UV/IR mixing, noncommutative instabilites and closed strings

Fortschr. Phys. 52, No. 6 – 7, 612 – 617 (2004) / DOI 10.1002/prop.200310152

UV/IR mixing, noncommutative instabilitesand closed strings

E. Lopez∗

Instituto de Fısica Teorica, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Received 15 December 2003, accepted 29 February 2004Published online 14 May 2004

The leading UV/IR mixing effects on noncommutative field theories modify the dispersion relations and,depending on their sign, may cause the appearance of unstable modes. For noncommutative gauge theorieson D-branes, this phenomenon is able to capture important information about the closed string spectrum ofthe parent string theory. We analyse noncommutative D-branes on nonsupersymmetric orbifolds and twistedcircle backgrounds. We find that the sign of the leading UV/IR mixing effects is governed by the mass gapbetween the lowest modes in the NSNS and RR closed string towers. For noncommutative D3-branes atorbifolds we obtain a stronger result: a one to one correspondence between noncommutative instabilitiesand closed string tachyons.

c© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Several arguments suggest the relevance of noncommutative geometry for the description of space-timeat short distances. It is thus important to study the implications of a noncommutative generalisation ofspace-time on the dynamics of field theories. In order to explore this issue, most works have consideredthe simple deformation of Rn

[xµ, xν ] = iθµν , (1)

where θµν is an antisymmetric matrix with constant entries. The reason to focus in (1) is that explicitcalculations can be done for field theories living on such spaces. The algebra (1) implies uncertaintyrelations in space-time and therefore the non decoupling of ultraviolet and infrared degrees of freedom.This mixing between UV and IR has drastic consequences in the nonplanar sector of field theories: It wasshown in [1] that it leads to the appearance of new infrared divergences.

The relations (1) are naturally realized on the world-volume of D-branes in a constant B-field background,where θµν ∼ 1/Bµν . Nonplanar field theory diagrams can then be related to nonplanar string diagrams.This suggests an important role of closed strings in the understanding of UV/IR mixing [1, 2]. At a morefundamental level, we could expect that if (1) is to capture relevant aspects of quantum gravity its effectsshould know about the closed string sector (although the decoupling of closed strings does not fail in thenoncommutative field theory limit [3]).

The leading IR divergences induced by UV/IR mixing, modify the dispersion relations of noncommuta-tive field theories as follows

E2 = �p2 − cg2

p2 , (2)

∗ E-mail: [email protected]


Fortschr. Phys. 52, No. 6 – 7 (2004) / www.fp-journal.org 613

where g is the coupling constant, pµ = θµνpν and c is a model dependent constant. For non-commutativegauge theories c ∼ Nb − Nf , with Nb and Nf the number of bosonic and fermionic degrees of freedomin the adjoint representation. This effect is absent in supersymmetric theories since then c = 0 [4]. WhenNb > Nf , (2) turns the low momentum modes unstable [5, 6].

The appearance of instabilities linked to the absence of supersymmetry is reminiscent of a similarphenomenon in string theory. Indeed, modular invariance relates the UV and IR contributions to the toruspartition function. As a consequence the absence of supersymmetry generically implies the presence oftachyons in the closed string spectrum [7]. It is then natural to wonder if there can be a relation betweennoncommutative instabilities and closed string tachyons for those noncommutative theories that can beembedded in string theory [8]. This question was answered affirmatively in [9] for gauge theories associatedto noncommutative D-branes in nonsupersymmetric orbifolds, and in [10] for D-branes in twisted circlebackgrounds [11]. We will present here a resume of those results.

