Light-like tachyon Light-like tachyon condensationcondensation

in OSFTin OSFT

Martin Schnabl Martin Schnabl (IAS)(IAS)

Rutgers University, Rutgers University, 02/26/2008 02/26/2008

Rutgers University, Rutgers University, 20082008

Much has been learned in the past decade about Much has been learned in the past decade about

tachyon condensationtachyon condensation (Sen’s conjectures, classical (Sen’s conjectures, classical

solutions, interesting toy-models)solutions, interesting toy-models)

Static vacua are now reasonably well understoodStatic vacua are now reasonably well understood

Numerically since 1999: Numerically since 1999: Sen, Zwiebach, Taylor and othersSen, Zwiebach, Taylor and others

Analytic progress since 2005, Analytic progress since 2005, M.S. and othersM.S. and others

One of the least understood issues is the One of the least understood issues is the dynamicsdynamics

of the condensation. Some new results from a of the condensation. Some new results from a

joint project with joint project with Simeon HellermanSimeon Hellerman will be presented. will be presented.

Plan of the talk:Plan of the talk:

Brief review of the rolling tachyon physicsBrief review of the rolling tachyon physics

New OSFT rolling tachyon solutions that New OSFT rolling tachyon solutions that

manifestly interpolate the vacuamanifestly interpolate the vacua

(in linear dilaton background)(in linear dilaton background)

P-adic toy modelsP-adic toy models

I.I.

II. II.

III.III.

In 2002 Sen studied the fate of the rolling tachyon,In 2002 Sen studied the fate of the rolling tachyon,

string-field-theoretic version of a equationstring-field-theoretic version of a equation

Whose simplest time-like solution is Whose simplest time-like solution is

Sen computed the corresponding boundary state Sen computed the corresponding boundary state (closed string state that has the same effect as having a boundary with (closed string state that has the same effect as having a boundary with

the above boundary interaction)the above boundary interaction)

(¤ + 1)T(X ) = 0

T(X ) = e§ X 0

The component of the boundary state for the profile The component of the boundary state for the profile

stays stays constantconstant, whereas the goes as, whereas the goes as

T00 eX 0

Ti j

Ti j »±i j

1+ eX 0

This, according to Sen implies existence of a new form This, according to Sen implies existence of a new form

of matter in string theory: of matter in string theory: tachyon mattertachyon matter, or , or tachyon dusttachyon dust

with potentially interesting applications.with potentially interesting applications.

This has been criticized by Strominger, Maldacena et al, This has been criticized by Strominger, Maldacena et al,

who argued this to be an artifact of perturbation theory that who argued this to be an artifact of perturbation theory that

does not survive to any order in finite does not survive to any order in finite gs

Later, it was found byLater, it was found by Karczmarek, Strominger, Liu and Karczmarek, Strominger, Liu and

Maldacena Maldacena that after all the that after all the tachyon dust might be existenttachyon dust might be existent

for a finite amount of time, at least in a subcritical string for a finite amount of time, at least in a subcritical string

theory with a theory with a space-like linear dilaton backgroundspace-like linear dilaton background. .

It is the goal of this work to understand better the tachyon It is the goal of this work to understand better the tachyon

matter issue from a Open String Field Theory standpoint.matter issue from a Open String Field Theory standpoint.

In OSFT this has been already studied, notably by In OSFT this has been already studied, notably by

Moeller and ZwiebachMoeller and Zwiebach in 2002. Their results can be best in 2002. Their results can be best

illustrated on the p-adic toy modelillustrated on the p-adic toy model

S =1g2

p

Zddx

·¡

12Áp¡ ®0¤ Á+

1p+ 1

Áp+1

¸:

Solution can be found in the formSolution can be found in the form

Á(t) = 1+1X

n=1

an ent

wherewhere

an =1

2n2 ¡ 2

n¡ 1X

k=1

akan¡ k

10 0 80 60 40 20 20 40

40 0

20 0

20 0

40 0

10 0 80 60 40 20 20 40

10

5

5

10

The tachyon starts slowly The tachyon starts slowly rolling down the potential,rolling down the potential,but eventually ends up in but eventually ends up in oscillations with ever oscillations with ever growing amplitude!growing amplitude!

