Microscopic Formulation of Puff Field Theory
aa r X i v : . [ h e p - t h ] J a n MAD-TH-08-03
Microscopic Formulation of Puff Field Theory
Sheikh Shajidul Haque and Akikazu HashimotoDepartment of Physics, University of Wisconsin, Madison, WI 53706
Abstract
We describe a generalization of Puff Field Theory to p + 1 dimensions where 0 ≤ p ≤ p = 0, “Puff Quantum Mechanics,” and construct aformulation independent of string theory. elvin twist, also known as the T-s-T transformation, is a powerful solution generatingtechnique in string and supergravity theories [1, 2, 3, 4, 5, 6]. Melvin twist of flat spacesretains the simplicity of the flat space, giving rise to string theory whose world sheet theoryis exactly solvable [7, 8, 9, 10, 11, 12]. Melvin twists, in a context of D-branes and their nearhorizon limits, can be used to formulate wide variety of exotic field theories, including non-commutative gauge theories [13, 14, 15, 16, 17, 18, 19], NCOS theories [20, 21, 22, 23, 24], dipoletheories [25, 26, 27], and β deformed superconformal theories [28].Recently, “Puff Field Theory,” a new class of decoupled non-local field theory based onMelvin universe, was introduced by Ganor [29]. In the construction of PFT, the Melvinbackground is supported by RR field strength, but the decoupled theory is distinct fromthe NCOS theories [23, 24]. In 3+1 dimensions, PFT preserves the spatial SO (3) subgroupof the SO (1 ,
3) Lorentz symmetry. The dual supergravity formulation of PFT, along thelines of [13], was constructed in [30], allowing physical feature of this model, such as theentropy as a function of temperature, to be computed at large ’t Hooft coupling. One ofthe main appeal of PFT is the fact that it is compatible with the symmetries of Freedman-Robertson-Walker cosmology. Some phenomenological aspects of PFT were studied recentlyin [31].The Melvin deformed field theories enumerated earlier: non-commutative field theory,NCOS, dipole field theories, and β -deformed superconformal field theories all have concreteformulations independent of string theory. In contrast, only definition available for PFT, forthe time being, is as a decoupling limit of fluctuations of D-branes in a Melvin geometry intype II string theory. The goal of this article is to provide an alternative definition of PFTwhich is independent of string theory. Our approach will closely parallel the formulation ofLittle String Theory and (0,2) superconformal field theory using deconstruction [32].Let us begin by reviewing the construction of PFT as a decoupled theory on a brane instring theory [29, 30]. A convenient place to start is flat 9+1 geometry in type IIA theory,with N coincident D0-branes. Let us ignore the gravitational back reaction of the D0-branesfor the time being. The M-theory lift of the IIA geometry is R , × S . Let us parameterizethis geometry with a line-element of the form ds = − dt + dr + r dφ + d~y + dz (1)where z is the M theory circle with periodicity z ∼ z + 2 πR , R = g s l s , and ~y is a vector inseven dimensions. The r , φ parameterize a plane spanned by the remaining two coordinatesin cylindrical coordinates.Now consider performing a Melvin twist on φ with respect to shift in z . This amounts See appendix A of [30] for a discussion of this point.
