VOLUME 84, NUMBER 10 PHYSICAL REVIEW LETTERS 6MARCH 2000 Holography and Noncommutative Yang-Mills Theory ☎ Miao Li✁ ✂ * and Yong-Shi Wu † ✁ Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China ✄ Department of Physics, National Taiwan University, Taipei 10764, Taiwan ✆ Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 29 September 1999) In this Letter a recently proposed gravity dual of noncommutative Yang-Mills theory is derived from the relations between closed string moduli and open string moduli recently suggested by Seiberg and Witten. The only new input one needs is a simple form of the running string tension as a function of energy. This derivation provides convincing evidence that string theory integrates with the holographical principle and demonstrates a direct link between noncommutative Yang-Mills theory and holography. PACS numbers: 11.25.Mj, 11.15.–q By now it becomes clear that any consistent theory that One might think that NCYM is relevant only in unifies quantum mechanics and general relativity requires nongeneric situations, such as the above-mentioned de- dramatically new ideas beyond what we have been famil- coupling limit in which the ✝ -background is brought to iar with. Two such ideas, the holographic principle [1] infinity, so it cannot shed much light on deep issues such and noncommutative geometry [2], have recently attracted as holography and quantum nature of spacetime. In this increasing attention in the string theory community. The Letter we show that this is not true. With an observation holographic principle [1], originally motivated by the area made on a direct link between the NCYM and its gravity dependence of black hole entropy, asserts that all infor- dual, we will try to argue for the opposite: Switching on mation on a quantum theory of gravity in a volume is en- a ✝ -background allows one to probe the nature of holog- coded in the boundary surface of the volume. Though this raphy with NCYM and will probably lead to uncovering principle seems to conflict with our intuition from local more, previously unsuspected links between a large-✞ quantum field theory, string theory as a promising candi- theory and its closed string dual. date of quantum gravity is believed [3] to integrate with One of the central observations in [11] is that the natural it. Indeed the Maldacena conjecture [4], motivated by the moduli to use in open string theory with ends on a set of ✞ D-brane models of black hole in string theory, is nothing coincident D-branes in the presence of a constant ✝ field but an embodiment [5] of the holographic principle: There are different from those defined for closed strings. The is an equivalence or correspondence between supergravity effective action for the system is more elegantly written if ✟ (or closed string theory) on an anti–de Sitter space (AdS), one uses the open string metric ✠✡ and an antisymmetric ✠✡ ☛ ☞ say of five dimensions, and a supersymmetric Yang-Mills tensor , and their relation to the closed string metric ✠✡ ✝ gauge theory on its four-dimensional boundary. and the antisymmetric tensor field ✠✡ is In a parallel development, Yang-Mills theory on a space ✑ ✂ ✓✁ ✝ ✟ ✝ ✌ ☞ ✍ ✎ ✏ ✒ ✎ ☞ ✒ ✔ ✠✡ ✠✡ with noncommutative coordinates [6], which we will call ✠✡ ✚ ✗ noncommutative Yang-Mills theory (NCYM), has been ✠✡ (1) ✘ ✑ ✠✡ ✔ ✌ ✕✖ ✏ found to arise naturally in string theory, first in the multi-D- ☛ ✑ ✙ ✝ ☞ ✏ brane description [7], then in matrix theory [8,9] or string ✛ theory [10] compactifications with nonvanishing antisym- where the subscript ✜ indicates the antisymmetric part. metric tensor background, and most recently in a special Our normalization of the ✝ field differs from that in [11] by limit that decouples closed string contributions from the a factor of ✕✖ . Seiberg and Witten noted that in the limit open string description for coincident D-branes with a con- ✑ ✢ ✣ ✢ ✣ ✝ ✏ ☞ ✠✡ and ✠✡ (assuming is nondegenerate), it ✠✡ stant rank-2 antisymmetric tensor ✝ -background (see [11], ✟ ✝ ☛ ✠✡ is possible to keep ✠✡ and fixed with a fixed . and references therein). Right now it is the last case that The tree level effective action surviving this limit is the is the focus of attention. A rather thorough discussion of noncommutative Yang-Mills action, with a star product of the aspects of NCYM from the open string versus closed ✠✡ functions defined using ☛ : string perspectives has been given in the work of Seiberg ✮ ✭ ✪ ✫ ✂ ✠ ★ ✩ ✬ ✬ ✪ ✫ ✤ ✦ ✯ ✰ ✴ ✤ ✥ ✦ ✧ ✎ ✒ ☞ ✎ ✒ ✒ ✌ and Witten [11] which, among other things, also clari- ☞ ✎ ✱ ✲✳ (2) fies several puzzles previously encountered in NCYM, in- ✠ ✡ ✦ ✥ ✦ cluding the one raised by one of us [12]. Moreover, the This is equivalent to the noncommutativity: ✍ ✡ ✠ ✠✡ ✦ ✥ ✦ ✵ ☛ supposed-to-be gravity duals of NCYM’s in the decou- ✌ [15]. ✟ pling limit, which generalize the usual Maldacena conjec- The open string coupling constant ✶ , proportional ✂ ✝ ture without -background, were also constructed [13,14]. ☞ to the Yang-Mills coupling ✷✸ , is also different from the ✹ ✹ 2084 0031-9007✹ 00 84(10) 2084(4)$15.00 © 2000 The American Physical Society VOLUME 84, NUMBER 10 PHYSICAL REVIEW LETTERS 6MARCH 2000 ✶ ✠ closed string coupling constant. The relation between the theory ✂ AdS CFT correspondence [4]. Note that there is ✍ ✍ ✘ two is an overall factor ✗ in (4) for the 4D geometry along ✡ ☛✌ ✍ D3-branes. This redshift factor can be interpreted as the det ✄ ✄ ✁ ✝ ✎ ☞ ☞ (3) effective string tension ✞ ✝ ✆ ☎ ✟ ✠ det✂ ✢ ✞ ✎ ✁ It is easy to see that in the “double scaling limit” when ☎ (7) ✹✺✺ ✍ ✍ ✗ ✘ ✞ ✏ ✑ ✝ ✏ ✑ ☎ , , keeping fixed the open string coupling ✞ ☎ ✄ ✝ Therefore ✹✺✺ also approaches zero in the UV limit. The ☞ ☞ (say for D3-branes), must be taken to zero too. ✝ Thus closed strings decouple from the open string sec- manner in which it approaches zero compared to ✷✸ agrees tor described by the NCYM. We will see that this per- with the limit taken in [11]. Note that here we differ from ✞ fectly matches with the closed string dual description of the philosophy of [14] in which ☎ itself is taken to zero, the NCYM. while we have set it to be 1. Using the conventions of [14], the Neveu-Schwarz (NS) Now we are ready to derive the NS fields in the closed fields in the gravity dual proposed for D3-branes with a string dual (4) by applying formulas (1) and (3). The way ✟ in which Seiberg and Witten derived these formulas is valid constant ✍✒ are [13,14] if we treat strings as effective strings at a fixed energy scale ✙ ✥ ✢ when loop effects are included. Thus we can take these ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✁ ✓ ✔ ✗ ✘ ✂ ✚✓✛ ✆ ✓ ✜ ✠ ✆ ✂✓✜ ✆ ✓ ✜ ✠ ☞ ☛ ✍ ✒ ✕✖ ✤ ✤ formulas as giving relations among the open string mod- ✢ ✆ ✣ ✘ ✳ ✁ ✘ ✍ uli and the closed string moduli at a fixed cutoff . ✓✘ ✍ ✍ ✍ ✮ ✆ ✗ ✓ ✦ ★ ✗ ✆ Bigatti and Susskind argued [17] that in the large- limit, ✧ ✍ ✘ the effective action of NCYM can be obtained by replac- ✤ ✤ (4) ✻ ✁ ✘ ✣ ing the usual product in the effective action of ✁ ✟ ✟ ★ ✍✒ ✤ ✤ ✬ SYM by the star product. This, in particular, implies ✢ ✆ ✣ ✘ that there is no renormalization for the open string metric, ✢ ✍✪ ✍ ✄ ✁ ✝ ✩ ★ the Yang-Mills coupling constant ☞ , and the noncommu- ✤ ✤ ✞ ✷✸ ✞ ✢ ✆ ✣ ✘ ✼ ☎ ☎ tative moduli . Now with replaced by ✹✺✺ at a fixed ✞ ✁ ✢ ✟ where we have set ☎ . The constant is the value of energy scale, the closed string moduli are renormalized. ✁ ✝ ✟ ✘ ✫ ✍✒ ★ ✜ ✠ at the boundary , and the constant is the closed ✂ ✜ ✒ Because of the rotational symmetry on the ✍ plane, ✤ ✁ ✑ ✁ ✬✭ ✝ ✮ ✗ string coupling in the infrared ✘ . Here , we introduce ansatz and the parameter ✣ is given by ✝ ✁ ✁ ✎ ✽ ✂✘ ✠✾ ★ ✟ ✿ ✂✘✠❀ ✷✸ ✷✸ ✷✸ ✷✸ (8) ✍ ✍ ✁ ✎ ✟ ✣ ✗ (5) The first equation in (1) yields ✍ ❁☛ ✤ ✤ ✤ ✰ ✱ ✁ ✶ ✾ ✾ ✂✂✂ ✽ ✆ ✿ ✽ ✂✗ ✘ ✠ ✠✠✠★ ✯ ✷✸ In addition to the usual induced by the presence of ✷✸ (9) ✍ ✰ ✱ D3-branes, there is also an induced ✯ field. Its presence or ✟ is quite natural for D3-branes. Recall that a constant ✍✒ ✍ ✿ on the branes can be replaced by a constant magnetic field. ✍ ✁ ✎ ✽ ✆ ✽ (10) ✤ ✤ ✗ ✘ Performing ✲ -duality transformation, this field becomes ✁ ✴ ✳ The second equation in (1) leads to ☛ the electric field ✵ . This electric field is defined ✡ ❁ ☛ ✍ ✍ ✯ ☛ using the dual quanta, thus it is equivalent to a ✵ . The ✢ ✿ ✿ ✣ ✍ ✭ ✁ ✭ ❂ ✽ ✆ ❀ ❀ ★ ❂ ✷✸ ✷✸ ✯ ✘ (11) ☛ -dependent ✵ is given in [14]: ✍ ✍ ✤ ✤ ✍ ✍ ✍ ✗ ✘ ✗ ✘ ✗ ✘ ✗ ✍ ✍ ✷✸ ✣ ✗ ✤ where we used the fact that ✼ is not renormalized and is ✁ ✎ ✯ ✘ ☛ ✵ (6) ✍ ✍ ✝ ✭ ✶ ✁ ✶ ✂ ❂ ✟ ✠❀ ✟ ✗ ✣ given by ✷✸ , and . This second equa- It is natural to interpret the fields appearing in the gravity tion is just ✡ ✍ dual (4) as closed string moduli. Note that apart from ✿ ✍ ✍ ✤ ✍ ✎ ✁ ✿ ✣ ✗ ✘ ✽ ✆ ✤ ✤ ✍ (12) ✢ ✶ ✢ ✂ ✆ ✣ ✘ ✠ ✤ ✤ the ✘ factor, there is an additional factor ✗ ✘ ✂ ✜ ★ ✜ ✠ ✒ in the closed string metric on the plane ✍ . Thus ✍ ✍ ✤ ✁ ✣ ✗ ✘ ✽ Combined with Eq. (10) we have ✿ , and sub- ✂✛ ★ ✜ ✠ if one is to hold the geometry on the plane ☛ fixed, stituting this into Eq. (10) we find ✂ ✜ ★ ✜ ✠ ✒ then the geometry on the plane ✍ shrinks when the ✢ boundary is approached. By the UV✶ IR relation [16], this ✁ ✎ ✽ ✂✘ ✠ (13) ✤ ✤ ✢ ✣ ✘ means that the closed string metric shrinks to zero in the ✆ UV limit from the open string perspective. The following This is precisely what appeared in (4) which is obtained as is our central observation. We identify the UV limit as a solution to classical equations of motion in closed string ✍ ✍ ✤ ✝ ✁ ✿ ✣ ✗ ✘ ✽ the double scaling limit of [11], thus that ✷✸ shrinks in theory. With