22 October 1998

Ž .Physics Letters B 438 1998 261–266

ž /Comments on non- chiral gauge theories and type IIB branes

Adi Armoni 1, Andreas Brandhuber 2

School of Physics and Astronomy BeÕerly and Raymond Sackler Faculty of Exact Sciences Tel AÕiÕ UniÕersity, Ramat AÕiÕ 69978, Israel

Received 6 April 1998; revised 27 July 1998Editor: H. Georgi

Abstract

We use type IIB brane configurations which were recently suggested by Hanany and Zaffaroni to study four dimensionalNs1 supersymmetric gauge theories. We calculate the one loop beta function and realize Seiberg’s duality using aparticular configuration. We also comment on the anomaly cancelation condition in the case of chiral theories and the betafunction in the case of chiral and SOrSp theories. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

In the last two years we have learned that branesare a useful tool in the study of supersymmetric

w xŽgauge theories 1 for a recent review and a completew x.list of references see 2 . Branes give a nice and

effective pictorial meaning to symmetries and pa-rameters of the low energy field theory.

Recently, there has been progress in the realiza-w xtion of chiral theories using branes 3–6 of type II

w xstring theory. In particular, Hanany and Zaffaroni 6constructed various classical Ns1 chiral four di-mensional field theories, using brane configurationsof type IIB string theory, in the limit of zero stringcoupling.

In this note we use the suggested construction inthe presence of non-zero string coupling. In this casethe fivebranes will bend, with asymptotic bending

w xwhich is dictated by supersymmetry 7,8 . The bend-

1 E-mail: armoni@post.tau.ac.il2 E-mail: andreasb@post.tau.ac.il

Ž .ing of a p,q fivebrane which is stretched in a lineŽ .in the x, y plane is

D x :D yspqqt , 1Ž .where ts irg qxr2p and g ,x are the strings s

coupling and type IIB axion. Moreover, it was ex-w xplained in Ref. 9 that the slope of each brane

should remain constant along the line even if thebrane crosses other fivebranes. This constraint leadsto anomaly cancelation of local symmetries in thefield theory.

The paper is organized as follows: In Section 2we use the bending of the branes to calculate thebeta function of Ns1 supersymmetric QCD. InSection 3 we suggest a brane configuration andbrane moves which reproduce Seiberg’s duality. InSection 4 we introduce a configuration for chiralgauge theories which is determined by anomaly can-celation in the field theory and bending constraints

w xsimilar to those in Ref. 9 . We also give an approxi-mate calculation of the beta function in these cases.We comment on the construction of brane configura-tions with symplectic and orthogonal groups in Sec-tion 5.

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00993-9

( )A. Armoni, A. BrandhuberrPhysics Letters B 438 1998 261–266262

Some of the issues that we discuss were previ-ously addressed in the framework of type IIA string

w xtheory and M-theory 10–14 . The issue of finitegauge theories, using type IIB branes was discussed

w xrecently in Ref. 15 .

2. Ns1 SQCD setup and the beta function

The brane configuration that we use is similar tow xthe one that was introduced in Ref. 6 . The configu-

Ž .ration Fig. 1 consists of two horizontal lines whichrepresent two NS branes with world volume 012367Ž .which we will denote by NS’ and two vertical lineswhich represent NS branes with world volume

Ž .012345 which we will denote by NS . In addition,there are D5 brane with world volume 012346 whichare attached to the NS and NS’ branes. This set ofbranes leads to Ns1 SQCD in four dimensionswith generically chiral matter. The numbers in theboxes of Fig. 1 correspond to the number of D5branes. The central D5 branes are finite in twodirections and represent gauge symmetries in thefour dimensional gauge theory. Semi-infinite boxesrepresent global symmetries. The intersection of theNS and NS’ brane might lead to additional lightstates, e.g. D1 strings connecting NS and NS’. Thesestates might affect the 4D field theory, but we be-lieve that the results which are presented in thispaper are not influenced by them. The direction of

w xallowed arrows are North, East and South-West 6 .Outgoing arrows are fundamentals and in-going ar-rows are anti-fundamentals of the gauge group. Thus

Fig. 1. Ns1 Theory with N s N q X qY flavors.f c

Fig. 1 represents four dimensional Ns1 supersym-Ž .metric SU N with N qXqY fundamentals andc

N qXqY anti-fundamentals. Notify that the globalc

symmetry factories into a left and right part. This isto be contrasted to other type IIA constructions ofNs1 SQCD where only the diagonal subgroup ofthe global flavor symmetry is manifest. Note also

Žthat in this construction N GN since X and Yf c.cannot be negative .

The construction guarantees that the bending ofthe four NS branes is constant along each brane. Thisis a necessary condition to ensure that the gauge

w xsymmetry is anomaly free 9 . In addition, note thatthe NS’ branes have zero bending, since we put thesame number of D5 branes on each side of them.

