Recent progress of lattice and non-lattice super Yang-Mills
Masanori Hanada, Issaku Kanamori, So Matsuura, Fumihiko Sugino
aa r X i v : . [ h e p - l a t ] N ov Recent progress of lattice and non-lattice superYang-Mills
Masanori Hanada ∗ Department of Physics, University of Washington, Seattle, WA 98195-1560, USAE-mail: [email protected]
Issaku Kanamori
Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, GermanyE-mail: [email protected]
So Matsuura
Department of Physics, and Research and Education Center for Natural Science, KeioUniversity, 4-1-1 Hiyoshi, Yokohama, 223-8521, JapanE-mail: [email protected]
Fumihiko Sugino
Okayama Institute for Quantum Physics, Kyoyama 1-9-1, Kita-ku, Okayama 700-0015, JapanE-mail: [email protected]
We report recent progress of non-perturbative formulation of supersymmetric Yang-Mills. Al-though lattice formulations of two-dimensional theories which are fine tuning free to all order inperturbation theory are known for almost ten years, however, there were only few evidence forthe validity at non-perturbative level. In this talk we argue that most numerical studies so far donot capture the physics in continuum, and add new evidence that lattice formulation works at non-perturbatively. We further point out that, by combining two-dimensional lattice and matrix modeltechniques inspired by D-brane dynamics in superstring theory, a non-perturbative formulation ofthe four-dimensional maximally supersymmetric Yang-Mills theory, which is free from the finetuning at least to all order in perturbation theory, is obtained.
The XXIX International Symposium on Lattice Field TheoryJuly 10-16, 2011Squaw Valley, Lake Tahoe, California ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ecent progress of lattice and non-lattice super Yang-Mills
Masanori Hanada
1. Introduction
Supersymmetric Yang-Mills (SYM) theories play prominent roles in theoretical particle physics.In particular, maximally supersymmetric theories are expected to describe the non-perturbative na-ture of superstring/M theory [1, 2, 3, 4, 5]. Lattice formulations of SYM theories are pursued be-cause many important question would be answered only via numerical simulations. However, it isnot a straightforward task, because the supersymmetry cannot be preserved completely on lattice:the SUSY algebra contains infinitesimal translation, which is broken on lattice by construction.Therefore, even if a given lattice theory converges to a supersymmetric theory at tree level, SUSYbreaking corrections may arise radiatively. In order to control the divergence one needs some exactsymmetries at discretized level. In 4d N = . In several extended SYM theories, it ispossible to keep a few supersymmetries unbroken. In two dimensions, those exact supersymme-tries, together with other global symmetries, forbid SUSY breaking radiative corrections at leastto all order in perturbation theory [8, 9, 10, 11]. (A similar statement holds for the Wess-Zuminomodel, and numerically tested in [12].)In one dimension (i.e. supersymmetric matrix quantum mechanics), the situation is much eas-ier. Because the theory is UV finite, one does not have to rely on exact symmetries and hence asimple momentum cutoff prescription works [13]. In fact, as demonstrated in [13], the momentumcutoff method is much more powerful than a usual lattice regularization and detailed Monte-Carlostudies have been done. In particular, the gauge/gravity duality between D0-brane quantum me-chanics and type IIA superstring theory [5] has been tested, and beautiful agreement including the a ′ correction has been confirmed [14, 15] . Furthermore long-disrance physics relevant for thematrix theory conjecture has also been studied [16].In order to study higher dimensions, one must establish fine-tuning free formulation of thosetheories. Obvious first step is non-perturbative test of two-dimensional theories. In particular oneshould see whether the fine-tuning-free nature persists to non-perturbative level. In § 2 we shownumerical results which strongly support the validity of these formulations at non-perturbativelevel. Based on the success of two-dimensional lattice formulation, in § 3 we provide a hybridformulation of two-dimensional lattice and matrix model techniques, which enables us to put four-dimensional maximally supersymmetric Yang-Mills theory on computer.
