Construction of BPS Solitons via Tachyon Condensation

1. Construction of BPS SolitonsviaTachyon Condensation So Matsuura @ RIKEN based on the work with T. Asakawa and K. Ohta hep-th/0603***

2. Introduction andMotivation

3. Solitons (Supersymmetric) Gauge Theory • non-trivial solutions of non-linear field equation • non-perturbative feature of gauge theory • non-trivial structure of the moduli space Atiya-Hitchin-Drinfeld-Manin(1978),Nahm(1980) Superstring Theory • BPS bound states of D-branes • non-perturbative feature of string theory • gauge/gravity duality • relation to black hole entropy Witten(1996), Douglas(1996) Ishibashi-Kawai-Kitatzawa-Tsuchiya(1997), Banks-Fishler-Shenker-Susskind(1997) Maldacena(1998)... Strominger-Vafa(1996)

4. How to read off information of solitons from the superstring theory?

5. D3-branes + D(-1)-branes with open strings D3-branes + open strings equivalent at the string level (today’s talk) one-to-one correspondence ADHM Construction instanton solution of 4D gauge theory some field configuration of 0-dim gauge theory Typical Example ~ instanton in string theory ~

6. ADHM Construction Atiyah-Hitchin-Drinfeld-Manin (1978) Watamura-san’s talk (bosonic) ADHM data ; N x k complex matrices ; k x k complex matrices the degrees of freedom of open strings on k D(-1)-branes ADHM constraint F-term and D-term conditions of the 0D SUSY gauge theory on D(-1)-branes Corresponding instanton gauge field ; N x (N+2k) matrix • self-dual field strength • instanton number k

7. Can we complete the following picture? cf) Hashimoto-Terashima(2005) tachyon condensation tachyon condensation Tachyon Condensation Kraus-Larsen (2001) Asakawa-Sugimoto-Terashima (2002) N Dp-branes + k D(-1)-branes tachyon complex tachyon condensation

8. CONTENTS • Introduction • Tachyon Condensation in Boundary State Formalism • Soliton Construction in Tachyon Condensation • Conclusion and Future Works

9. Tachyon Condensation in Boundary State Formalism (review) Neumann directions D-brane boundary of open strings modular transformation condensed state of closed strings In the closed string language, Dirichlet directions Neumann boundary condition Dirichlet boundary condition

10. Dp-brane as aboundary state Callan-Lovelace-Nappi-Yost (1989) tension of a Dp-brane string (super) coordinate boundary (super) coordinate : boundary coordinate : fermionic partner

11. Excitation of open strings insertion of vertex operators at the boundary (massless excitations) A Wilson loop operator is added; is called as the boundary interaction.

12. A system of (N+M) Dp-branes and M anti-Dp-branes Kraus-Larsen (2001) Takayanagi-Terashima-Uesugi (2001) Asakawa-Sugimoto-Terashima (2002) We can introduce complex tachyon; We call as the super-connection; The fate of this system depends on the tachyon profile .

13. Tachyon condensation (1) ~ pair annihilation of D-branes ~ N M M N M We set N M The boundary interaction in the NSNS sector becomes NOTE The RR sector is exactly same as the N Dp-brane because of the super-trace;

14. Tachyon condensation (2) ~ creation of D-instantons ~ Let us set the tachyon profile as where ; SO(4) gamma matrices We can show that this system becomes N D3-branes and k D(-1)-branes at the origin;

15. Technical preliminary (1) Sometimes it is convenient to integrate out θ If we decompose as we can carry out in the definition of the boundary interaction; supersymmetric path-ordered product usual path-ordered product example

16. (2) Gauge transformation of the boundary interaction Consider the system of (N+M) Dp-branes and M anti-Dp-branes. The boundary interaction, is invariant under the gauge transformation, or where, (ex)

17. Soliton Construction as Tachyon Condensation Summary of construction of solitons • Consider D-branes that construct a soliton as a bound state. • Realize individual D-branes by the tachyon condensation. • Add a fluctuation to the boundary interaction. • Carry out a gauge transformation and separate D-branes that vanish. • Read off the information of the moduli space of the soliton solution from the tachyon profile. ex) D3-branes + D(-1)-branes 4D instanton vanish ADHM construction is obtained.

18. (Example 1) Construction of 4D instantons • Consider D-branes that construct a soliton as a bound state. Consider k-instanton solution of U(N) gauge theory. N D3-branes + k D(-1)-branes • Realize individual D-branes by the tachyon condensation. Pauli matrices N D3-branes k D(-1)-branes

19. Add a fluctuation to the boundary interaction. Akhmedov-Gerasimov-Shatashivili (2001) Hashimoto-Terashima (2005) 2k N 2k Note are fluctuation from the profile, which expresses N D3-branes and k D(-1)-branes at the origin. corresponds to scalar fields on D(-1)-brane, thus, must be hermitian.

20. Let us define Then the tachyon profile can be rewritten as k k N k k This is nothing but the ADHM data. corresponding boundary state tachyon condensation

21. Carry out a gauge transformation and separate D-branes that vanish. Let us consider the gauge transformation by; vanish such that N 2k If we assume that is strictly positive definite, we can define and can be written as where V is a (N+2k)×N matrix which is a collection of zero vectors of

22. The gauge transformation of the super-connection is where corresponding boundary state tachyon condensation

23. formulae ・ ・ ・ ・ ・ ・ ・ if There appears a gauge field, on the remaining N Dp-branes after the tachyon condensation.

24. Read off the information of the moduli space of the soliton solution from the tachyon profile. self-dual part anti-self-dual part In order that this is an instanton solution, we must impose This is nothing but the ADHM condition.

25. correspondence at different low energy limit ADHM construction What have we done? full string level gauge tachyon condensation tachyon condensation equivalent

26. Comments • Tachyon configuration , corresponds to the small instanton singularity of the instanton moduli space. • The ADHM constraint is not necessary for this procedure. • The ADHM equations are parts of the tachyon potential • Another part determines the feature of the tachyon condensation. • The gauge transformation here is a large gauge transformation. D(-1)-branes appear at . deviation from the ADHM condition

27. (Example 2) Construction of 2D vortex Let us consider a bound state of D1-branes and D(-1)-branes. Tachyon condensation from (N+k) D1-branes and k anti-D1-branes with For U(1) (N=1), the field strength becomes Then the minimum of the Yang-Mills energy, is realized when H→∞. Well known result.

28. For 2n dimensional Yang-Mills theory, we must impose the “self-duality” for a maximal subgroup H of SO(2n); invariant tensor of H Then the Yang-Mills equation is trivial as a result of the Bianchi identity. For 8D Yang-Mills theory, For H=SO(4)xSO(4), the instanton is an intersection of the 4D instantons. Construction of other solutions is a future work. (Example 3) Construction of higher dimensional instantons I’m sorry, under construction m(__)m

29. Conclusion • We proposed a systematic way to construct a gauge field on D-branes by the tachyon condensation. • In particular, we can examine the structure of the moduli space of solitons in principle. • We applied it to the tachyon condensation of D3-branes and anti-D3-branes and showed that the ADHM construction can be understood as a gauge equivalence of two pictures of D-brane bound state.

30. Future Work • Techniques to be developed • This procedure is quite general one. • Relation to the supersymmetry • Usage of curved D-branes. • We want to impose the BPS condition at the level of the boundary state. • moduli space of non-trivial vortex solutions • construction of higher-dimensional instantons • What is the category of the gauge field that is constructed by this procedure? • At this stage, the role of supersymmetry is not clear. • Nekrasov’s formula by the tachyon condensation?