The Fokker-Planck formalism for closed bosonic strings

Talk at SFT@Cloud
Nobuyuki Ishibashi (University of Tsukuba)
March 9, 2023

1 / 39String Field Theory (SFT)• The amplitudes in string theory are expressed by Feynman diagrams =worldsheets∼Riemann surfaces• In order to construct an SFT, we should define a rule to cut all theworldsheets into propagators and vertices systematically.
• In general, we need infinitely many vertices to do so.
̵ +⋯S = φKφ + φ3 + φ4 + ⋯ + hφ
• Such a theory can be studied by using the homotopy algebra methods.
(Zwiebach, ...)

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SFT with only three string vertices

We would like to find out a way to construct an SFT as simple as
S = φKφ + φ3

• So far, there exist essentially two known rules for which the theory looks
like that.

• We would like to find out yet another rule.
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SFT with only three string vertices• SFT’s for bosonic strings were constructed using these rules.
S = φKφ + φ3
• Light-cone gauge SFT(Kaku-Kikkawa), α = p+ HIKKO (Kugo-Zwiebachtheory), covariantized light-cone• Witten’s SFT• These rules do not work for superstrings, because of the “spurioussingularity” problem.
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The pants decomposition• A Riemann surface with 2g − 2 + n > 0 admits a hyperbolic metric such thatthe boundaries are geodesics. (cf. Moosavian-Pius, Costello-Zwiebach)• It can be decomposed into pairs of pants whose boundaries are geodesics.

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An SFT based on the pants decomposition?

• We may be able to construct an SFT based on the pants decomposition atleast for closed bosonic strings
S = φKφ + φ3
• The SFT will be quite different from the usual ones.
• The string field ∣φ(L)⟩ depends on the length L of the string• We may consider ∣φ(L)⟩ as an operator from which we can derive various
properties of the particles.

• The kinetic term should be different from the conventional one⟨φ∣Qc+0 ∣φ⟩

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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.

• The decompositions are related by modular transformations.
• Most of the amplitudes diverge in the same way.

• We cannot construct the action.
We should take an alternative approach.Ð→the Fokker-Planck formalism

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The Fokker-Planck formalism• Euclidean field theory: action S[φ]⟨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: action IFP [φ, π]⟨φ(x1 )⋯φ(xn )⟩ =−I∫ [dπdφ]e FP φ(0, x1 )⋯φ(0, xn )−I∫ [dπdφ]e FP∞

IFP = ∫0dτ [− ∫ dxπ∂τ φ + HFP ]8 / 39In this talk• I would like to show that it is possible to construct an SFT for closed
bosonic strings based on the pants decomposition via the Fokker-Planckformalism.
IFP [φ, π, λ]∞=∫0∞dτ [− ∫dL⟨R∣πα (τ, L)⟩0∞

+∫0α

∂ α
∣φ (τ, L)⟩ + H(τ )∂τQαTdL (⟨R∣Q (τ, L)⟩∣λα (τ, L)⟩ + ⟨R∣T (τ, L)⟩∣λα (τ, L)⟩) ]T• λQ
α , λα : auxiliary fields• This action consists of kinetic terms and three string interaction terms.
• S[φ] is not well-defined in our setup.

Based on PTEP 2023,023B05 (2023)

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This talk

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Plan of the talk1. Mirzakhani recursion

2. A recursion relation for the off-shell amplitudes of closed bosonic strings

3. The Fokker-Planck formalism

4. BRST invariant formulation

5. Conclusions

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1. Mirzakhani recursion

1. Mirzakhani recursion• Reviews: Moosavian-Pius, Do arXiv:1103.4674 [math], Huang
arXiv:1509.06880 [math.GT]

12 / 39Mirzakhani recursion

The volume of the moduli space of Riemann surfaces with genus g and nboundaries (2g − 2 + n > 0) whose lengths are L1 , ⋯, Ln is given byVg,n (L1 , ⋯, Ln ) = ∫ ∏ [sls dls dθs]
2π

• The moduli space of Riemann surfaces (genus g, n boundaries) isparametrized by the coordinates (ls ; θs ) (s = 1, ⋯, 3g − 3 + n).
• ls denotes the length of a nonperipheral boundary and θs is the twist angle
in a pants decomposition.

