We show that the continuum limit of one-dimensional N=2 supersymmetric matrix
models can be described by a two-dimensional interacting field theory of a
massless boson and two chiral fermions. We interpret this field theory as a
two-dimensional N=1 supersymmetric theory of two chiral superfields, in which
one of the chiral superfields has a non-trivial vacuum expectation value.Comment: Latex document, 11 pages, preprint CERN-TH.7353/94, UPR-621

Brustein, R.

Faux, M.

Ovrut, B.

English

arXiv

We show that the continuum limit of one-dimensional N=2 supersymmetric matrix models can be described by a two-dimensional interacting field theory of a massless boson and two chiral fermions. We interpret this field theory as a two-dimensional N=1 supersymmetric theory of two chiral superfields, in which one of the chiral superfields has a non-trivial vacuum expectation value

Brustein, Ram

Faux, Michael

Ovrut, Burt A.

CERN Document Server

arXiv:hep-th/9407164v1  25 Jul 1994CERN-TH.7353/94, UPR-621TSUPERSYMMETRIC FIELD THEORYFROMSUPERMATRIX MODELSRAM BRUSTEIN∗)Theory Division, CERNCH-1211 Geneva 23, SwitzerlandMICHAEL FAUX∗∗) and BURT A. OVRUT∗∗)Department of PhysicsUniversity of PennsylvaniaPhiladelphia, Pa 19104-6396, USAABSTRACTWe show that the continuum limit of one-dimensional N = 2 supersymmetric matrixmodels can be described by a two-dimensional interacting field theory of a massless bosonand two chiral fermions. We interpret this field theory as a two-dimensional N = 1 su-persymmetric theory of two chiral superfields, in which one of the chiral superfields has anon-trivial vacuum expectation value.∗) Contribution to the Proceedings of the IV International Conference on MathematicalPhysics, String Theory and Quantum Gravity, Rakhov, Ukraine, 12-20 February 1994.∗∗) Work supported in part by DOE under Contract No. DOE-AC02-76-ERO-3071.CERN-TH.7353/94July 1994To use d = 1 matrix models [1, 2] for the purpose of understanding non-perturbativeeffects in superstring theory, it is essential to first construct the complete two-dimensionaleffective Lagrangian for the associated d = 2 superstring theory. We present the effectiveLagrangian and discuss its properties. Our presentation is based on [3]. More details andexplicit calculations are included there.A class of d = 1, N = 2 supermatrix models may be defined using a matrix superfield,Φij =Mij(t) + iθ1Ψ1ij(t) + iθ2Ψ2ij + iθ1θ2Fij(t), (1)where θ1 and θ2 are real anticommuting parameters,Mij and Fij areN×N bosonic Hermitianmatrices and Ψ1ij and Ψ2ij are N ×N fermionic Hermitian matrices. A manifestly invariantLagrangian,L =∫dθ1dθ2{12TrD1ΦD2Φ + iW (Φ)}, (2)can be written, using Φ and the covariant derivativesDI =∂∂θI+ iθI∂∂t, I = 1, 2. (3)The superpotential W is a real polynomial in Φ,W (Φ) =∑nbnTrΦn. (4)In terms of the component functions, Lagrangian (2) is the followingL =∑ij{12(M˙ijM˙ji + FijFji) +∂W (M)∂MijFij}− i2∑ij(Ψ1ijΨ˙1ji +Ψ2ijΨ˙2ji)− i∑ijklΨ1ij∂2W (M)∂Mij∂MklΨ2kl. (5)The supersymmetry transformations of the component functions areδMij = iη1Ψ1ij + iη2Ψ2ijδΨ1ij = η1M˙ij + η2FijδΨ2ij = η2M˙ij − η1FijδFij = iη2Ψ˙1ij − iη1Ψ˙2ij , (6)1where η1 and η2 are anticommuting constants.The classical theory possesses, in addition to supersymmetry, a global U(N) symmetry.To verify this fact note that Φij remains a Hermitian matrix of superfields under the trans-formation Φ → U †ΦU , where U is an arbitrary N × N matrix of complex numbers. TheLagrangian is invariant under such a transformation, provided that U ∈ U(N).We restrict our attention to the sector of the theory that is a singlet under the globalU(N) symmetry. After eliminating the auxiliary fields using their equations of motion, thesinglet sector can be described