Developments in String Field Theory
Ashoke Sen
Harish-Chandra Research Institute, Allahabad, India

Okinawa, June 2018

1

What is string field theory?
String field theory is a quantum field theory with infinite
number of fields in which perturbative amplitudes arecomputed by summing over Feynman diagrams.
Each Feynman diagram can be represented as an integral
over the moduli space of a Riemann surface with– the correct integrand (as in world-sheet description)– but only a limited range of integration.
Sum over all Feynman diagrams reproduces theintegration over the whole moduli space.
2

Why string field theory?
1. Since it is a quantum field theory with action, theS-matrix can be represented as path integral over string
fields– could possibly give a non-perturbative formulation ofstring theory(has not been successful so far)
2. Classical string field theory may be useful for finding
new classical solutions of string field theory that are noteasily describable in conventional world-sheet formulation(reasonably successful for open string field theory)33. For a given amplitude, the usual world-sheet descriptionof string perturbation theory gives one term at every loop
order
– usually considered an advantage, but this may notalways be the case.

4

Example: In a quantum field theory, self energy insertions
on external legs have to be resummed separately to findthe renormalized massl l lrequires– separating graphs with self-energy insertions on externallines from other graphs– off-shell 2-point function to look for renormalized poleposition
String field theory is well suited for this task.

5

Therefore string field theory is needed to fully define string
perturbation theory– by itself a strong motivation for the subject

6

PLAN
1. Formulation of string field theory (SFT)
2. Classical solutions in open string field theory
3. Background independence
4. Application to perturbation theory
We shall focus on covariant string field theory, and not
discuss light-cone string field theory.
7Formulation of covariant
string field theory8

Tree level bosonic SFT (open and closed)
Witten; Saadi, Zwiebach; Kugo, Kunitomo, Suehiro; Sonoda, Zwiebach; Zwiebach; · · ·For open strings,
H ≡ Vector space of boundary vertex operators / states ofmatter + ghost world-sheet CFT
For closed strings,
H ≡ Vector space of vertex operators / states of matter +ghost world-sheet theory subject to additional constraints:b−
0 |Ai = 0,b±
0 = b0 ± b̄0 ,L−
0 |Ai = 0L±
0 = L0 ± L̄0 ,for A ∈ Hc±
0 = (c0 ± c̄0 )/29String field Φ: An arbitrary vertex operator / state in H with– ghost number 1 for open strings– ghost number 2 for closed strings
Φ is– grassmann odd for open strings– grassmann even for closed strings
due to odd / even ghost number of Φ.
10

Action:Open :

S=

X1
1
hΦ|Q|Φi +
{Φn }
2
n
n

Closed :

S=

X 1
1
hΦ|c−
Q|Φi +
{Φn }
0
2
n!
n

Q: BRST operator in the matter-ghost CFT
For A1 , · · · An ∈ H, {A1 · · · An } is an n-point correlation
function of the vertex operators A1 , · · · An on the disk /sphere · · ·· · · integrated over some specific codimension 0 subspaceof the moduli space of disk / sphere with n punctures.
The vertex operators are inserted on the disk / sphere withsome specified local coordinates around the punctures.
11

{A1 · · · An } has– cyclic symmetry for open strings– full permutation symmetry for closed stringsfor grassmann odd / even A1 , · · · An
[A1 · · · An ] ∈ H is defined via
hC|[A1 · · · An ]i = {CA1 · · · An }hC|c−
0 |[A1 · · · An ]i = {CA1 · · · An }∀C ∈ H∀C ∈ Hfor open stringsfor closed strings

12Infinitesimal gauge transformation:XδΦ = QΛ +(−1)m+1 [Φm ΛΦn ] for open stringsm,nδΦ = QΛ +

X 1
n

n!

[ΛΦn ]

for closed strings

Gauge transformation parameter Λ: a state in H with– ghost number 0 for open strings– ghost number 1 for closed strings
13

Gauge fixing: Introduce ghosts
Set of all ghost fields + matter fields: A state in H with
arbitrary ghost number
Bochicchio; Thorn; Sonoda, Zwiebach
Action with matter + ghost has the same form as theoriginal action, but there is no constraint on the ghost
number of Φ
Natural framework: Batalin-Vilkovisky (BV) formalism

14Each tree Feynman diagram gives integration over part ofthe moduli space of punctured disk / sphere.
Sum over all Feynman diagrams gives integration over thefull moduli space.
Giddings; Giddings, Martinec, Witten; Saadi, Zwiebach15

