S-duality, noncritical open string and noncommutative gauge theory

doi:10.1016/S0370-2693(01)00029-6

Physics Letters B

Volume 499, Issues 1–2, 1 February 2001, Pages 215-222

https://doi.org/10.1016/S0370-2693(01)00029-6 Get rights and content

Abstract

We examine several aspects of S-duality of four-dimensional noncommutative gauge theory. By making duality transformation explicit, we find that S-dual of noncommutative gauge theory is defined in terms of dual noncommutative deformation. In ‘magnetic’ noncommutative U(1) gauge theory, we show that, in addition to gauge bosons, open D-strings constitute important low-energy excitations: noncritical open D-strings. Upon S-duality, they are mapped to noncritical open F-strings. We propose that, in dual ‘electric’ noncommutative U(1) gauge theory, the latters are identified with gauge-invariant, open Wilson lines. We argue that the open Wilson lines are chiral due to underlying parity noninvariance and obey spacetime uncertainty relation. We finally argue that, at high energy–momentum limit, the ‘electric’ noncommutative gauge theory describes correctly dynamics of delocalized multiple F-strings.

Introduction

Following the recent understanding concerning the equivalence between noncommutative and commutative gauge theories [1], an immediate question to ask is how the theory behaves under the S-duality interchanging the strong and weak coupling regimes. Indeed, this question has been addressed in several recent works [2], [3], [4], [5], [6]. (See also related works [7], [8], [9], [10], [11], [12].)

One expects an answer as simple as follows. The four-dimensional non-commutative theory describes the low-energy worldvolume dynamics of a D3-brane in the presence of a background of NS–NS two-form potential (Kalb–Ramond potential) B₂ but none others. Under the S-duality of the Type IIB string theory, the D3-brane is self-dual, while the NS–NS two-form potential B₂ is swapped with the R–R two-form potential C₂. Thus, after the S-duality, the dual theory appears to be the one describing the low-energy worldvolume dynamics of a D3-brane in the presence of a R–R two-form potential C₂, but none others. Since there is no NS–NS two-form potential present, noncommutative deformation via Seiberg–Witten map is not possible and the resulting theory ought to be a commutative theory. However, this is apparently not the answer one obtains [3], [4], [6]. Starting with a noncommutative U(1) gauge theory with coupling constant g and noncommutativity tensor θ and taking the standard duality transformation, one finds that the dual theory still remains a noncommutative U(1) gauge theory, but with coupling and noncommutativity parameters $g_{D} = 1 g and θ_{D} =−g^{2} θ ̃,$ where $θ ̃^{μν} = 12 ϵ^{μν}_{αβ} θ^{αβ}$ . Alternatively, as is done in [3], [4], one may utilize the gauge invariance of (F+B₂) to dial out the space–space noncommutativity and treat the theory as the standard gauge theory in a constant magnetic field. The S-duality would then turn this into a dual gauge theory, but now in a constant electric field, which is gauge equivalent to a theory with space–time noncommutativity. Actually, in the dual theory, it turns out to be impossible to take a field theory limit that the dual theory would be best described by a noncritical open string theory, whose tension is of the order of the noncommutativity scale.

The aim of this Letter is to understand the S-duality via the following routes: and, if possible, to reconcile these seemingly different results concerning the S-duality of the noncommutative gauge theory from various perspectives.

Section snippets

Naive S-duality

In this section, we shall be studying S-duality of noncommutative U(1) gauge theory.¹ The latter is defined by the following action: $S= 1 4g^{2} ∫d^{4} x Tr (F ★ F),$ where the noncommutative field strength is defined by $F = F + i {A_{μ}, A_{ν}}_{★}, F_{μν} =(∂_{μ} A_{ν} −∂_{ν} A_{μ})$ and Tr refers to “matrix notation” for spacetime index contractions, for instance, $Tr (FF)$ means $F_{μ}^{ν} F_{ν}^{μ}$ . Seiberg and Witten

Closer look — noncritical open string

We have seen, in the previous section, that the naive S-dual of a noncommutative gauge theory is again a noncommutative gauge theory, but with the noncommutativity parameter $θ_{D} =−g^{2} θ ̃$ . Thus, if the original theory were defined with ‘magnetic’ noncommutativity, the dual theory would have ‘electric’ noncommutativity and vice versa. We also note that, in performing the S-duality map, as we have expanded the original and the dual actions as in , , respectively, the result would be valid only for $θ F ⪡1$

Yet another look — strong noncommutativity limit

We would like to discuss yet another piece of physics associated with the S-duality, Eq. (1). In the previous section, we have seen that, due to the dual electric field background, the open F-string is oriented predominantly along the x⁰−x¹ directions. See Eq. (14). There, we have also argued that the open string is macroscopically stretched. According to Eq. (13), the open string is made out of the dual gauge field $B_{μ}$ as a sort of coherent state configuration. Thus, it ought to be possible to

Acknowledgements

We thank M.R. Douglas, J. Klusoň, U. Lindström and G. Moore for useful discussions. S.J.R. thanks warm hospitality of Marsaryk University, New High-Energy Theory Center at Rutgers University, and Theory Division at CERN during completion of the work.

