Analytic Results in Open String Field Theory Hawaii, 2006 Martin Schnabl (IAS) Joint Meeting of Pacific Particle Physics Communities

Open Bosonic String Field Theory has had a long history 1986 – 1990 SFT formulated Witten, Gross & Jevicky, Ohta, LeClair et al., Kostelecký & Samuel …..…. 1999 – 2002 SFT applied to tachyon condensation Sen, Zwiebach, Taylor, Rastelli, Hata, ……… With little or no activity in the mean time. Hopefully now we are entering new period in which SFT becomes a valuable tool and many new exciting things can be studied analytically.

Progress of the past 11 months in OSFT M.S. hep-th/0511286 Tachyon vacuum constructed , Sen’s first conjecture proved Okawa hep-th/0603159 many details elaborated pure-gauge like form, Fuchs & Kroyter hep-th/0603195 cubic term better understood Rastelli & Zwiebach hep-th/0606131 new solutions of SFT-like equations Ellwood, M.S. hep-th/0606142 Sen’s third conjecture proved Fuji, Nakayama, Suzuki hep-th/0609047 off-shell 4-point amplitude computed

Plan of the talk: I. II. III. IV. V. Brief review of the CFT techniques in SFT wedge states, Review of the tachyon solution Sen’s conjectures Pure gauge like form, partial isometries etc. Marginal deformations Open problems and new directions

Open String Field Theory (Witten 1986) We start with a string field j ª i = t ( X ) c 1 + A ¹ ® ¡ b ¢ Write a Chern-Simons-type Witten action S [ ª ] = 1 2 h ¤ Q i + 3 This action has an enormous gauge invariance ± ª = Q ¤ + ¡ provided that the star product is associative, BRST charge Q acts as a derivative, and the bracket like an integration

h ª ; i = f ± ( ) f ( z ) = t a ³ + r c ´ In the CFT language (LeClair et al., Rastelli et al.) the integration of a star product of N factors (N-vertex) is given by a CFT correlation function on glued world-sheet like here Normally we map the strips to half-disks h ª 1 ; 2 3 i = f ± ( ) U H P f n ( z ) = t a ³ 2 ¡ ¼ 3 + r c ´

Simplifying the Witten N-vertex Let us map the world-sheet from the UHP to a semi-infinite cylinder via (Rastelli et al., 2001) ~ z = a r c t n Create states by inserting local operators on the cylinder, their pullback to UHP is given by , where j ~ Á i = U t a n U t a n = e 1 3 L 2 ¡ 4 + 8 9 6 ¢ is a representation of the conformal map f ( z ) = t a n

Simplifying the Witten N-vertex ~ Á 1 ; 2 3 i = ³ ¼ ´ ( ) ¡ C

h ~ Á ; i = ¡ ¢ ( ) h ~ Â ; Á i = ¤ 8 L = I d ~ z 2 ¼ i T ( ) 1 + a r The two-vertex can be similarly written as h ~ Á 1 ; 2 i = ¡ ¼ ¢ ( ) C Using the two- and three-vertex, one can introduce the star product h ~ Â ; Á 1 2 i = ¤ 8 To relate both vertices one has to rescale the three-vertex cylinder by 2/3, this is generated by L = I d ~ z 2 ¼ i T ( ) 1 + a r c t n X k ¡ 4

~ Á ( ) j i ¤ = U ¡ ¢ U ~ Á ( x ) : j i U = ¡ ¢ ; We then easily find 1 ( ) j i ¤ 2 = U ? 3 ¡ ¼ 4 ¢ where U r = ¡ 2 ¢ L ; ? More generally, star product of n Fock states looks as U ? n + 1 ~ Á ( x ) : j i U ? r ~ Á 1 ( x ) : n j i ¤ s à y m = + ¡ ¼ 4 ¢ Manifestly associative ! [wedge states with insertions]