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Modular invarianceVg,n (L1 , ⋯, Ln ) = ∫ ∏ [sls dls dθs]
2π• Integrating over 0 < ls < ∞, the integral diverges.
• The pants decomposition is not unique. There are infinitely many pants
decomposition related by modular transformations.

• We should integrate over the fundamental domain F, which is very
complicated in general.
Vg,n (L1 , ⋯, Ln ) = ∫ ∏ [Fsls dls dθs]
2π14 / 39

McShane identity (g = n = 1, L = 0)2
• McShane identity (1998): for f (l) = 1+el

1=∑f (γ ⋅ l)γ∈modular group• V1,1 can be calculated multiplying this by ∫F ldldθ
(Mirzakhani)
2π
V1,1 (0)=∫F=∫ldldθldldθ= ∫ ∑ f (γ ⋅ l)
F γ2π2π∑ f (γ ⋅ l)

F γ=∫γ ⋅ ld(γ ⋅ l)d(γ ⋅ θ)ldldθ
= ∑∫f (l)γF2π2πγπ2dldθl 2=2π 1 + el
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Generalized McShane identity• Mirzakhani obtained identities for general g, n with 2g − 2 + n > 0.

16 / 39Mirzakhani recursion relation

MultiplyingnL1 =∑DL1 lγ lδ + ∑ ∑ (TL1 La lγ + DL1 La lγ ){γ,δ}∈C1

a=2 γ∈Cas dθs], we getby ∫F ∏s [ ls dl2π• One can calculate Vg,n (L1 , ⋯, Ln ) by solving this equation.
• The right hand side consists of quantities with less 2g − 2 + n.

17 / 39Mirzakhani recursion relation

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2. A recursion relation for the off-shell
amplitudes of closed bosonic strings

2. A recursion relation for the off-shell amplitudes of closed bosonic strings

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Amplitudes in string theory• In string theory, the amplitudes are given by integrals over the moduli
space of Riemann surfaces
Ag,n = ∫ ∏ [dls dθs ] ⟨∏ [b(∂ls )b(∂θs )] Vi1 ⋯Vin ⟩Fss

• It is conceivable that we can derive a recursion relation for theseamplitudes in the same way as we did for the recursion relation forVg,n (L1 , ⋯, Ln ) = ∫ ∏ [Fsls dls dθs]
2π20 / 39

The recursion relation

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Details 1:The off-shell amplitudes• The off-shell amplitudes on Σ can be defined using gr′∞ Σ.
(Costello-Zwiebach)Ag,n = ∫ ∏ [dls dθs ] ⟨∏ [b(∂ls )b(∂θs )] Vi1 ⋯Vin ⟩Fss

• We can use the coordinates ls , θs to parameterize the moduli space of thepunctured Riemann surface. (Mondello)• For states ∣ϕia ⟩ = Oia (0)∣0⟩ in the state space of the bosonic string (in anybackground), satisfying(b0 − b̄0 )∣ϕia ⟩ = (L0 − L̄0 )∣ϕia ⟩ = 022 / 39

Details 2: b-ghost insertions

Ag,n = ∫ ∏ [dls dθs ] ⟨∏ [b(∂ls )b(∂θs )] Vi1 ⋯Vin ⟩Fss• b(∂ls ), b(∂θs ) are constructed following the standard prescription. (Sen2015, Erbin’s book, ...)• They are expressed by the variations of the transition functions between
local patches.
• In our case, we can take the patches to be the pairs of pants.
• Since a pair of pants∼ C − ∪k Dk , we take z on C as the local coordinate.

