D-instanton Amplitudes in String Theory

Ashoke Sen

Harish-Chandra Research Institute, Allahabad, India

Paris, February 2021

1

Plan:

1. Introduction to the problem

2. Solution via string field theory

3. Final remarks

References:

arXiv:2012.11624, 2101.08566

2

The problem

3

Amplitudes in string perturbation theory are usually expressedas integrals over the moduli spaces of Riemann surfaces withpunctures.

However these integrals often run into divergences.

Example: Veneziano amplitude (in α′ = 1 unit)∫ 1

0dy y2p1.p2 (1− y)2p2.p3

– diverges for 2p1.p2 ≤ −1 or 2p2.p3 ≤ −1.

Conventional viewpoint: Define the amplitude for 2p1.p2 > −1,2p2.p3 > −1 and then analytically continue to the other kinematicregions.

String field theory (SFT) provides a different, but equivalentviewpoint. 4

SFT is a regular quantum field theory, designed so that theFeynman diagrams of this theory ‘formally’ reproduce theamplitudes based on the world-sheet formalism.

However, in SFT we have a clear physical interpretation of thedivergences.

Consider Schwinger parametrization of a propagator in QFT:

(k2 + m2)−1 =

∫ ∞0

ds e−s (k2+m2) =

∫ 1

0dq q(k2+m2)−1, q = e−s

In SFT, the role of k2 + m2 is played by the eigenvalues of L0, andwe write:

(L0)−1 =

∫ 1

0dq qL0−1

When we translate SFT Feynman diagrams to the world-sheetlanguage, q’s become moduli of Riemann surfaces.

The divergences arise from the q→ 0 limit. 5

(L0)−1 =

∫ 1

0dq qL0−1

– an identity for L0 > 0 states.

For L0 < 0 the l.h.s. is finite but the r.h.s. diverges.

SFT instructs us to use the l.h.s.

– equivalent to analytic continuation when the latter is available.

For L0 = 0 both sides diverge, reflecting the presence of on-shellstates in the internal lines of a Feynman diagram.

In such cases we can use SFT to identify the physical origin ofthese states and use insights from QFT to deal with thesedivergences. 6

Today we shall discuss a specific situation where analyticcontinuation fails and we need to invoke SFT to make sense ofthe divergences

– D-instanton contribution to string amplitudes.

7

D-instantons: D-branes with Dirichlet boundary condition on allnon-compact directions including (euclidean) time.

They are saddle points of the Euclidean path integral and givenon-perturbative contribution to string amplitudes.

Problem: Open strings living on the D-instanton do not carryany continuous momenta

⇒ we cannot move away from the singularities by varying theexternal momenta. Polchinski; Green, Gutperle; ...

Even if the divergent parts cancel after suitable choice ofregulators, the finite parts that remain after the cancellationbecome ambiguous. Fischler, Susskind

8

We shall study this in the context of bosonic string theory in twodimensions, but the method we describe can be applied to anystring theory.

World-sheet theory: A free scalar X describing time coordinateand a Liouville field theory with central charge 25.

Total central charge adds up to 26, cancelling anomalies on theworld-sheet.

In this case the closed string ‘tachyon’ is actually a masslessstate of the theory.

In arXiv:1907.07688 Balthazar, Rodriguez and Yin (BRY)computed the single D-instanton contribution to the two pointamplitude of closed string tachyons.

This model is interesting because there is a dual matrix modeldescription that gives the exact results.

Das, Jevicki; Gross, Klebanov; Sengupta, Wadia; Moore, Plesser, Ramgoolam; · · · 9

The leading contribution comes from the product of two diskone point functions.

× ×

Result:

N e−1/gs 2π δ(ω1 + ω2) 2 sinh(π|ω1|) 2 sinh(π|ω2|)

gs: string coupling constant

−ω1, ω2: energies of incoming / outgoing ‘tachyons’

N : An overall normalization constant given (naively) by:

N = i exp[annulus partition function

]= i exp

[∫ ∞0

dt2t

(e2πt − 1)

]– divergent in the world-sheet description of string theory. 10

× ×

Naively one might have expected this to be proportional toδ(ω1)δ(ω2).

However, for D-instanton boundary conditions, individual diskamplitudes do not conserve energy, since time translationinvariance is broken.

The energy conservation is restored at the end after integrationover the collective coordinate describing the position of theD-instanton

– will be discussed later.

11

At the next order, there are three contributions.

