Cosmological Unification of String Theories
Simeon Hellerman
based on :hep-th/0405041, S.H.
hep-th/0409071, S.H. and Xiao Liu
hep-th/0611317, S.H. and Ian Swanson
hep-th/0612051, S.H. and Ian Swanson
hep-th/0612116, S.H. and Ian Swanson
arXiv:0705.0980, S.H. and Ian Swanson
arXiv:0709.2166, S.H. and Ian Swanson
arXiv:0710.1628, S.H. and Ian Swanson

LPTHE, Paris, 5 Feb 2008

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

Preface
It would be useful to understand how strongly nonsupersymmetricstring theories – such as the bosonic string, as well as string
theories in supercritical dimensions – are related to the more
supersymmetric versions of string theory.

Preface
It would be useful to understand how strongly nonsupersymmetricstring theories – such as the bosonic string, as well as string
theories in supercritical dimensions – are related to the more
supersymmetric versions of string theory.
In this talk I will describe a class of α′ -exact cosmological solutionsof string theory that interpolate in time between theories withdifferent numbers of spacetime dimensions and different amounts
of worldsheet and spacetime supersymmetry.

Preface
It would be useful to understand how strongly nonsupersymmetricstring theories – such as the bosonic string, as well as string
theories in supercritical dimensions – are related to the more
supersymmetric versions of string theory.
In this talk I will describe a class of α′ -exact cosmological solutionsof string theory that interpolate in time between theories withdifferent numbers of spacetime dimensions and different amounts
of worldsheet and spacetime supersymmetry.
These cosmologies connect supercritical string theories to the morefamiliar string duality web in ten dimensions.

Preface
It would be useful to understand how strongly nonsupersymmetricstring theories – such as the bosonic string, as well as string
theories in supercritical dimensions – are related to the more
supersymmetric versions of string theory.
In this talk I will describe a class of α′ -exact cosmological solutionsof string theory that interpolate in time between theories withdifferent numbers of spacetime dimensions and different amounts
of worldsheet and spacetime supersymmetry.
These cosmologies connect supercritical string theories to the morefamiliar string duality web in ten dimensions.
They also provide a precise link between supersymmetric andpurely bosonic string theory.

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

Quintessent cosmology in D dimensions
Let’s start with a very general class of cosmological models.

Quintessent cosmology in D dimensions
Let’s start with a very general class of cosmological models.
The metric for a spatially flat (k = 0) FRW cosmology is
ds 2 = −dt 2 + a(t)2 dx i dx iwhere i = 1, · · · , D − 1.

Quintessent cosmology in D dimensions
Let’s start with a very general class of cosmological models.
The metric for a spatially flat (k = 0) FRW cosmology is
ds 2 = −dt 2 + a(t)2 dx i dx iwhere i = 1, · · · , D − 1.

The equation of motion for the scale factor a(t):ä
D − 3 + w (D − 1)=−ρa(D − 1)(D − 2)
where w ≡ p/ρ is the state equation.

Quintessent cosmology in D dimensions
Let’s start with a very general class of cosmological models.
The metric for a spatially flat (k = 0) FRW cosmology is
ds 2 = −dt 2 + a(t)2 dx i dx iwhere i = 1, · · · , D − 1.

The equation of motion for the scale factor a(t):ä
D − 3 + w (D − 1)=−ρa(D − 1)(D − 2)
where w ≡ p/ρ is the state equation.

Consider a theory of a real scalar field φ with Lagrangian11√Lφ = 2 − det G(∂µ φ)(∂ µ φ) − V(φ)
κ
2
where
V(φ) ≡ c exp (γφ) ,c, γ > 0

Linear dilaton backgroundAt this point, we adopt the ansatz that our solution exhibits aconstant equation of state w .

Linear dilaton backgroundAt this point, we adopt the ansatz that our solution exhibits aconstant equation of state w .
It follows that φ̇2 , H 2 and V all scale as t −2 , so we find thegeneral expressions
φ(t) = λ log(t/t1 )t αa(t) = a0
t0
for some α, λ.

Linear dilaton backgroundAt this point, we adopt the ansatz that our solution exhibits aconstant equation of state w .
It follows that φ̇2 , H 2 and V all scale as t −2 , so we find thegeneral expressions
φ(t) = λ log(t/t1 )t αa(t) = a0
t0
for some α, λ.
This amounts to a direct relation between ȧ and a, which can be
integrated to yield the following:
α =
γ2 =

2
(1 + w )(D − 1)
2(D − 1)(w + 1)
D −2

Linear dilaton background
Because c > 0, the energy density ρ is positive, and thecosmological scale accelerates as a function of FRW time if−1 ≤ w < wcrit , where
wcrit = −in D spacetime dimensions.

D −3
D −1

Linear dilaton background
Because c > 0, the energy density ρ is positive, and thecosmological scale accelerates as a function of FRW time if−1 ≤ w < wcrit , where
wcrit = −

D −3
D −1

in D spacetime dimensions.
The global causal structure of the solution therefore depends onwhether w is less than, greater than, or equal to the critical value
wcrit .

Linear dilaton background
Because c > 0, the energy density ρ is positive, and thecosmological scale accelerates as a function of FRW time if−1 ≤ w < wcrit , where
wcrit = −

D −3
D −1

in D spacetime dimensions.
The global causal structure of the solution therefore depends onwhether w is less than, greater than, or equal to the critical value
wcrit .
The spatial slice t = 0 defines an initial singularity in all threecases.

Linear dilaton background
Because c > 0, the energy density ρ is positive, and thecosmological scale accelerates as a function of FRW time if−1 ≤ w < wcrit , where
wcrit = −

D −3
D −1

in D spacetime dimensions.
The global causal structure of the solution therefore depends onwhether w is less than, greater than, or equal to the critical value
wcrit .
The spatial slice t = 0 defines an initial singularity in all threecases.
The precise nature of this singularity and the nature of theasymptotic future t → +∞, however, depend on the state
equation of the cosmology.

Linear dilaton backgroundWe can rewrite the metric in a canonical form for a conformallyflat spacetime. We define a new time coordinate t̄ via the equation2(D−1)w+(D−3)(D − 1)(1 + w )(D−1)(1+w) −1
t̄ ≡
t0
a0
t (D−1)(1+w)(D − 1)w + (D − 3)Linear dilaton backgroundWe can rewrite the metric in a canonical form for a conformallyflat spacetime. We define a new time coordinate t̄ via the equation2(D−1)w+(D−3)(D − 1)(1 + w )(D−1)(1+w) −1
t̄ ≡
t0
a0
t (D−1)(1+w)(D − 1)w + (D − 3)
In these coordinates, the metric takes the formhihids 2 = ω(t̄)2 −d t̄ 2 + dx i dx i = ω(t̄)2 −d t̄ 2 + dr 2 + r D−2 dΩ2D−2
where we have definedω(t̄) ≡ l
l



≡ a0

(D − 1)w + (D − 3)t̄
(D − 1)(1 + w )



a0
t0



2
(D−1)w+(D−3)

2(D−1)w+(D−3)

Linear dilaton backgroundTo construct Penrose diagrams, one ignores the (D − 2)-spherefibered over each diagram and conformally compactifies the (r , t)plane using the transformation
r≡

sin χ
cos χ + cos τ

t̄ ≡

sin τ
cos χ + cos τLinear dilaton backgroundTo construct Penrose diagrams, one ignores the (D − 2)-spherefibered over each diagram and conformally compactifies the (r , t)plane using the transformation
r≡

sin χ
cos χ + cos τ

t̄ ≡

sin τ
cos χ + cos τ

In these coordinates, the metric on the (r , t) plane becomesds 2 =

l2
4 cos(sin |τ |)2∆
2+2∆χ+τ 
cos χ−τ22×(−dτ 2 + dχ2 )(D − 1)w + (D − 3) 2∆2(D − 1)(1 + w )

with
∆≡

2
(D − 1)w + (D − 3)Linear dilaton backgroundFor −1 < w < wcrit , the constant ∆ is negative and the range ofthe τ and χ coordinates isτ ∈ [−π, 0]χ ∈ [0, τ + π]

(accelerating universe).

For w > wcrit , the quantity ∆ becomes positive, and we haveτ ∈ [0, π]χ ∈ [0, π − τ ](decelerating universe).

Linear dilaton background(τ,χ) = (π,0)

Futurenull
infinity(τ,χ) = (0,0)

Past spacelike singularity(τ,χ) = (0,π)

Penrose diagram of the decelerating universe (w > wcrit ). The
initial singularity is spacelike, and the future boundary is null.

Linear dilaton background
Future spacelike infinity(τ,χ) = (0,π)

FuturehorizonApparent
horizon

Past nullsingularity(τ,χ) = (−π,0)Penrose diagram of the accelerating (−1 < w < wcrit ) universe.
The initial singularity is null, and the future spacelike boundary is
obscured from observers by a horizon.

Linear dilaton background(τ,χ) = (π,0)

Futurenull
infinity

Past null singularity(τ,χ) = (0,0)

Penrose diagram of the universe with critical equation of state(w = wcrit ).

Linear dilaton background(τ,χ) = (π,0)

Futurenull
infinity

Past null singularity(τ,χ) = (0,0)

Penrose diagram of the universe with critical equation of state(w = wcrit ). The initial singularity is null, as is the futureboundary. It is conformally equivalent to Minkowski space(conventional big bang, asymptotic infinity, and ordinary final
states).

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

Supercritical string theory
For the bosonic string in D > 26, the effective action for the
metric and dilaton appears as
Seff =

1
2κ2

Z

dDx



p
D − 26(S)
2− det G (S) exp (−2Φ) −
+
R
+
4(∂Φ)
3α′

Supercritical string theory
For the bosonic string in D > 26, the effective action for the
metric and dilaton appears as
Seff =

1
2κ2

Z

dDx



p
D − 26(S)
2− det G (S) exp (−2Φ) −
+
R
+
4(∂Φ)
3α′

Higher dimension terms are dropped: such terms in the tree-level
action are suppressed by powers of α′ = 1/(2πTstring ), where
Tstring is the fundamental string tension.

Supercritical string theory
For the bosonic string in D > 26, the effective action for the
metric and dilaton appears as
Seff =

1
2κ2

Z

dDx



p
D − 26(S)
2− det G (S) exp (−2Φ) −
+
R
+
4(∂Φ)
3α′

Higher dimension terms are dropped: such terms in the tree-level
action are suppressed by powers of α′ = 1/(2πTstring ), where
Tstring is the fundamental string tension.
We may rewrite the action in terms of the Einstein metric using
the field redefinition


4Φ
(S)
(E )
Gµν
= exp
Gµν
D −2
√
We may also rescale Φ → 12 D − 2 φ. . .

Supercritical string theory
We obtain:
Seff =

1
2κ2

Z

dDX

p





2(D − 26)
2φ
(E )
2
√
+
R
−
(∂φ)
− det G (E ) −
exp
3α′
D−2

Supercritical string theory
We obtain:
Seff =

1
2κ2

Z

dDX

p





2(D − 26)
2φ
(E )
2
√
+
R
−
(∂φ)
− det G (E ) −
exp
3α′
D−2

This is the action for a quintessent cosmology, with coefficientsnow defined by the following values:
γ=√

2
D −2

c=

D − 263α′Supercritical string theory
We obtain:
Seff =

1
2κ2

Z

dDX

p





2(D − 26)
2φ
(E )
2
√
+
R
−
(∂φ)
− det G (E ) −
exp
3α′
D−2

This is the action for a quintessent cosmology, with coefficientsnow defined by the following values:
γ=√

2
D −2

c=

D − 263α′

We therefore recover a quintessent solution with equation of state
w =−

D −3
D −1

Supercritical string theory
The tree-level potential of the string theory gives rise to an
equation of state at the boundary between accelerating anddecelerating cosmological backgrounds.

