Quantum closed superstrings & hyperbolic geometry

Roji Pius
Perimeter Institute for Theoretical Physics

Strings 2018
OIST, Okinawa

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String field theory
The scattering amplitudes in string theory are defined as the sum over all possible string
diagrams, and string field theory decomposes this into a sum over Feynman diagrams.

Identifies certain string diagrams as elementary interactions, and the remaining string
diagrams can be constructed by gluing elementary string diagrams with string propagators.

Using the elementary string diagrams, SFT constructs an action for string theory.

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PlanDiscuss an explicit characterization of the elementary closed string diagrams, and describe
the effective expression of the quantum action for closed superstring field theory, byexploring the hyperbolic geometry of string diagrams.

arXiv:1703.10563, 1708.04977, 1706.07366: Seyed Faroogh Moosavian, R.P.
To appear: R.P.

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Elementary string diagrams
Elementary string diagrams can’t be constructed by gluing other diagrams with propagators.

It is equipped with a choice of local coordinates wi around the punctures and a distribution ofpicture changing operators (PCOs).

Local coordinates must be invariant under the permutation of the punctures, and the PCO
distribution must satisfy the following requirements:

I Invariant under the permutation of the punctures (NS and R separately).
I Avoid the occurrence of unphysical poles.
I Invariant under the action of the large diffeomorphisms (MCG) on the string diagram.

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Closed string propagatorpq

A closed string propagator is a specific gluing of closed string diagrams.

Identifying the local coordinates wp and wq in the neighbourhoods of two punctures p and q
on two different closed string diagrams using the following rule provide family of diagrams:

wp wq = tGluing for st=e−s+i θ≥ 0 is the closed string propagator having length s.
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String vertices
The set of all elementary string diagrams with h handles, m number of NS and n number ofR punctures is called as the string vertex

Vh,m,n .

Closed SFT can be consistently quantized using the Batalin-Vilkovisky formalism only if

Vh,m,n together with the diagrams F obtained by gluing elementary diagrams using string
propagators generate a single cover of Mh,m+n :SS S
Mh,m+n = Vh,m,n Fh1,m,n · · · FhQ,m,nJ

Fh,m,n : String diagram with J propagators. (Zwiebach)

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Gluing compatibility

t-channels-channelvertex

u-channel

String vertices decomposes the moduli space into cells, and the integral of string measureover each cell is the contribution from a specific Feynman diagram.

This consistency condition demands that the local coordinates and PCO distribution on thediagrams at the boundary of a string vertex must match with that induced on the glued
diagrams with zero length propagators.
7/21Minimal area metricComplete set of consistent string vertices can be constructed by using string diagrams withmetric of least possible area under the condition that lengths of all the nontrivial closedπcurves be longer than or equal to 2 . (Zwiebach)Current understanding of minimal area metrics is limited, and have to depend on numerics.
(N.Moeller; Wolf, Zwiebach; Headrick, Zwiebach)Claim: complete set of string vertices can be constructed using string diagrams with metrichaving−1 constant curvature.

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Hyperbolic geometry of string diagrams

glue

pull tighttwist

A hyperbolic string diagram with n punctures and h handles, can be obtained by considering
the proper discontinuous action of the Fuchsian group on the Poincaré upper half-plane.

Every such hyperbolic string diagrams can be obtained by the geometric sum of

2h − 2 + n number of hyperbolic pairs of pants.

` of the boundary and the twist
τ performed before gluing them. There are 3h − 3 + n attaching sites:
Each attaching site has two parameters: the geodesic length(τj , `j )1 ≤ j ≤ 3h − 3 + nτj ∈ R`j ∈ R+

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Naive string vertices
The unit area disc around a puncture on a hyperbolic diagram is isometric to a punctured
disc with coordinates w and metricString vertex|dw ||w |ln|w |2

Vh0,m,n : the set of all inequivalent hyperbolic string diagrams having hhandles, m- NS punctures and n- R punctures with no simple closed geodesic having length
π2less than c∗ . Each diagrams has local coordinates e c∗ w around the punctures and anconsistent distribution of 2h2m
number of PCOs.
2− +Does−

Vh0,m,n together with the Feynman diagrams provide a single cover of Mh,m+n ?
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Gluing incompatibility of local coordinates

SFT gluing produces a string diagram having a collar with curvature accumulated on a curve.

Metric around a puncture on a glued sting diagram with k zero length propagators:

2 2 c∗|dw |31+Y + O c∗
|w |ln|w |ln|w |Pk
1
Y is the leading term 3 ln|w |i =1 Ei ,1 + Ei ,2 around the puncture, whereEi ,1 , Ei ,2 denote the Eisenstein series associated with the punctures that are being glued.
Vh0,m,n is not gluing compatible ⇒ Vh0,m,n together
with the Feynman diagrams does not provide a single cover of Mh,m+nLocal coordinates on the diagrams in

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Gluing compatible local coordinates

Vh,m,nViolation of gluing compatibility is of the order

Assume that c∗

Mh,m+nO c∗2 .