Loops:Generically open string field theory requires coupling toclosed string field theory
Zwiebach
Closed string field theory makes sense by itself
Action has similar form except that definition of {A1 · · · An }requires additional contribution from integration oversubspaces of moduli space of higher genus surfaces– additional finite ‘local’ counterterms needed for gauge
invariance– also ensures that the sum over all Feynman diagrams
gives correct result for on-shell amplitudes

16

Exception: Cubic open string field theory
Witten– uses special choice of local coordinates for defining{A1 A2 A3 }1. No vertices other than cubic are needed either at tree orat loop level.
2. No need to couple to closed strings (for on-shell
amplitudes)(closed string poles appear automatically from the UV
region of open SFT)Special notation: [AB] = A ∗ B or just AB
17

The SFT’s constructed this way are not unique.
Different possible choice of local coordinate system at thepunctures lead to different string field theories . . .
. . . related by field redefinition

Hata, Zwiebach; · · ·Nevertheless some choice may be simpler than the otherse.g. the cubic open SFT
There is no analogous ‘simple choice’ for closed SFT
18

A particular proposal involves using local coordinates
induced by ‘minimal area metric’
Zwiebach– minimizes the area of the Riemann surface subject to alower bound on the length of all non-trivial closed
geodesics– hard to construct such metrics and very few explicit
examples are known

19

Recent progress:
1. Moosavian and Pius proposed a choice of local coordinates
based on constant negative curvature metric on the Riemannsurface– the metric needs to be modified near the boundaries of moduli
spaces but systematic construction is possible
2. Headrick and Zwiebach developed systematic tools for(numerically) constructing the minimal area metric
Both approaches are useful for providing explicit definition ofoff-shell string amplitudes

20Superstring field theory (in RNS formulation)
There are two equivalent approaches to perturbation
theory1. Integration over supermoduli space
2. Use of picture changing operators (PCO)Friedan, Martinec, Shenker

So far mostly the second approach has been used in theconstruction of SFT.
21

The Hilbert space becomes a direct sum of Hilbert spacesof different picture numbers
H = ⊕n Hn
n ∈ ZZ: NS sectorn ∈ ZZ + 12 : R-sector–

all n are equivalent on-shell–

all n are equivalent on-shell

Canonical choice of open string field: Φ ∈ H−1 + H−1/2
Canonical choice of closed string field:Φ ∈ H−1,−1 + H−1/2,−1 + H−1,−1/2 + H−1/2,−1/222For a g-loop amplitude of m NS and n R-sector states, weneed to insert (2g − 2 + m + n/2) PCO’s for picture number
conservation
Result is independent of the locations of the PCO’s if wechoose them avoiding ‘spurious singularities’– any singularity that depends on PCO locations e.g. those
associated with PCO collisions
There is a systematic procedure for avoiding thisA.S.; A.S., Witten

However off-shell amplitudes depend on PCO locations
just as they depend on the local coordinates at thepunctures.
23

For NS sector fields (NSNS for closed strings) the tree level
string field theory can be constructed following the sameprocedure as bosonic string field theory
Saroja, A.S.; Erler, Konopka, Sachs– PCO locations enter in the definition of {A1 · · · An }.
Different string field theories associated with different
choices of local coordinates and PCO locations are related
by field redefinition.

24

Ramond sector has additional complications associatedwith kinetic term.
Two Φ’s have total picture number −1 since Φ ∈ H−1/2
We need an additional picture number −1 at genus 0This is related to the absence of a covariant kinetic term for
type IIB supergravity due to self-dual 5-form field strength.

25

Resolution: In the R sector introduce another string fieldΨ ∈ H−3/2 and write the kinetic term as
A.S.
1− hΨ|Q G|Ψi + hΨ|Q|Φi
2
G: zero mode of PCO
The interaction terms involve the original fields Φ
Result: One combination of Ψ and Φ remains free field anddecouples from the theory
The rest of the degrees of freedom describe correctly theinteracting string field theory.
26

This construction works at the loop level exacty as in thecase of bosonic string theory.
The string field theory action depends on the choice of
local coordinates at the punctures and the choice of PCO
locations.
All such theories are related to each other by field
redefinition.

27

Question: Is there a ‘simple choice’ that is simpler than theothers?
Difficult for closed string theory since even the bosonicclosed string field theory does not have a simple choice.
However progress has been made in tree level open
superstring field theory.

