1. D-Brane Moduli in String Field Theory : Exact non-perturbative results for marginal deformations Carlo Maccaferri Torino University and INFN, sezione di Torino JHEP 1405(2014)004 (CM) & [To appear] with M. Schnabl NEW FRONTIERS IN THEORETICAL PHYSICS Cortona, 28/05/2014

2. Open String Field Theory (OSFT) is a microscopic theory for D-branes, formulated as a field theoretic description of open strings CARTOON-ANALOGY OSFT Yang-Mills Open strings Gauge fields D-branes Solitons, Instantons Closed strings Gauge invariant op. elusive! Gauge/Gravity?

3. Fix a bulk CFT (closed string background) Fix a reference BCFT 0 (open string background, D-brane’s system) The string field is a state in BCFT 0 There is a non-degenerate inner product (bpz) The bpz-inner product allows to write a target-space action Witten product: peculiar way of gluing surfaces through the midpoint in order to have associativity Equation of motion OPEN STRING FIELD THEORY: SNAPSHOT

4. OSFT CONJECTURE (once known as Sen’s conjecture) Key tool for connecting the two sets is the OSFT construction of the boundary state (Kiermaier, Okawa, Zwiebach (2008), Kudrna, CM, Schnabl (2012) ) The (KMS) boundary state is constructed from gauge invariant quantiities starting from a given solution Intriguing possibility of relating BCFT consistency conditions (Cardy-Lewellen, Pradisi-Sagnotti-Stanev) with OSFT equation of motion Classical Solutions Allowed D-branes Allowed D-branes

5. Today’s application: how does OSFT describes D-brane’s moduli space? Long-standing Puzzle: In 2000 Sen & Zwiebach wanted to see if it was possible to translate a D-brane a finite distance, in OSFT. On the BCFT side this means to add an exactly marginal deformation to the WS action On the OSFT side this means to search for a solution of the form Sen-Zwiebach strategy (2000): work in LEVEL TRUNCATION, plug the finite level ansatz in the action and solve ALL coefficients in terms of λ SFT. At any level this gives a set of algebraic quadratic equations which can be numerically solved. Improve the solution by increasing the level. Continuous family of conformal boundary conditions Continuous family of OSFT solutions

6. Plug into the OSFT action to see if it becomes constant (EOM) as a function of λ SFT. The marginal branch becomes flat (1-parameter family of solutions)… but it ends at finite λ SFT !!! Does this mean that the moduli space is not covered??? Examining the gauge invariant observables and constructing the boundary state it was possible to conclude that the critical λ SFT corresponds to finite λ CFT ((KMOSY, 2012) Therefore the numerically found branch of solutions doesn’t cover the whole moduli space and it only allows to shift the brane at a distance of the order of the string lenght. Courtesely taken from Kudrna, Masuda, Okawa, Schnabl, Yoshida JHEP 1301(2013)103 

7. Luckily level truncation is not the only tool we have! Since the 2005 discovery of the analytic tachyon vacuum solution by Martin Schnabl, Impressive analytic developments since then (Rastelli, Sen, Zwiebach, Kiermaier, Okawa, Erler, Fuchs, Kroyter, Potting, Bonora, Tolla, Giaccari, CM, Noumi, Murata, Masuda, Takahashi, Kishimoto, Kojita, Hata, Baba, Ishibashi, etc…..) In particular EXACT regular solutions for marginal deformations of the kind discussed here (singular self-OPE) has been written down. They are expressed as a perturbative expansion in the marginal parameter. ( Fuchs, Kroyter, Potting, Kiermaier, Okawa, Rastelli, Zwiebach, Schnabl) (2006-2008) Every term is known in closed form, but the resummation is elusive. Without a regular resummed form it is not possible to address the covering of moduli space (radius of convergence???) ANALYTIC REVOLUTION

8. Recently a new simple solution has been found (CM, JHEP 1405(2014)004 ) The new solution is, formally, gauge equivalent to the old “identity-based” TT solution. Contrary to TT, the new solution has well defined computable observables (CM). The solution is a rational (non commutative) function of the marginal parameter λ, already resummed form. We can compute analytically the KMS boundary state associated to it and find the expected result! From here it is clear that the full moduli space is covered! Now we have an explicit analytic solution which covers the full moduli space! NEW NON-PERTURBATIVE SOLUTION Old “formal” solution by Takahashi-Tanimoto (2002)

9. From the privileged perspective of an analytic solution we can now reconsider the puzzle found by Sen and Zwiebach. We can expand the solution in the Fock-space basis The Fock-space coefficients are well-defined and computable EXPLAINING SEN-ZWIEBACH PUZZLE (CM, Schnabl, to appear) λ SFT REACHES A MAXIMUM, THEN IT DECREASES AND FINALLY IT RELAXES TO ZERO!!! AMAZINGLY, ZWIEBACH PROPOSED THIS POSSIBILITY FROM A TOY-MODEL (2001). HE ALSO HYPOTHIZED THAT THE VEV OF THE TACHYON WOULD ASYMPTOTE THE TACHYON VACUUM.

