Noncommutativity and Holographic Entanglement Entropy
NNoncommutativity and Holographic Entanglement Entropy
Tuo Jia ∗ and Zhaojie Xu † Department of Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA
Abstract
In this paper we study the holographic entanglement entropy in a large N noncommutative gaugefield theory with two θ parameters by Ryu-Takayanagi prescription (RT-formula). We discusswhat contributions the presence of noncommutativity will make to the entanglement entropy intwo different circumstances: 1) a rectangular strip and 2) a cylinder. Since we want to investigatethe entanglement entropy only, we will not be discussing the finite temperature case in which theentropy calculated by the area of minimal surface will largely be the thermal part rather than theentanglement part. We find that divergence of the holographic entanglement entropy will be worsein the presence of noncommutativity. In future study, we are going to explore the concrete wayof computing holographic entanglement entropy in higher dimensional field theory and investigatemore about the entanglement entropy in the presence of black holes/black branes. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] D ec . INTRODUCTION Over the past two decades, tremendous progress has been made in AdS/CFT dualityand noncommutative geometry in string theory [1, 2]. Noncommutativity has been widelyexplored in various cases since 40’s in the last century [3–8]. In noncommutative geoemtryand nonocommutative field theory, the coordinates of spacetime do not commute, which hasmany interesting properties. Meanwhile, the duality between supergravity in anti-de Sitterspace and conformal field theory that lives on its boundary (AdS/CFT duality) has madegreat success [9–11], and large changed our way of thinking about the nature. AdS/CFThas become a powerful tool for us to solve problems in quantum chromodynamics (QCD)[12–16], condensed matter physics [17–20], etc. Noncommutative geometry shows up whenwe consider D-branes with B-field [2, 5, 21–23], and has many useful applications.Entanglement entropy, being a fascinating topic as always, has been enhanced to a holo-graphic version by Ryu and Takayanagi [24]. The famous Ryu-Takayanagi formula wasintroduced as a formula of computing entanglement entropy of a system by computing thearea of the minimal surfaces, which agrees the area law as in the calculation of the thermalentropy of black holes. However, a more concrete formula should be introduced when wedeal with the entanglement entropy of higher dimensional field theory, we will leave this forfuture study.What would be interesting is to consider combining holographic entanglement entropyand noncommutative field theory together. In [25], the holographic entanglement entropy ofa large N -strongly coupled noncommutative field theory with one θ parameter has been wellstudied. In our work, we will consider a noncommutative gauge theory defined on R θ × R θ ,which has two θ parameters. This is the case when we turn on two components of B-field, weare going to compute the holographic entanglement entropy of the noncommutative gaugetheory by Ryu-Takayanagi formula in two different circumstances: 1) a rectangular stripand 2) a cylinder. We will discuss what the additional noncommutativity contribute tothe entanglement entropy by comparing our results with the results by Fishler, Kundu andKundu [25]. In this article, we will only discuss the spacetime without black holes, becausewhen we introduce finite temperature by the presence of black holes, the entropy given bythe RT-formula will not be the entanglement entropy only, the thermal entropy will alsocontribute to our results from the area of the minimal surfaces. Consequently, we will not2iscuss the finite temperature cases and we conjecture that the presence of black holes willnot change the holographic entanglement entropy.In section II, we will review the holographic entanglement entropy of a large N stronglycoupled noncommutative gauge field theory in an infinite rectangular strip when there isonly one θ parameter, i.e., we turn on only one component of B-field. In section III, weturn on another component of B-field, and the resultant spacetime geometry will have anadditional θ paremeter, the metric has an additional factor of h ( u ). In order to computethe holographic entanglement entropy, we use RT-formula the compute the area of minimalsurface. The entropy is simply given by S RT = Area ( γ )4 G , (1)where