28

In 1995 Berkovits constructed a tree level open superstring
field theory with string field Φ in the large Hilbert space– contains states of arbitrary integer picture number.
1. Has many more fields but also extra gauge invariancethat reduces the number of degrees of freedom.
2. The action can be written in a compact form
It is hard (but possible for simple cases) to show that thesum of Feynman diagrams gives the correct amplitude
Berkovits, Echevarria29

In 2013, Erler, Konopka and Sachs constructed an open
superstring field theory for NS sector fields generalizing
an earlier construction of Witten.
also Iimori, Noumi, Okawa, Torii
1. Local coordinates at the punctures are similar to thoseused in cubic open bosonic string field theory.
2. PCO insertions are smeared so as to alleviate thesingularities arising from PCO collisions.
3. Need to add higher order vertices, but there are simple
algorithms to generate them maintaining gauge invariance.
4. Reproduces correctly tree amplitudes of NS sector
states.

Konpka30

Recently Berkovits open superstring field theory was
shown to reduce to the formulation based on PCO
insertions after partial gauge fixing
Iimori, Noumi, Okawa, Torii; Erler, Okawa, Takezaki; Erler– also proves indirectly that Berkovits theory correctly
reproduces the tree level NS sector amplitudes of open
superstring field theory31

Ramond sector:
The doubling trick can be used to get a simple version ofopen superstring field theory including Ramond sector
fieldsErler, Okawa, Takezaki; Konopka, Sachs
1. Uses local coordinate systems similar to the one for
cubic open bosonic SFT
2. Systematic construction of smeared PCO insertions

32

A different formulation of the Ramond sector was given by
Kunitomo and Okawa.
Instead of taking |Φi ∈ H−1/2 , we put additional constraintson Φ.
– analog of the L−
0 |Φi = 0 constraint for closed strings but
more complicated.
1. The action is quadratic in the Ramond sector fields but
has higher powers of NS sector fields.
2. Checking that Feynman diagrams reproduce theamplitudes is hard
– tested up to some tree level 5-point functions
33

Recently a new approach to classical open string field
theory has been developed
Ohmori, Okawa– based on integration over supermoduli space– still in its infancy

34

Classical solution in open
string field theory

1. Numerical (level truncation)
Kostelecky, Samuel; Kostelecky, Potting; A.S., Zwiebach; Moeller, Taylor; Gaiotto, Rastelli; · · ·2. Analytical

Schnabl, · · ·35

Numerical solution on D-branes
1. Expand the string field in a basis of L0 eigenstatesXΦ=an |φn in2. Truncate the expansion to those |φn i with L0 eigenvalue
below some fixed number L, and satisfying appropriate
symmetry restrictions
3. Evaluate the action, and extremize with respect to the
an ’s to find solution.
4. Hope that as we increase L, the solution converges.
36This has successfully generated many solutions in
Witten’s cubic bosonic open string field theory andBerkovits’ open superstring field theory.
Examples:
1. Tachyon vacuum describing vacuum without D-branes
2. Solitons describing lower dimensional D-branes
The energies carried by these solutions come quite close
to the expected energies of the D-brane configurations
they attempt to describe.
37

Can we use this approach to construct new solutions thatare not known otherwise?
– new boundary conformal field theories.
Recently some solutions have been found with no obvious
description as D-branes.
Kudrna, Schnabl, Vosmera, to appear38

Set-up: D2 brane of bosonic string theory wrapped on T2∃ numerical solution which do not have obvious CFTinterpretation except at special points in the moduli spaceof T2 .
At special point in the moduli space one can construct
boundary state analytically with the help of enhanced
chiral algebra.
Kudrna, Schnabl, Vosmera; also earlier work by Yi, Kane; Affleck, Oshikawa, Saleur

B = 0,

R1 = R2 =

√

2α0 ,

angle=2π/3

The energy density profile agrees between numerical andanalytical results.
39

Comparison between numerical and analytic results forenergy density profile
Kudrna, Schnabl, Vosmera, to appear– demonstrates the utility of open string field theory inlooking for new conformally invariant boundary conditions
40

Analytic solutions in cubic open string field theory
First solution due to Schnabl– describes the tachyon vacuum on the Dp-brane
The energy density of the vacuum= - the tension of the Dp-brane⇒ described vacuum without any D-brane
Solution is given in ‘Schnabl gauge’ and not in Siegel
gauge, and cannot be directly compared with earlier
analysis based on level truncation
41

Subsequent developments have provided better
understanding of this solution and also paved the way tofinding more analytic solutions.
Okawa; Rastelli, Zwiebach, Erler, · · ·Computation of Veneziano amplitude in the Schnabl gauge
gives the correct result confirming validity of this gauge
choiceRastelli, Zwiebach42

A rewriting of Schnabl’s solution:

