Localization of e ective actions in OSFT - Applications and Future Developments

Localization of e ective actions in OSFT - Applications and Future Developments Alberto Merlano In collaboration with: Carlo Maccaferri Università di Torino, INFN Sezione di Torino and Arnold-Regge Center Dipartimento di Fisica Teorica February, 08 Meeting on String Field Theory and String Phenomenology 08, Harish-Chandra Research Institute, Allahabad.

Summary of the presentation Recap of the general results (see Carlo s talk, 80.07607) Yang-Mills potential from D(n) branes system Yang-Mills instantons from D3 D( ) branes system Future Directions

Main results (see Carlo s talk) From Berkovits OSFT, the e ective action for a massless field A at the quartic level is S 4 ( A ) = + 8 Tr [[ 0 A, Q B A ] 0 b 0 L 0 [ 0 A, Q B A ]] 4 Tr [[ 0 A, A][ A, Q B A ]]. For consistent and interesting string theory background we can always promote the natural worldsheet N =ton =. (see Sen (986), Banks and Dixon et al.(987-88)) So there is a R-symmetry current j in the matter sector. Assume that the string field is the sum of opposite charged string fields A = (+) A + ( ) A, (±) A = c V (±).

The conservation of the R charge implies that the e ective action localizes in terms of unphysical auxiliary fields such as: µ h P 0[ A, Q B A ] = +A µa c p3 c i p @X i p3 A µa hc i p @X i p3 c = [A µ, A ]c µ (0) 0i. µ p3 where known OPEs of the ghost/superghosts have been used 0 (z) ( z) =+O(z), c(z) c( z) z c@c(0) +

The general structure of the auxiliary fields appearing at the quartic order is h (±±) (±) P 0 [ A, Q B (±) A ]= c H(±) bh (±±) (±) P 0 [ A, 0 (±) A ]=c@c e H (±) g (± ) (±) P 0 [ 0 A, Q B ( ) A ]= c H 0 bg (± ) (±) P 0 [ A, ( ) A ]= c@c @ e H 0. The explicit form of the above string fields is given by the leading OPE between the matter superconformal primaries V ( ) V (±) (z)v (±) (z)v (±) ( z) =H (±) (0) + ( z) V (±) (z)v ( ) ( z) = z H 0 +.

The localized e ective action The ghost/superghost CFT is universal and can be computed. Then we remain with apple S (4) e ( A)=tr hh (+) H ( ) i + 4 hh 0 H 0 i, This is the main result of 80.07607: Choose your string background; the e ective action at the quartic level for the massless fields is given by the BPZ product of the matter part of the auxiliary field. If not for this term, the e ective action would be zero. This matter vertex depends on the chosen string background. So let us analyze two important examples.

Yang-Mills potential:the worldsheet theory The worldsheet theory of a system of N coincident D(n) euclidean branes contains the µ superconformal fields which we rearrange according to a U(n) SO(n) decomposition = p ( j + i j )=e ih j J 0 =+, = p ( j i j )=e ih j J 0 =. with (,, j) =,...,n. The localizing current is given J(z) = nx i @h j (z). j=

Yang-Mills potential: the auxiliary fields The matter vertex is then given V (+) (z) =A (z), V ( ) (z) =A (z), The auxiliary fields are easily extracted from the leading term in the OPE H (+) (z) = [A ı, A ] ı (z), H ( ) (z) = [A ī, A ] ī (z), H 0 (z) = [A, A ]. where ı (z) : ı :(z). The e ective action is then easily computed from apple S (4) e ( A) = tr hh (+) H ( ) i + 4 hh 0 H 0 i

Yang-Mills potential apple S (4) e ( A) = tr [A ı, A ][Aī, A ]+ 4 [A, A ][Aī, A ı ]. The N N matrices A, A are related to the original A µ s by A = µ A µ A = µ A µ, where non-vanishing entries are j = p = j nx = j = i p = µ = ( µ i µ ). Thus, the Y-M potential is correctly reproduced S (4) e = j. 8 tr [[A µ, A ][A µ, A ]].

Y-M Instantons: the worldsheet theory of D3 D( ) The total massless string field is given by A! A(z) =c (z),! a where A(z) =A µ µ (z)+ p p (z),!(z) =! N k!(z) =! k N S (z), S (z), a(z) =a µ µ (z)+ p p (z). Greek indices µ label the directions along the D3 branes, roman indices p label the transverse directions while SO(4) spinor indices are (, )and(, ).

Benefits of the Localization Certainly we can compute this e ective action using the Berkovits quartic e ective action, but for the instanton potential it requires 4 point functions of (excited) twist fields which are not even well known in the literature. The most challenging one is roughly given: Tr [[ 0 A, Q B A ] 0 b 0 L 0 [ 0 A, Q B A ]] h µ (z ) (z ) (z 3 ) (z 4 )i. So it is easy to appreciate how localization avoids this problems reducing all the computation to a point function.

