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 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) Atiya-Hitchin-Drinfeld-Manin(1978),Nahm(1980) Ishibashi-Kawai-Kitatzawa-Tsuchiya(1997), Banks-Fishler-Shenker-Susskind(1997) Maldacena(1998)... Strominger-Vafa(1996)
How to read off information of solitons from the superstring theory?
Typical Example ~ instanton in string theory ~ some field configuration of 0-dim gauge theory D3-branes + open strings D3-branes + D(-1)-branes with open strings one-to-one correspondence ADHM Construction (today’s talk) equivalent at the string level instanton solution of 4D gauge theory
ADHM Construction (bosonic) ADHM data ; N x k complex matrices ; k x k complex matrices ADHM constraint the degrees of freedom of open strings on k D(-1)-branes 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 Atiyah-Hitchin-Drinfeld-Manin (1978) Watamura-san’s talk
Tachyon Condensation complex tachyon N Dp-branes + k D(-1)-branes condensation tachyon Kraus-Larsen (2001) Asakawa-Sugimoto-Terashima (2002) Can we complete the following picture? tachyon condensation tachyon condensation cf) Hashimoto-Terashima(2005)
CONTENTS 1.Introduction 2.Tachyon Condensation in Boundary State Formalism 3.Soliton Construction in Tachyon Condensation 4.Conclusion and Future Works
Tachyon Condensation in Boundary State Formalism (review) D-brane boundary of open strings condensed state of closed strings Neumann directions modular transformation Neumann boundary condition Dirichlet boundary condition In the closed string language, Dirichlet directions
: boundary coordinate : fermionic partner Dp-brane as a boundary state boundary (super) coordinate string (super) coordinate tension of a Dp-brane Callan-Lovelace-Nappi-Yost (1989)
Excitation of open strings insertion of vertex operators at the boundary A Wilson loop operator is added; is called as the boundary interaction. (massless excitations)
A system of (N+M) Dp-branes and M anti-Dp-branes The fate of this system depends on the tachyon profile. We can introduce complex tachyon; We call as the super-connection; Kraus-Larsen (2001) Takayanagi-Terashima-Uesugi (2001) Asakawa-Sugimoto-Terashima (2002)
Tachyon condensation (1) ~ pair annihilation of D-branes ~ We set 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; N M M NMMNNMMN
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;
Technical preliminary If we decompose as we can carry out in the definition of the boundary interaction; (1) Sometimes it is convenient to integrate out θ supersymmetric path-ordered product usual path-ordered product example
(2) Gauge transformation of the boundary interaction is invariant under the gauge transformation, Consider the system of (N+M) Dp-branes and M anti-Dp-branes. The boundary interaction, or where, (ex)
Soliton Construction as Tachyon Condensation Summary of construction of solitons ① Consider D-branes that construct a soliton as a bound state. ② Realize individual D-branes by the tachyon condensation. ③ Add a fluctuation to the boundary interaction. ④ Carry out a gauge transformation and separate D-branes that vanish. ⑤ 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.
(Example 1) Construction of 4D instantons ① Consider D-branes that construct a soliton as a bound state. N D3-branes k D(-1)-branes ② Realize individual D-branes by the tachyon condensation. Consider k-instanton solution of U(N) gauge theory. N D3-branes + k D(-1)-branes Pauli matrices
③ Add a fluctuation to the boundary interaction. Note which expresses N D3-branes and k D(-1)-branes at the origin. 2k N 2k are fluctuation from the profile, corresponds to scalar fields on D(-1)-brane, thus, must be hermitian. Akhmedov-Gerasimov-Shatashivili (2001) Hashimoto-Terashima (2005)
Let us define Then the tachyon profile can be rewritten as This is nothing but the ADHM data. k k NkkNkk corresponding boundary state tachyon condensation
Let us consider the gauge transformation by; such that If we assume that is strictly positive definite, we can define andcan be written as where V is a (N+2k)×N matrix which is a collection of zero vectors of vanish N 2k ④ Carry out a gauge transformation and separate D-branes that vanish.
The gauge transformation of the super-connection is where tachyon condensation corresponding boundary state
・ ・ ・ ・ ・ ・ ・ There appears a gauge field, on the remaining N Dp-branes after the tachyon condensation. formulae if
⑤ Read off the information of the moduli space of the soliton solution from the tachyon profile. self-dual partanti-self-dual part In order that this is an instanton solution, we must impose This is nothing but the ADHM condition.
What have we done? tachyon condensation gauge equivalent full string level correspondence at different low energy limit ADHM construction
Comments ① Tachyon configuration, corresponds to the small instanton singularity of the instanton moduli space. ② The ADHM constraint is not necessary for this procedure. ③ The ADHM equations are parts of the tachyon potential ④ Another part determines the feature of the tachyon condensation. ⑤ The gauge transformation here is a large gauge transformation. D(-1)-branes appear at. deviation from the ADHM condition
(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.
(Example 3) Construction of higher dimensional instantons 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. I’m sorry, under construction m(__)m
Conclusion 1.We proposed a systematic way to construct a gauge field on D-branes by the tachyon condensation. 2.In particular, we can examine the structure of the moduli space of solitons in principle. 3.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.
Future Work Techniques to be developed This procedure is quite general one. Relation to the supersymmetry 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? Usage of curved D-branes. We want to impose the BPS condition at the level of the boundary state.