Okawa

Φ = e−K/2 c K B (1 − e−K )−1 c e−K/2
K, B, c: String field constructed respectively from– line integral of stress tensor– line integral of b-ghost– insertion of c-ghostin specific coordinate system.
All products are ∗-products
43

Other analytic solutions
If the world-sheet theory has a boundary marginal
deformations by some operator V, then we have a family ofboundary CFT– provides consistent background for open string theory
There must be a family of solutions in open string field
theory describing the family of consistent backgound
Can we construct such solutions analytically?
Yes, if VV operator product has no singularity.
Schnabl; Kiermaier, Okawa, Rastelli, Zwiebach; · · ·44Absence of singularity in VV OPE gives a very limited
0
choice, containing eaX factors in V
Standard time independent operators will have (z − w)−2
singularities.
Remedy: Add a term proportional to i∂X0 to cancel thispole in OPE– corresponds to switching on a constant A0 (pure gauge)
There have been many subsequent developments
simplifying the construction that paved the route forfurther developments.
Fuchs, Kroyter, Potting; Kiermaier, Okawa

45

Question: Given a new solution, how do we know what it
represents?
Answer: See how a closed string feels the solution
A systematic procedure for carrying this out wasdeveloped by Ellwood
Ellwood invariant: Senses the coupling of a closed string
to the open string field
This was later used to give an algorithm for constructing aboundary state associated with a given solution of open
string field theory.
Kiermaier, Okawa, Zwiebach; Kudrna, Maccaferri, Schnabl
46

Can we construct analytic solutions associated with
non-marginal deformations?
e.g. take a D1 brane wrapped on a circle of radius R . . .
. . . and find a solution representing a D0 brane

47

Strategy: Make use of boundary condition changing
operators
Erler, Maccaferri, based on Kiermaier, Okawa, Soler

×
b.c.1

b.c.2

One can construct solutions in the string field theory
constructed using boundary CFT with b.c.1 that describes
boundary CFT with b.c.2
48

Solution takes a suggestive form:Φ = Ψtv − Ψ∗
Ψtv : tachyon vacuum solution in b.c.1Ψ∗ : tachyon vacuum solution in b.c.2, ‘written in thevariables of the string field theory around b.c.1’– as if we first create the tachyon vacuum in b.c.1 and thencreate the D-brane associated with b.c.2 from there49

Classical solution in superstring field theory
Still in its infancy

Erler

50

An open problem:
Given a string field configuration that satisfies equationsof motion, how do we know if the solution is singular or
non-singular?
There is no natural norm in the space of string fields thatcan be used to demand normalizability.
There are currently many solutions, including strange ones
with negative tension, whose interpretation is unclear due
to this issue.
So far all solutions found in Siegel gauge level truncation
seem to be sensible, but this is too restrictive.
51

Background independence
Construction of string field theory requires choosing aclassical background– corresponds to a choice of (boundary) CFT describing
the world-sheet theory.
The action is then given in terms of the correlation
functions and BRST operator in this world-sheet theory.
Question: How are string field theories constructed using
two different CFT’s related?
52

When the two CFT’s are related by marginal deformation
the answer is simple
The two string field theories are related by a field
redefinition.
For bosonic string field theories the explicit field
redefinition relating the two theories was constructed in
the 90’s
A.S., Zwiebach– constructed for infinitesimal deformation which could be
integrated to give the result for finite deformation

53

This construction has now also been generalized tosuperstring field theories, including the Ramond sector A.S.
One subtle point: Only the interacting part is background
independent.
The extra free string field action that one requires for theconstruction of the action is background dependent.

54

For open string field theory one can do better using
boundary condition changing operators– can actually construct explicit field redefinitions relating
string field theories formulated around finitely separated
backgrounds.
Erler, Maccaferri55

Application to perturbation theory
Using the description of amplitudes as sum over Feynman
diagrams, one can use techniques of quantum field theory
to prove general properties of these amplitudes.
Rudra, Pius, A.S.; Pius, A.S.; A.S

1. Unitarity of perturbative S-matrix
2. Systematic procedure for computing finite mass
renormalization of massive particles
3. Systematic procedure for computing string amplitudes
when the perturbative vacuum gets destabilized andsettles down to a nearby stable vacuum
56Summary
We now have a complete formulation of field theories of all
string theories, although the formulation of closed string field
theories remain complicated.
The most ambitious goal of string field theory would be to give a
non-perturbative definition of string theory– has not been realized so far.
Nevertheless on two fronts it has had reasonable success:1. Classical solutions in open string field theory
2. A systematic procedure for string perturbation theory in caseof finite mass renormalization or shift in the vacuum.
57