2. Non-perturbative test of two-dimensional lattice SYM
Numerical simulations of the supersymmetric gauge theories often suffer from the fermionsign problem . One of the exceptions is 4d N = N = ( , ) SYM, which is the dimensional reduction to two dimensions. The action is S = N l Z L x dx Z L y dy Tr (cid:26) F mn + ( D m X i ) − [ X i , X j ] −
12 ¯ y G m D m y − i y G i [ X i , y ] (cid:27) , (2.1) For recent numerical studies, see [7]. Other numerical studies in the context of the string theory can be found in [17, 18, 19], for example. In the case of maximally supersymmetric matrix quantum mechanics, agreement with the dual gravity prescriptionhas been observed by ignoring the phase of the Pfaffian, even when the sign fluctuates violently [14]. ecent progress of lattice and non-lattice super Yang-Mills Masanori Hanada where m and n run x and y , i and j run 1 and 2, and G I = ( G m , G i ) are gamma matrices in fourdimensions. X i are N × N hermitian matrices, y a are N × N fermionic matrices with a Majoranaindex a and the covariant derivative is given by D m = ¶ m − i [ A m , · ] . The only parameters ofthe model are the size of circles L x and L y . (Note that the coupling constant can be absorbed byredefining the fields and coordinates. Therefore we take the ’t Hooft coupling l to be 1. Then thestrong coupling corresponds to the large volume.)One obstacle for the simulation is the existence of the flat direction, along which two scalarfields X and X commute. In contrary to a theory on R , , there is no superselection of the moduliparameter in this case. That is, eigenvalues of scalars are determined dynamically. Therefore, somemechanism which restrict eigenvalues to a finite distribution is necessary for the stable simulation.In addition, to obtain an interesting dynamical system, having a (small) finite region for the eigen-values is important as well; if the eigenvalues of the scalar spread so large, the theory would runinto the abelian phase, which is just a free theory . In this work, we introduce soft SUSY-breakingmass to scalar fields m N Z d x (cid:229) i = , TrX i , (2.2)so that the flat direction is lifted. It is crucial to control the flat direction for various reasons. Wehave just mentioned two of them — stability of the simulation and interesting non-abelian phase.In the Weyl notation, with an appropriate choice of the gamma matrices, the Dirac operator iswritten as D ≡ i s m D m , (2.3)where s = − i and s i ( i = , , ) are Pauli matrices. By using s ( i s m ) s = ( i s m ) ∗ and the factthat D m is real in adjoint representation, we obtain s D s = D ∗ . (2.4)Therefore, if j is an eigenvector corresponding to an eigenvalue l , s j ∗ is also an eigenvector,with eigenvalue l ∗ . They are linearly independent and eigenvalues appear in a pair ( l , l ∗ ) . Thisassures the positivity of the determinant after removing l = in the SU ( ) Sugino model. Here wechose L x = L y = L and imposed periodic boundary condition for all fields along both directions.In the left panel we fixed the physical volume to be L = .
707 and used three lattice sizes, 4 × × ×
6. It turns out the peak around zero becomes sharper as the lattice becomes finer. Inthe right panel we have fixed the lattice size and changed the lattice spacing. It can be seen that Which phase is preferred is in fact a dynamical question. At large- N , the flat direction is lifted and the system staysnon-abelian phase [20, 21]. This phase is an analogue of the black 1-brane solution in type IIB supergravity. Here we adopt the Majorana fermion. Hence we calculate the pfaffian rather than the determinant. ecent progress of lattice and non-lattice super Yang-Mills Masanori Hanada
Figure 1: [Sugino] Argument of the Pfaffian in SU ( ) theory. The scalar mass is m = .
20. The left panelis for a fixed volume 0 . × .
707 and thus different lattice spacings. The right panel is for a fixed 4 × Figure 2:
The expectation value of the Wilson loop h| W |i at m = .
0. Extrapolation to the continuum limithas been performed. The gauge grope is U ( ) and SU ( ) , respectively. (Figure from [21]) smaller lattice spacing gives sharper peak. Both plots clearly show that the sign disappears as thecontinuum limit is taken. For more details, see [21].Since we have established the absence of the sign problem in 2d N = ( , ) lattice SYM, wecan perform various tests. So far, conservation of the supercurrent in the Sugino model has beenobserved in [23]. Here we add another result. In Fig. 2 we have plotted the expectation valuesof the Wilson loop winding on compactified circle, which are calculated in CKKU and Suginomodels. Two models give the identical result, which strongly suggests that they converge to thesame continuum limit without performing parameter fine tuning.Although theories with more supersymmetries suffer from the sign problem, it is a technicalproblem attached to the importance sampling and we can expect the restoration of all supersymme-tries in the continuum limit also for them. Further investigation along this line is desirable.
