Two-dimensional lattice for four-dimensional N=4 supersymmetric Yang-Mills
aa r X i v : . [ h e p - l a t ] A ug Two-dimensional lattice for four-dimensional N = 4supersymmetric Yang-Mills Masanori
Hanada , , ∗ ) , So Matsuura , ∗∗ ) and Fumihiko Sugino , ∗∗∗ )1 Department of Particle Physics and Astrophysics, Weizmann Institute of Science,Rehovot 76100, Israel Department of Physics, University of Washington, Seattle, WA 98195-1560, USA Department of Physics, and Research and Education Center for Natural Science,Keio University, 4-1-1 Hiyoshi, Yokohama, 223-8521, Japan Okayama Institute for Quantum Physics, Kyoyama 1-9-1, Kita-ku, Okayama700-0015, Japan
We construct a lattice formulation of a mass-deformed two-dimensional N = (8 ,
8) superYang-Mills theory with preserving two supercharges exactly. Gauge fields are represented bycompact unitary link variables, and the exact supercharges on the lattice are nilpotent up togauge transformations and SU (2) R rotations. Due to the mass deformation, the lattice modelis free from the vacuum degeneracy problem, which was encountered in earlier approaches,and flat directions of scalar fields are stabilized giving discrete minima representing fuzzy S .Around the trivial minimum, quantum continuum theory is obtained with no tuning, whichserves a nonperturbative construction of the IIA matrix string theory. Moreover, aroundthe minimum of k -coincident fuzzy spheres, four-dimensional N = 4 U ( k ) super Yang-Millstheory with two commutative and two noncommutative directions emerges. In this theory,sixteen supersymmetries are broken by the mass deformation to two. Assuming the breakingis soft, we give a scenario leading to undeformed N = 4 super Yang-Mills on R without anyfine tuning. As an evidence for the validity of the assumption, some computation of 1-loopradiative corrections is presented. §
1. Introduction
Nonperturbative aspects of supersymmetric Yang-Mills (SYM) theories playprominent roles in physics beyond the standard model as well as in superstring/Mtheory. However, to construct their nonperturbative formulations such as latticeis not a straightforward task because of the notorious difficulties of supersymmetry(SUSY) on lattice. So far, lattice formulations for SYM are constructed for one- andtwo-dimensional cases and N = 1 pure SYM in three and four dimensions, whereno requirement of fine tunings due to the ultra-violet (UV) effects can be shown atleast in perturbative arguments. For one-dimensional theory (matrix quantum me-chanics) more powerful “non-lattice” technique is applicable. (For correspondinglattice study, see Ref. 8).) For two-dimensional N = (2 ,
2) SYM, nonperturbative ev-idences for the lattice model presented in Ref. 9) to require no fine tuning have beengiven by numerical simulation for the gauge group G = SU (2) in Ref. 10) and for ∗ ) E-mail: [email protected] ∗∗ ) E-mail: [email protected] ∗∗∗ ) E-mail: fumihiko [email protected] typeset using
PTP
TEX.cls h Ver.0.9 i M. Hanada, S. Matsuura and F. Sugino G = SU ( N ) with N = 2 , , , ∗ ) . Combining such techniques with theplane wave deformation and the Myers effect, three-dimensional theory can beobtained as a theory on fuzzy sphere. Also, in the planar limit, four-dimensionaltheory can be obtained using a novel large- N reduction technique inspired by theEguchi-Kawai equivalence. However, four-dimensional theories of extended SUSYat a finite rank of a gauge group are still out of reach.We consider N = (8 ,
