The Fokker-Planck formalism for closed bosonic strings

JPS meeting at Tohoku University
Nobuyuki Ishibashi (University of Tsukuba)
Sep. 18, 2023

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String Field Theory (SFT)• Second-quantized theory of strings.
• In order to construct an SFT, we need a rule to decompose worldsheets(Riemann surfaces) into propagators and vertices.

• In other words, we need to find fundamental building blocks from which all
the Riemann surfaces are generated.
• In general, we need infnitely many such blocks and the SFT action tends
to be very complicated.
S = φKφ + φ3 + ⋯2 / 12

SFT

S = φKφ + φ3 + ⋯• SFT people nowadays are not afraid of such an action.
• Zwiebach, ... developed a sophisticated machinery to deal with such anaction.
• It seems that the action for superstring field theory should be like that todeal with the spurious singularity problem. (Sen, ...)

• It is very difficult to solve the equation of motion for S.
• There are two known rules (light-cone, Witten) for which the action
becomes very simple.
S = φKφ + φ3• SFT for bosonic strings were constructed and exact solutions for eom. werediscovered.
• These rules are not good for superstrings.

• It may be helpful to find out yet another rule to decompose Riemannsurfaces such that the SFT becomes simple.
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This talk
• We construct an SFT using the pants decomposition of Riemann surfaces.
• In order to do so, we employ the Fokker-Planck (FP) formalism.
• We get an SFT for bosonic strings with FP action which consists ofkinetic term and three string vertices.
PTEP 023B05(2023)
1. Pants decomposition
2. Mirzakhani’s idea
3. The Fokker-Planck formalism
4. Conclusions

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1. Pants decomposition

The pants decomposition• A Riemann surface with 2g − 2 + n > 0 admits a hyperbolic metric suchthat the boundaries are geodesics.
• It can be decomposed into pairs of pants whose boundaries are geodesics.
• This fact implies that we may be able to construct an SFT with
S = φKφ + φ3
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An SFT based on the pants decomposition?

S = φKφ + φ3
• This action does not work. (D’Hoker-Gross)• One-loop one point amplitudes diverge because the pants decomposition isnot unique. There are infinitely many pants decompositions related bymodular transformations.
• Most of the amplitudes diverge in the same way.

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2. Mirzakhani’s idea

2. Mirzakhani’s idea
• Mathematicians were interested in calculating the volume of the moduli
space of Riemann sufaces of genus g and with n boundaries.

• Integrating over 0 < l < ∞, the integral diverges.
• The pants decomposition is not unique. There are infinitely many pants
decompositions related by modular transformations.

• We should integrate over the fundamental domain F, which is very
complicated in general.
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McShane identity (g = n = 1, L = 0)2
• McShane identity: for f (l) = 1+el

1 = f (l1 ) + f (l2 ) + f (l3 ) + ⋯• V1,1 can be calculated using this identity (Mirzakhani)
ldldθ
dldθlπ2=∫f (l) =2π6F 2π

V1,1 (0)= ∫• Mirzakhani generalized the McShane identity and construct a recursion
relation for Vg,n .
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3. The Fokker-Planck formalism

3. The Fokker-Planck formalism• We would like to construct an SFT based on the pants decomposition
using the Mirzakhani’s idea.

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The Fokker-Planck formalism• 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

SFT• We have constructed an SFT for closed bosonic strings based on the pants
decomposition via the Fokker-Planck formalism.
=IFP∫0∞dτ [− ∫∞0+∫0H=∫0∞dL⟨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 [⟨R∣φ (L)⟩∣πα (L)⟩ − ⟨R∣πα (L)⟩∣π−α (L)⟩]123α−gs ∫ dL1 dL2 dL3 ⟨TL2 L3 L1 ∣B−α Bα Bα ∣φ 1 (L1 )⟩1 ∣πα2 (L2 )⟩2 ∣πα3 (L3 )⟩31231αα
123− gs ∫ dL1 dL2 dL3 ⟨DL3 L1 L2 ∣B−α B−α Bα ∣φ 1 (L1 )⟩1 ∣φ 2 (L2 )⟩2 ∣πα3 (L3 )⟩31232• 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 a nilpotent BRST transformation and we candefine the physical states using it.
• S[φ] diverges.

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4. Conclusions

5. Conclusions• We have constructed an SFT for closed bosonic strings based on the pants
decomposition 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)⟩) ]• 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 a nilpotent BRST transformation and we candefine the physical states using it.
• S[φ] diverges.

• SFT for superstrings?

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