2 Open Wilson lines versus closed strings

In this section we will consider noncommutative D3-branes at C3/ZN orbifolds. A C3/ZN orbifold actswith twist (a1, a2, a3, a4)/N and (b1, b2, b3)/N on SO(6) spinors and vectors respectively. The integersaα are subject to

∑aα = 0(modN) and are related to bl by b1 = a2 +a3, b2 = a1 +a3, b3 = a1 +a2. The

gauge theory on n D3-branes placed at the fixed point of the orbifold has gauge group G = ⊗Ni=1U(ni),

where∑

ni = n. The coupling constants of all gauge group factors coincide. The matter content is givenby ( i, i+aα) Weyl fermions and ( i, i+bl

) complex scalars. Both the gauge theory on the D3-branesand the closed string spectrum are supersymmetric if at least one aα = 0(modN).

Turning on a B-field background on two of the spatial directions of the D3-branes will render the world-volume noncommutative, i.e. [x1, x2] = iθ. In the generic nonsupersymmetric case the noncommutativegauge theory presents a complicated pattern of UV/IR mixing effects. The leading infrared contribution tothe nonplanar polarization tensor affects only U(1)i ∈ U(ni) degrees of freedom1 and is non-diagonal ingroup labels indices. However the linear combinations B

(k)µ = 1√

N

∑e2πi jk

N TrA(j)µ diagonalize it, with

the result [9]

Πµνk = εk

g2

π2

pµpν

p4 . (3)

The quantities εk, which play an analogous role to c in (2), have a simple expression in terms of the orbifoldtwist parameters

εk = 2

(1 −

4∑α=1

cos2πaαk

N+

3∑l=1

cos2πblk

N

). (4)

Remarkably, these quantities can be rewritten in terms of the masses of four low lying closed stringmodes in the NSNS kth twisted sector of the orbifold background

εk = −164∏

α=1

sinπα′m2

α

2. (5)

These masses satisfy the following properties: i) among them is the lowest mode in the NSNS kth twistedsector; ii) −1 ≤ α′m2

α < 2; iii) only one of the m2α can be negative (see [9] for details). This implies

a direct relation between the sign of εk and the presence or not of tachyons in the kth twisted sector of

1 The theories we are considering have in general mixed anomalies. However when noncommutativity is switched on the anomalyonly affects U(1) modes with p = 0 [12], while for p �= 0 the anomaly vanishes [13]. We will always consider the latter case.


614 E. Lopez: UV/IR mixing, noncommutative instabilites

the parent string theory: εk > 0 if the kth twisted sector contains tachyons; εk = 0 only if the kth twistis supersymmetry-preserving; εk < 0 when the twisted sector is nonsupersymmetric but does not includetachyons.

This remarkable relation between noncommutative instabilities and closed string tachyons calls for aderivation purely in terms of closed strings of the leading UV/IR mixing effects. As a first step let us comeback to the origin of UV/IR mixing, which are the uncertainty relations derived from (1): ∆x1∆x2 ≥ θ inour case. It would seem that the natural degrees of freedom of noncommutative theories must satisfy theprevious uncertainty relations. Contrary to this expectation, noncommutative field theories are formulatedusing local fields. This puzzle is solved by studying the effective action of the theory. It has been shownboth for scalar [14] and gauge theories [8, 15, 16] that the 1-loop nonplanar effective action, includingcontributions from all the N-point functions, can be rewritten in terms of straight open Wilson line operators[17]. Straight openWilson line operators exhibit the desired behaviour since their momentum, p, is correlatedwith their transversal extent p,

W (p) = Tr∫

d4x P∗(ei g

∫ 10 dσ pµAµ(x+p σ)

)∗ eipx . (6)

For the orbifold gauge theories above, the gauge invariant piece of the 1-loop effective action containing(3) has the simple expression [9]

∆S =1

2π2

N−1∑k=0

εk

∫d4p

(2π)41p4 W (N−k)(p) W (k)(−p) , (7)

where W (k) = 1√N

∑Nj=1 e2πi jk

N W (j), and W (j) denotes (6) with the vector field belonging to the jth

gauge group factor.Several facts suggest the interpretation of (7) in terms of a closed string exchange between D-branes.