Supercritical string theory
The tree-level potential of the string theory gives rise to an
equation of state at the boundary between accelerating anddecelerating cosmological backgrounds.
The resulting spacetime is globally conformally equivalent to
Minkowski space.

Supercritical string theory
The tree-level potential of the string theory gives rise to an
equation of state at the boundary between accelerating anddecelerating cosmological backgrounds.
The resulting spacetime is globally conformally equivalent to
Minkowski space.
In retrospect, this must have been the case: The worldsheet theoryof the string in this background is defined to have a target spacewith string frame metric ηµν , and coordinates X µ that are infinite
in extent.

Supercritical string theory
The tree-level potential of the string theory gives rise to an
equation of state at the boundary between accelerating anddecelerating cosmological backgrounds.
The resulting spacetime is globally conformally equivalent to
Minkowski space.
In retrospect, this must have been the case: The worldsheet theoryof the string in this background is defined to have a target spacewith string frame metric ηµν , and coordinates X µ that are infinite
in extent.
The equation of state w = − D−3
D−1 with positive energy also
describes negative curvature dominated cosmology.

Supercritical string theory
The tree-level potential of the string theory gives rise to an
equation of state at the boundary between accelerating anddecelerating cosmological backgrounds.
The resulting spacetime is globally conformally equivalent to
Minkowski space.
In retrospect, this must have been the case: The worldsheet theoryof the string in this background is defined to have a target spacewith string frame metric ηµν , and coordinates X µ that are infinite
in extent.
The equation of state w = − D−3
D−1 with positive energy also
describes negative curvature dominated cosmology.
Maximal SUSY breaking at weak coupling?

Supercritical string theory
The general quintessent solution is of the form


D −2
t
Φ = Φ0 −
ln
2
t0
a0
a =
t
t0

Supercritical string theory
The general quintessent solution is of the form


D −2
t
Φ = Φ0 −
ln
2
t0
a0
a =
t
t0
We have the relation
tFRW

2q tconf= t0 exp +D −2Supercritical string theory
The general quintessent solution is of the form


D −2
t
Φ = Φ0 −
ln
2
t0
a0
a =
t
t0
We have the relation
tFRW

2q tconf= t0 exp +D −2

So when we move to coordinates in which the metric is manifestly
conformally flat and we decanonicalize the scalar field, we find thatthe dilaton is logarithmic as a function of FRW time, and linear as
a function of conformal time:a2
2
ds 2 = 20 t 2 −dtconf+ dx i dx i = a2 ηµν dx µ dx ν(1)t0
Φ = Φ0 − q tconf

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

Stability
In time-dependent backgrounds there is no obvious or
natural definition of stability.

Stability
In time-dependent backgrounds there is no obvious or
natural definition of stability.
One must take into account the coupling of the background fields
to the string modes, which is suppressed by a factor of the string
coupling gs ∼ e φ .

Stability
In time-dependent backgrounds there is no obvious or
natural definition of stability.
One must take into account the coupling of the background fields
to the string modes, which is suppressed by a factor of the string
coupling gs ∼ e φ .
Exponentially growing modes will have decreasing effect on theremaining degrees of freedom in the theory if they grow more
slowly than gs−1 .

Stability
In time-dependent backgrounds there is no obvious or
natural definition of stability.
One must take into account the coupling of the background fields
to the string modes, which is suppressed by a factor of the string
coupling gs ∼ e φ .
Exponentially growing modes will have decreasing effect on theremaining degrees of freedom in the theory if they grow more
slowly than gs−1 .
A useful definition, therefore, is that unstable modes must growfaster than gs−1 at late times.

Stability
Consider a massless scalar coupled to the string:Lσ = −

1 p
1 p(S) e −2Φ (∂σ)2 = −−det
G− det G (E ) (∂σ)22κ2
2κ2

Stability
Consider a massless scalar coupled to the string:Lσ = −

1 p
1 p(S) e −2Φ (∂σ)2 = −−det
G− det G (E ) (∂σ)22κ2
2κ2The scalar field has solutions of the formσ = σ∞ − ξ t −(D−2)
where σ∞ and ξ are constants of motion that can take arbitrary
real values.

Stability
Consider a massless scalar coupled to the string:Lσ = −

1 p
1 p(S) e −2Φ (∂σ)2 = −−det
G− det G (E ) (∂σ)22κ2
2κ2The scalar field has solutions of the formσ = σ∞ − ξ t −(D−2)
where σ∞ and ξ are constants of motion that can take arbitrary
real values.
The modes of the field σ asymptote to the constant value σ∞ as
t → ∞.

Stability
Consider a massless scalar coupled to the string:Lσ = −

1 p
1 p(S) e −2Φ (∂σ)2 = −−det
G− det G (E ) (∂σ)22κ2
2κ2The scalar field has solutions of the formσ = σ∞ − ξ t −(D−2)
where σ∞ and ξ are constants of motion that can take arbitrary
real values.
The modes of the field σ asymptote to the constant value σ∞ as
t → ∞.
From the point of view of the Einstein frame, this effect is due to
Hubble friction; in the string frame this behavior is understood to
be caused by the drag force arising from the interaction between σ
and the linear dilaton.

Stability
Quanta of the string are necessarily coupled to the flat metric. To
recover fluctuations in such a form we must introduce a rescaled
field σ̃:σ̃ ≡ e −Φ σStability
Quanta of the string are necessarily coupled to the flat metric. To
recover fluctuations in such a form we must introduce a rescaled
field σ̃:σ̃ ≡ e −Φ σThis induces a mass term for the rescaled field that represents a
proper quantum of string:e −2Φ (∂σ)2 = (∂ σ̃)2 + σ̃ 2 (∂Φ)2 + 2σ̃(∂ σ̃) [(∂Φ)background + (∂Φ)fluctuation ]Stability
Quanta of the string are necessarily coupled to the flat metric. To
recover fluctuations in such a form we must introduce a rescaled
field σ̃:σ̃ ≡ e −Φ σThis induces a mass term for the rescaled field that represents a
proper quantum of string:e −2Φ (∂σ)2 = (∂ σ̃)2 + σ̃ 2 (∂Φ)2 + 2σ̃(∂ σ̃) [(∂Φ)background + (∂Φ)fluctuation ]◮ The fluctuation term represents a trilinear vertex that we

discard.
◮ The background term is constant, so its product with σ̃ ∂ σ̃amounts to a total derivative.

Stability
The mass term for the rescaled field is tachyonic, and equal to−q 2 :Lσ̃ ∼ −1 (∂ σ̃)2 − q 2 σ̃ 222κThe criterion for stability we proposed was that gs times the
canonical field σ̃ not increase exponentially with time.

Stability
The mass term for the rescaled field is tachyonic, and equal to−q 2 :Lσ̃ ∼ −1 (∂ σ̃)2 − q 2 σ̃ 222κThe criterion for stability we proposed was that gs times the
canonical field σ̃ not increase exponentially with time.
Since gs σ̃ is just σ (the original field appearing in the spacetime
action in front of exp (−2Φ)), the requirement for physical stability
is that modes normalized to have the factor exp (−2Φ) in theirkinetic term should shrink exponentially in the future (or at least
remain constant).

Stability
The mass term for the rescaled field is tachyonic, and equal to−q 2 :Lσ̃ ∼ −1 (∂ σ̃)2 − q 2 σ̃ 222κThe criterion for stability we proposed was that gs times the
canonical field σ̃ not increase exponentially with time.
Since gs σ̃ is just σ (the original field appearing in the spacetime
action in front of exp (−2Φ)), the requirement for physical stability
is that modes normalized to have the factor exp (−2Φ) in theirkinetic term should shrink exponentially in the future (or at least
remain constant).
So, canonical modes may grow exponentially in time, but they do
not necessarily represent a physical instability: the exponential
growth is countered by the shrinking string coupling.
[Hellerman, IS: hep-th/0611317; Aharony, Silverstein: hep-th/0612031]

Stability

We are interested in “truly tachyonic” perturbations of unstablestring theories. In tbe bosonic string, for instance, the tachyon
T (X ) couples to the worldsheet as a normal-ordered potential: T (X ) :.

Stability

We are interested in “truly tachyonic” perturbations of unstablestring theories. In tbe bosonic string, for instance, the tachyon
T (X ) couples to the worldsheet as a normal-ordered potential: T (X ) :.
We will now discuss a large class of solvable and exactly marginal
perturbations of this form.

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

Bubble of nothing
Consider a theory with stress tensor
1
T++ = − ′ : ∂σ+ X µ ∂σ+ Xµ : +∂σ2+ (Vµ X µ )α1: ∂σ− X µ ∂σ− Xµ : +∂σ2− (Vµ X µ )α′where colons represent normal ordering of the 2D theory. Here, σ ±are particular light-cone combinations of the worldsheet
coordinates σ 0,1 :T−− = −

σ ± = −σ 0 ± σ 1

Bubble of nothing
Consider a theory with stress tensor
1
T++ = − ′ : ∂σ+ X µ ∂σ+ Xµ : +∂σ2+ (Vµ X µ )α1: ∂σ− X µ ∂σ− Xµ : +∂σ2− (Vµ X µ )α′where colons represent normal ordering of the 2D theory. Here, σ ±are particular light-cone combinations of the worldsheet
coordinates σ 0,1 :T−− = −

σ ± = −σ 0 ± σ 1
Physical states of the string correspond to local operators U thatare Virasoro primaries of weight one. That is, their operator
product expansion (OPE) with the stress tensor satisfies:
T++ (σ)U(τ ) ≃and similarly for T−− ,U(τ )(σ + − τ + )2+∂+ U(τ )σ+ − τ +Bubble of nothing
A profile T (X ) for the tachyon corresponds to the vertex operator
UM ≡: T (X ) :and admits the following on-shell condition:∂µ ∂ µ T (X ) − 2V µ ∂µ T (X ) +4
T (X ) = 0
α′Bubble of nothing
A profile T (X ) for the tachyon corresponds to the vertex operator
UM ≡: T (X ) :and admits the following on-shell condition:∂µ ∂ µ T (X ) − 2V µ ∂µ T (X ) +4
T (X ) = 0
α′

For tachyon profiles of the form
T (X ) = µ2 exp (Bµ X µ )this condition is
B 2 − 2V · B = −4/α′Bubble of nothing
A profile T (X ) for the tachyon corresponds to the vertex operator
UM ≡: T (X ) :and admits the following on-shell condition:∂µ ∂ µ T (X ) − 2V µ ∂µ T (X ) +4
T (X ) = 0
α′

For tachyon profiles of the form
T (X ) = µ2 exp (Bµ X µ )this condition is
B 2 − 2V · B = −4/α′
A general value of Bµ will lead to a nontrivial interacting theory
when the strength µ2 of the perturbation is treated as
non-infinitesimal.

Bubble of nothing
There is a special set of choices for Bµ that renders the 2D theory
well-defined and conformal to all orders in perturbation theory.