3. Four-dimensional SYM out of two-dimensional lattice
In four dimensions, a few exact supersymmetries are not strong enough to control the radiativecorrection, although it does reduce the number of fine tuning parameters down to three [24]. In[25] it has been pointed out that the fine tuning problem can be circumvented by using the matrix4 ecent progress of lattice and non-lattice super Yang-Mills
Masanori Hanada model techniques . The basic idea is simple. Consider the Berenstein-Maldacena-Nastase matrixmodel [28], which is a deformation of the D0-brane matrix quantum mechanics keeping maximalsupersymmetry. It has a fuzzy sphere solution, which is BPS. If one considers k -coincident fuzzysphere, three-dimensional U ( k ) non-commutative super Yang-Mills (NCSYM) on fuzzy sphere isobtained. By taking appropriate large- N limit, NCSYM on flat non-commutative plane is obtained,and by turning off the non-commutativity, one arrives at usual SYM on R [29]. This method caneasily be generalized to four-dimensional theory; if one starts with ‘BMN-like’ two-dimensionaltheory with fuzzy sphere solutions, one can construct 4d theory. The crucial point is that suchtwo-dimensional theory can be regularized by using the lattice. Bosonic part of the action of thistwo-dimensional theory is given by [25] S d = g d Z d x Tr n F + ( D m X I ) − [ X I , X J ] + m (cid:229) a = X a + i m X [ X , X ] − m X [ X , X ] o . (3.1)The deformation by m breaks 14 out of 16 SUSY softly. As shown in [25], this model can beregularized by lattice keeping two supercharges unbroken, and the continuum theory is obtainedwithout parameter fine tuning, at least to all order in perturbation theory.The continuum action (3.1) has constant BPS fuzzy sphere solution X a ( x ) = m L a ( a = , , ) , X i ( x ) = ( i = , · · · , ) , (3.2)where L a are M × M matrices satisfying SU ( ) commutation relation [ L a , L b ] = i e abc L c . (3.3)By taking k -coincident fuzzy sphere solution, L a = L ( M / k ) a ⊗ k , where L ( M / k ) a is the ( M / k ) × ( M / k ) irreducible representation, we obtain 4d U ( k ) theory on fuzzy sphere. Essentially, adjoint action of L a is identified with the derivative and [ X a , · ] is regarded as the gauge covariant derivative [26].The noncommutativity is given by q ∼ k / ( m M ) and UV/IR momentum cutoffs along sphericaldirections are m M / k and m , respectively. 4d coupling is given by g d = pq g d . In order to getcontinuum 4d theory, we take large- M and small m limit while fixing k and g d . In that limit,maximal supersymmetry is restored because soft SUSY breaking parameter m goes to zero. Onecan take a limit with any value of non-commutativity q , and q → For another matrix model approach which works at large- N , see [27]. This background preserves exact supersymmetries at discretized level. ecent progress of lattice and non-lattice super Yang-Mills Masanori Hanada
4. Conclusion
We reported recent progress of the non-perturbative formulations of SYM. Lattice formu-lations of two-dimensional N = ( , ) SYM has been tested non-perturbatively and confirmedto work without parameter fine tuning beyond the perturbative level. It is also shown that four-dimensional SYM can be constructed by combining two-dimensional lattice and a matrix modelmethod.There are various problems one can attack based on above results. Firstly, 2d N = ( , ) SYM can have interesting physics on its own (see e.g. [22, 30, 31, 32, 33]) and further numericalstudy is desirable. 2d maximal SYM is even more interesting because it describes [34] the blackhole/black string phase transition [35]. It is important to establish the absence of fine tuning inthis case and perform a large-scale simulation along the line of [17]. Numerical study of four-dimensional SYM would be the most important. The formulation explained in § 3 would be usefulto understand the AdS/CFT duality further, especially away from the weak and strong coupling.Another interesting class of theories are SUSY QCD. At large- N , the large- N reduction [27] can beapplied to SUSY QCD [36]. It would provide us with a tool to study important problems e.g. thestudy of spontaneous SUSY breakdown or the test of Seiberg duality. It is interesting especiallybecause it may have a close connection to physics at LHC. Acknowledgments
The work of M. H. is supported by Japan Society for the Promotion of Science Postdoc-toral Fellowships for Research Abroad. I. K. is supported by the EU ITN STRONGnet and theDFG SFB/Transregio 55. F. S. is supported in part by Grant-in-Aid for Scientific Research (C),21540290.
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