[PPT] - Towards Field Theory of D-Branes Tamiaki Yoneya (University of PowerPoint Presentation

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Towards Field Theory of D-Branes

Tamiaki Yoneya

(University of Tokyo, Komaba)

⋄ Motivations: D-brane field theory ? ⋄ Free-fermion representation of single-charge 1/2-BPS operators: Puzzles and resolution ⋄ A Toy Theory: D3-brane field theory restricted to 1/2-BPS sector: ⋄ D-brane exclusion principle and its realization ⋄ D-brane fields and their bilinears ⋄ Holographic interpretaion and ‘Superstar’ entropy ⋄ Discussion

Based on T. Y. , hep-th/0510114 (JHEP12-028)

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I. Why D-brane field theory?

Basic motivation for second-quantized field theories for D-branes ⋄ Two main approaches to D-brane dynamics :

Open strings: Effective super Yang-Mills theories or open-string field theories of D-branes are

⋄ configuration-space (first-quantized) formulations of D-branes

Closed strings: D-branes ∼ soliton (or ‘lump’) solutions

(‘VSFT’ also belongs to this category) ⋄ difficult to treat fluctutations with respect to creation and annihilation of D-branes Desirable to develop a truly second-quantized formulation of D-brane dynamics ∼ field theory of D-branes ⇓ Is it possible to treat the whole set {N=0, 1, 2, 3, . . . } of U(N) super Yang-Mills theory by some ‘Fock space’-like representation?

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quantum-statistical symmetry (permutation of particles) in particle quantum mechanics (x1, x2) ↔ (x2, x1) ⇓ U(N) gauge symmetry (or Chan-Paton symmetry) of Yang-Mills theory Xij ↔ (UXU −1)ij continuous quantum-statistics ?! (For diagonal Xij = xiδij, gauge symmetry reduces to permutation symmetry) It is not easy to imagine workable (as physics) Fock space with continuous statistical symmetry. Mathematical notions such as K-theory, ... are useful for various topological characterizations of D-branes, but are not for discussing real quantum mechanical dynamics of D-branes.

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Old c = 1 matrix model suggests a toy model for this situation :

Gauge invariant Hilbert space of one (hermitian) N × N matrix Xij is equivalent to the Hilbert

space of N fermions

Z [dX]

[dU]Ψ1|XX|Ψ2 =

Z

N

Y

i=1

dxi

˜

Ψ1|XX| ˜ Ψ2 X| ˜ Ψ ≡

Y

i<j

(xi − xj)

|

{z }

Vandermonde determinant ×X|Ψ : completely antisymmetric Second quantization of this N fermion system gives the Fock space of the c = 1 matrix model. Recent development on 1/2-BPS states in AdS/CFT suggests that this viewpoint might be a useful starting point to a possible second-quantized field theory for D-branes.

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II. Free fermion representation of single-charge 1/2-BPS operators in AdS/CFT

⋄ Generic 1/2 BPS operators (on the Yang-Mills side) in AdS5/SYM4 O(k1,k2,...,kn)(x) ≡

h

Ok1(x)Ok2(x) · · · Okn(x)

i

(0,r,0)

Ok(x) ≡ Tr

φ{i1(x)φi2(x) · · · φik}(x)
φi

(i = 1, 2, . . . , 6) (∼ transverse coordinates of D3-branes)

(For n = 1, KK modes of hαβ, aαβγδ on the sugra side) r = k1 + k2 + · · · kn

⋄ Pick up 2 (i = 5, 6) directions, OJ

(k1,k2,...,kn)(x) ≡ Tr

Z(x)k1
Tr
Z(x)k2
· · · Tr
Z(x)kn
,

Z = 1 √ 2 (φ5 + iφ6), J = r = angular momentum in 5 − 6 plane For large J ∼ N (n ≫ 1), these correspond to the excitation of ‘Giant Gravitons’ large spherical D3-branes with dipole-like RR-fields in the bulk of AdS5 × S5

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Suppose we compute two-point functions O

J (ℓ1,ℓ2,...,ℓm)(x) OJ (k1,k2,...,kn)(y)

Nonrenormalization property of 2 (and 3-point) functions of 1/2 BPS operators allows us to use

the free-field limit of the SYM4.