This action seems to know about the different closed string twisted sectors since the quantities εk measurean independent property of each sector. Closed string modes in the kth twisted sector couple naturally tolinear combinations of field theory operators such as those that define Bk

µ and W k [18]. Moreover, it hasbeen shown in [19] that closed strings couple to straight Wilson line operators on noncommutative D-branes.In order to complete the argument, we only need to show that the term 1/p4 in (7) can be related to a closedstring propagator.

In the absence of B-field, scalar closed string modes couple to the brane tension TD3 ∼ Tr1/α′2 atleading order in α′. When B �= 0 the trivial field theory operator Tr1 gets promoted to the open Wilsonline operator (6). Hence, at leading order in α′, the contribution to the D3-brane effective action from theemission, propagation and posterior absorption of a scalar closed string mode ϕ in the kth twisted sector is

∆S ∼∫

d4p

(2π)4W (N−k)(p)W (k)(−p) f(p, u) , (8)

where the function f(p, u) denotes the closed string propagator

f(p, u) = α′− d+22

∫ddv

(2π)d

eivu

v2 + p2 + (2πα′mϕ)2. (9)

d is the number of dimensions transverse to the D-brane where the twisted field ϕ can propagate: d =0, 2, 4, 6depending on the particularC3/ZN orbifold. We have definedv = 2πα′p⊥, withp⊥ the transversalmomentum to the D-brane. u has been introduced in order to have a well defined closed string propagator; ithas the interpretation of an infrared regulator from the point of view of the field theory. In the denominatorwe have used the relation between open (η) and closed (g) string metrics g−1 = η−1−θηθ/(2πα′)2 [20],and discarded terms suppressed by two α′ powers. mϕ is the mass of ϕ. The factor of α′ in front of theintegral can be obtained just by dimensional analysis.



We want to extract from (8) a contribution to the noncommutative field theory effective action. We needto take the limit α′ → 0. In this limit f diverges due to the negative α′ power in front of the integral.However we should notice the following. If we had done the α′ → 0 limit of the standard annulus diagramassociated to each nonplanar N-point function, we would had obtained a result O(α′0) whose leading IRcontribution should reproduce the corresponding term in (7). The question is then whether we can directlydefine an O(α′0) contribution from (8), regarding as artifacts other α′ powers. We observe that p acts asan infrared regulator for the integral in (9). Therefore, for p �= 0, we can expand the integral in powers of(2πα′mϕ)2 ∼ α′ to the desired order. At O(α′0) we obtain

f(p, u)|O(α′0) ∼∫

ddv

(2π)d

eivu

(v2 + p2)d2 +2

∼u→01p4 . (10)

After removing the field theory infrared regulator u we recover 1/p4, independently of d. The possibilityto relate 1/p4 to a closed string propagator does not mean that the decoupling of closed strings fails in thenoncommutative field theory limit, since the IR singularities do not have kinetic part. Hence they do notforce the introduction of additional degrees of freedom.

In (8) we have considered the exchange of a single closed string mode. Any closed string mode ableto couple to W k will contribute the same 1/p4 up to a numerical factor depending on mϕ and the diskamplitude of ϕ with boundary conditions on the brane. The coefficients εk that appear in (7) will be thusa collective effect of the closed string towers. However, we know from (5) that εk is determined by afinite set of low lying string modes and, moreover, its sign just depends on the presence of tachyons. Toreconcile these two facts, notice that modes in the NSNS and RR sectors contribute with opposite sign tothe exchange between D-branes. In addition, when the theory is supersymmetric we have εk = 0, implyingthat the contribution from both towers must cancel. This suggests to consider εk as a measurement of themisalignment between the NSNS and RR towers. Tachyons can only belong to twisted NSNS sectors, whilethe lightest RR twisted modes are always massless in orbifold backgrounds. It seems then consistent thatthe presence of tachyons is linked to the sign of the mentioned misalignment inside each twisted sector.