Bubble of nothing
There is a special set of choices for Bµ that renders the 2D theory
well-defined and conformal to all orders in perturbation theory.
We choose the first term in the linearized tachyon equation of
motion to vanish separately.

Bubble of nothing
There is a special set of choices for Bµ that renders the 2D theory
well-defined and conformal to all orders in perturbation theory.
We choose the first term in the linearized tachyon equation of
motion to vanish separately.
This is tantamount to choosing the vector Bµ to be null. Thisrenders the vertex operator : exp (Bµ X µ ) : non-singular in thevicinity of itself.

Bubble of nothing
There is a special set of choices for Bµ that renders the 2D theory
well-defined and conformal to all orders in perturbation theory.
We choose the first term in the linearized tachyon equation of
motion to vanish separately.
This is tantamount to choosing the vector Bµ to be null. Thisrenders the vertex operator : exp (Bµ X µ ) : non-singular in thevicinity of itself.
We therefore put Bµ in the form√
B0 = B1 ≡ β/ 2
Bi

= 0,

i ≥2

Bubble of nothing
The initial singularity of the cosmology lies in the strong-couplingregion, and the tachyon increases into the future.

Bubble of nothing
The initial singularity of the cosmology lies in the strong-couplingregion, and the tachyon increases into the future.
This gives rise to a particularly simple quantum theory. The kinetic
term for X ± appears as
L∼−

i
1 h
+
−
+
−
(∂
0 X )(∂σ 0 X ) − (∂σ 1 X )(∂σ 1 X )σ
2πα′Bubble of nothing
The initial singularity of the cosmology lies in the strong-couplingregion, and the tachyon increases into the future.
This gives rise to a particularly simple quantum theory. The kinetic
term for X ± appears as
L∼−

i
1 h
+
−
+
−
(∂
0 X )(∂σ 0 X ) − (∂σ 1 X )(∂σ 1 X )
σ
2πα′



The propagator for the X ± fields is therefore oriented.

X+

X−

Bubble of nothing◮ The X field has oriented propagators.
◮ All the interaction vertices in the theory depend only on X + .
◮ There are no non-trivial Feynman diagrams in the theory.
◮ This constitutes an interacting quantum theory, without

quantum corrections.

Bubble of nothing◮ The X field has oriented propagators.
◮ All the interaction vertices in the theory depend only on X + .
◮ There are no non-trivial Feynman diagrams in the theory.
◮ This constitutes an interacting quantum theory, without

quantum corrections.
(In conformal gauge, prior to enforcing gauge constraints, thetheory is not unitary.)

Bubble of nothing◮ The X field has oriented propagators.
◮ All the interaction vertices in the theory depend only on X + .
◮ There are no non-trivial Feynman diagrams in the theory.
◮ This constitutes an interacting quantum theory, without

quantum corrections.
(In conformal gauge, prior to enforcing gauge constraints, thetheory is not unitary.)
The tachyon couples to the worldsheet in the term
L∼−


1 2
µ exp βX +2π

Bubble of nothing◮ The X field has oriented propagators.
◮ All the interaction vertices in the theory depend only on X + .
◮ There are no non-trivial Feynman diagrams in the theory.
◮ This constitutes an interacting quantum theory, without

quantum corrections.
(In conformal gauge, prior to enforcing gauge constraints, thetheory is not unitary.)
The tachyon couples to the worldsheet in the term
L∼−


1 2
µ exp βX +2πClassically, X + is harmonic, and acts as a source for X − .

Bubble of nothing
By writing the solution to the Laplace equation for X + as
X + = f+ (σ + ) + f− (σ − )the general solution for X − can be expressed as follows:
X − = g+ (σ+ ) + g− (σ− ) +α′ βµ2
4

Z ∞
σ+


dy + exp βf+ (y + ) Z ∞σ−dy − exp βf− (y − )Bubble of nothing
By writing the solution to the Laplace equation for X + as
X + = f+ (σ + ) + f− (σ − )the general solution for X − can be expressed as follows:
X − = g+ (σ+ ) + g− (σ− ) +α′ βµ2
4

Z ∞
σ+


dy + exp βf+ (y + ) Z ∞We thus see that the theory is exactly solvable.

σ−dy − exp βf− (y − )Bubble of nothing
By writing the solution to the Laplace equation for X + as
X + = f+ (σ + ) + f− (σ − )the general solution for X − can be expressed as follows:
X − = g+ (σ+ ) + g− (σ− ) +α′ βµ2
4

Z ∞
σ+


dy + exp βf+ (y + ) Z ∞σ−dy − exp βf− (y − )We thus see that the theory is exactly solvable.
All interaction vertices in the theory depend only on X + , andtherefore correspond to diagrams composed strictly from outgoinglines:

 
,

,

,

... ,



Physical interpretation
The solution can be thought of as a phase boundary in spacetime
between the T = 0 phase and the T > 0 phase.

Physical interpretation
The solution can be thought of as a phase boundary in spacetime
between the T = 0 phase and the T > 0 phase.
The spacetime picture is therefore a phase bubble expanding out
from a nucleation point:
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Time

Space

Physical interpretation
To see what happens to states in the neighborhood of the bubble
we can place a string state near the phase boundary.

Physical interpretation
To see what happens to states in the neighborhood of the bubble
we can place a string state near the phase boundary.
150
100
X0

50
0
-50

-60

-40
X1

-20

0

The state collides with the bubble wall and is forced out of theregion with nonzero tachyon. (The solution has µ2 = 1, β = .1,
and the trajectory corresponds to p + = 3, H⊥ ≡α′ pi2
2 = 4.)

Physical interpretation
We can also plot the velocity of the particle as a function of time:
0.5
0.25

Velocity

0
-0.25
-0.5
-0.75
-1
-200

0

200

400

X0

So the particle propagates until it hits the bubble wall, where theexponential term becomes important. At that point, the speed ofthe particle rapidly goes to −1.

Physical interpretation
Absolutely no matter (including gravitons) can enter the region ofnonzero tachyon.

Physical interpretation
Absolutely no matter (including gravitons) can enter the region ofnonzero tachyon.
The solution can be thought of as a bubble of nothing.
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Time

Space

Dimension-changing solutions in the bosonic string
Let’s now inroduce some dependence on a third direction:
T (X ) = µ20 exp βX + − µ2k cos(kX2 )exp βk X +with√qβk = 2



2
1
− k2
′
α
2



Dimension-changing solutions in the bosonic string
Let’s now inroduce some dependence on a third direction:
T (X ) = µ20 exp βX + − µ2k cos(kX2 )exp βk X +with√qβk = 2



2
1
− k2
′
α
2



This is exactly marginal when the wavelength k −1 of the tachyon
is long compared to the string scale:µ2
exp βX + : X22 : +T0 (X + )
′
2α
2 X+


µ
√ exp βX + + µ′2 exp βX +T0 (X + ) =′α q 2

T (X + , X2 ) = +Dimension-changing solutions in the bosonic string
States with modes of X2 excited are pushed out along the bubble
wall: the physics is essentially the same as the bubble of nothing.

Dimension-changing solutions in the bosonic string
States with modes of X2 excited are pushed out along the bubble
wall: the physics is essentially the same as the bubble of nothing.
At late times, the adiabatic theorem is satisfied to a better
approximation, and these modes become frozen in an excited state.

Dimension-changing solutions in the bosonic string
States with modes of X2 excited are pushed out along the bubble
wall: the physics is essentially the same as the bubble of nothing.
At late times, the adiabatic theorem is satisfied to a better
approximation, and these modes become frozen in an excited state.
So these string states are pushed out to infinity and disappear fromthe theory in the late-time limit:
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States with X2 excitedDimension-changing solutions in the bosonic string
There is a less generic class of states with no energy in the X2
direction.

Dimension-changing solutions in the bosonic string
There is a less generic class of states with no energy in the X2
direction.
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States with no X2 excitations
States with X2 excited

These propagate through the domain wall and into the bubble
region.

Dimension-changing solutions in the bosonic string
There is a less generic class of states with no energy in the X2
direction.
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States with no X2 excitations
States with X2 excited

These propagate through the domain wall and into the bubble
region.
The result is that the amount of matter on the worldsheet
decreases dynamically as a function of time.

Dimension-changing solutions in the bosonic string
In other words, the number of dimensions in the target spacedecreases as a function of time.

Dimension-changing solutions in the bosonic string
In other words, the number of dimensions in the target spacedecreases as a function of time.
Question: What happens to the central charge if the spacetime
dimension shifts? How can the perturbation be marginal?

Dimension-changing solutions in the bosonic string
In other words, the number of dimensions in the target spacedecreases as a function of time.
Question: What happens to the central charge if the spacetime
dimension shifts? How can the perturbation be marginal?
The theory is solvable, so we should be able to answer thisquestion exactly.

Dimension-changing solutions in the bosonic string
In other words, the number of dimensions in the target spacedecreases as a function of time.
Question: What happens to the central charge if the spacetime
dimension shifts? How can the perturbation be marginal?
The theory is solvable, so we should be able to answer thisquestion exactly.

 
 

In fact, quantum corrections in this theory truncate at one-loop
order:

=

+

+

+

+

+ (perms.)

Dimension-changing solutions in the bosonic string
The one-loop diagrams can be thought of as a set of effective
vertices for X + , associated with integrating out the massive fieldX2 .

Dimension-changing solutions in the bosonic string
The one-loop diagrams can be thought of as a set of effective
vertices for X + , associated with integrating out the massive field
X2 .
As you approach the domain wall, there are a number of massive
string modes that acquire expectation values, and there arehigher-derivative operators dressed with factors of exp (X + ).

Dimension-changing solutions in the bosonic string
The one-loop diagrams can be thought of as a set of effective
vertices for X + , associated with integrating out the massive field
X2 .
As you approach the domain wall, there are a number of massive
string modes that acquire expectation values, and there arehigher-derivative operators dressed with factors of exp (X + ).
Most of these decay exponentially in the future.

Dimension-changing solutions in the bosonic string
The one-loop diagrams can be thought of as a set of effective
vertices for X + , associated with integrating out the massive field
X2 .
As you approach the domain wall, there are a number of massive
string modes that acquire expectation values, and there arehigher-derivative operators dressed with factors of exp (X + ).
Most of these decay exponentially in the future.
In fact: in the far future, all corrections coming from integratingout X2 decay away, except for three contributions:◮ the effective tachyon,◮ the dilaton,◮ the string-frame metric.

Dimension-changing solutions in the bosonic string
The effective tachyon can be fine-tuned away in the future.



Dimension-changing solutions in the bosonic string
The effective tachyon can be fine-tuned away in the future.
The remaining contributions are always nonzero, coming from thefollowing diagrams:∆(∂+ Φ) =∆G++ =



Dimension-changing solutions in the bosonic string
The effective tachyon can be fine-tuned away in the future.
The remaining contributions are always nonzero, coming from thefollowing diagrams:∆(∂+ Φ) =∆G++ =

Write the renormalized dilaton gradient and string-frame metric as:V̂µ ≡ Vµ + ∆VµĜ µν≡ Gµν + ∆Gµν

Dimension-changing solutions in the bosonic string
We obtain:qV̂− = V− = − √2qβV̂+ = − √ +122Ĝ +− = Ĝ −+ = −1

Ĝ −− = −α′ β 224

Ĝ ++ = 0

Dimension-changing solutions in the bosonic string
We obtain:qV̂− = V− = − √2qβV̂+ = − √ +122Ĝ +− = Ĝ −+ = −1

Ĝ −− = −α′ β 224

Ĝ ++ = 0√

q and β take values such that q 2 = (D − 26)/6α′ and qβ = 2α′2 .