The free-field limit of SYM4 is further replaced by the (complex) 1 dimensional model with

S1d =

Z

dτ Tr

dZ(τ)

dτ dZ(τ) dτ + Z(τ)Z(τ)

⇓

O

J (ℓ1,ℓ2,...,ℓm)(τ1) OJ (k1,k2,...,kn)(τ2) = f(N) e−J(τ1−τ2)

if we make the following indentification eτ1−τ2 = |x − y|2 (τ ∼ radial time) to be regarded as an effective theory for spherical D3-branes travelling only along the 5-6 plane.

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By using the similar technique as for the hermitian 1-matrix model, this model can equivalently be

described by the Fock space of free fermions on a 1 (complex) dimensional base space. ψ(z, z∗) =

X

n=0

bnpn(z)e−|z|2 , ψ(z, z∗)† =

X

n=0

b†

npn(z∗)e−|z|2

pn(z) =

r

2n πn!zn, {bn, b†

m} = δnm,

bn|0 = 0 = 0|b†

n.

with the ‘lowest Landau level’ (LLL) condition : (z + ∂ ∂z∗)ψ(z, z∗) = 0

Matrix traces for the 1/2 BPS operators = fermion bilinears

Tr

Zn
↔

Z

dzdz∗ψ†(z, z∗)znψ(z, z∗) = 1 2n/2

∞

X

q=0

s

(n + q)! q! b†

n+qbq

⋄ sugra fields ∼ particle-hope pairs near the fermi sea (‘ripplons’) ⋄ giant gravitons ∼ higher (and deeper) excitations of particles and/or holes

Exact two-point functions are reproduced for arbitrary N with Hamiltonian

H =

Z

dzdz∗ψ(z, z∗)†

z ∂

∂z + zz∗

ψ(z, z∗) =

∞

X

n=0

nb†

nbn

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On the bulk sugra side, LLM (Lin-Lunin-Maldacena) showed

(singularity free) backgrounds satisfying the energy condition ∆ = J with symmetry SO(4)

| {z }

S3 in AdS5

× SO(4) × U(1)

| {z }

S3×R in S5

boundary condition defining droplets on a two-dimensional plane embedded in 10D.
classical approximation (fermi liquid) to the Hilbert space of the fermion fields ψ(z, z∗)

– holes deep in the fermi sea ∼ giants extended in S5 – particles excited high above the fermi sea ∼ giants extended in AdS5

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In particular, the Ground state = circular fermi sea |N ≡ b†

N−1b† N−2 · · · b† 0|0

corresponds to the unique AdS5× S5 geometry.

ground state

density of state z-plane

1

ground state droplet

Let us now return to our problem.

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Is this fermion field what we are seeking for? : Not quite! Puzzles

The ground state (J = 0) should be independent of the choice of the direction of angular momentum.

But, apparently, it looks as if the ground state has a nonzero angular momentum. Rotation in 5-6 plane z → eiθz ∼ b†

n → einθb† n

⇓ |N = b†

N−1b† N−2 · · · b† 0|0 → eiJ0θ|N

J0 = N(N − 1)/2 Also the vacuum |0 depends on the choice of the U(1) plane. We could absorb this phase by assuming that the vacuum is transformed as |0 → e−iJ0θ|0, but then the vacuum itself would be dependent of the number N of D3-branes.

More seriously, if we choose different directions for the angular momentum plane, one and the same

ground state ( AdS5×S5 itself) is represented by different Hilbert spaces with different excitation modes.