1. Two point function on the disk.

×1×2

Result:

8πN e−1/gs δ(ω1 + ω2) sinh(π|ω1|) sinh(π|ω2|) gs f(ω1, ω2)

f(ω1, ω2) =12

∫ 1

0dy y−2(1 + 2ω1ω2y) + ffinite(ω1, ω2)

Note the divergences from the y→ 0 limit

– cannot be tamed by deforming the ωi’s.

Writef(ω1, ω2) = Af + Bf ω1 ω2 + ffinite(ω1, ω2)

12

2. Product of disk one point function and annulus one pointfunction.

× ×1 2

+ (1⇔ 2)

Result:

8πN e−1/gs δ(ω1 + ω2) sinh(π|ω1|) sinh(π|ω2|) gs{g(ω2) + g(ω1)}

g(ω) =

∫ 1

0dv∫ 1/4

0dx{

v−2 − v−1

sin2(2πx)+ 2ω2 v−1

}+ gfinite(ω)

Note the divergences from x→ 0 and v→ 0 that cannot betamed by deforming ω.

Writeg(ω) = Ag + Bg ω

2 + gfinite(ω)

13

3. Product of two disk one point functions and zero pointfunction on a surface with Euler number −1

– a disk with two holes and a torus with one hole

× ×+1 more

8πN e−1/gs δ(ω1 + ω2) sinh(π|ω1|) sinh(π|ω2|) gs C

14

Total result to order gs:

8πN e−1/gs δ(ω1 + ω2) sinh(π|ω1|) sinh(π|ω2|)× [1 + gs{f(ω1, ω2) + g(ω1) + g(ω2) + C}]

= 8πN e−1/gs δ(ω1 + ω2) sinh(π|ω1|) sinh(π|ω2|)[1 + gs{ffinite(ω1, ω2) + gfinite(ω1) + gfinite(ω2) + Af + 2Ag + C + (Bf − 2Bg)ω1ω2}]

At this stage Af,Bf,Ag,Bg,C are unknown constants

– divergent in the world-sheet description. 15

Using the same functions we can also get the expressions forn-point function of closed string tachyon at order gs e−1/gs .

The leading contribution comes from product of n disk one pointfunctions.

Result:

N e−1/gs 2π δ(ω1 + · · ·ωn)n∏

i=1

{2 sinh(π|ωi|)}

= 2n+1 πN e−1/gs δ(ω1 + · · ·ωn)n∏

i=1

sinh(π|ωi|)

16

At next order there are 3 types of diagrams:

1. (n− 2) disk one point functions and one disk 2-point function

2. (n− 1) disk one point functions and oneannulus 1-point function

3. n disk one point functions and a zero point function onsurfaces of Euler number −1

Result

2n+1 πN e−1/gs δ(ω1 + · · ·ωn)

{n∏

i=1

sinh(π|ωi|)

}gs

∑j<k

f(ωj, ωk) +∑

j

g(ωj) + C

Use

f(ω1, ω2) = Af + Bf ω1 ω2 + ffinite(ω1, ω2), g(ω) = Ag + Bg ω2 + gfinite(ω)

17

D-instanton contribution to n tachyon amplitude to order gs:

2n+1πN e−1/gs δ(ω1 + · · ·ωn)

n∏

i=1

sinh(π|ωi|)

1 + gs

∑j<k

f(ωj, ωk) +∑

j

g(ωj) + C

= 2n+1πN e−1/gs δ(ω1 + · · ·ωn)

n∏

i=1

sinh(π|ωi|)

×

1 + gs

∑j<k

ffinite(ωj, ωk) +∑

j

gfinite(ωj) +n(n− 1)

2Af + n Ag + C +

{Bg −

Bf

2

}∑j

ω2j

The result is known from the dual matrix model.

For scattering of a tachyon of energy −ω1 into tachyons ofenergy ω2, · · · , ωn, the result is:

− 18π2 2n+1 π e−1/gs δ(ω1 + ω2 + · · ·ωn)

{n∏

i=1

sinh(π|ωi|)

}1− i gs

n∑j=2

ωj

(1−

n∑i=2

πωi coth(πωi)

) ,18

2n+1πN e−1/gs δ(ω1 + · · ·ωn)

n∏

i=1

sinh(π|ωi|)

gs

×

1 + gs

∑j<k

ffinite(ωj, ωk) +∑

j

gfinite(ωj) +n(n− 1)

2Af + n Ag + C +

(Bg −

Bf

2

)∑j

ω2j

= −1

8π22n+1

π e−1/gs δ(ω1 + ω2 + · · ·ωn)

n∏

i=1

sinh(π|ωi|)

1− i gs

n∑j=2

ωj

1−n∑

i=2

πωi coth(πωi)

.