The coupling constant of the gauge theory isrelated to the area of the middle box.

1 D x D x4 6s . 2Ž .2 2g g ls s

The running of the coupling constant is due to thebending of the branes that surround the middle box.In the type IIA construction of Ns1 gauge theories,one could not define a distance which is directlyrelated to the running coupling because the D4 branesend on branes whose world-volume extends in dif-ferent directions. However, since in our constructionthe two NS’ branes do not bend and their distance isconstant, we can relate the beta function to thechange of the distance between the two NS branes.The result is

b s XyN q YyN sN y3N , 3Ž . Ž . Ž .0 c c f c

which agrees with the field theory result!

3. Ns1 duality

In this section we would like to propose a branew xmotion that can be related to Seiberg’s duality 16 .

The realization of Ns1 duality using branes in typeIIA string theory was first demonstrated in Refs.w x10,11 . Here we show that a simple construction intype IIB leads to this duality, too. Consider thetheory in Fig. 2. This theory describes Ns1 super-

Ž .symmetric SU N gauge theory with N fundamen-c f

tals and N anti-fundamentals. We refer to this the-f

ory as the ‘‘electric’’ theory. Now attach an addi-

( )A. Armoni, A. BrandhuberrPhysics Letters B 438 1998 261–266 263

Ž .Fig. 2. The ‘‘electric’’ theory. SU N gauge theory with Nc f

flavors.

tional NS brane to the D5 branes at x sy`. As6

long as the additional NS brane is positioned atinfinity the theory on the branes remains unchanged.Now we move the NS brane to a finite x position.6

In addition, take the right NS brane and bring it tox s`. The resulting theory is described in Fig. 3.6

Ž .This theory is an SU N theory with N yN fla-c f c

vors. We refer to this theory as the ‘‘magnetic’’Žtheory. The intermediate theory the left and the right

.NS are at finite distance in the x position is not6

dual to either the electric or the magnetic theories. InŽ . Ž .fact this is a theory with an SU N =SU N yNc f c

gauge group and the matter content given in Table 1.

Ž .Fig. 3. The ‘‘magnetic’’ theory. SU N y N gauge theory withf c

N flavors.f

Table 1The matter content of the ’intermediate’ theory

Ž . Ž .SU N = SU N y Nc f c

N I 1f

N I 1cŽ .1 N y N If c

1 N If

I I

However, the magnetic and the electric theoriescan be viewed as two limits of this intermediatetheory.

While we cannot justify the motion of the braneswhich leads to the duality, it reproduces the knownfield theory results.

As a simple test of the above suggestion, we willprove that the relation between giving a mass termsin the electric theory and Higgsing in the magnetic

Ž .theory and vice versa holds in the brane configura-tion. Adding a mass term for one of the quarks in theelectric theory reduces the flavor symmetry by one.In the magnetic theory this corresponds to giving amass to one of the mesons which via the equationsof motion induces a vev for one of the magneticquarks reducing both colors and flavors by one unit.

This correspondence can be seen in the braneconstruction in the following way: reconnect the D5brane in the left-upper box of Fig. 2 to the two otherD5 branes in the left column. Now the D5 brane isinfinite in the x direction and can be lifted in the x4 5

direction. The x position is proportional to the mass5

term. The result is an electric theory with N y1f

flavors. Introducing the additional NS brane andmoving the right NS brane, we obtain the magnetictheory in which the middle column of Fig. 3 hasN yN y1 D5 branes. The x position is related tof c 5

the mass term for the meson. This magnetic theoryhas N yN y1 colors and N y1 flavors, as ex-f c f

pected.In the same way we can show the relation be-

tween Higgsing in the electric theory and mass termsin the magnetic theory. In this case we reconnect D5branes in the middle column of Fig. 2 to obtain oneD5 brane that is infinite in the x direction but finite4

in the x direction and lift it in the x direction. This6 5

reduces color and flavor by one and corresponds to

( )A. Armoni, A. BrandhuberrPhysics Letters B 438 1998 261–266264

Higgsing in the electric theory where the x position5˜² :is proportional to QQ . After the brane motion we

obtain the magnetic theory with flavor reduced byone which is related to giving a mass term to one ofthe quarks in the magnetic theory.

In the above discussion, we did not consider thesuperpotential in the two dual descriptions. Dualityof the field theories requires zero superpotential inthe electric theory and a superpotential of the formWsMqq in the magnetic theory. Since in our de-˜scription the flavor symmetry boxes are semi-in-finite, the mesons in both sides are not dynamical, incontrast to the situation in the magnetic theory infield-theory. This obstacle can probably be solved byadding an extra NS brane at large value of x in the6

magnetic brane configuration. We leave this issue asan open question.