23 / 39b-ghost insertions• The explicit forms of Wk (z) are given in terms of the hypergeometricfunction (Fırat, Hadasz-Jaskolski)• b(∂l ) has contributions from two adjacent pairs of pants.
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Details 3: The recursion relation

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The recursion relation• The string field is labeled by (i, L, α) ≡ I (α = ±)26 / 39

3. The Fokker-Planck formalism

3. The Fokker-Planck formalism

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The Fokker-Planck formalism

1 ⋯In• One can derive the amplitudes AIg,nperturbatively solving this equation.
• This equation can be regarded as the Schwinger-Dyson equation of string
theory.

• We may be able to construct an SFT from this equation.

• This equation can be turned into an SFT in the FP formalism via themethod developed by Kawai-NI, Jevicki-Rodrigues,
Fukuma-Kawai-Ninomiya-NI,
Ikehara-Kawai-Mogami-Nakayama-Sasakura-NI , Ikehara, .....

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The Fokker-Planck formalism for closed bosonic strings

ĤFP = T̂ I π̂I29 / 39

The Fokker-Planck formalism for closed bosonic strings

ĤFP=T̂ I π̂I=−Lπ̂I π̂I ′ GI I + Lφ̂I π̂I′′′′ ′′1− gs DII I GI ′′ K ′′ GI ′ K ′ φ̂K φ̂K π̂I2′′ ′′′′−gs T II I GI ′′ K ′′ φ̂K π̂I ′ π̂IT̂ I=′−Lπ̂I ′ GII + Lφ̂I′ ′′′′′
1− gs DII I GI ′′ K ′′ GI ′ K ′ φ̂K φ̂K2′ ′′′′

−gs T II I GI ′′ K ′′ φ̂K π̂I ′⟨⟨φI1 ⋯φIn ⟩⟩ = lim ⟨⟨0∣e−τ ĤFP φ̂I1 ⋯φ̂In ∣0⟩⟩τ →∞• The Hamiltonian consists of the kinetic terms and the three stringinteraction terms.

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The action S[φ]• It is possible to (formally) define the action S[φ].
e−S[φ]= lim ⟪0∣e−τ Ĥ δ(φ − φ̂)∣0⟫
−S[φ]
τ →∞
∫ [dφ]e−S[φ]InI

φ 1 ⋯φ∫ [dφ]e∫ [dφ]e−S[φ]= lim ⟪0∣e−τ Ĥ∫ [dφ]δ(φ − φ̂)φ= lim ⟪0∣e−τ Ĥφ̂ 1 ⋯φ̂ n ∣0⟫τ →∞τ →∞I1II⋯φ n ∣0⟫I• One can calculate S[φI ] perturbatively.
S[φI ]=′ ′′1gsJ ′′ J ′ JIφ φ
GIJ φI φJ − AII
0,3 GIJ GI ′ J ′ GI ′′ J ′′ φ
26gs II ′ I ′′+ TGI ′ I ′′ GIJ φJ + O(gs2 )L′′
δS[LGIJ + gs T IJI GI ′ J ′ φJ ] J
δφ′ ′′′′′′ ′′1= LφI − gs DII I GI ′ J ′ GI ′′ J ′′ φJ φJ + gs T II I GI ′ I ′′231 / 39

The action S[φ]• S[φI ] is divergent and ill defined.
• The 1 loop 1 point amplitude• S[φI ] includes infinitely many divergent counterterms.

• FP formalism breaks the modular invariance.