Quasi Logarithmic plotQuasi Logarithmic plot

This can be understood analytically. Up to some This can be understood analytically. Up to some numerical coefficients the functions behaves asnumerical coefficients the functions behaves as

This function obeys a replication formulaThis function obeys a replication formula

Once sufficiently large, the function changes sign after Once sufficiently large, the function changes sign after each and its magnitude grows quadratically each and its magnitude grows quadratically in the exponential. in the exponential.

Full-fledged string field theory exhibits similar behavior.Full-fledged string field theory exhibits similar behavior.Zwiebach & Moeller, Fujita & Hata and othersZwiebach & Moeller, Fujita & Hata and others

f (t) =1X

n=0

(¡ 1)ne¡ 12 n2

ent

f (t + 1) = 1¡ et+ 12 f (t)

¢ t = 1

Part IIPart II

Light-like rolling tachyon solutionsLight-like rolling tachyon solutions in Linear Dilaton backgroundsin Linear Dilaton backgrounds

Why light-like ? Why light-like ?

1)1) Solutions dependent only on a light-cone coordinate Solutions dependent only on a light-cone coordinate are typically easier to construct and under better controlare typically easier to construct and under better control

2)2) Spatially homogenous rolling tachyon seems rather Spatially homogenous rolling tachyon seems rather contrived. One can argue that initial destabilization would contrived. One can argue that initial destabilization would occur locally, and would then propagate as a spherical occur locally, and would then propagate as a spherical bubble, which at a distance would look locally light-like. bubble, which at a distance would look locally light-like.

Part IIPart II

Light-like rolling tachyon solutionsLight-like rolling tachyon solutions in Linear Dilaton backgroundsin Linear Dilaton backgrounds

Why Linear Dilaton background ? Why Linear Dilaton background ?

Without the LD, the light-like tachyon Without the LD, the light-like tachyon cannot ever be on-shell, and be thus a physical instability.cannot ever be on-shell, and be thus a physical instability.

In the LD background the dimension of In the LD background the dimension of equals equals and can be set to one by and can be set to one by appropriate choice of appropriate choice of

e X +

©= V¹ X ¹ e X +

h = ®0 V+

¯

Open String Field TheoryOpen String Field Theory in in Linear Dilaton backgroundLinear Dilaton background

Can be constructed from the knowledge of LD CFT in level Can be constructed from the knowledge of LD CFT in level truncation, just like for constant dilaton. truncation, just like for constant dilaton.

E.g. for the tachyon:E.g. for the tachyon:

S = ¡1g2

o

ZdD x e¡ V¹ x¹

·12®0(@t)2 ¡

12t2 +

13K ¡ 3+®0V 2 ~t3

¸:

Here Here andand ~t = K ¡ ®0¤ tK = 43p

3¼0:77

Looking for dependent solutions, Looking for dependent solutions, the equation of motion becomesthe equation of motion becomes

X + = 1p2

¡X 0 + X 1

¢

This can be easily solved in the form of a seriesThis can be easily solved in the form of a series (with infinite radius of convergence)(with infinite radius of convergence)

t(X +) =1X

n=1

an expµ

nX +

V+

¶

(V+@+ ¡ 1)t(X +) + K ¡ 3£t¡X + + 2V+ logK

¢¤2= 0

10 5 5

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

The tachyon slowly The tachyon slowly relaxes to the true vacuum !relaxes to the true vacuum !

