Light-like tachyon condensation in OSFT Martin Schnabl (IAS) Rutgers University, 02/26/2008 Rutgers University, 02/26/2008 Rutgers University, 2008

Much has been learned in the past decade about tachyon condensation (Sen’s conjectures, classical solutions, interesting toy-models) Static vacua are now reasonably well understood Numerically since 1999: Sen, Zwiebach, Taylor and others Analytic progress since 2005, M.S. and others One of the least understood issues is the dynamics of the condensation. Some new results from a joint project with Simeon Hellerman will be presented.

Plan of the talk: Brief review of the rolling tachyon physics New OSFT rolling tachyon solutions that manifestly interpolate the vacua (in linear dilaton background) P-adic toy models I. I. II. II. III.

In 2002 Sen studied the fate of the rolling tachyon, string-field-theoretic version of a equation Whose simplest time-like solution is Sen computed the corresponding boundary state (closed string state that has the same effect as having a boundary with the above boundary interaction) ( ¤ + 1 ) T ( X ) = 0 T ( X ) = e § X 0

The component of the boundary state for the profile stays constant, whereas the goes as T 00 e X 0 T ij T ij » ± ij 1 + e X 0 This, according to Sen implies existence of a new form of matter in string theory: tachyon matter, or tachyon dust with potentially interesting applications. This has been criticized by Strominger, Maldacena et al, who argued this to be an artifact of perturbation theory that does not survive to any order in finite g s

Later, it was found by Karczmarek, Strominger, Liu and Maldacena that after all the tachyon dust might be existent for a finite amount of time, at least in a subcritical string theory with a space-like linear dilaton background. It is the goal of this work to understand better the tachyon matter issue from a Open String Field Theory standpoint.

In OSFT this has been already studied, notably by Moeller and Zwiebach in Their results can be best illustrated on the p-adic toy model S = 1 g 2 p Z d d x · ¡ 1 2 Á p ¡ ® 0 ¤ Á + 1 p + 1 Á p + 1 ¸ : Solution can be found in the form Á ( t ) = X n = 1 a n e n t where a n = 1 2 n 2 ¡ 2 n ¡ 1 X k = 1 a k a n ¡ k

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

This can be understood analytically. Up to some numerical coefficients the functions behaves as This function obeys a replication formula Once sufficiently large, the function changes sign after each and its magnitude grows quadratically in the exponential. Full-fledged string field theory exhibits similar behavior. Zwiebach & Moeller, Fujita & Hata and others f ( t ) = 1 X n = 0 ( ¡ 1 ) n e ¡ 1 2 n 2 e n t f ( t + 1 ) = 1 ¡ e t f ( t ) ¢ t = 1

Part II Light-like rolling tachyon solutions in Linear Dilaton backgrounds in Linear Dilaton backgrounds Why light-like ? 1)Solutions dependent only on a light-cone coordinate are typically easier to construct and under better control are typically easier to construct and under better control 2)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 II Light-like rolling tachyon solutions in Linear Dilaton backgrounds in Linear Dilaton backgrounds Why Linear Dilaton background ? Without the LD, the light-like tachyon cannot ever be on-shell, and be thus a physical instability. In the LD background the dimension of equals and can be set to one by appropriate choice of e ¯ X + © = V ¹ X ¹ e ¯ X + h = ® 0 ¯ V + ¯

Open String Field Theory in Open String Field Theory in Linear Dilaton background Linear Dilaton background Can be constructed from the knowledge of LD CFT in level truncation, just like for constant dilaton. E.g. for the tachyon: S = ¡ 1 g 2 o Z d D xe ¡ V ¹ x ¹ · 1 2 ® 0 t ) 2 ¡ 1 2 t K ¡ 3 + ® 0 V 2 ~ t 3 ¸ : Here and ~ t = K ¡ ® 0 ¤ t K = 4 3 p 3 ¼ 0 : 77

Looking for dependent solutions, the equation of motion becomes X + = 1 p 2 ¡ X 0 + X 1 ¢ This can be easily solved in the form of a series (with infinite radius of convergence) t ( X + ) = 1 X n = 1 a n exp µ n X + V + ¶ ( V + ¡ 1 ) t ( X + ) + K ¡ 3 £ t ¡ X V + l og K ¢¤ 2 = 0

The tachyon slowly relaxes to the true vacuum !

There is a well-defined procedure for constructing solutions to OSFT given an exactly marginal matter operator Given a matter primary field of dimension one, the general solution can be constructed as follows: Write and insert into the e.o.m the equation becomes which can be solved recursively, starting with J ( z ) ª = 1 X n = 1 ¸ n Á n Q B Á n = ¡ [ Á 1 Á n ¡ 1 + Á 2 Á n ¡ 2 + ¢¢¢ + Á n ¡ 1 Á 1 ] Á 1 = c J ( 0 )j 0 i Exact Rolling-Tachyon solution Exact Rolling-Tachyon solution in Open String Field Theory in Open String Field Theory