References (20)

J.L.F. Barbon et al.
Phys. Lett. B
(2000)
J. Gomis et al.
Nucl. Phys. B
(2000)
N. Ishibashi et al.
Nucl. Phys. B
(2000)
S.R. Das et al.
Nucl. Phys. B
(2000)
D.J. Gross et al.
Observables of noncommutative gauge theories
N. Ishibashi
Nucl. Phys. B
(1999)
N. Ishibashi
Relation between commutative and noncommutative descriptions of D-branes
J.-X. Lu et al.
Nucl. Phys. B
(2000)
R. Gopakumar et al.
J. High Energy Phys.
(2000)
A. Schild
Phys. Rev. D
(1977)
N. Seiberg et al.
J. High Energy Phys.
(1999)
S.-J. Rey, R. von Unge, Unpublished work, December,...
N. Seiberg et al.
J. High Energy Phys.
(2000)
R. Gopakumar et al.
S-duality and noncommutative gauge theory

There are more references available in the full text version of this article.

Cited by (56)

Modified star-products beyond the large- B limit
2004, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Derivative corrections to the Wess–Zumino couplings of open-string effective actions are computed at all orders in derivatives, taking the open-string metric into account. This leads to a set of deformed star-products beyond the Seiberg–Witten limit, and allows to reinterpret the couplings in terms of a deformed integration prescription along a Wilson line in the non-commutative set-up. Moreover, the recursive definition of the star-products induces deformations of U(1) non-commutative Yang–Mills theory.
A note on duality symmetry in non-linear gauge theories
2003, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Softer hard scattering and noncommutative gauge-string duality
2003, Nuclear Physics B
We study exclusive scattering of ‘hadrons’ at high energy and fixed angle in (nonconformal) noncommutative gauge theories. Via gauge-string duality, we show that the noncommutativity renders the scattering soft, leading to exponential suppression. The result fits with the picture that, in noncommutative gauge theory, ‘fundamental partons’ consist of extremely soft constituents and hadrons are made out of open Wilson lines.
Duality symmetry and plane waves in non-commutative electrodynamics
2003, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
We generalise the electric–magnetic duality in standard Maxwell theory to its non-commutative version. Both spacespace and spacetime non-commutativity are necessary. The duality symmetry is then extended to a general class of non-commutative gauge theories that goes beyond non-commutative electrodynamics. As an application of this symmetry, plane wave solutions are analysed. Dispersion relations following from these solutions show that, in the presence of spacetime non-commutativity, non-commutative electrodynamics admits two waves with distinct polarisations propagating at different velocities in the same direction.
Quantum field theory on noncommutative spaces
2003, Physics Reports
A pedagogical and self-contained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the Weyl–Wigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative Yang–Mills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an in-depth study of the gauge group of noncommutative Yang–Mills theory. Some of the more mathematical ideas and techniques of noncommutative geometry are also briefly explained.
Anatomy of two-loop effective action in noncommulative field theories
2002, Nuclear Physics B
We compute, at two-loop order, one-particle-irreducible Green functions and effective action in noncommutative λ[Φ³]_★-theory for both planar (g=0, h=3) and nonplanar (g=1, h=1) contributions. We adopt worldline formulation of the Feynman diagrammatics so that relation to string theory diagrammatics is made transparent in the Seiberg–Witten limit. We argue that the resulting two-loop effective action is expressible via open Wilson lines: one-particle-irreducible effective action is generating functional of connected diagrams for interacting open Wilson lines.

View all citing articles on Scopus

^☆: Work supported in part by BK-21 Initiative in Physics (SNU Team-2), KRF International Collaboration Grant, KOSEF Interdisciplinary Research Grant 98-07-02-07-01-5, and KOSEF Leading Scientist Program 2000-1-11200-001-1. The work of RvU was supported by the Czech Ministry of Education under contract No. 144310006 and by the Swedish Institute.

View full text

S-duality, noncritical open string and noncommutative gauge theory☆

Abstract

Introduction

Section snippets

Naive S-duality

Closer look — noncritical open string

Yet another look — strong noncommutativity limit

Acknowledgements

Phys. Lett. B

Nucl. Phys. B

Nucl. Phys. B

Nucl. Phys. B

Nucl. Phys. B

Nucl. Phys. B

J. High Energy Phys.

Phys. Rev. D

J. High Energy Phys.

J. High Energy Phys.

S-duality and noncommutative gauge theory