There is a well-defined procedure for constructing There is a well-defined procedure for constructing solutions to OSFT given an exactly marginal matter solutions to OSFT given an exactly marginal matter operatoroperator

Given a Given a matter primary field of dimension onematter primary field of dimension one, , the general solution can be constructed as follows:the general solution can be constructed as follows:

Write and insert into the e.o.mWrite and insert into the e.o.m

the equation becomesthe equation becomes

which can be solved recursively, starting with which can be solved recursively, starting with

J (z)

ª =1X

n=1

¸nÁn

QB Án = ¡ [Á1Án¡ 1 + Á2Án¡ 2 + ¢¢¢+ Án¡ 1Á1]

Á1 = cJ (0)j0i

Exact Rolling-Tachyon solution Exact Rolling-Tachyon solution in Open String Field Theoryin Open String Field Theory

Án = ¡B0

L0[Á1Án¡ 1 + Á2Án¡ 2 + ¢¢¢+Án¡ 1Á1]

To do that, one has to `invert’ the BRST charge. The To do that, one has to `invert’ the BRST charge. The simplest choice is to impose our gaugesimplest choice is to impose our gauge

Since ,the recursion is to easy to Since ,the recursion is to easy to follow to all orders and we can guess (and subsequently follow to all orders and we can guess (and subsequently verify) that the solution is given byverify) that the solution is given by

B0

1=L0 =R1

0zL 0¡ 1

Án =³¡

¼2

´n¡ 1Z 1

0

n¡ 1Y

i=1

dri Á1 ¤B L1 jr1i ¤Á1 ¤¢¢¢¤B L

1 jrn¡ 1i ¤Á1:

Discovered independently byDiscovered independently by Kiermaier, Okawa, Rastelli, Zwiebach Kiermaier, Okawa, Rastelli, Zwiebach (hep-th/0701249),(hep-th/0701249),generalized recently to superstring bygeneralized recently to superstring by Erler and Okawa.Erler and Okawa.

Geometric representation of the solution is quite simple.Geometric representation of the solution is quite simple.It is important that has nonsingular operator product It is important that has nonsingular operator product with itself.with itself.

J

The solution can be summed upThe solution can be summed up

ª ¸ =¸

1+ ¼2¸

R10

dr Á1j0i ¤B L1 jri

¤Á1j0i

For the time-like tachyon rolling, the wild oscillations seemFor the time-like tachyon rolling, the wild oscillations seemto be still present at large to be still present at large

However, However, EllwoodEllwood in 2007 made an interesting observation: in 2007 made an interesting observation:By replacing all operators with their zero modesBy replacing all operators with their zero modeshe found at large the static vacuum constructed by he found at large the static vacuum constructed by myself in 2005.myself in 2005.

For time-like tachyon rolling, Ellwood’s trick cannot be For time-like tachyon rolling, Ellwood’s trick cannot be justified, for light-like rolling it can.justified, for light-like rolling it can.

eX 0 ex0

x0

Consider Consider light-like linear dilatonlight-like linear dilaton background with background with

Then for a specific choice of the operator is of Then for a specific choice of the operator is of dimension 1 and one can moreover ignore all normal ordering.dimension 1 and one can moreover ignore all normal ordering.

It turns out that one can explicitly perform the n-dimensionalIt turns out that one can explicitly perform the n-dimensionalintegrals, sum the geometric series over n and one finds for integrals, sum the geometric series over n and one finds for the level k coefficients in the ‘universal’ sector the level k coefficients in the ‘universal’ sector

The leading term agrees with the tachyon vacuum!The leading term agrees with the tachyon vacuum!