8) SYM with a mass deformation reminiscent of the “planewave matrix string” in the next section, and construct a lattice formulationof the theory preserving two supercharges ∗∗ ) in section 3. It is a modification of alattice formulation by one of the authors (F. S.). (For related constructions,see Refs. 23)–25).) Thanks to the deformation, it is free from the vacuum degeneracyproblem encountered in Ref. 22) as well as from the problem of flat directions. Ina perturbative argument, it is shown that the continuum theory is obtained with-out any fine tuning. If we turn off the mass parameter in the continuum theory,undeformed N = (8 ,
8) SYM in two dimensions, i.e. the IIA matrix string theory is obtained. In section 4, we furthermore present an intriguing scenario leadingto four-dimensional N = 4 SYM with a finite-rank gauge group G = U ( k ), start-ing with the lattice formulation of the two-dimensional theory. In order to realizethe four-dimensional theory, we consider a k -coincident fuzzy sphere solution in themass-deformed two-dimensional theory and take a large- N limit with scaling thedeformation parameter appropriately. The crucial point is that one takes the con-tinuum limit as two-dimensional theory first ∗∗∗ ) and then lifts the two-dimensionalcontinuum theory to four dimensions using a matrix model formulation of noncom-mutative space. Section 5 is devoted to discuss possible future directions. §
2. 2d continuum theory
We start with continuum N = (8 ,
8) SYM on R : S = 1 g d Z d x Tr (cid:26) F + (cid:0) D µ X I (cid:1) − (cid:2) X I , X J (cid:3) + Ψ T ( D + γ D ) Ψ + iΨ T γ I (cid:2) X I , Ψ (cid:3) (cid:27) , (2.1)where µ = 1 , I = 3 , · · · ,
10, and D µ = ∂ µ + i [ A µ , · ]. 16 ×
16 gamma matrices γ i ( i = 2 , · · · ,
10) satisfy { γ i , γ j } = − δ ij .We rewrite this action by using Hermitian scalars X i ( i = 3 , , B A ( A = 1 , , C , complex scalars φ ± , bosonic auxiliary fields H A , ˜ H µ , ˜ h i , and fermionic vari-ables ψ ± µ , ρ ± i , χ ± A and η ± . As presented in appendix A, scalars ( B A , C, φ ± ) and the ∗ ) Recently, Ref. 12) has shown that the model constructed in Ref. 13) is free from the signproblem and gives the same physics as that in Ref. 9) after an appropriate treatment of the overall U (1) modes. ∗∗ ) For simplicity, we focus on the gauge group G = U ( N ), although the similar argument isvalid also for G = SU ( N ). ∗∗∗ ) Similar asymmetric continuum limit is discussed on four-dimensional lattice in order toreduce the number of fine tunings.
D lattice for 4D N = 4 SYM X I ( I = 5 , · · · ,
10) and spinor components of Ψ bya simple rearrangement, respectively. There are appropriate supercharges Q (0) ± bywhich S can be written in exact form ∗ ) as S = Q (0)+ Q (0) − F (0) , (2.2)where F (0) = 1 g d Z d x Tr n − iB A Φ A − ǫ ABC B A [ B B , B C ] − ψ + µ ψ − µ − ρ + i ρ − i − χ + A χ − A − η + η − o , (2.3)and Φ = 2( − D X − D X ), Φ = 2( − D X + D X ), Φ = 2( − F + i [ X , X ]).Supercharges Q (0) ± transform fields as Q (0) ± A µ = ψ ± µ , Q ± ψ ± µ = ± iD µ φ ± , Q (0) ∓ ψ ± µ = i D µ C ∓ ˜ H µ ,Q (0) ± ˜ H µ = [ φ ± , ψ ∓ µ ] ∓
12 [
C, ψ ± µ ] ∓ i D µ η ± ,Q (0) ± X i = ρ ± i , Q (0) ± ρ ± i = ∓ [ X i , φ ± ] , Q (0) ∓ ρ ± i = −