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Note that the states with different SO(6) indices represent physically independent degrees of freedom of D3-branes, and also that the ground state corresponding to AdS5×S5 geometry with given N must corresponds to a uniquely fixed state in the Fock space of D-branes. ⇓

Desirable to develop an extended formalism in which all SO(6) directions are treated
n an equal footing, and the ground state is manifestly singlet under SO(6).
It would be a first step towards a quantum field theory of D3-branes (and of other cases).

⋄ According to the usual logic, however, the fermion picture seems to depend crucially on the reduction to the special U(1) plane.

Matrix models with two or more matrices (spatial dim.≥ 1) have never been

reduced to fermion theories.

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III. Extended fermion field theory of general (multi-charge) 1/2-BPS operators

Consider general 1/2-BPS operators OI

(k1,k2,...,kn)(x) ≡ wI i1···irTr

φi1 · · · φik1
· · · Tr
φir−kn+1 · · · φir
r = k1 + k2 + · · · + kn = ∆

{wI

i1···ir} = basis for totally symmetric traceless tensors

Due to the free-field contractions and traceless condition, two-point (three-point and also general extremal ) functions always take the factorized form M(τ). OI1

(k1,k2,...,kn)(τ1)OI2 (ℓ1,ℓ2,...,ℓn)(τ2) =

wI1wI2

| {z }

invariant product × G({k.ℓ}, N)e−r(τ1−τ2) G({k, ℓ}, N)e−r(τ1−τ2) = correlator of hermitian 1-matrix free-field theory = : Or

(k1,...,kn)(τ1)

| {z }

no contraction : : Or

(ℓ1,...,ℓn)(τ2)

| {z }

no contraction :M

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The single hermitian matrix field is completely inert against SO(6). Or

(k1,...,kn)(τ1)M ≡ Tr

M k1
Tr
M k2
· · · Tr
M kn
SM = 1

2

Z

dτTr( ˙ M 2 + M 2) Lessons:

Emergence of free-fermion picture for 1/2 BPS operators is essentially owing to this factorization,

not to the choice of a single U(1) plane.

The hermitian matrix degrees of freedom are actually invariant under SO(6).

However, to manage the normal ordering prescription, we can go to the coherent-state representation, by introducing complex field Z instead of the real field M. The 1-matrix hermitian matrix model can then be equivalently treated as the complex 1-matrix model restricted to the lowest Landau level (LLL). – ket states ↔ holomorphic wave functions – bra states ↔ anti-holomorphic wave functions Namely, the origin of LLL condition is nothing other than the normal ordering prescription!

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⇓ Our problem is now reduced to the following question: How to represent the factorization property (SO(6) factor × one-matrix model) in terms of a second quantized (extended) fermionic field theory? The factorization suggests that

D-brane creation and annihilation operators would also be some kind of composite operators,

(bn, b†

n) → (bn,I, b† n,I) ∼ (cI ⊗ bn, c† I ⊗ b† n)

such that the SO(6) index I ∼ (i1i2 · · · ir) are carried by cI, while the energy (conformal dimension) and particle number are carried by bn.

reminiscent of ‘composite fermion’ picture for fraction quantum Hall effect (FQHE)

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⋄ Generalized exclusion principle The operator algebras that give the desired property are

Cuntz algebra for cI:

cI1c†

I2 = δI1I2, ∞

X

I=0

c†

IcI = 1

Ordinary CAR for bn:

{bn, b†

m} = δnm

Multi-particle states of (spherical) D3-branes satisfy an exclusion principle that is much stronger than the ordinary Pauli principle. b†

n,I1b† n,I2 = c† I1c† I2(b† n)2 = 0 = bn,I1bn,I2

for any pair (I1, I2) ‘Dexclusion principle’ States with same ‘energies’ cannot be occupied by two branes simultaneously, irrespectively of their SO(6) states. An (almost) inevitable consequence of the factorization property! (should not be confused with the so-called ‘stringy exclusion principle’)

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not allowed ground state allowed

energy levels

However, as it stands, the property that the c-operators can be excited without any cost of energies would lead to an infinite degeneracy of each possible energy level.