1o deforming the line element by the amount η so that ds = − dt + dr + r ( dφ + ηdz ) + d~y + dz . (2)Reducing this to IIA gives rise to a Melvin geometry of the form ds IIA = (1 + η r ) / (cid:18) − dt + dr + r η r dφ + d~y (cid:19) (3)along with some RR 1-form and a dilaton. Recalling that there were N D0-branes in thebackground, perform a T-duality along 3 of the y i coordinates. The prescription of [29, 30]is to send α ′ → g Y M and ∆ = ηα ′ fixed. It is straight forward to reproduce thesupergravity background of [30] by repeating this procedure, but including the gravitationalback reaction of the D0-branes.One issue which was not emphasized in the discussions of [29, 30] is that one can justas easily construct a generalization of PFT in p + 1 dimensions for 0 ≤ p ≤ g Y Mp ∼ g s α ′ ( p − / to stay finitein the scaling limit. The resulting p + 1 dimensional field theory will preserve the SO ( p )subgroup of the Lorentz group.To demonstrate the decoupling limit of PFT for general p more concretely, let us workout the supergravity dual of the p = 0 case explicitly. With the gravitational back reactionof D0 taken into account, (2) becomes ds = − h − dt + h ( dz − vdt ) + dρ + ρ ( ds B (2) + ( dφ + ηdz + A ) ) + X i =1 dy i (4)where h ( ρ, y ) = 1 + gN α ′ / ( ρ + ~y ) / , v = h − (5)is the harmonic function of a D0-brane, and d Ω = ds B (2) + ( dφ + A ) (6)is the standard Hopf parameterization of S with the B (2) being the base S . In writingthis geometry, we generalized the twist from being along the angular coordinate φ in a planein (2), to being along the Hopf fiber of angular 3-sphere in R spanned by four of the ~y coordinates. This change essentially amounts to considering the F5 flux brane instead of F7flux brane in the language of [5]. The latter choice has the advantage of preserving half ofthe supersymmetries. Now, reduce to IIA and take a decoupling limit, by scaling α ′ → U = r/α ′ , ∆ = ηα ′ , and g Y M = g s α ′− / fixed. Note that ν = g Y M ∆ (7)2s a dimensionless and a finite quantity. This parameter will play an important role in thediscussions below.After taking the α ′ → ds α ′ = √ H + ∆ U − H − dt + dU + U ds B (2) + U (cid:18) dφ + A + ∆ H dt (cid:19) + d~Y ! Aα ′ = 1 H + ∆ U (cid:0) − dt + U ∆ dφ (cid:1) (8) e φ = g Y M ( H + ∆ U ) / where U = ρα ′ , ~Y = yα ′ , H ( U, ~Y ) = α ′ h ( ρ, ~y ) = g Y M N ( U + Y ) / (9)have finite α ′ → p . Let us refer to the decoupled theory for p = 0 as “Puff Quantum Mechanics.” If we set∆ = 0, the metric (8) precisely reduces to the near horizon limit of D0-branes [33]. From theform of (8), one can infer that ∆ deforms the matrix quantum mechanics of the decoupledD0-branes in the UV. It is also clear that the dynamics of the decoupled theory is somehowbeing modified by the RR 1-form potential in the background. In the remainder of thisarticle, we will provide a prescription to define PQM independent of string theory.One powerful tool in analyzing microscopic features of non-local field theories is the SL (2 , Z ) duality. In the case of non-commutative field theory on a torus, an SL (2 , Z )duality is realized in the form of Morita equivalence [34]. If the deformation parameter,e.g. the non-commutativity parameter 2 π ∆ , is expressed in a suitably dimensionless form,e.g. Θ = ∆ / Vol( T ), then for a rational value of Θ, one can find an SL (2 , Z ) element tomap this theory to a dual theory for which Θ = 0. For non-commutative field theories, theΘ = 0 theory corresponds to the standard non-abelian gauge theory on T with a ’t Hooftflux [35]. Various SL (2 , Z ) duals have non-overlapping regimes of validity as a function ofenergy, giving rise to a structure resembling a duality cascade [36]. For all rational values ofΘ, it is the SL (2 , Z ) dual with vanishing Θ which is effective in the deep UV.Analogous SL (2 , Z ) structure exists for PFT [30]. In the context of PQM, this structureis made most transparent by performing a modular transformation on the complex structure3f the torus defined by φ and z in (4), i.e. dφ dzR ! → a bc d ! dφ dzR ! (10)If the dimensionless parameter ν = g Y M ∆ = ηR = − bd (11)is a rational number, this SL (2 , Z ) will map (4) to ds = − h − / dt + h / dρ + ρ ds B (2) + ρ (cid:18) dφd + A (cid:19) + d~y ! A = − cRdφ − vdte φ = h / . Other than the seemingly innocent 1-form A = cRdφ , this is just a Z d orbifold of D0which defines a local theory in the decoupling