Properties of , and L L K ~ z L ¡ ~ z + " ( R e ) ¢ L K ´ L K µ ( R e L ? K 1 Useful operators associated to vector fields See RZ (2006) for generalizations ~ z @ L ¡ ~ z + ¼ 2 " ( R e ) ¢ @ L ? @ ~ z K 1 ´ L ¡ (star algebra derivative) K L 1 µ ( R e ~ z ) @ K R 1 µ ( ¡ R e ~ z ) @ Lie brackets give commutators and also [ L ; ? ] = + [ L ; K 1 ] = £ R ¤

b L = + £ L ; b ¤ = L b L j i L : j i U = e Let us introduce Thanks to the commutation relation we find rather unexpectedly new class of eigenstates with eigenvalues n. These states are NOT of of the form These states appear rather naturally in the star product of Fock states due to b L = + ? £ L ; b ¤ = L b L n j i L ¡ n 1 2 : k j i U ? n + 2 = e ¡ b L

b L j i ¤ = X C b L j I i ¤ = L j I i ¤ = X D ~ Á Using the star product formula we find MS (2005), Rastelli, Zwiebach (2006) b L n j i ¤ m = X k ¸ + C super-additivity b L n j I i ¤ m = + exact additivity L ? n j I i ¤ m = X k · + D sub-additivity Under certain assumptions, these formulas generalize to larger sectors involving modes of primary fields ~ Á n

Solving Equations of Motion

Solving Equations of Motion - Toy model Similar equation studied numerically in Gaiotto et al. (2002) ( L ¡ 1 ) © + ¤ = Given the algebra, a natural ansatz is © = X f n ! ¡ 1 2 ¢ b L j i Simple solution to the recursion is , where are the Bernoulli numbers. Can be summed to a closed form f n = B B n © = b L 2 1 ¡ e j i ? ³ + ¢ I X n ¸ @ Related by Euler-Maclaurin formula

Solving Equations of Motion - Witten’s theory We have seen the power of as opposed to L L Therefore, to solve the equation of motion leads us to consider instead of the usual Siegel gauge. Here Very natural ansatz appears to be Q ª + ¤ = B ª = B = H d ~ z 2 ¼ i b ( ) ª = X n ; p f b L ~ c j i + q B where b B = + ?

Thanks to the super-additivity of the star product, the e.o.m. leads to a solvable recursion. With a little bit of luck and help by Mathematica we discovered f n ; ¡ p = ( 1 ) ¼ 2 + ! B q 3 p odd p+q odd where are the Bernoulli numbers B n B = 1 ; ¡ 2 6 : 9 7 3

¡ K @ j n i ª = l i m " Ã ¡ X @ # ; c j ¤ B : Staring a bit at our solution and Euler-Maclaurin formula we realized that in fact ª = l i m N ! 1 " Ã ¡ X n @ # ; 2 ¼ c j ¤ B L : The derivative acts on a wedge state as ¡ ¼ 2 K L 1 @ n j n i

And this is how the solution looks like geometrically ª = 1 X n b T c P c where the distance of the two c-ghost insertions along the two connecting arcs is and respectively. ¼ 2 n ¼ Discovered independently by Okawa (2006)

Sen conjectures V ( ) ¡ T = E E The tachyon is a manifestation of instability of the D-brane, on which the open string ends. where is the D-brane tension There are nontrivial classical solutions corresponding to D-branes of lower dimensions. At the minimum, there are no perturbative degrees of freedom V ( ) ¡ T = E E Sen 1999

First conjecture V [ ª ] = ¡ L 1 L 1 Analytically Three ways to show that V [ ª ] = ¡ 1 2 ¼ g o Analytically Numerically in -level truncation, precision L 1 ¡ 3 L 1 ¡ 7 It would be nice to come up with a simpler analytic proof, and understand why the proof works

Third conjecture ª Q = + ¤ ² § f Q ; A g = j I i Q Á = Q ( A ) Ellwood, MS (2006) Expanding the SFT around the true vacuum produces a theory which looks just like the original one, but with a new BRST-like charge ª Q ª = B + ¤ ² § We construct a state which obeys f Q ª ; A g = j I i Existence of such a state proves that there is no cohomology Since all closed state are automatically exact Q ª Á = Q ª ( A )