Dimension-changing solutions in the bosonic string
We obtain:qV̂− = V− = − √2qβV̂+ = − √ +122Ĝ +− = Ĝ −+ = −1

Ĝ −− = −α′ β 224

Ĝ ++ = 0√

q and β take values such that q 2 = (D − 26)/6α′ and qβ = 2α′2 .
In the X + → ∞ limit, we therefore getc dilaton = 6α′ Ĝ µν V̂µ V̂ν = −(D − 26) + 1

Dimension-changing solutions in the bosonic string
We obtain:qV̂− = V− = − √2qβV̂+ = − √ +122Ĝ +− = Ĝ −+ = −1

Ĝ −− = −α′ β 224

Ĝ ++ = 0√

q and β take values such that q 2 = (D − 26)/6α′ and qβ = 2α′2 .
In the X + → ∞ limit, we therefore getc dilaton = 6α′ Ĝ µν V̂µ V̂ν = −(D − 26) + 1
The result is that the shift in cenral charge contribution from thedilaton precisely cancels the central charge shift due to the
reduction in spacetime dimension.

Dimension-changing solutions in the bosonic string
This mechanism of central charge transfer works equally well when
the tachyon has a quadratic minimum in several transverse
directions:n+1

T (X ) =T0 (X + ) = Xµ2
exp βX +: Xi2 : +T0 (X + )
′
2α
i =2

n µ2 X +α′ q√ exp βX + + µ′2 exp βX +2Dimension-changing solutions in the bosonic string
This mechanism of central charge transfer works equally well when
the tachyon has a quadratic minimum in several transverse
directions:n+1

T (X ) =T0 (X + ) = Xµ2
exp βX +: Xi2 : +T0 (X + )
′
2α
i =2

n µ2 X +α′ q√ exp βX + + µ′2 exp βX +2In this case the renormalization of the metric and dilaton leads toa central charge contribution in the X + → ∞ limit given by:c dilaton = 6α′ Ĝ µν V̂µ V̂ν = −6α′ q 2 +nqβα′nα′2 q 2 β 2√−= −(D − 26) + n
82

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

Dimension-changing solutions in the type 0 string
Let’s now consider the type 0 string theory in D greater than 10.
The magnitude of the dilaton gradient is now

Dimension-changing solutions in the type 0 string
Let’s now consider the type 0 string theory in D greater than 10.
The magnitude of the dilaton gradient is nowr

D − 104α′
and the linearized of motion for the tachyon isV0 = −q,q=Dimension-changing solutions in the type 0 string
Let’s now consider the type 0 string theory in D greater than 10.
The magnitude of the dilaton gradient is nowr

D − 104α′
and the linearized of motion for the tachyon isV0 = −q,q=∂ µ ∂µ T − 2V µ ∂µ T +

2
T =0
α′

Dimension-changing solutions in the type 0 string
Let’s now consider the type 0 string theory in D greater than 10.
The magnitude of the dilaton gradient is nowr

D − 104α′
and the linearized of motion for the tachyon isV0 = −q,q=∂ µ ∂µ T − 2V µ ∂µ T +One simple set of solutions is:

2
T =0
α′

Dimension-changing solutions in the type 0 string
Let’s now consider the type 0 string theory in D greater than 10.
The magnitude of the dilaton gradient is nowr

D − 104α′
and the linearized of motion for the tachyon isV0 = −q,q=∂ µ ∂µ T − 2V µ ∂µ T +

2
T =0
α′

One simple set of solutions is:

T = µ exp βX

+



,

√
2
βq = ′αDimension-changing solutions in the type 0 string
Let’s now consider the type 0 string theory in D greater than 10.
The magnitude of the dilaton gradient is nowr

D − 104α′
and the linearized of motion for the tachyon isV0 = −q,q=∂ µ ∂µ T − 2V µ ∂µ T +

2
T =0
α′

One simple set of solutions is:

T = µ exp βX

+



,

√
2
βq = ′
αWe will not consider these solutions yet – they are qualitatively
different from our previous examples!

Dimension-changing solutions in the type 0 string
Instead we will allow dependence on two transverse directions X2
and X3 :Dimension-changing solutions in the type 0 string
Instead we will allow dependence on two transverse directions X2
and X3 :T = µ exp βX+



X2 X3 ,

βq =

√

2
α′

Dimension-changing solutions in the type 0 string
Instead we will allow dependence on two transverse directions X2
and X3 :T = µ exp βX+



X2 X3 ,

βq =

√

2
α′

Note that this perturbation is strictly primary of weight (1,1) –without the addition of correction terms.

Dimension-changing solutions in the type 0 string
Instead we will allow dependence on two transverse directions X2
and X3 :T = µ exp βX+



X2 X3 ,

βq =

√

2
α′

Note that this perturbation is strictly primary of weight (1,1) –without the addition of correction terms.
The tachyon couples to the worldsheet as a (1, 1) superpotential:
Zi
Lint =
dθ+ dθ− T (X )2πDimension-changing solutions in the type 0 string
Instead we will allow dependence on two transverse directions X2
and X3 :T = µ exp βX+



X2 X3 ,

βq =

√

2
α′

Note that this perturbation is strictly primary of weight (1,1) –without the addition of correction terms.
The tachyon couples to the worldsheet as a (1, 1) superpotential:
Zi
Lint =
dθ+ dθ− T (X )2πThis gives rise to a potential and Yukawa term:
Lint = −

α′ MN
i α′
G
∂M T ∂N T +∂M ∂N T ψ̃ M ψ N
8π
4π

Dimension-changing solutions in the type 0 string
For our particular choice of profile, this gives an X + -dependent
mass term M ≡ µ exp (βX + ) to the bosons and fermions in theX2 and X3 multiplets:

 2
α′ µ2
exp 2βX +X2 + X328π
i α′ µ+exp βX +ψ̃2 ψ3 + ψ̃3 ψ2
4πLint = −Dimension-changing solutions in the type 0 string
For our particular choice of profile, this gives an X + -dependent
mass term M ≡ µ exp (βX + ) to the bosons and fermions in theX2 and X3 multiplets:

 2
α′ µ2
exp 2βX +X2 + X328π
i α′ µ+exp βX +ψ̃2 ψ3 + ψ̃3 ψ2
4πLint = −The spacetime dimension decreases by 2 in the limit of large X + !
The string theory in D-2 dimensions inherits the diagonal GSO
projection of the D-dimensional parent theory – that is, thefinal-state string theory is still type 0.

Dimension-changing solutions in the type 0 string
Once again, the domain wall acts as a filter for states with noexcitations in the modes of the X2,3 fields or their fermionicsuperpartners.

Dimension-changing solutions in the type 0 string
Once again, the domain wall acts as a filter for states with noexcitations in the modes of the X2,3 fields or their fermionicsuperpartners.
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States w/ no X 2,3 multiplets excited
States w/ X 2,3 multiplets excited

The domain wall is a boundary between type 0 in D dimensions
and type 0 in D-2 dimensions.

Dimension-changing solutions in the type 0 stringDimension-changing solutions in the type 0 string
Quantum corrections in the worldsheet theory truncate again at
one-loop order.

Dimension-changing solutions in the type 0 string
Quantum corrections in the worldsheet theory truncate again at
one-loop order.
All effective operators generated by integrating out the X2,3
multiplets decay exponentially as a function of X + , except for the
couplings to the dilaton Φ and string-frame metric GMN .
The renormalizations are:

Dimension-changing solutions in the type 0 string
Quantum corrections in the worldsheet theory truncate again at
one-loop order.
All effective operators generated by integrating out the X2,3
multiplets decay exponentially as a function of X + , except for the
couplings to the dilaton Φ and string-frame metric GMN .
The renormalizations are:

βX +
2
2 α′
β
∆G++ = −∆G −− = +2∆Φ = +

which gives∆c dilaton = +3

Dimension-changing solutions in the type 0 stringMore generally we can start with type 0 in an even number ofdimensions D = 10 + 2K , and consider a tachyon profile

Dimension-changing solutions in the type 0 stringMore generally we can start with type 0 in an even number ofdimensions D = 10 + 2K , and consider a tachyon profileT ≡ µ exp βX
with Yj ≡ X9+K +j .


+




n
X
j=1



X9+j Yj Dimension-changing solutions in the type 0 stringMore generally we can start with type 0 in an even number ofdimensions D = 10 + 2K , and consider a tachyon profileT ≡ µ exp βX
with Yj ≡ X9+K +j .


+




n
X
j=1



X9+j Yj 

Again, the increase dilaton central charge precisely compensates
the loss of matter central charge. Integrating out 2n dimensions inthis way renormalizes the dilaton central charge byDimension-changing solutions in the type 0 stringMore generally we can start with type 0 in an even number ofdimensions D = 10 + 2K , and consider a tachyon profileT ≡ µ exp βX


+




n
X
j=1



X9+j Yj with Yj ≡ X9+K +j .

Again, the increase dilaton central charge precisely compensates
the loss of matter central charge. Integrating out 2n dimensions inthis way renormalizes the dilaton central charge by∆c dilaton = +3n

Dimension-changing solutions in the type 0 stringMore generally we can start with type 0 in an even number ofdimensions D = 10 + 2K , and consider a tachyon profileT ≡ µ exp βX


+




n
X
j=1



X9+j Yj with Yj ≡ X9+K +j .

Again, the increase dilaton central charge precisely compensates
the loss of matter central charge. Integrating out 2n dimensions inthis way renormalizes the dilaton central charge by∆c dilaton = +3n
In the case n = K , the final state is critical, 10-dimensional type 0
string theory with lightlike linear dilaton rolling to weak coupling in
the future.

Dimension-changing solutions in the type 0 string
Note that no effective tachyon vev is generated inside the bubble –the vanishing of the tachyon at large X + is natural.

Dimension-changing solutions in the type 0 string
Note that no effective tachyon vev is generated inside the bubble –the vanishing of the tachyon at large X + is natural.
There is a discrete chiral R-symmetry (−1)FLW which acts as:
Y :
X :
G̃ :
G:

−

+

−

+

Dimension-changing solutions in the type 0 string
Note that no effective tachyon vev is generated inside the bubble –the vanishing of the tachyon at large X + is natural.
There is a discrete chiral R-symmetry (−1)FLW which acts as:
Y :
X :
G̃ :
G:

−

+

−

+

which is unbroken by the tachyon profile in the D-dimensional
theory. Also an unbroken non-R symmetry reflecting Yn+1,··· ,K .

Dimension-changing solutions in the type 0 string
Note that no effective tachyon vev is generated inside the bubble –the vanishing of the tachyon at large X + is natural.
There is a discrete chiral R-symmetry (−1)FLW which acts as:
Y :
X :
G̃ :
G:

−

+

−

+

which is unbroken by the tachyon profile in the D-dimensional
theory. Also an unbroken non-R symmetry reflecting Yn+1,··· ,K .
These symmetries forbid the generation of any term

∆L ∝

Z

dθ+ dθ− f (X , Yn+1,··· ,K )in the D − 2n dimensional theory on the right of the domain wall.