¯ e X +

f (0)k

¡X +

¢=

dk

d®k

¸e X +

1+ ¼2¸e X + e®¡ 1

®

¯¯¯¯¯®=0

=2¼

Bk +µ

2¼

¶2

((k ¡ 1)Bk + kBk¡ 1) ¸ ¡ 1e¡ ¯ X ++ ¢¢¢;

©= V¹ X ¹

More generally, we can show that all other coefficients More generally, we can show that all other coefficients go to zero, such as those containing go to zero, such as those containing

@X +;@2X +; (@X +)2; : : :

The proof rests on an identity for power series with The proof rests on an identity for power series with polynomial coefficientspolynomial coefficients

1X

n=1

P (n)qn = ¡1X

n=0

P (¡ n)q¡ n

and a curious Euler - Maclaurin type identityand a curious Euler - Maclaurin type identitynX

j =1

j k =k+1X

m=0

Bm

m!k!

(k ¡ m+ 1)!

£(n + 1)k¡ m+1 ¡ 1

¤

Part IIIPart III

P-adic toy modelsP-adic toy models

No one knows, how to couple p-adic string to dilaton No one knows, how to couple p-adic string to dilaton background. Let us try two simples possibilities.background. Let us try two simples possibilities.

VSFT motivated p-adic string actionVSFT motivated p-adic string action

The equation of motion for fields which depend only on The equation of motion for fields which depend only on simplifies dramaticallysimplifies dramatically

Demanding perturbative vacuum in the far past, the equationDemanding perturbative vacuum in the far past, the equationhas a unique solution has a unique solution

The resulting The resulting agrees with the boundary state !agrees with the boundary state !

S =1g2

p

Zddx e¡ V¹ X ¹

·¡

12

³p¡ ®0¤ =2Á

´2+

1p+ 1

Áp+1

¸:

X +

Á(X + + V+ logp) = pV 2=2Áp(X +):

Á(X +) = p¡ ®0V 22(p¡ 1) e¡ e¯ X +

:

T¹ º

‘‘Logistic’ p-adic string actionLogistic’ p-adic string action

Again, the equation of motion for fields which depend Again, the equation of motion for fields which depend only on simplifies toonly on simplifies to

For a discrete set of this is nothing but a logistic map For a discrete set of this is nothing but a logistic map

With parameter With parameter

S =1g2

p

Zddxe¡ V¹ X ¹

·¡

12Áp¡ ®0¤ Á+

1p+ 1

Áp+1

¸

Á(X +) + p¡ 12 V 2

Á(X + +V+ logp) = 2Áp(X +)

X +

xn+1 = rxn(1¡ xp¡ 1n );

r = ¡ p+ 12 V 2

X +

Continuous solutions can be uniquely constructed Continuous solutions can be uniquely constructed (up to a translation)(up to a translation)

Time-like dilaton in supercritical theoryTime-like dilaton in supercritical theory

r = ¡ 1=2

Light-like dilaton in the critical string theoryLight-like dilaton in the critical string theory

r = ¡ 1

Space-like dilaton in the subcritical string theorySpace-like dilaton in the subcritical string theory Now we are beyond the first bifurcationNow we are beyond the first bifurcation

r = ¡ 1:2

Space-like dilaton in the subcritical string theorySpace-like dilaton in the subcritical string theory

r = ¡ 1:5

Space-like dilaton in the subcritical string theorySpace-like dilaton in the subcritical string theory True chaos regimeTrue chaos regime

r = ¡ 1:57

Maximum value of for which the oscillations are Maximum value of for which the oscillations are bounded. Corresponds tobounded. Corresponds to D=14 D=14 for the bosonic string or for the bosonic string or D=2D=2 for the superstring. for the superstring.

r = ¡ 2

V2

Space-like dilaton in the subcritical string theorySpace-like dilaton in the subcritical string theory

ConclusionsConclusions

We did find the We did find the tachyon vacuum as the endpoint tachyon vacuum as the endpoint of light-like tachyon rollingof light-like tachyon rolling

The energy-momentum tensor for such a rolling The energy-momentum tensor for such a rolling tachyon is compatible with the existence of the tachyon is compatible with the existence of the tachyon mattertachyon matter

The way the tachyon approaches the vacuum The way the tachyon approaches the vacuum can be quite fun!can be quite fun!