From the field theory point of view it looks liketwo different degenerations of a field theory with aproduct gauge group. By decoupling one of thefactors we turned a local symmetry into global flavorsymmetry. One also would like to check that themoduli spaces in the electric and magnetic theorymatch. We have given some evidence for this byrelating massterms and Higgsing in the dual theories.Maybe the introduction of D7 branes is necessaryŽ .similar to the use of D6 branes in type IIA tointroduce matter to make this match complete. In the

w xtype IIA brane construction of Seiberg’s duality 10a certain brane creation process was a crucial ingre-dient. In this approach the moduli space could bematched precisely and the meson was identified. Butthis approach also has disadvantages e.g. only adiagonal subgroup of the flavor symmetry is mani-fest and an Fayet-Ilioupoulos term had to be intro-duced to avoid singularities in the brane motion.Such a singularity was never a problem the type IIBapproach. In summary it must be said that any braneconstruction of Seiberg’s duality presented so farcaptures many features of the field theory duality buthas also some problems.

4. Anomaly cancelation in chiral theories

Ž .In this section we study SU N gauge theoriesc

with an symmetricranti-symmetric tensor, N funda-f

mentals and N anti-fundamentals. Anomaly cancela-f

Ž .tion requires N yN sN q4 or N yN sN y4f f c f f cŽfor theories including the symmetric or anti-sym-

.metric representation.We cannot derive this result from string theory

but the configuration is fixed by the requirement thatNS branes have constant bending and that the fieldtheory anomaly is canceled. The relevant brane con-figuration for the symmetric case is described in Fig.4 In addition to the three NS branes, the two NS’branes and the D5 branes we added on top of themiddle NS brane an orientifold-sevenplane withworld volume 01234789. The presence of the orien-tifold induces a Z symmetry with respect to the x2 5

and x directions. An orientifold plane with RR6Ž . y Ž q.charge y8 q8 will be denoted by O7 O7

Ž .and gives rise to a SU N gauge theory with anŽ .anti-symmetric symmetric representation.

In order to ensure that the bending of each NSbrane remains constant along the brane, we addedadditional D5 branes around the N D5 branes in thec

middle. The additional branes are responsible forextra matter in the fundamental and anti-fundamentalof the gauge group. Above the upper NS’ brane thereshould be 2 extra D5 brane at each side of the mirrorŽand similarly two extra D5 branes below the lower

.NS’, at each side , to cancel the chiral anomaly infield theory. Although we have no string theoryargument for this rule, it should be possible to deriveit directly from string theory. Note that it is notsufficient to require a constant bending. There shouldbe two extra D5 branes above and below the middle

Ž .Fig. 4. Ns1 SU N theory with symmetric, fundamentals andanti-fundamentals. The dashed line denote the O7q.

( )A. Armoni, A. BrandhuberrPhysics Letters B 438 1998 261–266 265

row. Any other number of D5 branes leads to ananomaly in the field theory. We assume that thiseffect is due to the presence of the orientifold plane.

Let us count the number of fundamentalsranti-fundamentals which interact with the color group.The in-going arrows are anti-fundamentals and theoutgoing arrows are fundamentals. There is one ex-ception: The South-West arrow which crosses theO7q plane is counted as anti-fundamental and not asfundamental. The argument for this is the following:

˜ ˜A superpotential term AQQ is formed by the sym-Žmetric the horizontal arrow which connect the mid-

. Ždle box to its mirror , the anti-fundamentals thevertical arrow which connects the lower box to the

.middle box in the mirror and the South-West arrow.Since the superpotential is a scalar, the diagonalarrow should represent anti-fundamentals. Hence wehave N s2 N qXq6 and N sN qXq2.f c f c

We conclude that

N yN sN q4 . 4Ž .f f c

Similarly, in the case of anti-symmetric matter andO7 plane we obtain

N yN sN y4 . 5Ž .f f c

We can also calculate, approximately, the betafunction of these theories. As we explained in Sec-tion 2, the relation between the one-loop beta func-tion and the bending of the branes, is meaningfulonly when two of the branes that surround the gaugegroup box remain parallel. In the current configura-tion this is not the case, since the two NS’ branes arenot parallel. The D5 and the O7q bend the NS’brane in different directions and therefore the NS’can never be made flat in all directions. However, ifthe bending of the right NS brane is much largerthan the bending of the NS’ brane, we expect to getan approximate answer. By approximate we meanthat the dependence on N and N is correct.c f

The bending of the right NS is XyN , hencec

b fN y2 N y2 . 6Ž .0 f c

Note that the field theory result is

b sN y2 N q3 , 7Ž .0 f c

which indeed agrees when 2 N yN 41. Similarly,c f

the approximate value of the beta function can be

calculated in the case of the chiral theory withanti-symmetric matter.