Transitions from type 0 to type II string theory
Instead of starting with type 0 on a smooth space, we can consider
starting on a ZZ2 orbifold of flat space.

Transitions from type 0 to type II string theory
Instead of starting with type 0 on a smooth space, we can consider
starting on a ZZ2 orbifold of flat space.
In particular, let us orbifold by the symmetry (−1)FLW we justdefined for the type 0 initial state in D = 10 + 2K dimensions.
Apart from acting on G̃ , the generator (−1)FLW reflects all K of
the Y coordinates.

Transitions from type 0 to type II string theory
Instead of starting with type 0 on a smooth space, we can consider
starting on a ZZ2 orbifold of flat space.
In particular, let us orbifold by the symmetry (−1)FLW we justdefined for the type 0 initial state in D = 10 + 2K dimensions.
Apart from acting on G̃ , the generator (−1)FLW reflects all K of
the Y coordinates.
The orbifold singularity has real codimension K, with massless
spacetime fermions propagating on the 10 + K dimensional fixed
locus.

Transitions from type 0 to type II string theory
Instead of starting with type 0 on a smooth space, we can consider
starting on a ZZ2 orbifold of flat space.
In particular, let us orbifold by the symmetry (−1)FLW we justdefined for the type 0 initial state in D = 10 + 2K dimensions.
Apart from acting on G̃ , the generator (−1)FLW reflects all K of
the Y coordinates.
The orbifold singularity has real codimension K, with massless
spacetime fermions propagating on the 10 + K dimensional fixed
locus.
The boundary conditions at the orbifold force the tachyon T tovanish at Y = 0:

Transitions from type 0 to type II string theory
Instead of starting with type 0 on a smooth space, we can consider
starting on a ZZ2 orbifold of flat space.
In particular, let us orbifold by the symmetry (−1)FLW we justdefined for the type 0 initial state in D = 10 + 2K dimensions.
Apart from acting on G̃ , the generator (−1)FLW reflects all K of
the Y coordinates.
The orbifold singularity has real codimension K, with massless
spacetime fermions propagating on the 10 + K dimensional fixed
locus.
The boundary conditions at the orbifold force the tachyon T tovanish at Y = 0:

T (X , Y ) = −T (X , −Y )

Transitions from type 0 to type II string theory
Starting with type 0 in D = 10 + 2 K, we can consider a tachyon
profile which describes the behavior of a generic tachyon vev near
the orbifold fixed locus:

Transitions from type 0 to type II string theory
Starting with type 0 in D = 10 + 2 K, we can consider a tachyon
profile which describes the behavior of a generic tachyon vev near
the orbifold fixed locus:

T ≡ µ exp βX
with Yj ≡ X9+K +j .


+




K
X
j=1



X9+j Yj Transitions from type 0 to type II string theory
Starting with type 0 in D = 10 + 2 K, we can consider a tachyon
profile which describes the behavior of a generic tachyon vev near
the orbifold fixed locus:

T ≡ µ exp βX


+




K
X
j=1



X9+j Yj with Yj ≡ X9+K +j .
Since the tachyon is odd under reflection of the Y coordinates, theworldsheet mass matrix can at most pair up K of the(unorbifolded) X coordinates with the K (orbifolded) Y
coordinates.

Transitions from type 0 to type II string theory
Starting with type 0 in D = 10 + 2 K, we can consider a tachyon
profile which describes the behavior of a generic tachyon vev near
the orbifold fixed locus:

T ≡ µ exp βX


+




K
X
j=1



X9+j Yj with Yj ≡ X9+K +j .
Since the tachyon is odd under reflection of the Y coordinates, theworldsheet mass matrix can at most pair up K of the(unorbifolded) X coordinates with the K (orbifolded) Y
coordinates.
The minimum dimension of the final state is therefore 10!

Transitions from type 0 to type II string theory
To discern the effective physics of the ten-dimensional final state,note that the GSO projection of the X + → ∞ theory is now
generated by two elements.

Transitions from type 0 to type II string theory
To discern the effective physics of the ten-dimensional final state,note that the GSO projection of the X + → ∞ theory is now
generated by two elements.
The orbifold symmetry (−1)FLW acts on the remaining fields as:

Transitions from type 0 to type II string theory
To discern the effective physics of the ten-dimensional final state,note that the GSO projection of the X + → ∞ theory is now
generated by two elements.
The orbifold symmetry (−1)FLW acts on the remaining fields as:
X :

+

G̃ :

−

G:

+

Transitions from type 0 to type II string theory
To discern the effective physics of the ten-dimensional final state,note that the GSO projection of the X + → ∞ theory is now
generated by two elements.
The orbifold symmetry (−1)FLW acts on the remaining fields as:
X :

+

G̃ :

−

G:

+

We also have the generator (−1)FW of the type 0 GSO projection.
The product (−1)FRW ≡ (−1)FW · (−1)FLW acts as:

Transitions from type 0 to type II string theory
To discern the effective physics of the ten-dimensional final state,note that the GSO projection of the X + → ∞ theory is now
generated by two elements.
The orbifold symmetry (−1)FLW acts on the remaining fields as:
X :

+

G̃ :

−

G:

+

We also have the generator (−1)FW of the type 0 GSO projection.
The product (−1)FRW ≡ (−1)FW · (−1)FLW acts as:
X :

+

G̃ :

+

G:

−

Transitions from type 0 to type II string theory
We thus have the usual GSO projection of critical type II string
theory. The worldsheet theory X + → ∞ is therefore identical to
the worldsheet theory of the type II superstring.

Transitions from type 0 to type II string theory
We thus have the usual GSO projection of critical type II string
theory. The worldsheet theory X + → ∞ is therefore identical to
the worldsheet theory of the type II superstring.
The background values of all fields are trivial, save for the dilaton,
which has a lightlike gradient, rolling to weak coupling in the
future.

Transitions from type 0 to type II string theory
We thus have the usual GSO projection of critical type II string
theory. The worldsheet theory X + → ∞ is therefore identical to
the worldsheet theory of the type II superstring.
The background values of all fields are trivial, save for the dilaton,
which has a lightlike gradient, rolling to weak coupling in the
future.
A type II background with flat string-frame metric and lightlike
linear dilaton actually preserves sixteen Killing spinors.

Transitions from type 0 to type II string theory
We thus have the usual GSO projection of critical type II string
theory. The worldsheet theory X + → ∞ is therefore identical to
the worldsheet theory of the type II superstring.
The background values of all fields are trivial, save for the dilaton,
which has a lightlike gradient, rolling to weak coupling in the
future.
A type II background with flat string-frame metric and lightlike
linear dilaton actually preserves sixteen Killing spinors.
Our final state is therefore a half-BPS vacuum of type II string
theory.

Transitions from type 0 to type II string theory
We thus have the usual GSO projection of critical type II string
theory. The worldsheet theory X + → ∞ is therefore identical to
the worldsheet theory of the type II superstring.
The background values of all fields are trivial, save for the dilaton,
which has a lightlike gradient, rolling to weak coupling in the
future.
A type II background with flat string-frame metric and lightlike
linear dilaton actually preserves sixteen Killing spinors.
Our final state is therefore a half-BPS vacuum of type II string
theory.
This exact solution establishes conclusively that the type 0 theory
in supercritical dimensions can relax by tachyon condensation to asupersymmetric ground state in D=10!

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

Lightlike tachyon condensation in type 0
In all examples so far, the basic kind of string theory is unchanged
between the initial and final configurations.

Lightlike tachyon condensation in type 0
In all examples so far, the basic kind of string theory is unchanged
between the initial and final configurations.
We now turn to a related model of lightlike tachyon condensation
in type 0 string theory, where the tachyon depends only on X + ,and is independent of the D − 2 dimensions transverse to X ± .

Lightlike tachyon condensation in type 0
In all examples so far, the basic kind of string theory is unchanged
between the initial and final configurations.
We now turn to a related model of lightlike tachyon condensation
in type 0 string theory, where the tachyon depends only on X + ,and is independent of the D − 2 dimensions transverse to X ± .
We start with the Lagrangian for a timelike linear dilaton theory on
a flat worldsheet, describing D free, massless fields and theirsuperpartners:
1
Lkin =GMN
2π

2(∂+ X M )(∂− X N ) − iψ M (∂− ψ N ) − i ψ̃ M (∂+ ψ̃ N )α′Lightlike tachyon condensation in type 0
The dilaton gradient VM must satisfy 4α′ V 2 = −(D − 10), so wetakeq
V+ = V− = − √2Vi = 0,i = 2, · · · , D − 1
r
D − 10q ≡4α′assuming the dilaton rolls to weak coupling in the future.

Lightlike tachyon condensation in type 0
The dilaton gradient VM must satisfy 4α′ V 2 = −(D − 10), so wetakeq
V+ = V− = − √2Vi = 0,i = 2, · · · , D − 1
r
D − 10q ≡4α′assuming the dilaton rolls to weak coupling in the future.
We would like to consider solutions for which the type 0 tachyon
condenses, growing exponentially in the lightlike direction X + .

Lightlike tachyon condensation in type 0
The dilaton gradient VM must satisfy 4α′ V 2 = −(D − 10), so wetakeq
V+ = V− = − √2Vi = 0,i = 2, · · · , D − 1
r
D − 10q ≡4α′assuming the dilaton rolls to weak coupling in the future.
We would like to consider solutions for which the type 0 tachyon
condenses, growing exponentially in the lightlike direction X + .
We again takeT ≡ µ̃ exp βX +Lightlike tachyon condensation in type 0
The linearized equation of motion∂ 2 T − 2V · ∂T +
fixes
βq =

√

2
α′

2
T =0
α′

Lightlike tachyon condensation in type 0
The linearized equation of motion∂ 2 T − 2V · ∂T +

2
T =0
α′

fixes
βq =

√

2
α′

Remember that the tachyon couples to the worldsheet as a (1, 1)
superpotential, giving rise to a potential and Yukawa term:
Lint = −

α′ MN
i α′
G
∂M T ∂N T +∂M ∂N T ψ̃ M ψ N
8π
4π

Lightlike tachyon condensation in type 0
We also get a modified supersymmetry transformation for thefermions:{Q− , ψ M }

=

FM

≡

−{Q+ , ψ̃ M } = F M
rα′ MNG ∂N T−
8

Lightlike tachyon condensation in type 0
We also get a modified supersymmetry transformation for thefermions:{Q− , ψ M }

=

FM

≡

−{Q+ , ψ̃ M } = F M
rα′ MNG ∂N T−
8

Since the gradient of the tachyon is null, the worldsheet potential
α′ MN
G∂M T ∂N T
16π

is zero.

Lightlike tachyon condensation in type 0
We also get a modified supersymmetry transformation for thefermions:{Q− , ψ M }

=

FM

≡

−{Q+ , ψ̃ M } = F M
rα′ MNG ∂N T−
8

Since the gradient of the tachyon is null, the worldsheet potential
α′ MN
G∂M T ∂N T
16π

is zero.
But there is a nonvanishing F -term and Yukawa coupling between
the lightlike fermions:F−=

LYukawa=

where µ ≡ β 2 α′ µ̃.

√q α′ µexp βX +2iµexp βX + ψ̃ + ψ +4π+The M → ∞ limit
We want to determine the X + → ∞ limit of this theory.

The M → ∞ limit
We want to determine the X + → ∞ limit of this theory.
There is no worldsheet potential, so no string states are expelledfrom the interior of the bubble.