Area ( γ ) is the minimal value of an area functional and G is Newton constant. Wefind that additional noncommutativity will make divergence of the holographic entanglemententropy worse in both cases: 1) an infinite noncommutative rectangular strip and 2) aninfinite noncommutative cylinder. In section IV, we draw our conclusion and discuss somedirections for future explorations. II. R × R θ In this section, we review some results from [25], we study a large N -strongly couplednoncommutative field gauge theory with one θ parameter, i.e., on R θ × R . The noncom-mutative parameter will only be on R θ plane. Moyal algebra shows up in R θ , which is[ x , x ] = iθ. (2)We can describe this noncommutative gauge field theory by holographic dictionary, and thenwe obtain the following metric in string frame [1, 2] ds = R (cid:20) u f ( u ) dx + u dx + u h ( u )( dx + dx ) + du f ( u ) u + d Ω (cid:21) , (3)where f ( u ) = 1 − (cid:18) u H u (cid:19) , h ( u ) − = 1 + a u . (4)Here f ( u ) denotes the presence of a black hole. h ( u ), which is caused by B-field, representsthe existence of noncommutativity. 3n the other hand, we can compute the holographic entanglement entropy of a theoryby RT-formula [24] S RT = Area ( γ )4 G , (5)where
Area ( γ ) is the minimal value of an area functional, G is the Newton constant. Wecan see that RT-formula has a lot in common with black hole entropy formula. Next, we willreview the holographic entanglement entropy with the region being an infinite rectangularstrip, we are going to omit the discussion of the noncommutative cylinder with one θ param-eter and directly study the noncommutative cylinder with two θ parameters the compareour results with results from [25]. Noncommutative rectangular strip
We consider an infinite rectangular strip, which is parameterized by X = x ∈ [ − l , l , x , x ∈ [ − L , L L → ∞ . The area of the surface in the bulk is given by A = L R g s (cid:90) duu (cid:115) ( X (cid:48) + 1 h ( u ) f ( u ) u ) . (7)The Lagrangian is a function of X , then the corresponding integrals of motion would giveus l u c (cid:90) u b u c duu (cid:113) (1 − u c u ) hf . (8)Plug the equation of constant motion back in the area functional, we get A = L R g s (cid:90) udu (cid:113) (1 − u c u ) h ( u ) f ( u ) . (9) III. R θ × R θ In this section, we are going to turn on another component of B-field, i.e., now there aretwo θ parameters in the geometry and two functions h ( u ) and h ( u ) in the metric. Themetric in the bulk is given by ds = R (cid:20) u h ( u )( dx + dx ) + u h ( u )( dx + dx ) + du u (cid:21) , (10)4 c la au H (cid:61) FIG. 1: Variation of l with u c for U shaped profile. We set au H = 1 and find that l min ∼ . a which is the result of the spacetime noncommutativity and finite temperature. where h − i = 1 + a i u . (11)We make the convention a = b, a = a . A. Noncommutative rectangular strip
The strip is parameterized by X = x ∈ [ − l , l , x , x ∈ [ − L , L L → ∞ . The area of the surface in the bulk is given by A = L R g s (cid:90) duu (cid:114) ( X (cid:48) + 1 h u ) 1 h . (13)The constant of motion would give us l u c h ( u c ) (cid:90) u b u c duu (cid:115) h ( u )(1 − u c u ) h ( u ) , (14)where u c represents the point closest to the extrmal surface. Finally, we obtain the areafunctional, which is A = 2 L R g s (cid:90) u b u c du u (cid:115) h ( u ) h ( u ) (cid:18) u c h ( u )( u − u c ) h ( u c ) (cid:19) . (15)5 c la b (cid:61) (cid:144) a (cid:61) (cid:144) a (cid:61) (cid:144) a (cid:61) (cid:144) a (cid:61) FIG. 2: Variation of l with u c for U shaped profile. For b=0 this reduce to the result of [25]. For b (cid:54) = 0 the curves converge at large u c The divergence part of holographic entanglement entropy given by A div = 2 L R g s (cid:20) a b u b a b + b a ) u b (cid:21) . (16)As a comparison, A div with one h ( u ) is A (cid:48) div = 2 L R g s (cid:20) u b a ln ( au b ) (cid:21) . (17)We find that ∆ A div ≡ A div − A (cid:48) div > . (18)Therefore, the divergence is even worsen by the additional noncommutativity. B. Noncommutative cylinder
Now let’s consider an infinite noncommutative cylinder A = { ( x , x ) | x + x ≤ r } , x ∈ [ − L/ , L/ , L → ∞ , (19)where A is the circle that we construct in the { x , x } -plane.We adopt the polar coordinates dx + dx = dρ + ρ dθ . The area functional is thusgiven by A = 2 πLR g s (cid:90) duu ρ ( u ) (cid:115)(cid:18) ρ (cid:48) ( u ) + 1 u h ( u ) (cid:19) h ( u ) . (20)6he equation of motion is ddu (cid:18) u ρ ( u ) ρ (cid:48) ( u ) L (cid:19) = u L , (21)where L = (cid:115)(cid:18) ρ (cid:48) ( u ) + 1 u h ( u ) (cid:19) h ( u ) . (22)For arbitrary h ( u ) and h ( u ), the equation of motion is nearly impossible to solve. There-fore, we adopt the approximation a u (cid:29) b u (cid:28)