in terms of the eigenvalues λi of the bosonic matrix M , andtheir fermionic superpartners χi. χi are the diagonal elements of the matrix χ = UΨU†,where U is the matrix used to diagonalize M . Note that U diagonalizes M , but that χ isnot diagonal. The Lagrangian for the singlet sector is given by [4, 3]L =∑i{12λ˙2i −12(∂W∂λi)2− ∂w∂λi∂W∂λi− 12(∂w∂λi)2− i2(χ¯iχ˙i − ˙¯χiχi)}−∑ij{∂2W∂λi∂λjχ¯iχj +∂2w∂λi∂λjχ¯iχj}. (7)The induced superpotential,w = −∑j 6=iln |λi − λj |, (8)represents a repulsive interaction between the bosonic eigenvalues.In preparation for taking the continuum limit, it is useful to change variables, thusdefining three collective fields,ϕ(x, t) =∑iΘ(x− λi(t))ψ(x, t) = −∑iδ(x− λi(t))χi(t)ψ¯(x, t) = −∑iδ(x− λi(t))χ¯i(t). (9)Note that (9) is nothing but a change of variables. It does not increase or decrease thenumber of dynamical variables. In terms of the collective fields,L =∫dx{ϕ˙22ϕ′− 12ϕ′W ′(x)2 +W ′′(x)ϕ′ψ¯ψ2− 12ϕ′(ψ¯ ˙¯ψ + ˙¯ψψ¯) +i2ϕ˙ϕ′2(ψ¯ψ′ − ψ¯′ψ)}+13∫−dxdydzϕ′(x)ϕ′(y)ϕ′(z)(x− y)(x− z) ,+∫−dxdyϕ′(x)ϕ′(y)(x− y) W′(x)+∫− 1(x− y){ψ¯(x)ψ′(y)− ϕ′′(y)ϕ′(x)ψ¯(x)ψ(x)}. (10)The measure of the path integral has, of course, to be changed accordingly. We will not dothat explicitly here, since we will be interested in regions of parameter space in which themeasure takes a simple form.The continuum limit of bosonic matrix models is known to be a two-dimensional fieldtheory [5, 6]. We expect, therefore, that the continuum limit of supermatrix models is atwo-dimensional field theory as well. However, we will see that the number of fields thatsurvive in the continuum limit is larger in the supersymmetric case. Taking the continuumlimit consists of a few separate and independent steps which supply the original supermatrixmodels with additional information and should be considered as part the definition of thetheory. At each step some choices have to be made, each determines essential properties ofthe resulting models. It is at this juncture that the field content and specific background arechosen. Previously, some attempts were made, with varying degree of success, to obtain thecorrect two-dimensional continuum field theory [4, 7–10]. The first step, necessary to ensurethat the number of dynamical variables is enough to describe a two-dimensional field theoryis simplyN →∞. (11)It is useful to think about N as the cutoff, in momentum space, of the theory. Then aregularization procedure has to be chosen to ensure that all terms in the Lagrangian arefinite as the cutoff is taken to infinity. We use−∫ +∞−∞dxφ(x)x− a = ±ipiφ(a) (12)to define our regularization scheme. In addition, some dependence of the coupling parametersin the superpotential W , on the cutoff N , has to be chosen, placing the theory within a3specific universality class. On general grounds, W = NW ( x√N). Our choice is the followingW (x) =√Nc1x+16c3√Nx3 + · · ·c1c3 < 0. (13)The terms denoted by · · · are of higher power in x/√N and do not change the universalityclass for non-vanishing ci, i = 1, 3. For completeness we list some useful expressions for thederivatives of WW ′(x) =√Nc1 +12c3√Nx2 + · · · ,W ′(x)2 = Nc21 + c1c3x2 +13c1c4√Nx3 + · · · ,W ′′(x) =c3√Nx+ · · · . (14)The result of applying all the steps above to the Lagrangian (10), taking (14) into account,is the continuum LagrangianL =∫dx{ϕ˙22ϕ′± pi26ϕ′3 +12ω2x2ϕ′− i2ϕ′(ψ1ψ˙1 + ψ2ψ˙2)± ipi2ψ1ψ′1 ±ipi2ψ2ψ′2+i2ϕ˙ϕ′2(ψ1ψ′1 + ψ2ψ′2)}. (15)Note that there are still three ambiguous signs in the previous Lagrangian, related to the signambiguity in (12). For the first sign we choose a minus sign, corresponding to Minkowskispacetime. Our choice is a minus sign for the second and a plus sign for the third. Thischoice determines the chiralities