The M → ∞ limit
We want to determine the X + → ∞ limit of this theory.
There is no worldsheet potential, so no string states are expelledfrom the interior of the bubble.
The Lagrangian for the light-cone multiplets X µ , ψ µ , ψ̃ µ is:
LLC =

i +
i
iM + +ψ̃ ∂+ ψ̃ − + ψ + ∂− ψ − +
ψ̃ ψ
π
π
2π
−

1
1
(∂+ X + )(∂− X − ) −(∂+ X − )(∂− X + )πα′πα′where M ≡ µ exp (βX + ).

The M → ∞ limit
The stress tensor of the light-cone sector of the theory is
µ

µ

T LC

=

TX + Tψ

TX

µ

≡

−

Tψ

µ

=

i
+ Gµν : ψ µ ∂+ ψ ν :2

1
Gµν : ∂+ X µ ∂+ X ν : +Vµ ∂+2 X µα′with supercurrent
G LC (σ + )

≡
=

√
2
ψµ (∂+ X µ ) − 2α′ Vµ ∂+ ψ µ
′
α
r
r
√
√
2 +
2 −
−
−
ψ ∂+ X −ψ ∂+ X + + α′ q ∂+ ψ + + α′ q ∂+ ψ −α′α′rAnalogous equations apply for the left-moving stress tensor andsupercurrent, replacing ψ with ψ̃ and ∂+ with ∂− .

The M → ∞ limit

As M → ∞, the massive interaction becomes large and the theory
is strongly coupled in the variables X µ , ψ µ , ψ̃ µ .

The M → ∞ limit

As M → ∞, the massive interaction becomes large and the theory
is strongly coupled in the variables X µ , ψ µ , ψ̃ µ .

We would like to define an effective field theory useful for
analyzing the large-M regime, described by free effective fields
whose interactions are proportional to negative rather than positive
powers of M.

The M → ∞ limit

We will not integrate out degrees of freedom. Instead, we will
perform a canonical change of variables such that the new set ofvariables has interaction terms inversely proportional to M.

The M → ∞ limit

We will not integrate out degrees of freedom. Instead, we will
perform a canonical change of variables such that the new set ofvariables has interaction terms inversely proportional to M.

Nothing is integrated out and no information is lost as M → ∞,but the theory becomes free in this limit, when expressed in termsof the new variables.

The M → ∞ limit
First, consider an approximation in which the perturbation M istreated as a fixed constant M0 .

The M → ∞ limit
First, consider an approximation in which the perturbation M istreated as a fixed constant M0 .
As M0 → ∞, the conformal invariance of the original ψ ± , ψ̃ ±theory is broken.

The M → ∞ limit
First, consider an approximation in which the perturbation M istreated as a fixed constant M0 .
As M0 → ∞, the conformal invariance of the original ψ ± , ψ̃ ±theory is broken.
We would like to find a new set of variables in which the theory isapproximately conformal, with corrections that vanish in the
M0 → ∞ limit:
ψ + = 2c ′ 5 − M0−1 b̃5ψ̃ + = −2c̃ ′ 5 + M0−1 b5

ψ − = M0 c̃5ψ̃ − = −M0 c5

The M → ∞ limit

This change of variables is canonical, but not manifestly Lorentz
invariant.

The M → ∞ limit

This change of variables is canonical, but not manifestly Lorentz
invariant.
The Lagrangian becomes
Lfermi=iii− b̃5 ∂+ c̃5 − b5 ∂− c5 −b5 b̃5
π
π
2πM0
−

1
1
(∂+ X + )(∂− X − ) −(∂+ X − )(∂− X + )πα′πα′The M → ∞ limit

This change of variables is canonical, but not manifestly Lorentz
invariant.
The Lagrangian becomes
Lfermi=iii− b̃5 ∂+ c̃5 − b5 ∂− c5 −b5 b̃5
π
π
2πM0
−

1
1
(∂+ X + )(∂− X − ) −(∂+ X − )(∂− X + )πα′πα′

Enforcing the equations of motion, the change of variables isψ + = 2∂+ c5 ,

ψ − = M0 c̃5 ,ψ̃ + = 2∂− c̃5 ,ψ̃ − = −M0 c5

The M → ∞ limit

This change of variables is canonical, but not manifestly Lorentz
invariant.
The Lagrangian becomes
Lfermi=iii− b̃5 ∂+ c̃5 − b5 ∂− c5 −b5 b̃5
π
π
2πM0
−

1
1
(∂+ X + )(∂− X − ) −(∂+ X − )(∂− X + )πα′πα′

Enforcing the equations of motion, the change of variables isψ + = 2∂+ c5 ,

ψ − = M0 c̃5 ,ψ̃ + = 2∂− c̃5 ,ψ̃ − = −M0 c5

The transformation is therefore Lorentz invariant if we assign to b5
a Lorentz weight of 3/2, and to c5 a weight of −1/2.

The M → ∞ limit

So the M0 → ∞ limit of the original theory has a renormalizationgroup flow to a ghost system with spins (3/2, −1/2).

The M → ∞ limit

So the M0 → ∞ limit of the original theory has a renormalizationgroup flow to a ghost system with spins (3/2, −1/2).
The RG flow induced by the massive perturbation M0 ψ + ψ̃ +decreases the central charge by 12 units.

The M → ∞ limit

So the M0 → ∞ limit of the original theory has a renormalizationgroup flow to a ghost system with spins (3/2, −1/2).
The RG flow induced by the massive perturbation M0 ψ + ψ̃ +decreases the central charge by 12 units.

The central charge of the original ψ ± system is 1, but the central
charge of a bc ghost system with weights (3/2, −1/2) is −11.

Promoting M to a dynamical object
We now want to find a canonical change of variables thatgeneralizes what we have done to the case for which M is defined
as µ exp (βX + ), where X + is a dynamical field.

Promoting M to a dynamical object
We now want to find a canonical change of variables thatgeneralizes what we have done to the case for which M is defined
as µ exp (βX + ), where X + is a dynamical field.
We define a new set of variables b4 , c4 , b̃4 , c̃4 :
ψ+

=

ψ−

=

ψ̃ +

=

ψ̃ −

=

2c4′ − M −1 b̃4 + 2β(∂+ X + )c4Mc̃4−2c̃4′ + M −1 b4 + 2β(∂− X + )c̃4
−Mc4

Promoting M to a dynamical object
We now want to find a canonical change of variables thatgeneralizes what we have done to the case for which M is defined
as µ exp (βX + ), where X + is a dynamical field.
We define a new set of variables b4 , c4 , b̃4 , c̃4 :
ψ+

=

ψ−

=

ψ̃ +

=

ψ̃ −

=

2c4′ − M −1 b̃4 + 2β(∂+ X + )c4Mc̃4−2c̃4′ + M −1 b4 + 2β(∂− X + )c̃4
−Mc4

Perform a corresponding redefinition of the bosons X ± :
X+
X

−

≡
≡

Y+

Y − + iβα′ µ exp βX + c4 c̃4Promoting M to a dynamical object
This yields the following Lagrangian
L

=

i
i
i
− b̃4 ∂+ c̃4 − b4 ∂− c4 −b4 b̃4
π
π
2πM
−

1
1
(∂+ Y + )(∂− Y − ) −(∂+ Y − )(∂− Y + )πα′πα′Promoting M to a dynamical object
This yields the following Lagrangian
L

=

i
i
i
− b̃4 ∂+ c̃4 − b4 ∂− c4 −b4 b̃4
π
π
2πM
−

1
1
(∂+ Y + )(∂− Y − ) −(∂+ Y − )(∂− Y + )πα′πα′

The stress tensor becomes:µµ

TY + Tψ = −13ii
Gµν ∂+ Y µ ∂+ Y ν + Vµ ∂ 2 Y µ − ∂+ c4 b4 − c4 ∂+ b4
α′22Promoting M to a dynamical object
This yields the following Lagrangian
L

=

i
i
i
− b̃4 ∂+ c̃4 − b4 ∂− c4 −b4 b̃4
π
π
2πM
−

1
1
(∂+ Y + )(∂− Y − ) −(∂+ Y − )(∂− Y + )πα′πα′

The stress tensor becomes:µµ

TY + Tψ = −13ii
Gµν ∂+ Y µ ∂+ Y ν + Vµ ∂ 2 Y µ − ∂+ c4 b4 − c4 ∂+ b4
α′2
2

As M grows, the stress tensor becomes free in canonical variables,with all interaction terms going to zero as M −1 .

Promoting M to a dynamical object
We refer to the variables Y µ , b4 , c4 , b̃4 , c̃4 as the IR variables,
and the X µ , ψ µ , ψ̃ µ as the UV variables.

Promoting M to a dynamical object
We refer to the variables Y µ , b4 , c4 , b̃4 , c̃4 as the IR variables,
and the X µ , ψ µ , ψ̃ µ as the UV variables.
The IR fields are legitimate, weakly interacting variables, suitablefor describing the X + → +∞ limit of the theory.

Promoting M to a dynamical object
We refer to the variables Y µ , b4 , c4 , b̃4 , c̃4 as the IR variables,
and the X µ , ψ µ , ψ̃ µ as the UV variables.
The IR fields are legitimate, weakly interacting variables, suitablefor describing the X + → +∞ limit of the theory.
There is an exact duality between the UV description and the IR
description.

Promoting M to a dynamical object
We refer to the variables Y µ , b4 , c4 , b̃4 , c̃4 as the IR variables,
and the X µ , ψ µ , ψ̃ µ as the UV variables.
The IR fields are legitimate, weakly interacting variables, suitablefor describing the X + → +∞ limit of the theory.
There is an exact duality between the UV description and the IR
description.
In the case at hand, loop corrections are trivial on both sides, andthe duality inverts the expansion parameter for conformal
perturbation theory rather than for the loop expansion.

Promoting M to a dynamical object
However, the central charge of the fermion theory has dropped
from its original value of 1 in the ψ ± description, to a central
charge of −11 for a bc ghost system with weights (3/2, −1/2).

Promoting M to a dynamical object
However, the central charge of the fermion theory has dropped
from its original value of 1 in the ψ ± description, to a central
charge of −11 for a bc ghost system with weights (3/2, −1/2).
In fact, this is a quantum effect.

Promoting M to a dynamical object
However, the central charge of the fermion theory has dropped
from its original value of 1 in the ψ ± description, to a central
charge of −11 for a bc ghost system with weights (3/2, −1/2).
In fact, this is a quantum effect.
This is a subtle point, since the theory has no nontrivial dynamical
Feynman diagrams that might generate quantum corrections.

Promoting M to a dynamical object
However, the central charge of the fermion theory has dropped
from its original value of 1 in the ψ ± description, to a central
charge of −11 for a bc ghost system with weights (3/2, −1/2).
In fact, this is a quantum effect.
This is a subtle point, since the theory has no nontrivial dynamical
Feynman diagrams that might generate quantum corrections.
Question: How does this work?

Renormalization of the dilaton gradient

It turns out that the natural normal-ordering prescription for theUV variables agrees only up to finite terms with the naturalorderings for composite operators in the IR variables.

Renormalization of the dilaton gradient

It turns out that the natural normal-ordering prescription for theUV variables agrees only up to finite terms with the naturalorderings for composite operators in the IR variables.

The effect of these finite differences will be to renormalize thedilaton gradient of the system by an amount ∆V+ = β, ∆V− = 0.