1. This seemingly unreasonableassumption that we make b very small turns out to be well be reasonable, and it has some-thing to do with electromagnetic duality where a is the magnetic part and b is the electricpart. Electromagnetic duality (S-duality) requires that when one of them is very large, theother one has to be very small. However, it needs more concrete derivations, we leave it forfuture study.In our approximation, we can compute the solution ρ ( u ) in a series form. We are goingto use the ansatz ρ ( u ) = ku (cid:18) c u + c u + c u + . . . (cid:19) , (23)where k, c , c , c , . . . are constants. We find out that in the approximation b u (cid:28) b willnot contribute to our series solution ρ ( u ), the solution ρ ( u ) is the same as the case whichwe only have one B field! Therefore, the computation has been significantly simplified. Thesolution is ρ ( u ) = a √ u (cid:20) a u − a u + O (cid:18) a u (cid:19)(cid:21) , (24)which dose not contain any contributions from b . In spite of that, we could try to solve theequation of motion with large b , we will find that the solution can not exist in any seriesform. The only way to solve the equation of motion is to assume b to be very small, andthis agrees with electromagnetic duality as well.We now insert the solution ρ ( u ) above to the area functional, we obtain the entanglemententropy S ( a, b ) = N L (cid:20) a b u b
27 + (2 b / a ) u b
15 + 13 (cid:18) − b a (cid:19) u b (cid:21) . (25)We compare it with the entropy when we only turn on one component of B field S ( a ) = N L (cid:20) a u b
15 + u b (cid:21) , (26)7e find out that the divergence of the entropy has only been worsen∆ S ≡ S ( a, b ) − S ( a ) = N Lb (cid:18) a u b
27 + u b − u b a (cid:19) > . (27)This indicates that when we turn on another component of B field, the entanglement entropywill only be more divergent.As we can see from above, in both cases, ∆ S >
0, which means additional noncommu-tativity will only worsen the divergence of the holographic entanglement entropy. This is avery interesting fact.
IV. CONCLUSION
In this article, we first review the holographic entanglement entropy in a large N stronglycoupled noncommutative gauge field theory with only one θ parameter, i.e., we have onlyone h ( u ) in the metric. Then we generalize the noncommutative field theory to the two θ parameter case. We have two h ( u )’s in this case. By Ryu-Takayagagi formula, we computethe minimal surface for both regions: 1) an infinite noncommutative rectangular strip and2) an infinite noncommutative cylinder. We find that in both cases, the divergence of theholographic entanglement entropy is worse in the presence of additional noncommutativity.Another interesting fact in computing the holographic entanglement entropy in an infinitecylinder is that, we have assumed b to be very small b u (cid:28) , a u (cid:29) . (28)This assumption agrees with electromagnetic duality with a being the magnetic part and b being the electric part. Electromagnetic duality (S-duality) makes this approximationnatural and reasonable. However, the solid relation between electromagnetic duality andthe holographic entanglement entropy in a noncommutative gauge field theory still needsmore investigations in the future.We haven’t discussed the finite temperature noncommutative gauge field theory. Thepresence of a black hole will introduce a factor of f ( u ) in the metric and surely will changethe area of minimal surface, but the change of the entropy will largely be thermal part ratherthan entanglement part. We conjecture that the presence of black holes will not change theholographic entanglement entropy, it will contribute to the thermal entropy. For a detaileddiscussion, we will leave for future study. 8ast but not least, we still need a more concrete formulation of computing (holographic)entanglement entropy in higher dimensional field theory. This requires fundamentally newideas in both physics and mathematics. Despite that, we have shown that the additionalnoncommutativity will only worsen the divergence of the holographic entanglement entropyin a noncommutative field theory. We leave the above issues for future study. ACKNOWLEDGMENTS
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