Onogi-san as a string theorist

Nobuyuki Ishibashi (University of Tsukuba)
Sept. 22, 2022

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Outline

1. Onogi-san started his career as a string theorist.

2. Mirzakhani’s idea

3. Fokker-Planck formalism for closed bosonic strings

4. Conclusion

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1. Onogi-san started his career asa string theorist.

Graduate school days (1984-1989)• Onogi-san was my classmate in University of Tokyo and weentered the graduate school of University of Tokyo in 1984.
• 1984 is the year of “the first superstring revolution”. We(almost all the students at that time) studied and worked on
string theory.
• Onogi-san had a good reason to start his career as a string
theorist.
• His mentor was Yoneya-san.
• Yoneya-san is one of the fathers of string theory. He discoveredthat string theory can be used to describe quantum gravity.

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His first paper• Onogi-san’s first paper is with Ito-san (TIT).
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One-loop amplitude for strings• Amplitudes are calculated using the “Feynman rule”• One-loop Feynman graph• The amplitude would be A ∼R∞0dT

R 2π0dθhV1 · · · Vn i
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Modular invariance

Z ∞A∼

Z 2πdθhV1 · · · Vn idT00• This is divergent because of the modular invarianceτ→aτ + b, a, b, c, d ∈ Z, ad − bc = 1cτ + d
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Modular invariance• modular transformation τ → −1
τ=
• We should mod out this symmetry

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Light-cone gauge string field theory (SFT)• The procedure in the previous slides is rather ad hoc.
• It is better to have a formulation in which we get the rightanswer without modding out the symmetry.
• The light-cone gauge SFT (Kaku-Kikkawa)
ZS = [φ(∂x+ − H)φ + φ3 ]8 / 24

Ito-Onogi• One-loop Feynman diagram for the light-cone gauge SFT• They find the map (T 0 , θ0 ) → τ and show
Z ∞dT00Z 2π0Zdθ hV1 V2 i =0

hV1 V2 iM• The light-cone gauge SFT yields the right answer.
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Yoneya-san’s oracle• String theory we have at present is like the “old quantumtheory”.
• We should discover the “Schrödinger equation” for stringtheory.
• The light-cone gauge SFT will provide some insight into the“Schrödinger equation” .
• Yoneya-san himself wrote a paper on space-time local
symmetry of the light-cone gauge SFT.

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Light-cone gauge SFT• Onogi-san studied the Lorentz algebra of the light-cone gaugeSFT. (unpublished)• from Kugo-san’s paper “Lorentz Transformation in theLight-Cone Gauge String Field Theory”

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Dynamics• Besides SFT things, Onogi-san worked on various aspects ofstring theory.
• We wrote two papers together on open string theory.
• After getting Ph.D, we both went to KEK as JSPS posdocs.
• At KEK, we had a lot of seminars and journal club talks onphenomenology, lattice theory, cosmology etc..
• Onogi-san was more and more inclined to study dynamics,
probably influenced by people like Higashijima-san andOkawa-san.
• He started working on Nambu-Goto string on lattice.
Hashimoto-san may tell us the rest of the story.

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2. Mirzakhani’s idea

Yoneya-san’s oracle: 35+ years later• String theory we have at present is like the “old quantumtheory”.
• We should discover the “Schrödinger equation” for stringtheory.
• Have we got the “Schrödinger equation” for string theory?
−→Not yet (in my opinion)• The problem Onogi-san tackled is important and fundamental
but actually very difficult.