Normal ordering in the UV variables
Using the properties of Feynman diagrams and the equations of
motion, we can derive modified OPEs for the UV fields.

Normal ordering in the UV variables
Using the properties of Feynman diagrams and the equations of
motion, we can derive modified OPEs for the UV fields.
The natural basis for operators in the UV description is a basis ofnormal-ordered products: X µ1 (ρ1 ) · · · X µm (ρm ) ψ ν1 (σ1 ) · · · ψ νn (σn )ψ̃ π1 (τ1 ) · · · ψ̃ πp (τp ) :◮ The normal-ordered operator is nonsingular when any of thearguments in the normal ordering symbol approach one
another;◮ The normal-ordered operators obey the equations of motion.
For instance:∂τ + ∂τ − : X − (σ)X − (τ ) : = −
iβα′ µ: X − (σ) exp βX + (τ ) ψ̃ + (τ )ψ + (τ ) :4◮ The normal ordered product of two “+” operators is equal tothe ordinary product;

Normal ordering in the UV variables◮ The normal ordered product of a “+” field and a “−” field isdefined with the subtraction prescription of the free theory;◮ The normal ordered product of two “−” fields has only “+”

fields on the right-hand side, and scales as a single power ofM;◮ In the limit M → 0, the structure of the algebra of theoperators becomes that of the free theory (this property isimplied by the three previous properties).

Given these properties, we can derive the full structure of the OPE
for UV fields.

Normal ordering in the IR variables
The normal-ordering prescription defined for UV fields is not usefulfor the IR description.

Normal ordering in the IR variables
The normal-ordering prescription defined for UV fields is not usefulfor the IR description.
The UV normal ordering : : subtracts terms from the time-ordered
product that are proportional to M, which is very large in the IR.

Normal ordering in the IR variables
The normal-ordering prescription defined for UV fields is not usefulfor the IR description.
The UV normal ordering : : subtracts terms from the time-ordered
product that are proportional to M, which is very large in the IR.
Define a second normal ordering prescription, appropriate to the IR
limit of the theory. In this case we take our basis of operators to be◦ µ1◦
Y (ρ1 ) · · · Y µm (ρm ) b4 (σ1 ) · · · b4 (σn )b̃4 (τ1 ) · · · b̃4 (τp )c4 (ζ1 ) · · · c4 (ζq )c̃4 (ω1 ) · · · c̃4 (ωr ) ,◦◦◮ The normal-ordered operator is nonsingular when any of thearguments of operators in the normal ordering symbol
approach one another;◮ The normal-ordered operators obey the equations of motion.
For instance:∂τ + ∂τ −◦ −◦iβα′ ◦ −◦Y (σ)Y − (τ ) = −Y (σ) exp −βY + (τ ) b4 (τ )b̃4 (τ ) ;
◦
◦
4µ ◦
◦

Normal ordering in the IR variables◮ The normal ordered product of two operators from the set
b4 , b̃4 , Y + is equal to the ordinary product;◮ The normal ordered product of a field from the set c4 , c̃4 , Y −with a field from the set b4 , b̃4 , Y + is defined with thesubtraction prescription of the free theory;◮ The normal ordered product of two fields from the set
c4 , c̃4 , Y − has only fields from the set b4 , b̃4 , Y + on theright-hand side, and scales as a single power of M −1 ;◮ In the limit M → ∞, the structure of the algebra of theoperators becomes that of the free theory of the IR fields.

Normal ordering in the IR variables
The bosonic stress tensor turns out to transform unproblematically,
but the fermionic stress tensor picks up a quantum correction due
to the mismatch between : : and ◦◦ ◦◦ normal ordering
prescriptions.

Normal ordering in the IR variables
The bosonic stress tensor turns out to transform unproblematically,
but the fermionic stress tensor picks up a quantum correction due
to the mismatch between : : and ◦◦ ◦◦ normal ordering
prescriptions.
The corrections amounts to a renormalization of the dilaton
gradient:V̂µ ≡ Vµ + ∆Vµ∆V+ = +β∆V− = 0

Normal ordering in the IR variables
The bosonic stress tensor turns out to transform unproblematically,
but the fermionic stress tensor picks up a quantum correction due
to the mismatch between : : and ◦◦ ◦◦ normal ordering
prescriptions.
The corrections amounts to a renormalization of the dilaton
gradient:V̂µ ≡ Vµ + ∆Vµ∆V+ = +β∆V− = 0We are left with a contribution to the central charge equal to√c dilaton = 6α′ η µν V̂µ V̂ν = −6α′ q 2 + 6 2α′ βq3D= 27 −
2

Quantum corrections
We have the remaining central charge contributions◮ +2 from the Y µ◮ −11 from the b4 c4 system◮ 3 (D − 2) from the transverse degrees of freedom X i , ψ i
2◮ total free-field contribution of 3D − 122Quantum corrections
We have the remaining central charge contributions◮ +2 from the Y µ◮ −11 from the b4 c4 system◮ 3 (D − 2) from the transverse degrees of freedom X i , ψ i
2◮ total free-field contribution of 3D − 122

The total central charge in the theory is therefore equal to 15.

Quantum corrections
We have the remaining central charge contributions◮ +2 from the Y µ◮ −11 from the b4 c4 system◮ 3 (D − 2) from the transverse degrees of freedom X i , ψ i
2◮ total free-field contribution of 3D − 122

The total central charge in the theory is therefore equal to 15.
As one moves in the target space from the original theory to
X + = +∞, twelve units of central charge are transferred from thelight cone fermions ψ ± to the dilaton gradient.

Quantum corrections
We have the remaining central charge contributions◮ +2 from the Y µ◮ −11 from the b4 c4 system◮ 3 (D − 2) from the transverse degrees of freedom X i , ψ i
2◮ total free-field contribution of 3D − 122

The total central charge in the theory is therefore equal to 15.
As one moves in the target space from the original theory to
X + = +∞, twelve units of central charge are transferred from thelight cone fermions ψ ± to the dilaton gradient.
The central charge being transferred to the dilaton gradient does
not occur through a loop diagram of massive fields being
integrated out.

Quantum corrections
We have the remaining central charge contributions◮ +2 from the Y µ◮ −11 from the b4 c4 system◮ 3 (D − 2) from the transverse degrees of freedom X i , ψ i
2◮ total free-field contribution of 3D − 122

The total central charge in the theory is therefore equal to 15.
As one moves in the target space from the original theory to
X + = +∞, twelve units of central charge are transferred from thelight cone fermions ψ ± to the dilaton gradient.
The central charge being transferred to the dilaton gradient does
not occur through a loop diagram of massive fields being
integrated out.
Instead, the central charge is transferred through a mismatch ofnormal-ordering prescriptions appropriate to the free field theories
in the two limits X + → ±∞.

Quantum corrections
Break up the supercurrent: G LC = 1 + 2 + 3 + 4, with
r

2 +
1≡−
ψ (∂+ X − )α′√
3 ≡ α′ q ∂+ ψ +

r

2 −
2≡−
ψ (∂+ X + )α′√
4 ≡ α′ q ∂+ ψ −Quantum corrections
Break up the supercurrent: G LC = 1 + 2 + 3 + 4, with
r

r

2 +
1≡−
ψ (∂+ X − )α′√
3 ≡ α′ q ∂+ ψ +2 −2≡−
ψ (∂+ X + )α′√
4 ≡ α′ q ∂+ ψ −The full transformation of the supercurrent from UV to IR
variables isr1classical=−2

1quantum

=

−β

2+4

=

3

=

r

2 ◦
α′ ◦



(∂+ c4 )(∂+ Y − ) + βc4 (∂+ Y + )(∂+ Y − ) −α′ 2∂ c4 − 2β 2
2 +

r

α′
2 +
c4 ∂+
Y
2

i βα′
(∂+ c4 )b4 c4
2

q√ ′
α b4
2
r
r
√
2
2
2
+
2 +
2q α′ (∂+c4 ) + 2
(∂
Y
)(∂
c
)
+
2
c4 (∂+
Y )
+
+
4
α′
α′



◦
◦

Quantum corrections

Expressed in b4 , c4 , Y variables, the supercurrent is manifestly
finite in the limit X + → +∞ (as is the stress tensor).

Quantum corrections

Expressed in b4 , c4 , Y variables, the supercurrent is manifestly
finite in the limit X + → +∞ (as is the stress tensor).

The b4 , c4 , Y fields can indeed be regarded as dual variables
that render the theory free in the M → ∞ limit.

The IR limit
We now focus strictly on the limiting regime of the IR theory.

The IR limit
We now focus strictly on the limiting regime of the IR theory.
In practice, this means that, when written in IR variables, we
discard the exp (−βY + ) b̃4 b4 term in the action, as well as any
exp (−βY + ) terms in the supercurrent and stress tensor.

The IR limit
We now focus strictly on the limiting regime of the IR theory.
In practice, this means that, when written in IR variables, we
discard the exp (−βY + ) b̃4 b4 term in the action, as well as any
exp (−βY + ) terms in the supercurrent and stress tensor.
Rescale the b4 field so that the new b fermion appears in the
supercurrent with unit normalization. To preserve all canonicalcommutators, however, we will rescale the c4 field oppositely:√2b4 = √ b3 = β 2α′ b3
q α′
√
q α′
1
c4 =
c3 = √c32β 2α′The IR limit
The invariance properties of the system under spatial reflection are
still unclear.

The IR limit
The invariance properties of the system under spatial reflection are
still unclear.
The stress tensor is invariant under the discrete symmetryreflecting the spacelike vector orthogonal to V̂µ .

The IR limit
The invariance properties of the system under spatial reflection are
still unclear.
The stress tensor is invariant under the discrete symmetryreflecting the spacelike vector orthogonal to V̂µ .
The supercurrent is not, however, since Vµ and ∆Vµ appear
independently in G LC .

The IR limit
The invariance properties of the system under spatial reflection are
still unclear.
The stress tensor is invariant under the discrete symmetryreflecting the spacelike vector orthogonal to V̂µ .
The supercurrent is not, however, since Vµ and ∆Vµ appear
independently in G LC .
We would like to find field variables that render this discrete
symmetry more manifest, such that only the vector V̂µ enters G LC .

The IR limit
The invariance properties of the system under spatial reflection are
still unclear.
The stress tensor is invariant under the discrete symmetryreflecting the spacelike vector orthogonal to V̂µ .
The supercurrent is not, however, since Vµ and ∆Vµ appear
independently in G LC .
We would like to find field variables that render this discrete
symmetry more manifest, such that only the vector V̂µ enters G LC .
We therefore define new variables b2 , c2 , Z µ by:
Y±

=

b3

=

c3

=

i
c2 ∂+ c2
2β




2
1
(∂+ c2 ) ∂+ Z + − ∂+ Z − −c2 ∂+2 Z + − ∂+2 Z −
b2 −
′
′
βα
βα
i
2
+ 2 ′ c2 (∂+ c2 )(∂+ c2 )
2β α
c2
Z± ±

The IR limit
The worldsheet supersymmetry is now realized nonlinearly.