12 [ X i , C ] ∓ ˜ h i ,Q (0) ± ˜ h i = [ φ ± , ρ ∓ i ] ∓
12 [
C, ρ ± i ] ±
12 [ X i , η ± ] ,Q (0) ± B A = χ ± A , Q (0) ± χ ± A = ± [ φ ± , B A ] , Q (0) ∓ χ ± A = −
12 [ B A , C ] ∓ H A ,Q (0) ± H A = [ φ ± , χ ∓ A ] ±
12 [ B A , η ± ] ∓
12 [
C, χ ± A ] ,Q (0) ± C = η ± , Q (0) ± η ± = ± [ φ ± , C ] , Q (0) ∓ η ± = ∓ [ φ + , φ − ] ,Q (0) ± φ ± = 0 , Q (0) ∓ φ ± = ∓ η ± . (2.4)One can see the nilpotency (cid:16) Q (0)+ (cid:17) = (cid:16) Q (0) − (cid:17) = { Q (0)+ , Q (0) − } = 0 up to gaugetransformations.2.1. Mass deformation
Next, we introduce a mass M to deform these charges as Q ± = Q (0) ± + ∆Q ± , (2.5)where non-vanishing ∆Q ± transformations are ∆Q ± ˜ H µ = M ψ ± µ , ∆Q ± ˜ h i = M ρ ± i , ∆Q ± H A = M χ ± A ,∆Q ± η ± = 2 M φ ± , ∆Q ∓ η ± = ± M C. (2.6) ∗ ) This is obtained from BTFT formulation of four-dimensional N = 4 SYM in Ref. 21) bydimensional reduction. Here, we redefine H A + ǫ ABC [ B B , B C ], φ , ¯ φ in (4.13) in Ref. 21) as H A , φ + , φ − , respectively. M. Hanada, S. Matsuura and F. Sugino
Then Q ± satisfy the anti-commutation relations, Q = M J ++ , Q − = − M J −− , { Q + , Q − } = − M J , (2.7)up to gauge transformations, where J , J ++ and J −− are generators of SU (2) R symmetry acting to fields as J ++ = Z d x " ψ α + µ ( x ) δδψ α − µ ( x ) + χ α + A ( x ) δδχ α − A ( x ) − η α + ( x ) δδη α − ( x )+2 φ α + ( x ) δδC α ( x ) − C α ( x ) δδφ α − ( x ) (cid:21) ,J −− = Z d x " ψ α − µ ( x ) δδψ α + µ ( x ) + χ α − A ( x ) δδχ α + A ( x ) − η α − ( x ) δδη α + ( x ) − φ α − ( x ) δδC α ( x ) + C α ( x ) δδφ α + ( x ) (cid:21) ,J = Z d x " ψ α + µ ( x ) δδψ α + µ ( x ) − ψ α − µ ( x ) δδψ α − µ ( x )+ χ α + A ( x ) δδχ α + A ( x ) − χ α − A ( x ) δδχ α − A ( x ) + η α + ( x ) δδη α + ( x ) − η α − ( x ) δδη α − ( x )+2 φ α + ( x ) δδφ α + ( x ) − φ α − ( x ) δδφ α − ( x ) (cid:21) . (2.8)( α is an index for the gauge group generators.) The eigenvalues of J are ± ± , ± φ ± , and zero for the other bosonic fields. Note that φ ± and C form an SU (2) R triplet and each pair of ( ψ + µ , ψ − µ ), ( χ + A , χ − A ), ( η + , − η − )and ( Q + , Q − ) forms a doublet. In particular,[ J ±± , Q ± ] = 0 , [ J ±± , Q ∓ ] = Q ± , [ J , Q ± ] = ± Q ± . (2.9)Using the modified supercharges, we can define Q ± -closed action as ∗ ) S = (cid:18) Q + Q − − M (cid:19) F , (2.10)where F = F (0) + ∆ F , ∆ F = 1 g d Z d x Tr X A =1 a A B A + X i =3 c i X i ! . (2.11)That the action (2.10) is Q ± -closed can easily be seen by using (2.7), (2.9) and SU (2) R invariance of F . After integrating out the auxiliary fields, B A and X i havepositive mass terms as long as the parameters a A and c i all lie in the open interval ∗ ) This kind of deformation is extended to various SYM theories in Ref. 27).
D lattice for 4D N = 4 SYM − M/ , a = a = a = − M and c = c = − M for convenience.Then the action reads S = S + ∆S, (2.12)where ∆S = 1 g d Z d x Tr n M (cid:0) B A + X i (cid:1) + M (cid:18) C φ + φ − (cid:19) − M C [ φ + , φ − ]+ 2 M ψ + µ ψ − µ + 2 M ρ + i ρ − i + 4 M χ + A χ − A − M η + η − − iM B ( F + i [ X , X ]) o . (2.13)From this expression one can see some similarity to the plane wave matrix model and to PP wave matrix strings. The first two terms in the first line and the termsin the second line in (2.13) give mass terms to scalars and to fermions, respectively.The third term represents the so called Myers term.