10. Let’s then have a look at the tachyon coefficient It is extracted from When the brane is pushed far away in the moduli space we are locally left with empty closed string vacuum (the Tachyon Vacuum). Clear! (13 years later…) AS PREDICTED BY ZWIEBACH THE TACHYON COEFFICIENT RELAXES TO THE TACHYON VACUUM!

11. Discussion New simple “non-perturbative” solution for marginal deformations. The BCFT moduli space is entirely covered for the first time in a clear way. String field theory is able to describe open string backgrounds which are far, without going “so far” in the Fock space of the starting background. Small values of the moduli are covered by the marginal part of the string field, but later non-linearities enter the game and the motion in moduli space is collectively described by the whole string field. The VEV of the marginal field is not a global coordinate in moduli space (cfr Zwiebach 2001) Far away in the moduli space the solution approaches the Tachyon Vacuum.  Search for the new branches in level truncation ( challenging “experiment” )!  Can we also describe other backgrounds in a similar way?  Can we prove that OSFT is, at the end of the day, background independent?  Can we do closed string physics using OSFT???  Third SFT “revolution” is behind the corner… stay tuned! Thank you

12. To stay tuned: 28/7  1/8/2014 conference @ SISSA To get into: 21/7  26/7/2014 school @ SISSA L. Bonora (String Field Theory) C. Maccaferri (String Field Theory) M. Schnabl (String Field Theory) X. Bekaert (Higher Spins) D. Francia (Higher Spins) I. Areefeva (Lebedev Inst., Moscow, Russia) I. Bars (Univ. South Cal., LA, USA) X. Bekaert (Tours Univ., France) N. Berkovits (Inst. Fis. Teorica, Univ. Paulista, S.Paulo, Brasil) N. Boulanger (Mons Univ., Belgium) B. Dragovic (Belgrade Univ, Serbia) T. Erler (Inst. of Physics AS CR, Prague, Czech Republic) D. Francia (SNS, Pisa) K. Fredenhagen (Hamburg Univ., Germany) N. Ishibashi (Tsukuba Univ., Japan) I. Kishimoto (Niigata Univ. Japan) A. Koshelev (Vrije Univ., Bruxelles) M. Kroyter (Tel Aviv Univ., Israel) H. Kumitomo (YITP, Kyoto, Japan) C. Maccaferri (Torino Univ., Italy) H. Nielsen* (Niels Bohr Inst. Univ. Copenhagen, Denmark) Y. Okawa (Univ. Tokyo, Komaba, Japan) D. Polyakov (CQUeST, Seoul, S.Korea) I. Sachs (Munich Univ., Germany) M. Schnabl (Inst. of Physics AS CR, Prague, Czech Republic) A. Sen (HCRI, Allahabad, India) E.D. Skortsov (Lebedev Inst., Russia) T. Takahashi (Women's Nara Univ, Japan) B. Zwiebach (MIT, Boston, USA) M.A. Vasiliev* (Lebedev Inst., Moscow, Russia)

13. (Selected) Facts in OSFT 1985: Witten writes down the action: star product, associative non- commutative algebra 1990: Zwiebach proves that the Feynmann rules gives a complete covering of the moduli space of Riemann surfaces with boundary (loop-amplitudes are guaranteed to be reproduced), on shell closed strings are automatically accounted for (Shapiro-Thorn) (more on this later…) 1999: Sen uses OSFT to formulate his conjectures on the tachyon vacuum (empty closed string background, no D-branes) 2005: Schnabl solves the equation of motion and finds the first analytic solution describing the tachyon vacuum 2005-Today: Analytic revolution, new analytic solutions are found (general marginal deformations) Schnabl, Kiermaier, Okawa, Rastelli, Zwiebach, Fuchs, Kroyter, Potting, Noumi, Erler. Progresses for other backgrounds (relevant deformations, Bonora, CM, Tolla, Erler, Giaccari, multiple branes, Murata, Schnabl, Kojita, Hata, Masuda etc…), Non trivial gauge structure (Okawa, Ellwood), topological localization of observables (phantom terms), (Erler, CM), solutions are organized in a category, (Erler, CM) Numerically a large landscape of solutions is seen to emerge, (multiple lower dimensional branes, intersecting branes, minimal models) Kudrna, Rapchak, Schnabl Need to relate the parameters of the solutions with the correspondig BCFT moduli

14. The Landscape: CFT’s and BCFT’s 2D CFT Conformal Boundary conditions (isomorphic left/right moving sectors) BCFT Closed String Background Closed String Landscape D-Branes on a closed string background Open String Landscape For a given CFT, a choice of BCFT is encoded in a peculiar closed string state, called the BOUNDARY STATE It is defined as a path integral on the disk with given boundary conditions The boundary state is the stringy generalization of the energy momentum tensor, source for closed strings (gravity) CLOSED STRING FIELD THEORYOPEN STRING FIELD THEORY

15. The space of OSFT solutions Erler, CM (2012) BCFT 0 BCFT 1 Tachyon Vacuum Solutions Gauge Transformations Gauge Orbits (BOUNDARY STATES) Singular Gauge Transformations (no inverse), they exist because of the b- antighost, BCC OPERATORS IN BCFT