Noncommutative D-branes Research Articles

Commutative geometry for non-commutative D-branes by tachyon condensation

There is a difficulty in defining the positions of the D-branes when the scalar fields on them are non-abelian. We show that we can use tachyon condensation to determine the position or the shape of D0-branes uniquely as a commutative region in spacetime together with non-trivial gauge flux on it, even if the scalar fields are non-abelian. We use the idea of the so-called coherent state method developed in the field of matrix models in the context of the tachyon condensation. We investigate configurations of noncommutative D2-brane made out of D0-branes as examples. In particular, we examine a Moyal plane and a fuzzy sphere in detail, and show that whose shapes are commutative $\mathbb{R}^2$ and $S^2$, respectively, equipped with uniform magnetic flux on them. We study the physical meaning of this commutative geometry made out of matrices, and propose an interpretation in terms of K-homology.

D-brane on Poisson manifold and generalized geometry

The properties of the D-brane fluctuations are investigated using the two types of deformation of the Dirac structure, based on the B-transformation and the β-transformation, respectively. The former gives the standard gauge theory with two-form field strength. The latter gives a nonstandard gauge theory on the Poisson manifold with bivector field strength and the vector field as a gauge potential, where the gauge symmetry is a diffeomorphism generated by the Hamiltonian vector field. The map between the two gauge theories is also constructed with the help of Moser's Lemma and the Magnus expansion. We also investigate the relation to the gauge theory on the noncommutative D-branes.

Noncommutative D-branes from covariant AdS superstring

We study noncommutative (NC) D-branes on AdS 5 × S 5 from κ-invariance of covariant Green–Schwarz action of an open string with a non-trivial world-volume flux. Finding boundary conditions to ensure the κ-invariance, we can see possible configurations of the NC D-branes. With this method 1/4 BPS NC D-branes are discussed. The resulting NC D p-branes are 1/4 BPS at arbitrary position other than the p = 1 case. The exceptional D-string is 1/2 BPS at the origin and 1/4 BPS outside the origin. Those are reduced to possible 1/4 BPS or 1/2 BPS AdS D-branes in the commutative limit. The same analysis is applied to an open superstring in a pp-wave and leads to 1/4 BPS configurations of NC D-branes. These D-branes are consistently obtained from AdS D-branes via the Penrose limit.

Explicit construction of BRST charge of noncommutative D-brane system

In the BRST–BFV scheme for noncommutative D-branes with constant NS B-field, introducing ghost degrees of freedom we construct the gauge-fixed Hamiltonian and corresponding effective Lagrangian invariant under nilpotent BRST charge. It is also shown that the presence of auxiliary variables introduced via the improved Dirac formalism plays a crucial role in the construction of the BRST invariant Lagrangian.

The shape of nonabelian D-branes

We evaluate bulk distribution of energies, pressures and various D-brane/F-string charges generated by nontrivial matrix configurations in nonabelian D-brane effective field theories, using supergravity source density formulas derived originally in Matrix theory. Off-diagonal elements of worldvolume nonabelian fields, especially transverse scalar fields, induce various interesting bulk structures exhibiting the shape of branes. First, we study the energy distribution of string-brane networks generated in the bulk by the Yang-Mills monopoles and the 1/4 BPS dyons, and confirm force balance of them. An application to the Yang-Mills description of recombination of intersecting D-branes gives results indicating presence of the tachyon matter. Second, we analyse the shape of fuzzy D-branes given by nonabelian scalar fields which are mutually noncommutative. We employ fuzzy S^2, fuzzy S^4 and fuzzy cylinder/supertube as matrix configurations of N D0-branes representing higher dimensional noncommutative D-branes. We find that in the continuum (large N) limit the D-brane charge distributions become in the expected shape of a sphere or a cylinder with an infinitesimal thickness. However, the distributions found for finite N are difficult to interpret, which leaves a puzzle for a possible dual description in terms of higher dimensional D-branes. A resolution is provided with use of an ordering ambiguity in the charge density formulas.

On tachyon condensation of intersecting noncommutative branes in M(atrix) theory

The interaction between the intersecting noncommutative D-branes (or membranes) is investigated within the M(atrix) theory. We first evaluate the spectrum of the off-diagonal fluctuation and see that there is a tachyon mode, which signals the instability of the intersecting branes. We next explain in detail how the tachyon modes (which are the off-diagonal fluctuations) can be combined with the original branes (which are the diagonal elements) to become a new diagonal matrix, which then represents the new recombined configurations after the tachyon condensation. The tachyon condensation configurations of the noncommutative branes we find are different from those of the commutative cases studied in the previous literatures.

Comment on “Constraint quantization of open string in background B field and noncommutative D-brane”

In the paper “Constraint quantization of open string in background B field and non-commutative D-brane”, [Nucl. Phys. B 568 (2000) 447], it is claimed that the boundary conditions lead to an infinite set of secondary constraints and Dirac brackets result in a non-commutative Poisson structure for D-brane. Here we show that contrary to the arguments in that paper, the set of secondary constraints on the boundary is finite and the non-commutativity algebra cannot be obtained by evaluating the Dirac brackets.

Noncommutative D-brane and open string in pp-wave background with B-field

The open string ending on a D-brane with a constant B-field in a pp-wave Ramond–Ramond background is exactly solvable. The theory is controlled by three dimensionful parameters: α′, the mass parameter (RR background times the lightcone momentum) and the B-field. We quantize the open string theory and determine the full noncommutative structure. In particular, we find a fully noncommutative phase space whose noncommutativity depends on all these parameters. The lightcone Hamiltonian is obtained, and as a consequence of the nontrivial commutation relations of the theory, new features of the spectrum are noted. Various scaling limits of the string results are considered. Physical implications are discussed.