Thanks to these terms, fuzzy S configurations satisfying[ φ + , φ − ] = M C, [ C, φ ± ] = ± M φ ± , B A = X i = 0 (2.14)give the minima of the action ( S = 0) preserving Q ± SUSYs. Note that the lastterm in (2.13) is purely imaginary. Also, we should recognize that the mass-deformedaction preserves only two supercharges ( Q ± ) but the other 14 charges are softlybroken by the deformation. §
3. Lattice formulation
In this section we put the deformed theory on a two-dimensional square lattice.We use link variables U µ = e iaA µ belonging to the gauge group U ( N ), where a isthe lattice spacing. Other lattice fields, defined on sites, are made dimensionless bymultiplying suitable powers of a to the continuum counterparts:(scalars) lat = a (scalars) cont , (fermions) lat = a / (fermions) cont ,Q lat ± = a / Q cont ± . (3.1)Also, dimensionless coupling constants on the lattice are g = ag d , M = aM. (3.2)The supersymmetry transformations are realized as Q ± U µ ( x ) = iψ ± µ ( x ) U µ ( x ) ,Q ± ψ ± µ ( x ) = iψ ± µ ( x ) ψ ± µ ( x ) ± iD µ φ ± ( x ) ,Q ∓ ψ ± µ ( x ) = i { ψ + µ ( x ) , ψ − µ ( x ) } + i D µ C ( x ) ∓ ˜ H µ ( x ) , M. Hanada, S. Matsuura and F. Sugino Q ± ˜ H µ ( x ) = − h ψ ∓ µ ( x ) , φ ± ( x ) + U µ ( x ) φ ± ( x + ˆ µ ) U µ ( x ) † i ± h ψ ± µ ( x ) , C ( x ) + U µ ( x ) C ( x + ˆ µ ) U µ ( x ) † i ∓ i D µ η ± ( x ) ±
14 [ ψ ± µ ( x ) ψ ± µ ( x ) , ψ ∓ µ ( x )]+ i h ψ ± µ ( x ) , ˜ H µ ( x ) i + M ψ ± µ , (3.3)for the lattice fields U µ , ψ ± µ and ˜ H µ . D µ is a covariant forward difference operatordefined by D µ A ( x ) ≡ U µ ( x ) A ( x + ˆ µ ) U µ ( x ) † − A ( x ) , (3.4)for any adjoint field A ( x ). Transformation of the other fields is the same as the one incontinuum with the obvious replacement M → M . Then the anti-commutation re-lation (2.7) holds on the lattice with M → M . In order to construct a correspondinglattice action, we take lattice counterparts of Φ A as Φ ( x ) = 2 ( − D X ( x ) − D X ( x )) ,Φ ( x ) = 2 ( − D ∗ X ( x ) + D ∗ X ( x )) ,Φ ( x ) = i ( U ( x ) − U ( x ))1 − ǫ − || − U ( x ) || + 2 i [ X ( x ) , X ( x )] , (3.5)where D ∗ µ is a covariant backward difference operator, D ∗ µ A ( x ) ≡ A ( x ) − U µ ( x − ˆ µ ) † A ( x − ˆ µ ) U µ ( x − ˆ µ ) , (3.6) U µν ( x ) = U µ ( x ) U ν ( x + ˆ µ ) U µ ( x + ˆ ν ) † U ν ( x ) † is a plaquette variable, ǫ is a positiveconstant satisfying 0 < ǫ <
2, and the norm of a matrix is defined by || A || = p Tr( AA † ). The first term of the r.h.s. of Φ ( x ) is a lattice counterpart of thefield strength F . It is the same as the situation in the lattice formulation for two-dimensional N = (2 , U ( N ) SYM in Ref. 9). Q ± -invariant lattice action is givenas S lat = (cid:18) Q + Q − − M (cid:19) F lat (3.7)with F lat being the same form as F in (2.11) under the trivial replacement g d R d x → g P x , M → M , when the admissibility condition || − U ( x ) || < ǫ is satisfied for ∀ x . Otherwise, S lat = + ∞ .Note that in Eqs. (3.5) the covariant forward difference is used for Φ , while thecovariant backward difference is used for Φ . With this choice, no species doublerappears in both of bosonic and fermionic kinetic terms. Note also that the fuzzysphere solution of the lattice version of the equations (2.14) C = 2 M L , φ ± = M L ± iL ) , B A = X i = 0 (3.8)preserves Q ± SUSYs at regularized level, where L a ( a = 1 , ,
3) belong to an N -dimensional representation of SU (2) generators satisfying [ L a , L b ] = iǫ abc L c . D lattice for 4D N = 4 SYM
No unphysical degenerate minima
Here, we will check that the lattice action has the minimum only at the puregauge configuration U ( x ) = 11 N , which guarantees that the weak field expansion U µ ( x ) = 1 + iaA µ ( x ) + ( ia ) A µ ( x ) + · · · is allowed in the continuum limit so thatthe lattice theory converges to the desired continuum theory at the classical level.After integrating out the auxiliary fields, bosonic part of the action S lat takesthe form S ( B )lat = 1 g X x tr (cid:20) M (cid:0) X i ( x ) + B A ( x ) (cid:1) − i M B ( x ) Φ ( − )3 ( x ) (cid:21) + S PDT (3.9)where Φ ( − )3 ( x ) is Φ ( x ) in (3.5) with the sign of the first term flipped, and S PDT denotes positive (semi-)definite terms. We will treat the second term, which is purelyimaginary, as an operator in the reweighting method, and consider the minimum ofthe remaining part of S ( B )lat . The mass terms in (3.9) fix the minimum at X i ( x ) = B A ( x ) = 0 , (3.10)which is independent of S PDT . At the minimum (3.10), S PDT becomes S PDT = 1 g X x tr "X µ ( D µ X p ( x )) + (cid:18) i [ X p ( x ) , X q ( x )] + M ǫ pqr X r ( x ) (cid:19) + 14 g X x tr (cid:2) − ( U ( x ) − U ( x )) (cid:3)(cid:0) − ǫ || − U ( x ) || (cid:1) (3.11)with C = 2 X , φ ± = X ± iX and p, q, r = 8 , ,
10. Since the last term representingthe gauge kinetic term is the same as in the case of two-dimensional N = (2 ,
2) SYMdiscussed in Ref. 9), the admissibility condition with 0 < ǫ < U ( N ) singles out the trivial minimum U ( x ) = 11 N . It shows that the lattice actionhas a stable physical vacuum and unphysical degeneracies of vacua seen in the formerformulation do not appear. The mass deformation preserving Q ± SUSYs is crucialto stabilize flat directions of scalars as well as to remove the unphysical minima forgauge fields.3.2.