of the fermions.A classical static solution of the equations of motion derived from (15) is given byψ10 = 0ψ20 = 0ϕ′0 =1pi√ω2x2 − 1/g. (16)4We expand around that classical solutionϕ = ϕ0(x) +1√piζψ+ =21/4√piψ1ψ− =21/4√piψ2 (17)and change coordinates,τ ′(x) =1pi(ϕ′0(x))−1=1√ω2x2 − 1/g, (18)to obtainL =∫dτ{12(ζ˙2 − ζ ′2)− i√2(ψ+ψ˙+ − ψ+ψ′+)−i√2(ψ−ψ˙− + ψ−ψ′−)−12g(τ)ζ˙2ζ ′1 + g(τ)ζ ′− 16g(τ)ζ ′3+i√2g(τ)ζ ′1 + g(τ)ζ ′(ψ+ψ˙+ + ψ−ψ˙−)+i√2g(τ)ζ˙(1 + g(τ)ζ ′)2(ψ+ψ′+ + ψ−ψ′−)}+13∫dτ1g(τ)2. (19)The coupling parameter of the theory varies in spaceg(τ) = 4√pig1κe−2ω(τ−τ0)(1− 1κe−2ω(τ−τ0))2. (20)We are now in a position to take stock of the field content of the theory. Looking atthe quadratic terms in the first line of Eq.(19), we observe that the theory contains onemassless bosonic field ζ , and two chiral massless fermions ψ±. The chiralities of the fermionsare determined by the choice of signs in (15). If we choose them as we did they haveopposite chiralities and the field content can be fitted within a chiral superfield of a (1, 1)two-dimensional supersymmetry.The Lagrangian (19) is not supersymmetric. It is not even Poincare invariant. Moti-vated by the expected relation to string theory, and based on our experience in interpreting5the bosonic theory [12, 11], we interpret it as follows. We assume that the theory reallystarted out as a two-dimensional supersymmetric theory, containing two superfields, Φ1 andΦ2. The superfield Φ2 obtains a non-trivial vacuum expectation value (VEV). The VEVbreaks Poincare invariance as well as supersymmetry. Our task then becomes to reconstructthe original theory as best as we can. As will become obvious, it is not possible to recon-struct the theory completely. We can, however, capture enough of its features to make thereconstruction an interesting enterprise.Of the two chiral superfields of (1, 1) supersymmetry,Φ1 = ζ + iθ+ψ+ + iθ−ψ− + iθ+θ−ZΦ2 = α+ iθ+χ+ + iθ−χ− + iθ+θ−A, (21)it is Φ1 that contains the degrees of freedom in the original Lagrangian. It is straightforwardto write a manifestly invariant kinetic term for Φ1,L(eff)01 =∫dθ+dθ−D+Φ1D−Φ1=12(ζ˙2 − ζ ′2)− iψ+∂−ψ+ − iψ−∂+ψ− + Z2. (22)Doing the same for the other superfield is a little bit more involved procedure. We expect thesuperfield Φ2 to acquire a non-trivial VEV. Based on our experience in the interpretationof the bosonic theory and motivated by the expected relation with other formulations oftwo-dimensional superstring theory, we expect the components of Φ2 to obtain the followingVEV< α > = e−ω|τ−τ0|< χ± > = 0< A > = 0. (23)Furthermore, we impose that the most singular term in the Lagrangian has a 1/α4 depen-dence, in agreement with classical string theory. We now desire a manifestly supersymmetricLagrangian that (i) has (23) as a solution to its equations of motion, (ii) when the VEV,(23), of Φ2 is substituted into the Lagrangian, it reduces to the constant term in (19), and6(iii) the most singular term in the Lagrangian has a 1/α4 dependence. A solution with thedesired properties is given byL(eff)02 =∫dθ+dθ−{F1(Φ2)D+Φ2D−Φ2 − 1ω2F2(Φ2)∂−D+Φ2∂+D−Φ2}= F1(α)∂+α∂−α− 1ω2F2(α)(∂+∂−α)2+F1(α)A2 − 1ω2F2(α)∂+A∂−A−iF1(α)χ+∂−χ+ − iω2F2(α)∂−χ+∂+∂−χ+−iF1(α)χ−∂+χ− − iω2F2(α)∂+χ−∂−∂+χ−, (24)whereF1(Φ2) = − 148piκω2g2(115κ3Φ62− 283κ2Φ42+ 18κΦ22− 4 + 53Φ22κ)F2(Φ2) = − 148piκω2g2(−25κ3Φ62+83κ2Φ42− 12 κΦ22− 8 + 23Φ22κ). (25)This Lagrangian indeed has (23) as a solution of its equation of motion. Obviously, (24)is not the unique Lagrangian with the desired properties. However, it is the Lagrangian withthe least number of terms. We therefore choose to present it. The fact that we were able tofind any solution to our requirements is not at all trivial.So far, we were able to construct a