limit. The SL (2 , Z ) also changed the rank ofthe gauge group from U ( N ) to U ( d N ) /Z d , as well as acting on g Y M .Let us refer to the RR deformed D0-brane quantum mechanics as “Twist Quiver QuantumMechanics.” Our goal is to determine how the RR background affects the dynamics ofTQQM.In order to access the effect of form fields on the dynamics of open string states, it isconvenient to go to a dual frame where the form field in question is mapped to an NSNS2-form so that one has access to an explicit NSR sigma model. In order to accomplishthis, let us momentarily embed the Z d ALE orbifold spanned by ρ , B (2), and φ in a Taub-NUT. This is a UV modification which can be removed later. The reason for embeddinginto the Taub-NUT is to facilitate the T-duality along φ . This T-duality will map theTaub-NUT to NS5-branes, and the D0-brane to a D1-brane. They are oriented as follows:0 1 2 3 4 5 6 7 8 9D1 • • NS5 • • • • • •
The system is to be visualized as a system of dN D1-branes sprinkled with d NS5 impurities,which is illustrated in figure 1. The RR 1-form potential A = cd ( dR ) dφ becomes the RRaxion χ = cd under this duality.In order to map the RR axion into NSNS 2-form, we further compactify two directions,parallel to the NS5-brane world volume but orthogonal to the D1. This compactificationwill also deform the theory in the UV, which we will remove at the end of the construction.T-dualizing along these two directions, followed by S-duality will lead to a system consisting4igure 1: Configuration of dN D1-brane and d NS5 impurities obtained by T-dualizing thebackground (4) along the φ direction. There is also a RR axion χ = cd in this background.of dN D3-branes on T , with d D5 impurities extended along the T and localized in the re-maining spatial coordinate of the D3, in a background a constant NSNS 2-form along the T .0 1 2 3 4 5 6 7 8 9 B -field ≡ ≡ D3 • • • • D5 • • • • • • Except for the compactification and the B -fields, this is precisely the impurity model ofKarch and Randall [37] whose detailed microscopic formulation was given in [38].Of course, to isolate the PQM/TQQM dynamics, we are only interested in the deep IRwhere only the dimensionally reduced dynamics matters. We can take advantage of this factto further reformulate this system by performing additional dualities.Consider T-dualizing along the world volume of the D3-brane in the x direction alongwhich the D1 was originally oriented. This will map the D3-brane to a D2-brane lo-calized in a circle. Its covering space is an infinite array. The impurity D5-branes aremapped to an extended D6. The T along which the B -field was oriented is left intact.0 1 2 3 4 5 6 7 8 9 B -field ≡ ≡ D2 • • • D6 • • • • • • • This configuration is illustrated in figure 2.a. What we have done is to exchange the Kaluza-Klein mode associated with the x direction to a tower of massive W bosons in a U ( ∞ ) /Z gauge theory along the lines of [39].Let us now employ a trick of presenting U ( ∞ ) /Z as a limit of U ( N ) /Z N . Simply consider5 , , (a) (b)Figure 2: (a) T-dual of Defect Field Theory. The D2-branes is localized in a circle. In thecovering space, it corresponds to a U ( ∞ ) /Z theory corresponding to an array of D2-branesin a background of D6-branes. (b) The infinite array can be arrived as a limit of a circularlattice by scaling the radius and the number of branes keeping the linear density of thebranes fixed.. For the sake of illusteration, the D2-branes appear separated from the D6-branes along the 7,8,9 directions in the figure. The actual configuration we consider, the D2and the D6 branes are coincident along the 7,8,9 directions.arranging N D2 in a circular, instead of the linear, pattern as is illustrated in figure 2.b. Thisis essentially a technique to simulate T-duality via deconstruction along the lines of [32].What we have now is a configuration of D2 in Coulomb branch, in a background ofD6-branes. This is essentially the configuration which gives rise to 2+1 SYM with flavor,considered in [40]. The only novelty here is the fact that the world volume of D2 is com-pactified on a torus, and that there is a B -field oriented along it.Following the duality chain, it should be clear that each dot in figure 2 should correspondto dN D2-branes. It is quite natural therefore to interpret the c/d units of B -flux as givingrise to ’t Hooft’s fractional flux on the D2-brane world volume gauge theory [35, 41, 42, 43].There is one subtlety with this interpretation. In order for the ’t Hooft flux to exist asa consistent field configuration, it is necessary for all the fields in the theory to be invariantwith respect to the center of the gauge group. The flavor matter which arise in our setupdue to the presence of the D6-brane, is in the fundamental representation with respect tothe gauge group and does not satisfy this requirement.A moment’s thought, however, suffices to address this issue. One simply needs to recallthat the 2-6 strings are actually in a bifundamental representation of the D2 and the D6-6rane gauge fields. Since there are d D6-branes in the configuration illustrated in figure 2,it can also support a flux in the amount of R B = c/d . From the point of view of the 2+1dimensional Yang-Mills theory, this amounts to twisting with respect to gauge and the flavorgroup of the bifundamentals. In other words, we parameterize the color and flavor indices ofthe bi-fundamental as Φ i,c ; f where i = 1 . . . N , c = 1 . . . d , and f = 1 . . . d . As the notationsuggests, ( i, c ) are the color indices and f is the flavor index. To these bi-fundamental fields,we impose the boundary conditionΦ i,c ; f ( x + L , x ) = U c,c ′ Φ i,c ′ ; f ′ ( x , x ) U − f ′ ,f Φ i,c ; f ( x , x + L ) = V c,c ′ Φ i,c ′ ; f ′ ( x , x ) V − f ′ ,f (12)where U and V are d × d ’t Hooft matrices [35] satisfying U V U − V − = e πic/d . (13)In fact, precisely this form of twisted matter theory have been used before by Sumit Das inthe context of large N twisted reduced models [44]. See also [45, 46, 47, 48] for discussons onrelated issues.Of course, since we have performed various UV deformation of the original PFT to get tothis stage, one must take the appropriate scaling limit to decouple these effects. The fact thatthe decoupled supergravity dual solution (8) exists provides us with the assurance that sucha limit does exist. Because the chain of duality involved S-duality at one point, what we aredoing is similar in spirit to defining NCOS as the strong coupling limit of NCSYM [20,21,22].In summary, we have shown that PQM is a scaling limit, of a large N deconstructionlimit, of 2+1 dimensional SYM, with ’t Hooft flux, and matter in the fundamental repre-sentation, with twisted flavor. The derivation relied on a lengthy chain of dualities andmanipulations in string theory. Nonetheless, the formulation of the theory in its final formdoes not rely on any string theory concepts. While this definition is not especially useful formost practical applications, it does provide a concrete formulation of the theory whose onlyother formulation known today is as a decoupling limit of D-branes in a Melvin universebackground. [29, 30].The color/flavor twisted 2+1 dimensional theory (12) might be an interesting theoryin its own right to explore further. These theories are related via Morita equivalence tonon-commutative field theories with matter [49, 50], whose dual supergravity solution wasbriefly described in [40]. It might also be interesting to explore how the twists and non-commutativities modify the Intriligator Seiberg mirror symmetry of three dimensional gaugetheories [51]. 7 cknowledgements We would like to thank Sharon Jue, Bom Soo Kim, Anthony Ndirango, and especially OriGanor for collaboration which motivated this project, and for useful discussions. We alsothank Sumit Das for bringing his early work on large N twist reduced models to our attention.This work was supported in part by the DOE grant DE-FG02-95ER40896 and funds fromthe University of Wisconsin. References [1] F. Dowker, J. P. Gauntlett, D. A. Kastor, and J. H. Traschen, “Pair creation ofdilaton black holes,”
Phys. Rev.
D49 (1994) 2909–2917, hep-th/9309075 .[2] F. Dowker, J. P. Gauntlett, S. B. Giddings, and G. T. Horowitz, “On pair creation ofextremal black holes and Kaluza-Klein monopoles,”
Phys. Rev.
D50 (1994)2662–2679, hep-th/9312172 .[3] K. Behrndt, E. Bergshoeff, and B. Janssen, “Type II duality symmetries in sixdimensions,”
Nucl. Phys.
B467 (1996) 100–126, hep-th/9512152 .[4] M. S. Costa and M. Gutperle, “The Kaluza-Klein Melvin solution in M-theory,”
JHEP (2001) 027, hep-th/0012072 .[5] M. Gutperle and A. Strominger, “Fluxbranes in string theory,” JHEP (2001) 035, hep-th/0104136 .[6] M. S. Costa, C. A. R. Herdeiro, and L. Cornalba, “Flux-branes and the dielectriceffect in string theory,” Nucl. Phys.