No need of fine tuning
Next, we discuss in the perturbation theory that the desired quantum continuumtheory is obtained without any fine tuning.In the theory near the continuum limit with the auxiliary fields integrated out,let us consider local operators of the type: O p ( x ) = M m ϕ ( x ) α ∂ β ψ ( x ) γ , p ≡ m + α + β + 3 γ (3.12)where ϕ denotes scalar fields as well as gauge fields, ψ fermionic fields, and ∂ deriva-tives. The mass dimension of O p is p , and m, α, β, γ = 0 , , , · · · . From dimensionalanalysis, it can be seen that radiative corrections from UV region of loop momenta M. Hanada, S. Matsuura and F. Sugino to O p have the form (cid:18) g d c a p − + c a p − + g d c a p + · · · (cid:19) Z d x O p ( x ) , (3.13)up to possible powers of ln( aM ). c , c , c are dimensionless numerical constants.The first, second and third terms in the parenthesis are contributions from tree,1-loop and 2-loop effects, respectively. The “ · · · ” is effects from higher loops, whichare irrelevant for the analysis.Since relevant or marginal operators generated by loop effects possibly appearfrom nonpositive powers of a in the second and third terms in (3.13), we should seeoperators with p = 0 , ,
2. They are ϕ , M ϕ and ϕ . (Note that 11, M , M and ∂ϕ are not dynamical.) Candidates for ϕ are linear combinations of tr X i and tr B A from gauge and SU (2) R symmetries. But, all of them are not invariant under Q ± SUSYs, and thus are forbidden to appear. Similarly,
M ϕ and ϕ are not allowed tobe generated.Therefore, in the perturbative argument, we can conclude that any relevantor marginal operators except nondynamical operators do not appear radiatively,meaning that no fine tuning is required to take the continuum limit.3.3. Matrix String theory
The mass-deformed N = (8 ,
8) SYM in two dimensions can be obtained fromthe constructed lattice theory around the trivial minimum C = φ ± = 0 as seen inthe previous section. Since M is a soft mass breaking 16 SUSYs to Q ± , undeformedtheory, which is nothing but the IIA matrix string theory, can be defined by turningoff M after the continuum limit. §
4. 4d N = 4 SYM In this section, we discuss a scenario to obtain four-dimensional N = 4 SYMfrom the lattice formulation given in the previous section.Let us consider the lattice theory expanded around the minimum of k -coincidentfuzzy S given by (3.8) with L a = L ( n ) a ⊗ k ( a = 1 , ,
3) and N = nk. (4.1) L ( n ) a are SU (2)-generators of an n (= 2 j + 1)-dimensional irreducible representationof spin j .First, we take the continuum limit of the two-dimensional lattice theory. Then,we obtain four-dimensional N = 4 U ( k ) SYM on R × (Fuzzy S ) with 16 SUSYsbroken to Q ± by M ∗ ) . The fuzzy S has the radius R = 3 /M , and its noncommu-tativity (fuzziness) is characterized by the parameter Θ = M n . UV cutoff in the S directions is set at Λ = M · j . These properties of the fuzzy S are seen by doinga similar calculation as presented in Refs. 16), 17), 28). In particular, momentum ∗ ) Due to the infinite volume of R , tunnelling among discrete minima of various fuzzy spheresolutions is suppressed to stabilize each fuzzy sphere background. D lattice for 4D N = 4 SYM B A , on two dimensions are expanded further by fuzzy sphericalharmonics: ˜ B A ( q ) = j X J =0 J X m = − J ˆ Y ( jj ) J m ⊗ b A , J m , (4.2)corresponding to the expression (4.1). The fuzzy spherical harmonic ˆ Y ( jj ) J m is an n × n matrix whose elements are given by Clebsch-Gordon (C-G) coefficients as ˆ Y ( jj ) J m = √ n j X r,r ′ = − j ( − − j + r ′ C