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⋄ The algebras must be accompanied by a symmetry requirement to prevent infinite degeneracy. ‘S-charge’ symmetry b†

n → einθb†,

c†

I → e−ik(I)θc† I,

k(I) = rank of the representation I The ground state is characterized as the highest ‘S-charge’ state among an infinite set of states with the lowest-possible total energy E = N(N − 1)/2 of N particles. |N ≡ (c†

0)N ⊗ b† N−1b† N−2 · · · b† 0|0

If we demand that the S-charge be conserved, the excitations of nontrivial SO(6) tensors must always be accompanied by appropriate excitations of energies.

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Fundamental assumption: the S-charge is strictly conserved. ⇓ Would-be disastrous infinite degeneracy is avoided by restricting the system to the superselection sector of same charge as the ground state with respect to the S-symmetry. Later we will see that the S-charge symmetry is directly related to the scale invariance of SYM4.

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⋄ D-brane fields and their bilinears Field operators of spherical D3-branes (in the 1/2-BPS sector) as functions of real coordinates φi and a complex coordinate α. (f(k) = k!) Ψ(+)

n [φ, α, α]

=

8 < : P∞

k=0

q

2n+k (n+k)!e−|α|2−|φ|2/4f(k) √ k! φi1φi2 · · · φikαn+k bn+k ci1i2···ik,

(n ≥ 0)

P∞

k=−n

q

2n+k (n+k)!e−|α|2−|φ|2/4f(k) √ k! φi1φi2 · · · φikαn+k bn+k ci1i2···ik,

(n < 0) Ψ(−)

n [φ, α, α]

=

8 > < > : P∞

k=0

q

2n+k (n+k)!e−|α|2−|φ|2/4 1 f(k) √ k!φi1φi2 · · · φikαn+k b† n+k c† i1i2···ik,

(n ≥ 0)

P∞

k=0

q

2n+k (n+k)!e−|α|2−|φ|2/4 1 f(k) √ k!φi1φi2 · · · φikαn+k b† n+k c† i1i2···ik.

(n < 0)

The base space has effectively 6+1 (real) dimensions:

{φi} ∼ S5 harmonics + coherent state representation of AdS radial coordinate + time (suppressed here) reflecting the spherical approximation (10 → 7) for D3-branes.

Have definite S-charges, QS = n, −n, respectively.

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The superposition of these fields

Ψ(+)[φ, α, α] ≡

∞

X

n=−∞

Ψ(+)

n [φ, α, α],

Ψ(−)[φ, α, α] ≡

∞

X

n=−∞

Ψ(−)

n [φ, α, α]

satisfy a locality condition Ψ(+)[φ, α, α]Ψ(−)[φ′, α′, α′]|0 = |0 × δ[φ, α, α; φ′, α′, α′]

| {z }

δ function in the space (φ, α, α) ⇒The D-brane fields with definite S-charges are necessarily nonlocal !

Bilinear operators corresponding to the single-trace invariants (∼ elementary sugra fields)

wI

i1i2···ikTr(φi1φi2 · · · φik)

n SYM4 side are, in terms of the extended fermions,

Z

[d6φ|dα|2]

∞

X

n=−∞

Ψ(−)

n [φ, α, α]wI i1i2···ikφi1φi2 · · · φikαkΨ(+) n [φ, α, α]

⇒ closed string field theory ∼ ‘bosonization’ of D-brane field theory

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Generic giant gravitons with arbitrary momentum along S5 are created or annihilated by multiple

products of these bilinear operators. wI

(k1+k2+···+kn)B(k1)B(k2) · · · B(kn),

(k1 + k2 + · · · + kn) B(k) =

Z

[d6φ|dα|2]

∞

X

n=−∞

Ψ(−)

n [φ, α, α]φ(i1φi2 · · · φik)αkΨ(+) n [φ, α, α].