Yang-Mills Instantons: Gauge Boson The worldsheet theory contains the µ superconformal fields which we rearrange as before A = A (+) + A ( ) + (+) + ( ), a = a (+) + a ( ) + (+) + ( ), where using the same notation of previous example A (+) = A, A ( ) = A, (+) = m m, ( ) = m m, The same also for a. Here =, and =, indices denote respectively the fundamental and antifundamental representation of SU() SO(4), while m =,, 3and m =,, 3 indices denote respectively the fundamental and antifundamental representation of SU(3) SO(6).

Yang-Mills Instantons: Stretched strings The worldsheet theory contains also the composite S and S which are superconformal primaries of weight.spin fields are defined through bosonization by the scalars h, h S (, ) = e i (h +h ), S (, ) = e i (h +h ) J 0 S (, ) =+S (, ), J 0 S (, ) = S (, ). Finally we can decompose the o -diagonal part of V where! =! (+) +! ( ),! =! (+) +! ( )! (+) =! S (, ),! ( ) =! S (, ) as! (+) =! S (, ),! ( ) =! S (, ).

Yang-Mills Instantons - Auxiliary fields The auxiliary fields H 0, H (+), H ( ) along the D3 branes are H (+)D3 [A, A = ]+!! 0 0 [a, a ]!! 0i, H ( )D3 = [A, A ]+!! 0 0 [a, a ]!! 0i, H D3 0 = [A, A ] (!! +!! ) 0 0i. 0 [a, a ]+(!! +!! ) Note the absence of o -diagonal terms due to the OPE µ (z) S (w) ( µ ) S (w) p. (z w)

Yang-Mills Instantons - Covariant auxiliary fields At first sight maybe they do not look significant, but after algebraic manipulations involving the ladder t Hooft symbols: µ + µ + i µ, µ µ i µ, it is possible to see that they carry a SU() representation H (+)D3 = i 4 µ T µ 0i,, H ( )D3 =+ i 4 µ + T µ 0i, H D3 0 = i µ 3 T µ 0i. The covariant tensor T µ is given by 0 T µ = [A B µ,a ]+ @! ( µ )! 0 0 [a µ,a ]! ( µ )! C A.

The ADHM constraint and the e ective action On the D3-branes worldvolume, we get the quartic potential given by a perfect square apple S T = tr hh (+)D3 H ( )D3 i + 4 hhd3 0 H D3 0 i = 6 tr [D ad a ], where the auxiliary field D a = µ a T µ, contains through its EOM the 3 ADHM constraints (on the D( ) slot). apple a µ T µ =0! a µ [a µ, a ]! ( µ )! =0

Yang-Mills Instantons transverse auxiliary fields The auxiliary fields in the transverse directions are less interesting and have more complicated structure: 0 H (+)T B = @ [ m, n] mn +[A, m] m ( m!! m) m S (, ) C A 0i, ( m!! m) m S (, ) [ m, n] mn +[a, m] m H ( )T = H (+)T (m! m,! ), H T 0 = [ m, m] 0 0i, 0 [ m, m]

Yang-Mills Instantons e ective action The complete e ective action is apple S (4) e ( A) = tr hh (+) H ( ) i + 4 hh 0 H 0 i = and plugging all the auxiliary fields, the action is given by h = 8 tr [A µ, A ] +[a µ, a ] +[ p, q] +[ p, q] i apple 4 tr [A µ, p] +[a µ, p] + 4 (! µ!) + 4 (! µ!) 4 tr [[A µ µ, A ]!! [a µ, a ]! µ! ] + h i h i tr!!!! tr p! p!. The full e ective action is correctly reproduced (see Lerda, Frau, Billo, Pesando and Liccardo 00).

Higher Orders - Some comments on the future directions While the quintic order is trivially zero(by c ghost counting for example), the sixth order is definitely more challenging for several reasons: The number of involved terms considerably grows. The contact term is identically zero (c-ghost counting) and there is a clear hierarchy of propagator terms but it is not likewise evident how to proceed e ciently. The combinatorial nature of the vertices gets complicated due to presence of, or 3 propagators and 0 in the amplitudes.

Higher Orders - Some comments on the future directions However from brute force computation one can observe that at any order g n in the expansion = g A + P m= g m R m we have schematically that n Tr [R m (EOM(R n m ))], < m apple n and then it is zero since we solve the equations of motion perturbatively. For m = n we remain with only the kinetic term of R n : h i Tr ( 0 R n )(Q B R n ) This fact reduces a lot the number of involved terms. We think this is a good starting point and we hope to report on these progress soon.

Thank you for your Attention!