J mj r j − r ′ | j r ih j r ′ | (4.3)with an orthonormal basis | j r i representing L ( n ) a in the standard way: (cid:16) L ( n )1 ± iL ( n )2 (cid:17) | j r i = p ( j ∓ r )( j ± r + 1) | j r ± i ,L ( n )3 | j r i = r | j r i , (4.4)and the modes b A , J m are k × k matrices. It is seen that the fuzzy spherical harmonicsare eigen-modes of the Laplacian on the fuzzy S : X a =1 (cid:18) M (cid:19) [ L ( n ) a , [ L ( n ) a , ˆ Y ( jj ) J m ]] = (cid:18) M (cid:19) J ( J + 1) ˆ Y ( jj ) J m , (4.5)giving the rotational energy with the angular momentum J on the sphere of theradius R = 3 /M . The UV cutoff Λ = M · j can be read off from the upperlimit of the sum of J in the expansion (4.2). The fuzzy S is a two-dimensionalnoncommutative (NC) space, which is analogous to the phase space of some one-dimensional quantum system, and the noncommutativity Θ to the Planck constant ~ . The quantum phase space is divided into small cells of the size 2 π ~ , whose numberis equal to the dimension of the Hilbert space. Correspondingly, the area of the S is divided into n cells of the size 2 πΘ :4 πR = n · πΘ, (4.6)leading to the value Θ = M n .Notice, differently from the two-dimensional case, it is not clear whether theSUSY breaking by M is soft, because M appears not only in mass terms in theaction but also in the geometry of the fuzzy S . Let us proceed assuming thatthe breaking is soft ∗ ) . We will give some argument later for the validity of theassumption.Next, we take successive limits by following the two steps: • Step 1 : Take large n limit with Θ and k fixed. Namely, M ∝ n − / → Λ ∝ n / → ∞ . • Step 2 : Send Θ to zero. ∗ ) The assumption is plausible from the viewpoint of the mapping rule between matrix modeland Yang-Mills theory on noncommutative space. M. Hanada, S. Matsuura and F. Sugino
Step 1
At the step 1, the fuzzy S is decompactified to the NC Moyal plane R Θ . Fromthe assumption, the theory becomes N = 4 U ( k ) SYM on R × R Θ with
16 SUSYsrestored . The gauge coupling constant of the four-dimensional theory is given in theform g d = 2 πΘg d . (4.7)In the limit, the expansion (4.2) by the fuzzy spherical harmonics can be essentiallytranscribed to the one by plane waves on R Θ :˜ B A ( q ) = Z d ˜ q (2 π ) e i ˜ q · ˆ x ⊗ ˜ b A ( q ) , (4.8)where q and ˜ q are two-momenta on R and R Θ respectively, the position operator ˆ x =(ˆ x , ˆ x ) on R Θ satisfies [ˆ x , ˆ x ] = iΘ , and q ≡ ( q, ˜ q ) represents a four-momentum.The modes ˜ b A ( q ) in the four-dimensional space are k × k matrices. It is easy tocalculate the inner product between plane waves on R Θ :Tr (cid:16) e i ˜ p · ˆ x e i ˜ q · ˆ x (cid:17) = 2 πΘ δ (˜ p + ˜ q ) , (4.9)which leads to the Θ -dependence of the relation (4.7).Let us discuss radiative corrections in four-dimensional SYM on R × (Fuzzy S ).We give an argument below that there is no radiative correction which prevents fromthe full 16 SUSYs being restored after the step 1.In quantum field theory defined on NC space with a constant noncommutativity,there are two kinds of Feynman diagrams. One is planar diagrams. They have no NCphase factors depending on loop momenta, and their behavior is the same as that inthe corresponding theory on the ordinary space. The other is nonplanar diagrams.They have NC phase factors, which improve the UV behavior of the diagrams. But,when some of the NC phases vanish in the infra-red (IR) region of external or loopmomenta, singularities may arise, whose origin is the UV singularities in the corre-sponding theory on the ordinary space (UV/IR mixing).