Á n = ¡ B 0 L 0 [ Á 1 Á n ¡ 1 + Á 2 Á n ¡ 2 + ¢¢¢ + Á n ¡ 1 Á 1 ] To do that, one has to `invert’ the BRST charge. The simplest choice is to impose our gauge Since,the recursion is to easy to follow to all orders and we can guess (and subsequently verify) that the solution is given by B 0 1 = L 0 = R 1 0 z L 0 ¡ 1 Á n = ³ ¡ ¼ 2 ´ n ¡ 1 Z 1 0 n ¡ 1 Y i = 1 d r i Á 1 ¤ B L 1 j r 1 i ¤ Á 1 ¤ ¢¢¢ ¤ B L 1 j r n ¡ 1 i ¤ Á 1 : Discovered independently byKiermaier, Okawa, Rastelli, Zwiebach (hep-th/ ), Discovered independently by Kiermaier, Okawa, Rastelli, Zwiebach (hep-th/ ), generalized recently to superstring by Erler and Okawa.

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

The solution can be summed up ª ¸ = ¸ 1 + ¼ 2 ¸ R 1 0 d r Á 1 j 0 i ¤ B L 1 j r i ¤ Á 1 j 0 i For the time-like tachyon rolling, the wild oscillations seem to be still present at large However, Ellwood in 2007 made an interesting observation: By replacing all operators with their zero modes he found at large the static vacuum constructed by myself in For time-like tachyon rolling, Ellwood’s trick cannot be justified, for light-like rolling it can. e X 0 e x 0 x 0

Consider light-like linear dilaton background with Then for a specific choice of the operator is of dimension 1 and one can moreover ignore all normal ordering. It turns out that one can explicitly perform the n-dimensional integrals, sum the geometric series over n and one finds for the level k coefficients in the ‘universal’ sector The leading term agrees with the tachyon vacuum! ¯ e ¯ X + f ( 0 ) k ¡ X + ¢ = d k d ® k ¸ e ¯ X ¼ 2 ¸ e ¯ X + e ® ¡ 1 ® ¯ ¯ ¯ ¯ ¯ ® = 0 = 2 ¼ B k + µ 2 ¼ ¶ 2 (( k ¡ 1 ) B k + k B k ¡ 1 ) ¸ ¡ 1 e ¡ ¯ X + + ¢¢¢ ; © = V ¹ X ¹

More generally, we can show that all other coefficients go to zero, such as those X + 2 X + ; X + ) 2 ;::: The proof rests on an identity for power series with polynomial coefficients 1 X n = 1 P ( n ) q n = ¡ 1 X n = 0 P ( ¡ n ) q ¡ n and a curious Euler - Maclaurin type identity n X j = 1 j k = k + 1 X m = 0 B m m ! k! ( k ¡ m + 1 ) ! £ ( n + 1 ) k ¡ m + 1 ¡ 1 ¤

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

VSFT motivated p-adic string action The equation of motion for fields which depend only on simplifies dramatically Demanding perturbative vacuum in the far past, the equation has a unique solution The resulting agrees with the boundary state ! S = 1 g 2 p Z d d xe ¡ V ¹ X ¹ · ¡ 1 2 ³ p ¡ ® 0 ¤= 2 Á ´ p + 1 Á p + 1 ¸ : X + Á ( X + + V + l ogp ) = p V 2 = 2 Á p ( X + ) : Á ( X + ) = p ¡ ® 0 V 2 2 ( p ¡ 1 ) e ¡ e ¯ X + : T ¹º

‘Logistic’ p-adic string action Again, the equation of motion for fields which depend only on simplifies to For a discrete set of this is nothing but a logistic map With parameter S = 1 g 2 p Z d d xe ¡ V ¹ X ¹ · ¡ 1 2 Á p ¡ ® 0 ¤ Á + 1 p + 1 Á p + 1 ¸ Á ( X + ) + p ¡ 1 2 V 2 Á ( X + + V + l ogp ) = 2 Á p ( X + ) X + x n + 1 = rx n ( 1 ¡ x p ¡ 1 n ) ; r = ¡ p V 2 X +

Continuous solutions can be uniquely constructed (up to a translation) Time-like dilaton in supercritical theory r = ¡ 1 = 2

Light-like dilaton in the critical string theory r = ¡ 1

Space-like dilaton in the subcritical string theory Now we are beyond the first bifurcation Now we are beyond the first bifurcation r = ¡ 1 : 2

Space-like dilaton in the subcritical string theory r = ¡ 1 : 5

True chaos regime True chaos regime r = ¡ 1 : 57

Maximum value of for which the oscillations are bounded. Corresponds to D=14 for the bosonic string or D=2 for the superstring. r = ¡ 2 V 2 Space-like dilaton in the subcritical string theory

Conclusions We did find the tachyon vacuum as the endpoint We did find the tachyon vacuum as the endpoint of light-like tachyon rolling of light-like tachyon rolling The energy-momentum tensor for such a rolling tachyon is compatible with the existence of the tachyon matter The energy-momentum tensor for such a rolling tachyon is compatible with the existence of the tachyon matter The way the tachyon approaches the vacuum can be quite fun! The way the tachyon approaches the vacuum can be quite fun!