CONSISTENT QUANTIZATION OF NONCOMMUTATIVE D-BRANE IN NON-CONSTANT B FIELD BACKGROUND

We perform the consistent quantization of open string D-brane in non-constant NS–NS closed string B field background by directly imposing the worldsheet conformal symmetry. In addition to the previous noncommutative D-brane coordinates quantization with constant noncommutative parameter θ, we obtain a set of constraints on the B field which can be interpreted as a quantum consistent deformation of a curved noncommutative D-brane.

A note on noncommutative D-brane actions

We use the nonabelian action of N coincident D(−1) branes in constant background fields, in the N→∞ limit, to construct noncommutative D-brane actions in an arbitrary noncommutative description and comment on tachyon condensation from this perspective.

T-duality and actions for noncommutative D-branes

We show how the T-duality is realized for D-branes with noncommutative world-volume coordinates. We discuss D-branes wrapped on tori and the result is that the recently found noncommutative actions form a consistent collection due to the T-duality mapping between noncommutative D-branes and rotated commutative D-branes on deformed tori.

Formula omitted]-Trek III: the search for Ramond–Ramond couplings

We give a detailed discussion of the disk amplitudes with one closed string insertion, which we used to construct the supergravity couplings of noncommutative D-branes to the RR potentials, given in hep-th/0104139. We prove the inclusion of Elliott's formula, the integer-valued modification of the noncommutative Chern character, to all orders in the gauge field. We also give a detailed comparison between the form of the result in which Elliott's formula is manifest, and the form expressed in Matrix model variables.

Supergravity couplings of noncommutative D-branes

We discuss the supergravity couplings of noncommutative D-branes by considering the disk amplitudes with one closed string insertion. The result confirms a recent proposal for the general form of the noncommutative Yang–Mills operators coupling to the massless closed string modes. The construction involves smearing Yang–Mills field variables along an open Wilson line. For multiple D-branes interacting with background supergravity fields, this prescription reduces, in the B=0 limit, to the “symmetrized trace” prescription, and the supergravity fields are seen to be functionals of the nonabelian scalar fields on the branes.

Gauge-invariant couplings of noncommutative branes to Ramond-Ramond backgrounds

We derive the couplings of noncommutative D-branes to spatially varying Ramond-Ramond fields, extending our earlier results in hep-th/0009101. These couplings are expressed in terms of *n products of operators involving open Wilson lines. Equivalence of the noncommutative to the commutative couplings implies interesting identities as well as an expression for the Seiberg-Witten map that was previously conjectured. We generalise our couplings to include transverse scalars, thereby obtaining a Seiberg-Witten map relating commutative and noncommutative descriptions of these scalars. RR couplings for unstable non-BPS branes are also proposed.

Supergravity couplings to noncommutative branes open Wilson lines and generalised star products

Noncommutative gauge theories can be constructed from ordinary $U(\infty)$ gauge theories in lower dimensions. Using this construction we identify the operators on noncommutative D-branes which couple to linearized supergravity backgrounds, from a knowledge of such couplings to lower dimensional D-branes with no $B$ field. These operators belong to a class of gauge invariant observables involving open Wilson lines. Assuming a DBI form of the coupling we show, to second order in the gauge potential but to all orders of the noncommutativity parameter, that our proposal agrees with the operator obtained in terms of ordinary gauge fields by considering brane actions in backgrounds and then using the Seiberg-Witten map to rewrite this in terms of noncommutative gauge fields. Our result clarify why a certain {\it commutative} but {\it non-associative} ``generalized star product'' appears both in the expansion of the open Wilson line, as well as in string amplitude computations of open string - closed string couplings. We outline how our procedure can be used to obtain operators in the noncommutative theory which are holographically dual to supergravity modes.

Intersecting noncommutative D-branes and baryons in magnetic fields

We study supersymmetric intersecting configurations of D-branes with B-field backgrounds. Noncommutative D-brane or M-brane pairs can intersect supersymmetrically over a $(p\ensuremath{-}1)$-brane, as well as over $(p\ensuremath{-}2)$-brane-like ordinary branes. $d=10$ and $d=11$ supergravity solutions are obtained and the supersymmetry projection rule is examined. As an application we study a noncommutative D7-brane probe in a noncommutative D3-brane background, intersecting at a noncommutative plane, which describes Bogomol'nyi--Prasad--Sommerfield baryons of noncommutative gauge theory in the context of anti--de Sitter conformal field theory correspondence.

Closed strings interacting with non-commutative D-branes

Closed string dynamics in the presence of non-commutative D p-branes is investigated. In particular, we compute bulk closed string two-point scattering amplitudes; the bulk space-time geometries encoded in the amplitudes are shown to be consistent with the recently proposed background space-time geometries dual to non-commutative Yang–Mills theories. Three-point closed string absorption/emission amplitudes are obtained to show some features of non-commutative D p-branes, such as modified pole structures and exponential phase factors linearly proportional to the external closed string momentum.

Constrained quantization of open string in background B field and non-commutative D-brane

In a previous paper we provided a consistent quantization of open strings ending on D-branes with a background B field. In this letter, we show that the same result can also be obtained using the more traditional method of Dirac's constrained quantization. We also extend the discussion to the fermionic sector.