3 Generalisation: D-branes on twisted circles

We would like to know whether the relation we have obtained between UV/IR mixing effects and propertiesof the closed string spectrum can be applied to more general situations. An interesting example to analyseis that of D-branes on twisted circle backgrounds. These are R9 × S1 space-times where shifts along thecircle are combined with rotations on several 2-planes [11]. We will only consider rotations which coincidewith those defining a C3/ZN orbifold. When the radius of the circle is send to infinity we recover Type IIstring theory on R10. For zero radius the Type IIA(B) twisted circle backgrounds reduce, after T-duality,to the associated C3/ZN ⊗ R4 Type IIB(A) orbifold model [21].

The spectrum of closed strings on twisted circles was derived in [11]. For the cases we are interested in,the modes with winding number w along the S1 are closely related to the spectrum of the associated orbifoldmodel in the twisted sector k = w modN . In particular, the spectrum of modes with zero momentum alongS1 coincides precisely with that of the orbifold model, except for a positive mass shift due to the windingenergy. This shift stabilises otherwise tachyonic modes for sufficiently large radius.

Let us analyse what this implies for the UV/IR mixing effects on noncommutative D-branes in twistedcircle backgrounds. The closed string derivation implies that the signs of the leading UV/IR mixing effectsare linked to the misalignment between the NSNS and RR towers of the parent string theory. We willfocus in situations where only closed string modes with zero momentum on S1 can contribute (see [10]for details). Since the winding energy affects equally the NSNS and RR towers, we should expect that thesigns of the leading UV/IR mixing effects are independent of the radius of the circle, R. Moreover, theyshould coincide with those of the associated orbifold background.


616 E. Lopez: UV/IR mixing, noncommutative instabilites

In order to check these predictions we consider Type IIB D3-branes wrapped on the twisted circle andType IIA D2-branes transversal to it. The appropriate field theory limits for these branes differ. The intrinsicscale of the D3-brane theory is set by the KK masses m ∼ 1/R. Hence we should send α′ → 0 keeping Rfixed. In this regime the winding energy dominates the closed string spectrum and there are no tachyonicmodes. Contrary, the natural scale of the D2-brane theory is R/α′, governing the mass of strings endingon the brane and with non-zero winding along the circle. The interesting field theory limit in this case isα′ → 0 with R/α′ fixed. This corresponds to the limit of negligible winding energy, where the closedstring spectrum contains tachyons. Therefore these two examples allow us to explore opposite regimes ofthe closed string background.

An analysis of the gauge theory on noncommutative D2- and D3-branes shows [10] that the signs of theUV/IR mixing effects for both cases are governed by the same quantities εk which, as we expected, coincidewith those of the orbifold background (4). The functional dependence of the leading UV/IR mixing terms onp is however different for D2- and D3-branes, and in both cases can be related to a closed string propagator.We found IR finite corrections for D3-branes and IR divergent ones for D2-branes. For D2-branes there is aone to one correspondence between noncommutative instabilities and closed string tachyons, as it was thecase for D3-branes at orbifolds. It is tempting to speculate that a direct relation between noncommutativeinstabilities associated to IR divergences and closed string tachyons might hold in general. For D3-branesat twisted circles we have a weaker but still interesting result: Leading UV/IR mixing terms are governedby the mass gap between the lowest modes in the NSNS and RR sectors. Thus destabilising effects areabsent for those sectors related to closed strings which are stable for any radius.

At a more fundamental level, we are seeing that the gauge theory on the world-volume of D-branescontains very specific information about the closed string spectrum. This information is somehow hiddenin an ordinary situation, but becomes manifest through UV/IR mixing when noncommutativity is turnedon. It would be very interesting to determine how much information about the string theory spectrum canbe extracted in this way from the low energy theory on D-branes.

Acknowledgements It is a pleasure to thank Adi Armoni and Angel M. Uranga for their collaboration in this project.

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UV/IR mixing, noncommutative instabilites and closed strings