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String field theory• For bosonic strings, we have SFT with a simple action
S = φQφ + φ3
• (α = p+ ) HIKKO, covariantized light-cone• Witten’s SFT• Zwiebach proposed a SFT for closed bosonic strings withinfinitely many interactions:
S = φQφ + φ3 + φ4 + · · · + ~φ3 + · · ·• If one tries to formulate SFT for superstrings, one runs into
the “spurious singularity” problem. One should follow
Zwiebach’s formulation. Sen constructed such a SFT.
• We may need some new idea.
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Volume of the moduli space• In string theory, amplitudes are given by integrals over themoduli space of Riemann surfaces
ZAg,n =dXhV1 · · · Vn iMg,n• Mg,n = Tg,n /modular group• Mathematicians were interested in calculating the volume ofthe moduli space
ZVg,n =dXMg,n15 / 24

Volume of the moduli space• Computing Vg,n is very difficult for general g, n, because thefundamental domain of the modular group is very complicated.
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Mirzakhani’s idea

• Suppose we have a function f (X) satisfying
Xf (γX) = 1

γ∈modular group

ZdXf (X)=TXZ
γdXf (X) =γM

Z=dX
M

XZ
γX
γd(γX)f (γX)

M

Zf (γX) =dX = VM

• The integral over M is unfolded to that over T .
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McShane identity (g = n = 1)

• Cut the surface along a closed geodesic (hyperbolic metric)• T ←→ l, θ plane.
2• McShane identity (1998): for f (X) = f (l, θ) = 1+elXf (γX) = 1γ∈modular group

ZV =dldθl 2π2=2π 1 + el
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Mirzakhani’s idea

• Mirzakhani generalized the McShane identity to general (g, n)
and derived a recursion relation for Vg,n . (2007)• Solving the recursion relation, one can compute Vg,n .

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3. Fokker-Planck formalism forclosed bosonic strings

String field theory• So far, we were looking for SFT which yields amplitudes as
integrals over M.
• Is it possible to construct an SFT which yields amplitudes as
integrals over T following Mirzaknani’s idea?
• Yes, but we need to use the Fokker-Planck formalism.
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Fokker-Planck formalism• Euclidean field theoryRhφ(x1 ) · · · φ(xn )i =[dφ]e−S[φ] φ(x1 ) · · · φ(xn )R[dφ]e−S[φ]• Fokker-Planck formalism
hφ(x1 ) · · · φ(xn )i = lim h0|e−τ ĤFP φ̂(x1 ) · · · φ̂(xn )|0i
τ →∞[π̂(x), φ̂(y)] = δ(x − y) , [π̂, π̂] = [φ̂, φ̂] = 0h0|φ̂(x) = π̂(x)|0i = 0ZĤFP = − dx π̂(x) +δS[φ̂] π̂(x)δφ(x)• path integral[dπdφ]e−IFP φ(0, x1 ) · · · φ(0, xn )R[dπdφ]e−IFPZ ∞Z
IFP =dτ − dxπ∂τ φ + HFP
Rhφ(x1 ) · · · φ(xn )i =021 / 24

IFP for closed bosonic strings

Z ∞IFP=0 Z ∞∂αdτ −dLhR|πα (L)i|φ (L)i∂τ0Z ∞α+dLL hR|φ (L)i|πα (L)i − hR|πα (L)i|π−α (L)i0Z− gs000α00

00

0dLdL dL 123 hTLL0 L00 |b−α00 bα0 bα |φ(L )i1 |πα0 (L )i2 |πα (L)i3Z
1α00α00
00000(L )i2 |πα (L)i3dLdL dL 123 hDLL0 L00 |b−α0 b−α00 bα |φ (L )i1 |φ− gs2Z ∞αQ
αT+dL hR|Q (τ, L)i|λα (τ, L)i + hR|T (τ, L)i|λα (τ, L)i0Z ∞IFP ∼dτ −π∂τ φ + πφ + π 2 + φ2 π + φπ 2 + λJ

0

• λ:auxiliary fields• J ∼ φ + π + φπ + φ2

• IFP is invariant under nilpotent BRST symmetry
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S[φ] for strings?
• In the conventional field theory, we have two formalisms.
R[dφ]e−S[φ] φ(x1 ) · · · φ(xn )R
hφ(x1 ) · · · φ(xn )i =[dφ]e−S[φ]R[dπdφ]e−IFP φ(0, x1 ) · · · φ(0, xn )R=[dπdφ]e−IFP• S[φ] is not well-defined in our setup.

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4. Conclusion

Conclusion• Onogi-san started his career working on string field theory.
• There is still much to be done in this field of study.

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