Comparing the two results, one gets (BRY):

N = − 18π2 (analytical)

Af ' −0.50, Ag ' 0.00, C ' 0.00, 2Bg − Bf ' −1.40 (numerical)

Can we determine them using string field theory?

19

Solution via string fieldtheory

20

Strategy

1. Take the world-sheet expression for an amplitude and focuson the divergent part associated with open string degeneration,e.g. t→∞, y→ 0, v→ 0, x→ 0.

2. Make appropriate change of variables to express the moduli interms of SFT Schwinger parameters q1,q2, · · · .

3. Then we replace divergent integrals of the form∫ 1

0dq q−β−1, β 6= 0

by −1/β, interpreting this as due to states with L0 = −β.

β = 0 will require special treatment. 21

4. Divergent integrals of the form:∫ 1

0dq q−1

can be traced to the zero modes (L0 = 0 states).

These cannot be treated using Feynman diagrams and the firststep is to remove these contributions by hand

– corresponds to ‘unintegrating them’ in the path integral.

We then have to find the physical origin of these modes, andtreat them accordingly

– involves doing the path integral over them explicitly, instead oftreating them via Feynman diagrams.

22

Results from string field theory (analytical)

N = − 18π2 , Af = −1

2, Ag = −1

2, C = 0(indirect), 2Bg − Bf ' −ln 4 ' −1.386

Compare this with the matrix model result of BRY:

N = − 18π2 (analytical)

Af ' −0.50, Ag ' 0.00, C ' 0.00, 2Bg − Bf ' −1.40 (numerical)

We hope that the discrepancy in the value of Ag will be resolvedsoon, but at present it remains an enigma.

23

We shall now illustrate this by describing the calculation of theoverall normalization constant N

– technically simpler but conceptually most subtle

The naive world-sheet expression for this is given by:

i exp[A]

A: annulus partition function

A =

∫ ∞0

dt2t

Tr(e−2πt L0 ) =

∫ ∞0

dt2t

∑i

e−2πt hbi −

∑j

e−2πt hfj

{hb

i } and {hfj} are the L0 eigenvalues of the bosonic and

fermionic open string states.

Due to ghost zero modes, there is a projection operator b0c0

hidden in the trace and the sum runs over b0|s〉 = 0 states.

A diverges in string theory due to states with L0 ≤ 0. 24

For the two dimensional string theory:

A =

∫ ∞0

dt2t[e2πt − 1

]=

∫ ∞0

dt2t[e2πt + 1− 2

]Compare this with the general result:

A =

∫ ∞0

dt2t

Tr(e−2πtL0

)=

∫ ∞0

dt2t

∑i

e−2πt hbi −

∑j

e−2πt hfj

This shows that in the two dimensional string theorycontribution from L0 > 0 states all cancel

⇒ we can focus on the finite number of L0 ≤ 0 modes:

boson : c1|0〉, c1α−1|0〉, fermion : |0〉, c−1c1|0〉

α−1: oscillator of the euclidean time coordinate 25

A =

∫ ∞0

dt2t

∑i

e−2πt hbi −

∑j

e−2πt hfj

Typically in all consistent string theories the [· · · ] vanishes ast→ 0

⇒ effective number of bosonic and fermionic modes are equal(including ghosts).

In this case we can carry out the integral analytically and write:

A =12

ln

∏j hf

j∏i hb

i

Also fermionic modes come in pairs with equal hfj .

⇒ eA =

√∏j hf

j∏i hb

i=

∫ ∏i

dbi√2π

∏j

dfj d̃fj exp

−12

∑i

hbi b2

i +∑

j

′hf

j f̃j fj

bi: bosonic mode, fj, f̃j: fermionic modes with same hf

j 26

⇒ eA =

∫ ∏i

dbi√2π

∏j

dfj d̃fj exp

−12

∑i

hbi b2

i +∑

j

′hf

j f̃j fj

Our next task is to

1. Interpret this as the Siegel gauge fixed path integral of agauge invariant string field theory.

2. Show that the ambiguities associated with the hbi = 0 and

hfj = 0 states arise due to breakdown of the Siegel gauge.

3. Use the gauge invariant path integral to evaluate eA.

Since open strings on a D-instanton do not carry any continuousmomenta, the string field theory is a zero dimensional fieldtheory. 27

Define ghost number of open string states by assigning

c, c̄ : ghost number 1, b, b̄ : ghost number −1, matter : ghost number 0

{|φ(n)r 〉}: A basis of states of ghost number n.