 1 ⇒ make string vertex very large such that it cover most of themoduli space.

Modify the local coordinates continuously on the elementary string diagrams that belong to a0
thin neighbourhood of the boundary h,m,n , such that at the boundary local coordinates 12/21

Vmatches to2O c∗ with that on the glued diagrams.

Consistent distribution of PCOs
The PCO distribution on an elementary string diagram:

I Must be invariant under the action of elements in MCG.
I Must be invariant under the permutation of the punctures (NS and R separately).
I Must be compatible with the gluing.

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PCO distribution on a thrice-punctured sphere
L3

p1

p2

L1

L1
L2

R

Need to insert a PCO on a thrice-punctured sphere if all the three punctures carry NS states.

Thrice-punctured sphere is a pair of pants with vanishing border lengths. PCO distribution ona pair of pants can be described by specifying the locations in it’s fundamental domain: unionof two hyperbolic hexagons inH.

Distribution symmetric with respect boundaries, and interchange of right and left hexagons:

X (p1 ) + X (p2 )2

p1 , p2 :14/21

the centroids of right and left hexagons

PCO distribution on a once-punctured torus

γ1
γ2
Fundamental domain of a once-punctured torus is obtained by first cutting it along a simple
closed geodesic, then scissor the resulting once punctured-sphere with two borders.

Infinite number of fundamental domains by choosing different simple closed geodesics. All ofthem can be mapped to each other by the action of MCG.

Distributing PCO by choosing points in the fundamental domain also require specifying thesimple closed geodesic along which we cut the surface, which is not an MCG invariant data.

PCO distribution obtained by choosing locations in a fundamental domain is notinvariant under MCG.

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MCG invariant PCO distributionFirst distribute PCO in one fundamental domain and then average over all the fundamental
domains obtained by cutting the surface along different simple closed geodesic that are MCG
equivalent. The following distribution is manifestly MCG invariance:(XG(lh·γ , τh·γ )Q (c∗ )χ(p1h·γ ) + χ(p2h·γ )2h∈MCG

2whereG(lγ , τγ ) =2 sinc (τγ )l1+e γ)11 2Q (c∗ ) = 1 + 2 c∗ + 4 c∗

Infinite sums often diverge, but McShane Identity for a simple closed geodesic

γ on aonce-punctured torus makes the distribution well defined:

X 2 sinc2 (τh·γ )lh∈MCG1 + e h·γ=116/21

Gluing compatible PCO distribution

γ1γ2Figure: lγ

When lγ2→ e1/c∗ when lγ1 → c∗

→ c∗ the PCO distribution becomes that on the once-punctured torus with zero

length NS propagator:

Xh∈MCGG(lh·γ , τh·γ )Q (c∗ )χ(p1h·γ ) + χ(p2h·γ )2→χ(p1γ ) + χ(p2γ )2 3+O c∗
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Improved string vertices
It is possible to systematically construct manifestly MCG invariant PCO distributions K on
any hyperbolic string diagrams which is gluing compatible up to orderO c∗2 , using thegeneralization of McShane identity:

K=X X

XTG

TG h∈MCGFirst sum is over the set of all trivalent graphs TG with h handles and m+ n external legs.

Vh2,m,n provide a consistent closed superstring field theory for infintesimal c∗ .
Making c∗ very small a valid operation, since none of the physical quantities in string field
theory depends on it.
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Effective expression for quantum BV action
2h−2+m+n(∞,∞)

gsX

S = Kinetic term +(m + n, h)=(1, 0)

m + n!

{Ψm + n }h{· · · }h is obtained by integrating the h loop off-shell superstring measure Ω(· · · ) over22
the region inside the moduli space MV h,m,n covered by the string vertex Vh,m,n :Z{· · · }h =Ω(· · · )MV 2h,m,n=

X X Zh∈MCG
XZ ∞ Z ∞TG=ΩTG (· · · )MV 2TG`1 =c∗, ,h m nτ1 =−∞Z ∞ Z ∞···
`Q =c∗τQ =−∞ΩTG (· · · )19/21

String perturbation theory
The on-shell string amplitude is obtained by integrating the h loop on-shell superstring
measureZΛ(· · · ) over the region inside the moduli space Mh,m+n :Z ∞ Z ∞XZ ∞ Z ∞Λ(· · · ) =···Mh,m+nTG` 1 =0τ1 =−∞`Q =0

ΛTG (· · · )τQ =−∞

This form is very convenient for analyzing the field theory limit of superstring amplitudes.

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Summary
Hyperbolic geometry can be used to construct a calculable closed superstring field theory.

Provide efficient tools for computing the amplitudes using superstring perturbation theory.

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