Therefore, we can saythat UV behavior of planar and nonplanar diagrams in the theory on NC spaceis not worse than that in the corresponding theory on the ordinary space. Let usconsider the superficial degree of UV divergences of Feynman diagrams in ordinaryfour-dimensional gauge theory: D = 4 − E B − E F , (4.10)where E B ( E F ) is the number of the external lines of bosons (fermions). In our case,the divergence of D = 3, that is from E B = 1, is absent since the operator ϕ isforbidden by Q ± SUSYs as in the two-dimensional case. Thus, the possible mostsevere divergences are of the degree D = 2. The leading Λ terms are expected tocancel each other by 16 SUSYs under the assumption that M is soft. For radiativecorrections to gauge invariant observables, divergences possibly originate from the D lattice for 4D N = 4 SYM ∗ ) M p (cid:18) ln ΛM (cid:19) q = O ( M p (ln n ) q ) ( p = 1 , , q = 1 , , · · · ) . (4.11)However, such terms disappear in the limit of the step 1.Hence, there appears no radiative correction preventing restoration of the fullSUSYs after the step 1, leading to N = 4 U ( k ) SYM on R × R Θ with 16 super-charges.4.2. Step 2
In four-dimensional N = 4 SYM on NC space, the commutative limit ( Θ → that is, desired N = 4 U ( k ) SYM on usual flat R should be obtained with no fine tuning after the step 2.4.3. Check of the scenario
As a check of the scenario presented in the above, we computed 1-loop radiativecorrections to scalar kinetic terms of B A =1 , . Although details are presented in aseparate publication, contribution from planar diagrams to the kinetic terms inthe 1-loop effective action finally becomes1 g d Z d q (2 π ) X A =1 , tr k h q ˜ b ( R ) A ( − q )˜ b ( R ) A ( q ) i (cid:26) g d k π (cid:18) −
12 ln q µ R + 1 (cid:19) + O ( g d ) (cid:27) (4.12)after the limit of the step 1. ˜ b ( R ) A ( q ) are renormalized momentum modes which arerelated to the modes ˜ b A ( q ) of the bare fields B A as˜ b ( R ) A ( q ) = (cid:18) g d k π ln Λµ R (cid:19) / ˜ b A ( q ) , (4.13)where µ R is the renormalization point ∗∗ ) . The logarithmic nonlocal term in (4.12)has a definite physical meaning contributing to anomalous scaling dimension of theoperator. Since the result does not depend on Θ , the limit of the step 2 is trivial.Note that four-dimensional rotational symmetry is restored in (4.12), which canbe regarded as an evidence of the restoration of the full 16 SUSYs and of the softnessof M . Furthermore, we found that nonplanar contribution is essentially the same asthe planar contribution, supporting the smoothness of the commutative limit Θ → ∗ ) We should note that the behavior (4.11) is not valid for gauge-dependent divergences whichcan be absorbed into wave function renormalization. (4.11) is derived based on UV finiteness of theundeformed four-dimensional N = 4 SYM, where the finiteness holds except such divergences. ∗∗ ) Although four-dimensional N = 4 SYM is said to be UV finite, divergence of the wave functionrenormalization can appear as a gauge artifact. In fact, a modified light-cone gauge fixing inRefs. 32) leads to no UV divergence even in the part concerning the wave function renormalization,differently from the 1-loop computation in Feynman gauge fixing.
The point is that even if UVdivergences appear in radiative corrections, all of them can be absorbed by rescaling the fields.
We adopted a Feynman-like gauge fixing in the calculation. M. Hanada, S. Matsuura and F. Sugino §
5. Discussions
We constructed a lattice formulation of two-dimensional N = (8 ,
8) SYM witha mass deformation, which preserves two supercharges. It serves a basis of nonper-turbative investigation of the IIA matrix string theory. Also, it gives an intriguingscenario to obtain four-dimensional N = 4 U ( k ) SYM with arbitrary k , requiring nofine tuning.It is interesting to extend such construction to theories coupled to fundamen-tal matters. Although it is difficult to introduce fundamental fields directly, bi-fundamental fields can easily be incorporated. For instance, let us start with atwo-dimensional system with SUSYs, which is obtained by dimensional reductionfrom the corresponding theory in four dimensions. The action S = S ,g + S ,g ′ + S m is • S ,g is the action of U ( N ) SYM with gauge field A µ and adjoint matters X I ( I = 1 , · · · , ℓ a ): S ,g = 1 g d Z d x tr N (cid:20) F + ( D µ X I ) −