B619 (2001) 155–190, hep-th/0105023 .[7] J. G. Russo and A. A. Tseytlin, “Constant magnetic field in closed string theory: anexactly solvable model,”
Nucl. Phys.
B448 (1995) 293–330, hep-th/9411099 .[8] J. G. Russo and A. A. Tseytlin, “Exactly solvable string models of curved space-timebackgrounds,”
Nucl. Phys.
B449 (1995) 91–145, hep-th/9502038 .[9] J. G. Russo and A. A. Tseytlin, “Heterotic strings in uniform magnetic field,”
Nucl.Phys.
B454 (1995) 164–184, hep-th/9506071 .[10] J. G. Russo and A. A. Tseytlin, “Magnetic flux tube models in superstring theory,”
Nucl. Phys.
B461 (1996) 131–154, hep-th/9508068 .811] A. A. Tseytlin, “Melvin solution in string theory,”
Phys. Lett.
B346 (1995) 55–62, hep-th/9411198 .[12] A. A. Tseytlin, “Exact solutions of closed string theory,”
Class. Quant. Grav. (1995) 2365–2410, hep-th/9505052 .[13] A. Hashimoto and N. Itzhaki, “Non-commutative Yang-Mills and the AdS/CFTcorrespondence,” Phys. Lett.
B465 (1999) 142–147, hep-th/9907166 .[14] O. Aharony, J. Gomis, and T. Mehen, “On theories with light-like noncommutativity,”
JHEP (2000) 023, hep-th/0006236 .[15] A. Hashimoto and S. Sethi, “Holography and string dynamics in time-dependentbackgrounds,” Phys. Rev. Lett. (2002) 261601, hep-th/0208126 .[16] L. Dolan and C. R. Nappi, “Noncommutativity in a time-dependent background,” Phys. Lett.
B551 (2003) 369–377, hep-th/0210030 .[17] A. Hashimoto and K. Thomas, “Dualities, twists, and gauge theories withnon-constant non-commutativity,”
JHEP (2005) 033, hep-th/0410123 .[18] A. Hashimoto and K. Thomas, “Non-commutative gauge theory on D-branes inMelvin universes,” JHEP (2006) 083, hep-th/0511197 .[19] D. Dhokarh, S. S. Haque, and A. Hashimoto, “Melvin Twists of global AdS × S andtheir Non- Commutative Field Theory Dual,” arXiv:0801.3812 [hep-th] .[20] N. Seiberg, L. Susskind, and N. Toumbas, “Strings in background electric field,space/time noncommutativity and a new noncritical string theory,” JHEP (2000)021, hep-th/0005040 .[21] R. Gopakumar, J. M. Maldacena, S. Minwalla, and A. Strominger, “S-duality andnoncommutative gauge theory,” JHEP (2000) 036, hep-th/0005048 .[22] J. L. F. Barbon and E. Rabinovici, “Stringy fuzziness as the custodian of time-spacenoncommutativity,” Phys. Lett.
B486 (2000) 202–211, hep-th/0005073 .[23] R.-G. Cai, J.-X. Lu, and N. Ohta, “NCOS and D-branes in time-dependentbackgrounds,”
Phys. Lett.
B551 (2003) 178–186, hep-th/0210206 .[24] R.-G. Cai and N. Ohta, “Holography and D3-branes in Melvin universes,”
Phys. Rev.
D73 (2006) 106009, hep-th/0601044 . 925] A. Bergman and O. J. Ganor, “Dipoles, twists and noncommutative gauge theory,”
JHEP (2000) 018, hep-th/0008030 .[26] A. Bergman, K. Dasgupta, O. J. Ganor, J. L. Karczmarek, and G. Rajesh, “Nonlocalfield theories and their gravity duals,” Phys. Rev.
D65 (2002) 066005, hep-th/0103090 .[27] O. J. Ganor and U. Varadarajan, “Nonlocal effects on D-branes in plane-wavebackgrounds,”
JHEP (2002) 051, hep-th/0210035 .[28] O. Lunin and J. M. Maldacena, “Deforming field theories with U (1) × U (1) globalsymmetry and their gravity duals,” JHEP (2005) 033, hep-th/0502086 .[29] O. J. Ganor, “A new Lorentz violating nonlocal field theory from string- theory,” Phys. Rev.