Two-point and general n-point extremal correlation functions of these operators reproduce precisely

the factorized form for arbitrary N. Note: if curvature radius R = (gsN)1/4 is fixed (string unit), gs ∝ 1/N.

Due to the ‘free’ nature of the Cuntz algebra,

the operators B(k) themselves do not obey any simple commutation relations, but becomes commutative upon acting on the ground state |N.

Note: The Cuntz algebra may be regarded as being more fundamental than CAR, since the latter can be embedded into the former. All the previous applications (known to T. Y.) of the Cuntz algebra in field theory are related to planar limit (large N limit). By contrast to this, our formalism does not assume the planar limit.

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Hamiltonian is given as

H =

Z

[d6φ|dα|2]

∞

X

n=−∞

Ψ(−)

n [φ, α, α](α ∂

∂α + αα)Ψ(+)

n [φ, α, α]

=

∞

X

(k)=0

c†

(k)c(k)

|

{z }

=identity

∞

X

n=0

nb†

nbn

=

∞

X

n=0

nb†

nbn

⇓ Ordinary Heisenberg equation of motion is valid O(τ) = eHτO(0)e−Hτ Similarly, the number operator is N =

Z

[d6φ|dα|2]

∞

X

n=−∞

Ψ(−)

n [φ, α, α]Ψ(+) n [φ, α, α]

=

∞

X

(k)=0

c†

(k)c(k)

∞

X

n=0

b†

nbn

=

∞

X

n=0

b†

nbn

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⋄ Spacetime meaning of the S-charge symmetry

For generic time dependent state of the theory in terms of extended fermion operators,

2−k/2c†

(k)wI (k)c0 ∞

X

ℓ1,ℓ2,...,ℓn=0

n

Y

i=1

s

(ℓi + ki)! ℓi!

n

Y

i=1

b†

ℓi+kibℓi

|N ekτ

(k = k1 + k2 + · · · + kn) S-charge transformation c†

(k) → e−ikθc† (k),

c(k) → eikθc(k), b†

n → einθb† n,

bn → e−inθbn θ → iσ ⇓ ‘Wick rotation’ scale transformation of SYM4 (λ ≡ eσ) φi → λφi, eτ → λ−1eτ (eτ ∼ r = |x|) consistent with the spacetime uncertainty relation ∆T ∆X > ∼ ℓ2

s (∆T ∼ ∆|x|, ∆X ∼ ∆|φ|)

S-charge symmetry is a ‘Unitary-tricked’ disguise of conformal symmetry for spherical D3-branes

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⋄ Interpretation of Dexclusion principle on the bulk side: The sheet of droplet does never bifurcate.

ground state not allowed allowed

Our results suggest that on the supergravity side LLM-type analysis might be extended to a much more general case with smaller symmetries. → ‘covariant’ treatment of all SO(6) directions

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⋄ Superstar entropy At this time, only known 1/2-BPS solutions with multiple charges along S5 are ‘superstar’ solutions.

Myers-Tafjord, hep-th/0109127

– smaller symmetry : SO(4) × R → U(1)3 – three independent angular-momentum charges J1, J2, J3 – (naked) singularities in the extremal (1/2-BPS) limit – Mass in the extremal limit : M = π 4G5

X

i

|Ji| = π 4G5 ∆ Naively, suggests that the 3 directions would represent independent dynamical degrees of freedom. – D3-brane charge distribution is consistent with the condensation of giant gravitons – In the single charge case (J2 = J3 = 0), the droplet density is lower than 1, in terms of the LLM classification.

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There exist a large number of nonsingular (LLM) solutions with the same single charge J = ∆. It seems natural to interpret the superstar solutions as objects corresponding to a statistical average of the large number of nonsingular solutions. Perhaps, α′-corrections would ‘smooth out’ the singularity and produce horizon with a finite area. For such an example, see Dabholker-Kallosh-Maloney, hep-th/0410076;

Also in accord roughly with Mathur’s ‘fuzzball’ conjecture

⇓ entropy of superstar = statistical entropy associated with the microstates of D3-brane field theory.

