)⟩2 ]gs ∫0∞

dL3 ∫L30dL1 ∫L3 −L10dL2 (L3 − L1 − L2 )123αα× ⟨G0,3,L ∣B−α B−α Bα ∣ϕ 1 (L1 )⟩1 ∣ϕ 2 (L2 )⟩2 ∣πα3 (L3 )⟩31−gs ∫∞0dL2 ∫∞dL3 ∫L2 +L3∣L2 −L3 ∣023dL1 (L2 + L3 − L1 )123α× ⟨G0,3,L ∣B−α Bα Bα ∣ϕ 1 (L1 )⟩1 ∣πα2 (L2 )⟩2 ∣πα3 (L3 )⟩31−gs ∫0∞

dL2 ∫∞0dL3 ∫0∣L2 −L3 ∣23dL1 min(L2 , L3 )123α
× ⟨G0,3,L ∣B−α Bα Bα ∣ϕ 1 (L1 )⟩1 ∣πα2 (L2 )⟩2 ∣πα3 (L3 )⟩3 ,123

• This action consists of kinetic terms and three string interaction terms.
• One can calculate the amplitudes perturbatively starting from this action.
• It is manifestly invariant under the BRST transformation and we can define
the physical states using it.
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3. Conclusions

5. Conclusions• We have constructed an SFT for closed bosonic strings based on theStrebel differentials via the Fokker-Planck formalism.
IFP=

∫0∞

dτ [− ∫∞0+∫0dL⟨R∣πα (τ, L)⟩∞α∂ α∣ϕ (τ, L)⟩ + H(τ )∂τQαTdL (⟨R∣Q (τ, L)⟩∣λα (τ, L)⟩ + ⟨R∣T (τ, L)⟩∣λα (τ, L)⟩) ]• AdS/CFT?
• Our method may be used to study the classical solutions of the
nonpolynomial SFT. (hyperbolic case: Firat-Valdes-Meller)
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