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4. BRST invariant formulation

4. BRST invariant formulation

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BRST symmetry on the worldsheet• We need the worldsheet BRST symmetry to define the physical states withpositive norm.
Q∣phys.⟩=0∣⟩∼∣ ⟩ + Q∣ ⟩′• In order to discuss this symmetry, we change the notationα∣φ (L)⟩≡Ic∑ φ̂ ∣ϕi ⟩i∣πα (L)⟩≡∑ ∣ϕi ⟩π̂IiĤFP=∫∞0αdLL [⟨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 )⟩3123−12123ααgs ∫ dL1 dL2 dL3 ⟨DL3 L1 L2 ∣B−α B−α Bα ∣φ 1 (L1 )⟩1 ∣φ 2 (L2 )⟩2 ∣πα3 (L3 )⟩3123• The BRST transformation−+δ ∣φ+ (L)⟩ = 12 c−0 b0 P Q∣φ (L)⟩+δ ∣φ− (L)⟩ = Q∣φ− (L)⟩ − b−0 P ∂L ∣φ (L)⟩δ ∣π+ (L)⟩ = Q∣π+ (L)⟩ − b−0 P ∂L ∣π− (L)⟩−δ ∣π− (L)⟩ = 21 c−0 b0 P Q∣π− (L)⟩34 / 39Ĥ is not BRST invariant• ĤFP is not BRST invariant.
• If it were, FP formalism would be modular invariant• Let Q̂ be the generator of the BRST transformation
δ ĤFP = [Q̂, ĤFP ] = ∫∞0dL (⟨R∣Qα (L)⟩∣πα (L)⟩ + ⟨R∣T α (L)⟩[Q̂, ∣πα (L)⟩])∞ĤFP = ∫0ααdL⟨R∣T (L)⟩∣πα (L)⟩α∣Q (L)⟩ ≡ [Q̂, ∣T (L)⟩]• The amplitudes are invariant, because ∣Qα (L)⟩, ∣T α (L)⟩ are “nullquantities” satisfying[ lim ⟪0∣e−τ ĤFP ] ∣T α (L)⟩ = 0τ →∞[ lim ⟪0∣e−τ ĤFP ] ∣Qα (L)⟩ = 0τ →∞and do not contribute in limτ →∞ ⟨⟨0∣e−τ ĤFP φ̂I1 ⋯φ̂In ∣0⟩⟩.
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BRST invariant formulation• We can modify the Hamiltonian by introducing the auxiliary fields
T∣λQα (L)⟩, ∣λα (L)⟩ so that it becomes BRST invariant and still yields thecorrect amplitudes.
∞ĤFP → ĤFP + ∫∞

δ ĤFP = ∫αα

QTdL (⟨R∣Q (L)⟩∣λα (L)⟩ + ⟨R∣T (L)⟩∣λα (L)⟩)

0ααdL (⟨R∣Q (L)⟩∣πα (L)⟩ + ⟨R∣T (L)⟩[Q̂, ∣πα (L)⟩])0• The action
∞IFP=

∫0∞

dτ [− ∫dL⟨R∣πα (τ, L)⟩0∞

+∫0α

∂ α
∣φ (τ, L)⟩ + H(τ )∂τQαTdL (⟨R∣Q (τ, L)⟩∣λα (τ, L)⟩ + ⟨R∣T (τ, L)⟩∣λα (τ, L)⟩) ]• This action is invariant under the BRST transformation.
• It consists of kinetic terms and three string interaction terms.

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5. Conclusions

5. Conclusions

IFP [φ, π, λ]∞=∫0∞dτ [− ∫dL⟨R∣πα (τ, L)⟩0∞

+∫

0α

∂ α
∣φ (τ, L)⟩ + H(τ )∂τQαTdL (⟨R∣Q (τ, L)⟩∣λα (τ, L)⟩ + ⟨R∣T (τ, L)⟩∣λα (τ, L)⟩) ]• We have constructed an SFT for closed bosonic strings based on the pants
decomposition via the Fokker-Planck formalism.
• The action consists of kinetic terms and three string interaction terms.
• It is manifestly invariant under a nilpotent BRST transformation and we candefine the physical states using it.

• How can one interpret the procedure to select the physical states in termsof the 2nd quantized language?
• SFT for superstrings?

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