manifestly supersymmetric theory, to lowest orderin the coupling g(τ), using the two superfields and their covariant derivatives. Amazinglyenough, there exist a manifestly supersymmetric Lagrangian that reduces to the full non-linear interacting two-dimensional field theory. The details of the derivation are given in [3].We give the final result here,L(eff) =∫dθ+dθ−{D+Φ1D−Φ1+F1(Φ2)D+Φ2D−Φ2 − 1ω2F2(Φ2)∂−D+Φ2∂+D−Φ2−f(Φ2)ω3Φ32∂(+Φ1∂−)Φ2∂[+Φ1∂−]Φ21 + f(Φ2)ωΦ2∂(+Φ1∂−)Φ2D(+Φ1D−)Φ2+13f(Φ2)ω5Φ52(∂[+Φ1∂−]Φ2)3D+Φ2D−Φ27−f(Φ2)ω5Φ52(∂[+Φ1∂−]Φ2)2∂(+Φ1∂−)Φ21 + f(Φ2)ωΦ2∂(+Φ1∂−)Φ2D+Φ2D−Φ2}, (26)wheref(Φ2) = 4√pig1κΦ22(1− 1κΦ22)2. (27)In components, (26) is given byL(eff) = +∂+ζ∂−ζ + Z2 − iψ+∂−ψ+ − iψ−∂+ψ−+F1(α)∂+α∂−α− 1ω2F2(α)(∂+∂−α)2+F1(α)A2 − 1ω2F2(α)∂+A∂−A−iF1(α)χ+∂−χ+ − iω2F2(α)∂−χ+∂+∂−χ+−iF1(α)χ−∂+χ− − iω2F2(α)∂+χ−∂−∂+χ−+∑nO(αnχ+χ− + αn−1Aχ+χ−).−12f(α)ζ˙2ζ ′1 + f(α)ζ ′− 16f(α)ζ′3+i√2f(α)ζ ′1 + f(α)ζ ′(ψ+ψ˙+ + ψ−ψ˙−) +i√2f(α)ζ˙[1 + f(α)ζ ′]2(ψ+ψ′+ + ψ−ψ′−)+O{∂ζ(ψχ+ χχ + Zψχ+ Zχχ+ Aψψ + Aψχ) + ψψχ+ ψχχ}. (28)As can be checked, the general solution of the equations of motion derived from (26) is thefollowing,< α > = exp{ω[|t− t0| sinh θ0 − |τ − τ0| cosh θ0]}< ζ > = constant< χ± > = η±0 < α >< ψ± > = 0< A > = 0< Z > = 0. (29)8If this solution is substituted back into (26) and the auxiliary fields are eliminated throughtheir equations of motion, the result exactly reproduces (19).The Lagrangian (26) has some interesting properties. First, the superfield Φ1 has onlyderivative interactions, and so, in particular, has no superpotential. The interactions of thesuperfield Φ2 always contain some derivatives, therefore the superfield Φ2 has no superpo-tential as well. The coupling parameter of the theory is field-dependent. This is a typicalsituation in low-energy effective field theories of string theory. The overall coupling strengthis determined by the parameter g, which is sometimes called the “string coupling constant”.However, if Φ2 has a space-dependent VEV, as in (23), the coupling strength varies in space-time and even blows up at some finite point, signalling the possible existence of new physicalphenomena.References[1] D. J. Gross and N. Miljkovic, Phys. Lett. B238 (1990) 217;P. Ginsparg and J. Zinn-Justin, Phys. Lett. B240 (1990) 333 ;E. Brezin, V. Kazakov and Al. Zamolodchikov, Nucl. Phys. B338 (1990) 673.[2] E. Marinari and G. Parisi, Phys. Lett. B247 (1990) 537.[3] R. Brustein, M. Faux and B. Ovrut, CERN preprint, CERN-TH.7013/93, Talk pre-sented at the international Europhysics Conference on High Energy Physics, Marseille,France, July 22-28, 1993;R. Brustein, M. Faux and B. Ovrut, Nucl. Phys. B421 (1994) 293;R. Brustein, M. Faux and B. Ovrut, CERN preprint, CERN-TH.7051/93, Talk pre-sented at the International Workshop on Supersymmetry and Unification of Fundamen-tal Interactions (SUSY93), Boston, MA, Mar. 29 - Apr. 1, 1993.[4] A. Dabholkar, Nucl. Phys. B368 (1992) 293.[5] S. R. Das and A. Jevicki, Mod. Phys. Lett. A5 (1990) 1639.9[6] A. Jevicki, Brown preprint, HET-918/TA-502 (1993).[7] A. Jevicki and J. P. Rodrigues, Phys. Lett. B268 (1991) 53.[8] J. P. Rodrigues and J. van Tonder, Int. J. Mod. Phys.A8 (1993) 2517.[9] J. Cohn and H. Dykstra, Mod. Phys. Lett. A7 (1992) 1163.[10] J. Feinberg, Technion preprint, TECHNION-PH-92-35 (1992).[11] R. Brustein and S. DeAlwis, Phys. Lett. B272 (1991) 285.[12] R. Brustein and B. Ovrut, Phys. Lett. B309 (1993) 45;R. Brustein and B. Ovrut, Pennsylvania preprint, UPR-523T (1992);R. Brustein and B. Ovrut, Proceedings of the 26th International Conference on HighEnergy Physics (ICHEP 92), Dallas, TX, 6-12 Aug 1992, pp. 1485-1490.10

Supersymmetric Field Theory from Supermatrix Models

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