The IR limit
The worldsheet supersymmetry is now realized nonlinearly.
The bosons Z µ transform into their own derivatives, times agoldstone fermion:[Q, Z µ ] = ic2 ∂+ Z µ{Q, c2 } = 1 + ic2 ∂+ c2

where
1
Q≡
2π

Z

dσ1 G (σ)

The IR limit
The worldsheet supersymmetry is now realized nonlinearly.
The bosons Z µ transform into their own derivatives, times agoldstone fermion:[Q, Z µ ] = ic2 ∂+ Z µ{Q, c2 } = 1 + ic2 ∂+ c2

where
1
Q≡
2π

Z

dσ1 G (σ)

In the sector involving the transverse fields Xi , ψ i , supersymmetry
is realized in the usual linear fashion:r
α′ i[Q, Xi ] = i
ψ
2
r
2
i
{Q, ψ } =∂+ Xiα′The IR limit
At first sight, our realization of supersymmetry in the full theory is
unfamiliar, with worldsheet supersymmetry realized linearly in one
sector and nonlinearly in another.

The IR limit
At first sight, our realization of supersymmetry in the full theory is
unfamiliar, with worldsheet supersymmetry realized linearly in one
sector and nonlinearly in another.
However, it turns out that this realization is equivalent to one forwhich worldsheet supersymmetry is realized completely nonlinearlyin all sectors.

The IR limit
At first sight, our realization of supersymmetry in the full theory is
unfamiliar, with worldsheet supersymmetry realized linearly in one
sector and nonlinearly in another.
However, it turns out that this realization is equivalent to one forwhich worldsheet supersymmetry is realized completely nonlinearlyin all sectors.
We now perform a final transformation on the system. Defining
the Hermitian infinitesimal generatorZig ≡−dσ1 c2 (σ)G ⊥ (σ)
2πwe transform all operators in the theory according to
O → U O U −1
withU ≡ exp (ig )The IR limit
The total final supercurrent G ≡ G LC + G ⊥ is then


1 ⊥ 1
′
mat
′′
′ 2
G = b1 + ic1 b1 c1 − c1 T
+ c1 − c − + α q
6
2


i
i
i
+c1 c1′ c1′′ − α′ q 2 − + c ⊥4
2 24

The IR limit
The total final supercurrent G ≡ G LC + G ⊥ is then


1 ⊥ 1
′
mat
′′
′ 2
G = b1 + ic1 b1 c1 − c1 T
+ c1 − c − + α q
6
2


i
i
i
+c1 c1′ c1′′ − α′ q 2 − + c ⊥4
2 24
And the total transformed stress tensor is
T = T mat + T b1 c1
withT b1 c1

= −

3i
i
i
2
∂+ c1 b1 − c1 ∂+ b1 + ∂+ (c1 ∂+
c1 )
2
2
2

The IR limit
The total final supercurrent G ≡ G LC + G ⊥ is then


1 ⊥ 1
′
mat
′′
′ 2
G = b1 + ic1 b1 c1 − c1 T
+ c1 − c − + α q
6
2


i
i
i
+c1 c1′ c1′′ − α′ q 2 − + c ⊥4
2 24
And the total transformed stress tensor is
T = T mat + T b1 c1
withT b1 c1= −Plugging in q =

q

3i
i
i
2
∂+ c1 b1 − c1 ∂+ b1 + ∂+ (c1 ∂+
c1 )
2
2
2

D−10
3
⊥
4α′ and c = 2 (D − 2):5
G = b1 + ic1′ b1 c1 − c1 T mat − c1′′
2

The IR limit

The X + → ∞ limit of our solution is described by a freeworldsheet theory with a bc ghost system of weights (3/2, −1/2),
i
D free scalars Z M and D − 2 free fermions ψ Z .

The IR limit

The X + → ∞ limit of our solution is described by a freeworldsheet theory with a bc ghost system of weights (3/2, −1/2),
i
D free scalars Z M and D − 2 free fermions ψ Z .
i

The total central charge of the Z M , ψ Z system is 26, and thecontribution of −11 from the b1 c1 system brings the total central
charge to 15.

The IR limit

The X + → ∞ limit of our solution is described by a freeworldsheet theory with a bc ghost system of weights (3/2, −1/2),
i
D free scalars Z M and D − 2 free fermions ψ Z .
i

The total central charge of the Z M , ψ Z system is 26, and thecontribution of −11 from the b1 c1 system brings the total central
charge to 15.
The theory has critical central charge for a SCFT interpreted as
the worldsheet theory of a RNS superstring in conformal gauge.

Berkovits-Vafa construction
This type of superconformal field theory belongs to a class ofconstructions introduced by Berkovits and Vafa, in which thebosonic string is embedded in the solution space of the superstring.
[hep-th/9310170]

Berkovits-Vafa construction
This type of superconformal field theory belongs to a class ofconstructions introduced by Berkovits and Vafa, in which thebosonic string is embedded in the solution space of the superstring.
[hep-th/9310170]

For a conformal field theory T mat with a central charge of 26, it ispossible to construct a corresponding superconformal field theory
defined by G , T with central charge 15.

Berkovits-Vafa construction
This type of superconformal field theory belongs to a class ofconstructions introduced by Berkovits and Vafa, in which thebosonic string is embedded in the solution space of the superstring.
[hep-th/9310170]

For a conformal field theory T mat with a central charge of 26, it ispossible to construct a corresponding superconformal field theory
defined by G , T with central charge 15.
Upon treating the superconformal theory as a superstring theory,
the resulting physical states and scattering amplitudes are identicalto those of the theory defined by T mat when treated as a bosonicstring theory.

Berkovits-Vafa construction
The construction can be summarized as follows:◮ Given a conformal stress tensor T mat with central charge 26,

a ghost system b1 c1 can be introduced with weights(3/2, −1/2) and stress tensor T b1 c1 .

Berkovits-Vafa construction
The construction can be summarized as follows:◮ Given a conformal stress tensor T mat with central charge 26,

a ghost system b1 c1 can be introduced with weights(3/2, −1/2) and stress tensor T b1 c1 .
◮ This gives rise to a fermionic primary current of weight 3/2:

5
G ≡ b1 + ic1′ b1 c1 − c1 T mat − c1′′
2

Berkovits-Vafa construction
The construction can be summarized as follows:◮ Given a conformal stress tensor T mat with central charge 26,

a ghost system b1 c1 can be introduced with weights(3/2, −1/2) and stress tensor T b1 c1 .
◮ This gives rise to a fermionic primary current of weight 3/2:

5
G ≡ b1 + ic1′ b1 c1 − c1 T mat − c1′′
2
◮ This closes on the stress tensor of the theory:

G (σ)G (τ ) ≃

10i
2i
+
T total (τ )(τ + − σ + )3 (τ + − σ + )whereT total ≡ T mat + T b1 c1

Berkovits-Vafa construction◮ This defines a superconformal theory of central charge 15.

Berkovits-Vafa construction◮ This defines a superconformal theory of central charge 15.
◮ To construct physical states of the corresponding superstring

theory, one starts with a Virasoro primary state |U i of weight
1 in the theory defined by T mat :Lmat|U i = 0
n
Lmat
|U i
0

= 1

n≥1

Berkovits-Vafa construction◮ This defines a superconformal theory of central charge 15.
◮ To construct physical states of the corresponding superstring

theory, one starts with a Virasoro primary state |U i of weight
1 in the theory defined by T mat :Lmat|U i = 0
n
Lmat
|U i
0

= 1

n≥1

We have an exact solution describing a dynamical transition
between string theories that differ from one another in theirworldsheet gauge algebra.

Transition to bosonic string theory
This transition follows an instability in an initial D-dimensional
type 0 theory.

Transition to bosonic string theory
This transition follows an instability in an initial D-dimensional
type 0 theory.
The dynamics then spontaneously break worldsheet
supersymmetry, giving rise to a bosonic string theory in the same
number of dimensions deep inside the tachyonic phase.
111111111111111111111111111111
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Bosonic string theory
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Bubble nucleation

Time

X1

Type 0 superstring theory

Outline
Overview of quintessent cosmology and linear dilaton backgrounds
Supercritical string theory: spacetime effective action
Stability in time-dependent backgrounds
Bosonic string solutions with nonzero tachyon
Dimension-changing solutions in the type 0 string
Transitions from type 0 to bosonic string theory
Conclusions

A partial catalog of exact transitions
start
bos
0
0 (orb)
0

Dinit
D
D
D
D

exp (−βX + ) Tµ2 X22µX2 X3µXi +1 Yi
µ

end
bos
0
II
bos

UHE
HO(+1)
HO(+1)/
HO(+1)/
(orb)
N
=2
N
=2

10
11
11
11

µX2
µX2
µX2
µX2

HE9
HO
HO/
HO

2 Dc
-1
2 Dc
-1

µφ2 φ3

N
=2
bos

µ

Dfin
D-1
D-2
10
D
+ 12 (D-2)
9
10
10
10

comments
tuned
naturalstable
tuned

2 Dc
-5
3 Dc
-2

naturalstablestablenaturalstabletuned

The Big Picture – Part I
Type 0B
even D

Type 0A
even D

Type 0D−1Type 0A

Type 0B

D−2

D−2

Type 0A
12D

Type 0B
12D

Type 011D

Type 0A
10D

Type 0B
10D

Type 09D

Type 0A
8D

Type 0B
8D

Type 0A
4D

Type 0B
4D

Natural transition

Type 03D

Type 0A
2D

Type 0B
2D

T−duality

Tuned,µ>0

Tuned,

µ<0

....

....

....

....

The Big Picture – Part II

0B
14D + 2orb.

0A
14D + 2orb.

0B
12 D + 1orb.

0A
12 D + 1orb.
Scherk−Schwarz+ T−duality

IIA
10D

IIB
10D

....

....

Wilson Line

....

....

The Big Picture – Part III

Type 0A
12D

12D Bosonic String

Type 0B
12D

Type 0
11D

11D Bosonic String....

10D Bosonic String

Type 0A
4D

4D Bosonic String....

Type 0B
10D

Type 0A
10D

Type 0B
4D

Type 0
3D

3D Bosonic String
Type 0A
2D

2D Bosonic String

Type 0B
2D

2D Bosonic String
2 copies

Conclusions◮ Supercritical string theory has some surprising and interesting

properties.

Conclusions◮ Supercritical string theory has some surprising and interesting

properties.
◮ We see that the supercritical string can be connected to theduality web of critical string theory.

Conclusions◮ Supercritical string theory has some surprising and interesting

properties.
◮ We see that the supercritical string can be connected to theduality web of critical string theory.
◮ We have found solutions that interpolate between superstring

theory and purely bosonic string theory.

Conclusions◮ Supercritical string theory has some surprising and interesting

properties.
◮ We see that the supercritical string can be connected to theduality web of critical string theory.
◮ We have found solutions that interpolate between superstring

theory and purely bosonic string theory.
◮ The surprising feature of these connections is the crucial roleof time dependence.

Conclusions◮ Supercritical string theory has some surprising and interesting

properties.
◮ We see that the supercritical string can be connected to theduality web of critical string theory.
◮ We have found solutions that interpolate between superstring

theory and purely bosonic string theory.
◮ The surprising feature of these connections is the crucial roleof time dependence.
◮ There may be other interesting links between theories that wehave yet to discover.

Conclusions◮ Supercritical string theory has some surprising and interesting

properties.
◮ We see that the supercritical string can be connected to theduality web of critical string theory.
◮ We have found solutions that interpolate between superstring

theory and purely bosonic string theory.
◮ The surprising feature of these connections is the crucial roleof time dependence.
◮ There may be other interesting links between theories that wehave yet to discover.
◮ Thank you!