ADHM is a Tachyon Condensation --- Stringy Derivation of the ADHM Construction --- Koji Hashimoto (U. of Tokyo, Komaba) 30 Oct Hawaii meeting hep-th/ , hep-th/ with Seiji Terashima (Rutgers, YITP)

N D4-branes SU(N) YM at low energy N D4-branes k D0-branes = Equality? instanton gauge field D-brane interpretation of 4d YM instantons 1. Question

Reasons for “believing in” the equality In the D4-brane effective action, the CS term reads # of SUSY is the same. : 1/2 SUSY. D0 + D4 : 1/2 SUSY on D4. Mass. But these are merely supporting evidences. [Douglas]

Difficulty for proving the equality D0 pictures is valid around small instanton singularity In general, computing corrections is quite difficult and almost impossible to perform that to all order in General belief is that the SUSY may protect the instanton configurations against the corrections. Ex. [Thoracius] Our result : We show the equality. In showing the equivalence, all the ADHM procedures naturally emerge! Instanton picture is valid for slowly varying fields, Cf. [Witten] [Douglas]

ADHM construction : From given ADHM data, multi-instanton config. can be constructed algebraically. ADHM equation ADHM data Self-dual gauge field configuration 1 to 1 [Atiyah-Drinfeld-Hitchin-Manin]

ADHM construction : From given ADHM data, multi-instanton config. can be constructed algebraically. ADHM equation = BPS equation on D0 at low energy ADHM data = massless excitations of strings ADHM data Self-dual gauge field configuration 1 to 1 = massless modes of D0-D0 string = massless modes of D0-D4 string (eigenvalues are location of D0s) [Witten] [Douglas]

Procedures of the ADHM construction 1. Define a “Dirac operator” matrix, 2. Solve the zero mode equation, 3. Write the gauge field, This solves the self-dual equation Mystery: What are the physical meaning of these procedures? We show that these are exactly tachyon condensation

2. Proof of the equality Only D4D4+D0 Dimensions are different.. Let us represent D0s by D4s! Tachyon Condensation Technique (K-theory) [Townsend] [Sen] [Witten].... All the D-brane dynamics can arise from tachyon condensation of higher dimensional unstable D-branes (Brane-antibranes / non-BPS D-branes). We may unify D0s and D4s.“Brane Democracy”

We’ll show the following equalities among boundary states N D4 + k D0 + ADHM data = (N+2k) D4 + 2k D4bar + tachyon = N D4 with the gauge field which is constructed by the ADHM method (i) is Sen’s descent relation. (ii) is just a change of the basis of the Chan-Paton factor of D4s. ADHM construction = just a change of the basis in the tachyon condensation We prove these equalities exactly in string theory, thus this holds beyond the corrections. (i) (ii) (i) (ii) D4 D4bar

Equivalence (i) can be shown by the following tachyon profile on D4D4bar boundary state, (i) part naturally appears as the location of the D0s, in the ABS construction. part has charges identical with the D0D4 string : bi-fundamental rep. in in [Asakawa,Sugimoto,Terashima] Tachyon is the Dirac operator, N+2k 2k Ref. [Hori] [Hellerman,Kachru,Lawrence,McGreevy] Ref. [Akhmedov,Gerasimov,Shatashvili] (i)

(ii) Let us see the equivalence (ii). To see which D4 disappears, we diagonalize the tachyon by the gauge symmetry, Then the whole matrix becomes This is a change of the basis of the Chan-Paton factor. (ii)

The tachyon condensation limit means that CP factors with nonzero eigenvalue “disappear” CP factors with zero eigenvalue remain remains Write the unitary matrix then the remaining gauge field is written as And satisfies which is the Dirac zero mode!

The tachyon on D4D4bar is the Dirac operator, Remaining D4s = tachyon zeromodes of the CP factor. The basis rotation induces the nontrivial connection. This is a part of the pure gauge. Essence of the derivation ( = physical meaning of ADHM) : All procedures are done in off-shell boundary states. Two systems are equivalent to all orders in.

3. Summary We have derived the ADHM construction of instantons in string theory. The tachyon condensation is nothing but the ADHM construction, showing the equivalence of the D0D4 system and gauge fields on the D4. The equivalence is exact in string theory. This method can be applied also to 1.Derivation of inverse ADHM. 2.Easy proof of completeness. 3.Derivation of Nahm construction of monopoles. 4. Derivation of non-commutative ADHM(N) construction.

At on-shell? We haven’t imposed the ADHM equation for the ADHM data and the self-dual equation for the Yang-Mills field. (Off-shell correspondence) However, we know that D0D4 gives a CFT, and applying our derivation of ADHM, this means that Yang-Mills instanton configuration with sub-stringy size (small instantons) gives also an equivalent CFT. Self-dual Yang-Mills configuration solves stringy EOM corrected to all orders in. This is known at, and was checked at by Koerber and Sevrin.