5. SO and Sp gauge groups

We would like to comment on the one-loop betafunction in the case of Ns1 SQCD with gauge

Ž . Ž . Ž .groups SO N and Sp N with N even . Thec c c

realization of these models was worked out in Ref.w x6 and it is similar to the theory in Fig. 4. The brane

Ž .configuration for SO N is given in Fig. 5. Notec

that in contrast to the previous case, there is no NSbrane at x s0 and the color box should be under-6

stood as one box. Here we assume that the orien-tifold plane imposes the same constraints on thenumber of branes in the upper and lower row as inthe previous section. This brane rule can be sup-ported by field theory arguments. Chiral theories andorthogonalrsymplectic theory are closely related by

w x Ž . Ž .duality 17 . A theory with Sp N y4 =SU Nc c

gauge group can be constructed in a similar way tothe theory in Fig. 5, but with an O7 plane andadditional NS branes at finite x and yx . By6 6

moving the middle NS brane and its mirror towardsthe orientifold plane and moving one of them to

Ž .x s` we obtain an SU N theory with antisym-7 c

metric tensor and extra matter. This process wasw xalready described in Ref. 6 . Here we conclude that

two extra D5 branes above and below the gauge boxŽ .in the chiral case Fig. 4 , are also necessary in the

orthogonal and symplectic cases.

Ž . qFig. 5. Ns1 SO N theory. The dashed line denote the O7 .c

( )A. Armoni, A. BrandhuberrPhysics Letters B 438 1998 261–266266

According to the rules of the previous section, wewill not be able to get the accurate beta function ofthese theories since a completely flat NS’ branecannot be achieved. But it is again possible to get thecorrect dependence on N and N . The number off c

flavors in this case is

2 N s4 Xq2 N q8 . 8Ž .f c

The one-loop beta function is obtained, approxi-mately, by 2 Xy2 Nc

b fN y3N y4 , 9Ž .0 f c

in agreement with the field theory result

b sN y3N q6 . 10Ž .0 f c

A similar result can be obtained for the Sp case. Inthis case we replace the O7q by O7, and we againcapture the N y3N dependence.f c

Acknowledgements

We thank Nissan Itzhaki, Udi Fuchs and espe-cially Shimon Yankielowicz for discussions.

References

w x Ž .1 A. Hanany, E. Witten, Nucl. Phys. B 492 1997 152,hep-thr9611230.

w x2 A. Giveon, D. Kutasov, Brane Dynamics and Gauge Theory,hep-thr9802067.

w x3 K. Landsteiner, E. Lopez, D.A. Lowe, Duality of ChiralNs1 Supersymmetric Gauge Theories via Branes, hep-thr9801002.

w x4 I. Brunner, A. Hanany, A. Karch, D. Lust, Brane Dynamics¨and Chiral non-Chiral Transitions, hep-thr9801017.

w x5 S. Elitzur, A. Giveon, D. Kutasov, D. Tsabar, Branes, Orien-tifolds and Chiral Gauge Theories, hep-thr9801020.

w x6 A. Hanany, A. Zaffaroni, On the Realization of Chiral Four-Dimensional Gauge Theories Using Branes, hep-thr9801134.

w x Ž .7 O. Aharony, A. Hanany, Nucl. Phys. B 504 1997 239,hep-thr9704170.

w x Ž .8 O. Aharony, A. Hanany, B. Kol, Webs of p,q 5-branes,Five Dimensional Field Theories and Grid Diagrams, hep-thr9710116.

w x9 E.G. Gimon, M. Gremm, A note on brane boxes at finitestring coupling, hep-thr9803033.

w x Ž .10 S. Elitzur, A. Giveon, D. Kutasov, Phys. Lett. B 400 1997269, hep-thr9702014.

w x11 S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici, A.Ž .Schwimmer, Nucl. Phys. B 505 1997 202, hep-thr9704104.

w x12 K. Hori, H. Ooguri, Y. Oz, Strong Coupling Dynamics ofFour Dimensional Ns1 Gauge Theories from M TheoryFivebrane, hep-thr9706082.

w x Ž .13 E. Witten, Nucl. Phys. B 507 1997 658, hep-thr9706109.w x14 A. Brandhuber, N. Itzhaki, V. Kaplunovsky, J. Sonnen-

Ž .schein, S. Yankielowicz, Phys. Lett. B 410 1997 27, hep-thr9706127.

w x15 A. Hanany, M.J. Strassler, A.M. Uranga, Finite Theories andMarginal Operators on the Brane, hep-thr9803086.

w x Ž .16 N. Seiberg, Nucl. Phys. B 435 1995 129, hep-thr9411149.w x Ž .17 M. Berkooz, Nucl. Phys. B 452 1995 513, hep-thr9505067.