For the open string states with L0 ≤ 0, this list is finite:

c1|0〉, |0〉, c−1c1|0〉, c1α−1|0〉, c0c1|0〉, c0|0〉, c0c−1c1|0〉, c0c1α−1|0〉

α−1: oscillator of the Euclidean time X

28

In open string field theory the classical string field |ψ〉 is anarbitrary state of ghost number 1.

If we expand:|ψ〉 =

∑r

ψr|φ(1)r 〉

then {ψr} are the dynamical degrees of freedom.

The quadratic part of the gauge invariant classical action is:

Sgi =12〈ψ|QB|ψ〉

QB: BRST charge of the world-sheet theory

29

The gauge transformation parameters of the classical openstring field theory are described by an arbitrary state |θ〉 of ghostnumber 0.

If we expand|θ〉 =

∑u

θu|φ(0)u 〉

then {θu} are the gauge transformation parameters.

δ|ψ〉 = QB|θ〉

Partition function in the gauge invariant formalism is defined as:

Igi ≡∫ ∏

r

dψr e−Sgi

/∫ ∏s

dθs

The normalization has been fixed arbitrarily and will not affectthe final result. 30

Igi =

∫ ∏r

dψr e−Sgi

/∫ ∏s

dθs

One can show, using standard Fadeev-Popov formalism, that ifwe formally gauge fix to Siegel gauge b0|ψ〉s = 0, then thisreduces to∫ ∏

i

dbi

∏j

dfj d̃fj exp

−12

∑i

hbi b2

i +∑

j

′hf

j f̃j fj

= (2π)nb/2 eA

nb: number of bosonic modes in the Siegel gauge

In the present case:

|ψ〉 = ψ0c1|0〉+ ψ1c0|0〉+ φc1α−1|0〉, |θ〉 = θ|0〉

Sgi =12〈ψ|QB|ψ〉 = −1

2(ψ0)2 − (ψ1)2

This gives nb = 2 and

eA =1

2π

∫dψ0dψ1 dφe

12 (ψ

0)2+(ψ1)2

/∫dθ

31

eA =1

2π

∫dψ0dψ1 dφe

12 (ψ

0)2+(ψ1)2

/∫dθ

The ψ0 and ψ1 integration can be carried out by choosing theintegration contour along the steepest descent contour(imaginary axis) yielding a factor of −π

√2.

eA = − 1√2

∫dφ

/∫dθ

32

eA = − 1√2

∫dφ

/∫dθ

φ is related to the D-instanton location φ̃ along the time direction.

θ is related to the rigid U(1) symmetry transformation parameterθ̃ on the D-instanton

– gives a phase eiθ̃ to open strings connecting the D-instanton toa second D-brane.

The precise relation between these modes can be found bycomparing their amplitudes.

Results: π√

2 go φ = φ̃, θ = iθ̃/go

go: coupling constant of open string theory. 33

eA = − 1√2

∫dφ

/∫dθ

π√

2 go φ = φ̃, θ = iθ̃/go

Since θ̃ has period 2π we have∫

dθ̃ = 2π.

The integration over the D-instanton position φ̃ generates theenergy conserving δ-function 2πδ(ω) and is not usually includedin N .

This gives

eA = − 1√2

1π√

2 go

go

2πi=

i4π2

But we are not done yet.

34

The multiplier factor:

We have computed the leading contribution from the steepestdescent contour around the instanton.

How does the steepest descent contour around the instanton fitinside the full contour along which the ‘path integral’ is to beperformed?

For this we look at the open string tachyon mode ψ0.

Tachyon potential has a maximum at 0 representing theD-instanton and a local minimum at some value 1/a representingthe vacuum without any D-instanton.

ψ0

V

1/a35

ψ0

V

1/a

The potential goes to −∞ as ψ0 → −∞

This shows that the ψ0 integration contour cannot be taken to bealong the real axis since e−V approaches∞ as ψ0 → −∞.

36

The correct choice of integration contour:

×⇒ ⇒×⇑

This covers the full steepest descent contour of the perturbativevacuum at ψ0 = 1/a but only half of the steepest descentcontour of the instanton at ψ0 = 0.

This gives:

N = i eA 12

= ii

4π212

= − 18π2

in agreement with the matrix model result.

37

Final remarks

38

In this talk we have focussed on D-instantons in twodimensional bosonic string theory.

However D-instantons are present in many string theories, andthe problems we discussed are universal problems for allD-instantons.

This includes for example the breakdown of the Siegel gauge,translational zero modes, rigid U(1) transformation etc.

String field theory for superstrings is well established by nowand can be used to address these issues.

This gives a fully systematic method for computing D-instantoncontribution to string amplitudes

– provides a window into non-perturbative aspects of stringtheory using perturbative techniques. 39