12 [ X I , X J ] + (fermions) (cid:21) (5.1)with D µ X I = ∂ µ X I + i [ A µ , X I ]. • S ,g ′ is the action of U ( N ′ ) SYM with gauge field A ′ µ and adjoint matters X ′ I ( I = 1 , · · · , ℓ a ): S ,g ′ = 1( g ′ d ) Z d x tr N ′ (cid:20) ( F ′ ) + ( D µ X ′ I ) −
12 [ X ′ I , X ′ J ] + (fermions) (cid:21) (5.2)with D µ X ′ I = ∂ µ X ′ I + i [ A ′ µ , X ′ I ]. It is essentially the same as (5.1) except thechange g d → g ′ d , N → N ′ . • S m is the action of U ( N ) × U ( N ′ ) bi-fundamental matters Φ i ( i = 1 , · · · , ℓ f )coupled to the above two systems: S m = Z d x tr N ′ h ( D µ Φ i ) † D µ Φ i + (cid:0) X I Φ i − Φ i X ′ I (cid:1) † (cid:0) X I Φ i − Φ i X ′ I (cid:1) + (fermions) i (5.3)with D µ Φ i = ∂ µ Φ i + iA µ Φ i − iΦ i A ′ µ .We consider the situation that both of S ,g and S ,g ′ allow a deformation by mass M preserving some SUSYs as discussed in section 2, and that deformed actions S g and S g ′ have classical solutions of k - and k ′ -coincident fuzzy S : X a = M L ( n ) a ⊗ k ( N = nk ) ,X ′ a = M L ( n ) a ⊗ k ′ ( N ′ = nk ′ ) , (5.4)with all the other fields nil, respectively. (We labelled the index I so that scalarswith I = 1 , , S configurations.) Then, for k and k ′ general,vanishing the bi-fundamental fields gives the minima of the zero total action S ≡ D lattice for 4D N = 4 SYM S g + S g ′ + S m = 0. Expanding S around the background (5.4) leads to two systemsof gauge and adjoint fields with gauge groups U ( k ) and U ( k ′ ), which are coupledby U ( k ) × U ( k ′ ) bi-fundamental matters. They are defined on R × (Fuzzy S ),analogous to the situation seen in section 4. Finally, after turning off the coupling g ′ d ,we obtain the system of U ( k ) gauge and adjoint fields coupled to k ′ ℓ f fundamentalmatters (with U ( k ′ ) gauge and adjoint fields becoming free and decoupled) on R × (Fuzzy S ). Therefore, if the system of the action S can be realized on lattice, andif successive limits analogous to those discussed in section 4 can be taken safely, itis expected to obtain the desirable quantum system on R . (Similar constructionusing the Eguchi-Kawai equivalence can be found in Ref. 37).)Using our formalism, many interesting theories will be realized on computer. Weexpect new insights into nonperturbative dynamics of supersymmetric theories willbe obtained in near future. Acknowledgements
The authors are deeply grateful to Hiroshi Suzuki for his joining to their loop cal-culation. They would like to thank Ofer Aharony, Adi Armoni, Masafumi Fukuma,Hikaru Kawai, Jun Nishimura, Adam Schwimmer, Hidehiko Shimada, Asato Tsuchiya,Mithat ¨Unsal and Kentaroh Yoshida for stimulating discussions and comments. F. S.would like to thank Weizmann Institute of Science for hospitality during his stay.S. M. would like to thank Jagiellonian University for hospitality during his stay. Thework of S. M. is supported in part by Keio Gijuku Academic Development Funds,and the work of F. S. is supported in part by Grant-in-Aid for Scientific Research(C), 21540290.
Appendix A
Notations
We give the relation between field variables in (2.1) and those in (2.2), (2.3).For scalars, B = − X , B = X , B = X , C = 2 X , φ ± = X ± iX . (A.1)For fermionic variables, Ψ = U Ψ (0) , ( Ψ (0) ) T ≡ (cid:18) ρ +3 , ρ +4 , ψ +2 , ψ +1 , − χ +1 , χ +2 , χ +3 , η + ,ρ − , ρ − , ψ − , ψ − , − χ − , χ − , χ − , η − (cid:19) , (A.2)where U is a 16 ×
16 unitary matrix of the form U = 12 (cid:18) A A A A (cid:19) (A.3)4 M. Hanada, S. Matsuura and F. Sugino with A ≡ − i i − − i − i i − i i − i , A ≡ − i − i − i − − i − − − i i − i i − , A ≡ i i − i − i i − − i − − i i , A ≡ − i − i i − i − i i − i − i . (A.4)Also, the explicit form of the gamma matrices we used is γ = − i ⊗ ⊗ ⊗ σ ,γ = − i ⊗ ⊗ ⊗ σ ,γ = + i ⊗ ⊗ σ ⊗ σ ,γ = − iσ ⊗ σ ⊗ σ ⊗ σ ,γ = + iσ ⊗ ⊗ σ ⊗ σ ,γ = − iσ ⊗ σ ⊗ σ ⊗ σ ,γ = − iσ ⊗ σ ⊗ σ ⊗ σ ,γ = + i ⊗ σ ⊗ σ ⊗ σ ,γ = + iσ ⊗ σ ⊗ σ ⊗ σ (A.5)with σ a ( a = 1 , ,
3) being the Pauli matrices.
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