D75 (2007) 025002, hep-th/0609107 .[30] O. J. Ganor, A. Hashimoto, S. Jue, B. S. Kim, and A. Ndirango, “Aspects of puff fieldtheory,”
JHEP (2007) 035, hep-th/0702030 .[31] G. Minton and V. Sahakian, “A new mechanism for non-locality from string theory:UV-IR quantum entanglement and its imprints on the CMB,” arXiv:0707.3786 [hep-th] .[32] N. Arkani-Hamed, A. G. Cohen, D. B. Kaplan, A. Karch, and L. Motl,“Deconstructing (2,0) and little string theories,” JHEP (2003) 083, hep-th/0110146 .[33] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, “Supergravity andthe large N limit of theories with sixteen supercharges,” Phys. Rev.
D58 (1998)046004, hep-th/9802042 .[34] B. Pioline and A. S. Schwarz, “Morita equivalence and T-duality (or B versus Θ),” JHEP (1999) 021, hep-th/9908019 .[35] G. ’t Hooft, “Some twisted selfdual solutions for the Yang-Mills equations on ahypertorus,” Commun. Math. Phys. (1981) 267–275.[36] A. Hashimoto and N. Itzhaki, “On the hierarchy between non-commutative andordinary supersymmetric Yang-Mills,” JHEP (1999) 007, hep-th/9911057 .[37] A. Karch and L. Randall, “Localized gravity in string theory,” Phys. Rev. Lett. (2001) 061601, hep-th/0105108 . 1038] O. DeWolfe, D. Z. Freedman, and H. Ooguri, “Holography and defect conformal fieldtheories,” Phys. Rev.
D66 (2002) 025009, hep-th/0111135 .[39] W. Taylor, “D-brane field theory on compact spaces,”
Phys. Lett.
B394 (1997)283–287, hep-th/9611042 .[40] S. A. Cherkis and A. Hashimoto, “Supergravity solution of intersecting branes andAdS/CFT with flavor,”
JHEP (2002) 036, hep-th/0210105 .[41] P. van Baal, “Some results for SU ( N ) gauge fields on the hypertorus,” Commun.Math. Phys. (1982) 529.[42] Z. Guralnik and S. Ramgoolam, “Torons and D-brane bound states,” Nucl. Phys.
B499 (1997) 241–252, hep-th/9702099 .[43] A. Hashimoto and W. Taylor, “Fluctuation spectra of tilted and intersecting D-branesfrom the Born-Infeld action,”
Nucl. Phys.
B503 (1997) 193–219, hep-th/9703217 .[44] S. R. Das, “Quark fields in twisted reduced large N QCD,”
Phys. Lett.
B132 (1983)155.[45] T. Eguchi and H. Kawai, “Reduction of dynamical degrees of freedom in the large N gauge theory,” Phys. Rev. Lett. (1982) 1063.[46] A. Gonzalez-Arroyo and M. Okawa, “A twisted model for large N lattice gaugetheory,” Phys. Lett.
B120 (1983) 174.[47] H. Levine and H. Neuberger, “A quenched reduction for the topological limit ofQCD,”
Phys. Lett.
B119 (1982) 183.[48] E. Cohen and C. Gomez, “A Computation of Tr( − F in supersymmetric gaugetheories with matter,” Nucl. Phys.
B223 (1983) 183.[49] J. Ambjorn, Y. M. Makeenko, J. Nishimura, and R. J. Szabo, “Nonperturbativedynamics of noncommutative gauge theory,”
Phys. Lett.
B480 (2000) 399–408, hep-th/0002158 .[50] J. Ambjorn, Y. M. Makeenko, J. Nishimura, and R. J. Szabo, “Lattice gauge fields anddiscrete noncommutative Yang- Mills theory,”
JHEP (2000) 023, hep-th/0004147 .[51] K. A. Intriligator and N. Seiberg, “Mirror symmetry in three dimensional gaugetheories,” Phys. Lett.
B387 (1996) 513–519, hep-th/9607207hep-th/9607207