The solution turns out to be quite simple A = B L j I i b Z 2 1 d r Surprisingly it is very close in form to the conjectured of Siegel gauge A A = b L j I i

A = g d ª = U Q V U V = I U V = ; Q ( ) Pure gauge like form, partial isometries etc. It has long been suspected that just as in ordinary Chern-Simons theory any solution to the equations of motion can be brought locally to the pure gauge form also in SFT solutions should be possible to write in the pure-gauge-like form We do not need the e.o.m. are satisfied under weaker assumptions, e.g.: A = g d ¡ 1 ª = U Q B V U V = I U V = ; Q B ( )

A = ­ ª = Ã ­ ( 1 ¡ P ) P n S ; S = j i h 1 + 2 ¢ S = 1 ¡ P Early evidence came from study of the SFT in the large NS-NS B-field background Witten factorization: (Witten; M.S. 2000) Allows for simple construction of solutions in the form where is a rank projector in the Moyal algebra. Interestingly such projectors can be very efficiently constructed using partial isometries A = S F T ­ M o y a l ª = Ã ­ ( 1 ¡ P ) P n S ; y S = j i h 1 + 2 ¢ S n y = 1 ¡ P M.S.; Harvey, Kraus, Larsen (2000)

ª = l i m ( I ¡ © ) Q © = B c j i I = j i U = I ¡ © It was found by Okawa (2006) that the solution can be written in the pure-gauge-like form ª = l i m ¸ ! 1 ( I ¡ © ) Q B where is the identity and © = B L 1 c j i I = j i How is this possible ? is perfectly regular string field, its inverse however is more tricky U = I ¡ ©

L V = ( I ¡ © ) X ¸ ; 3 L X ¸ ; ¸ = 1 Ã = c j i + ¢ U Q V In level expansion contains terms of the form whereas in level expansion it contains terms so it is more singular, and even though the singularities are simpler, the value at cannot be defined. This is welcome since can never be written as L V = ( I ¡ © ) 1 1 X n = ¸ k ; 3 L 1 X n = ¸ k ; ¸ = 1 Ã = 2 ¼ c 1 j i + ¢ U Q B V

ª = U Q V S [ ª ] = ¡ h U Q V i S [ U Q V ] = S [ V Q U ] = ¡ V Q U Ordinarily the Chern-Simons action gives quantized values for pure (large) gauge configurations. Similar property can be shown to hold more generally Let Then ª = U Q B V S [ ª ] = ¡ 1 6 h U Q B V i S [ U n Q B V ] = In particular, formally S [ V Q B U ] = ¡ This suggests that should be interpreted as two D-brane solution ! Currently we are trying to test this conjecture by computing the energy rigorously and by working out the spectrum. Ian Ellwood, M.S. in progress V Q B U

J ( z ) ª = 1 + R d r c J ( ) j i ¤ B Marginal deformations Given any exactly marginal operator of the matter CFT we can construct SFT solution J ( z ) ª ¸ = 1 + ¼ 2 R d r c J ( ) j i ¤ B L One can also easily work out formally the spectrum of fluctuations around the solution. The real challenge, however, is to make the solution explicit in some basis especially when the OPE between two J‘s is nontrivial.

Summary Open string field theory is simple in the ‘cylinder’ coordinate. gauge is very natural in this context Having found the tachyon solution, Sen’s first and third conjectures were proved. Work is currently under progress to find out whether the multi brane solutions really exist. Marginal deformation solutions found and are being analyzed. B

Open problems and new directions Construct lump solution and prove Sen’s second conjecture Construct general rolling tachyon solutions and prove Sen’s rolling tachyon conjectures Find more solutions (some progress in multi-brane solutions, Wilson line deformations) Compute systematically off-shell amplitudes (see e.g. recent paper by Fuji, Nakayama, Suzuki ) Study closed strings (open-closed duality, boundary states, etc.) Everything above in super-OSFT (e.g. the Berkovits’ theory)