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1

a plausible interpretation of classical superstar solutions

microscopic quantum state ensemble-averaged classical state superstar ads ground state

Counting the microstates of 3-charge superstars gives their entropy as S = f(

X

i

|Ji|) in terms of the entropy f(J) of the single-charge superstar.

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The Dexclusion principle says that the microscopic degrees of freedom for different directions

are not independent in the usual sense, but are mutually exclusive if they have same energies. ⇒ f(

X

i

|Ji|) <

X

i

f(|Ji|)

The entropy is computed through the partition function, S = ln dn(m)

Z(u, v) = Tr

uNvH
=

∞

X

n=0,m=0

dn(m)unvm+n(n−1)/2 dN(∆) =

N

Y

r=1

1 (1 − vr) ∆ = m = total excitation energy of a generic giant graviton configuration In our case, ∆ =

P

i |Ji|

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The case of single charge superstar has already been studied (see, e.g. Suryanarayana, hep-th/0411145; Shepard, hep-th/0507260;V. Balasubramanian et al, hep-th/0508023 ) S = ln dN(∆ = J) = f(J) →

2π2

3

1/2√

J, (J ≫ 1, J/N 2 → 0, N → ∞) indeed satisfies the above inequality. If we did not take into account the Dexclusion principle, we would have incorrectly concluded the equality f(

X

i

|Ji|) =

X

i

f(|Ji|) → f(J) ∝ J

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IV. Discussion

Summary: From the viewpoint of second-quantized Hilbert-space interpretaion of multi D-brane states, ⋄ we have pointed out puzzles in the familiar free-fermion picture for (single-charge) 1/2-BPS operators. ⋄ To resolve these puzzles, we have shown that the free fermion picture for 1/2-BPS operators can be extended such that all directions transverse to spherical D3-branes are treated on an equal footing. ⋄ We have then succeeded in constructing a field theoretic representation of the multi-body system of spherical D3-branes in 1/2 BPS sector. This new extended fermion-field theory is characterized by

‘D’-exclusion principle

– natural holographic interpretation – consistent with superstar entropy

nonlocality consistent with the spacetime uncertainty principle ← S-charge superselection rule

– a disguise of scale symmetry (∈ conformal symmetry) that is apparently lost in the 1D matrix model

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Hopefully, our ‘toy field theory’ of D3-branes in the 1/2-BPS sector may provide a starting point towards general quantum field theory for D-branes and towards a deeper (non-perturbative) understanding on the structure of string/M theory.

pen-string field theories

effective Yang-Mills theories D-brane field theories closed-string field theories

pen-closed duality

first quantization

r

second quantization

bosonization

r

Mandelstam duality

Principles for D-brane field theory? Does our composite operator algebra capture the essence of “D-brane quantum statistics” ?

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Directions:

extension to non-spherical branes
higher-point correlators and fermion interactions
supersymmetric completion
extension to general Dp-branes, especially to D(D)-particles
extension to non-BPS states

(would require to extend the base spacetime to some noncommutative spacetime) → pp-wave limit, spin chain and integrability, etc

inclusion of anti-D-branes → covariantization

→ nonlinear realizaton of maximal N = 2 susy in 10 D related to the issue raised in T. Y., hep-th/9912255, hep-th/0010173 (with T. Hara).

deformation of backgrounds ⇒ moduli, fluxes, ...

In principle, we can deform the Hamiltonian by adding arbitrary bilinear terms corresponding to sugra fields.

case of M-theory; in particular, extension to AdS4(7) × S7(4) → supermembranes, M5 branes ...
formulation of T-duality and S-duality
general formulation of ‘bosonization’ → quantum closed string field theory

(amounts to a derivation of ‘emergent’ gravity from second- (or rather ‘third-’) quantized gauge field theory

classical approximation, large N limit, ....