From Lattice Gauge Theory to Black Holes Pos(LATTICE 2015)017 (1.1) Ab-Initio Daisuke Kadoh ], and Provide Deep ] 6 9 ] and Gauge/Gravity Duality 91 , ]

Home , Lattice gauge theory

PoS(LATTICE 2015)017 http://pos.sissa.it/ ∗ [email protected] Speaker. Supersymmetry (SUSY) is a fascinatingcounterintuitive topic properties. in It theoretical is physics, expected becauseand to it of emerge is as its also new unique a physics and building beyondobtained block the via for standard SUSY supergravity model, and theories superstring provide theory.SUSY insights A theories into number are field of not theory. exact yet results However, fullytice the understood, simulations dynamics and is of numerical promising many study as ofthe regards SUSY furthering current theories this status through understanding. of lat- lattice Inreviewing SUSY this some by simple paper, discussing models. I its overview In development addition,duality, in I which chronological discuss is the order, one numerical and of verification the by of recent gauge/gravity significant developments in this field. ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). The 33rd International Symposium on Lattice14 Field -18 Theory July 2015 Kobe International Conference Center, Kobe, Japan Daisuke Kadoh Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521,E-mail: Japan Recent progress in lattice supersymmetry: from lattice gauge theory to black holes PoS(LATTICE 2015)017 (1.1) ab-initio Daisuke Kadoh ], and provide deep ] 6 9 ] and gauge/gravity duality 91 , ]. However, such a “brute-force” ], AdS/CFT[ 90 5 44 , 41 , m 1 supersymmetric Yang-Mills (SYM) has P ˙ β = m α σ N which is broken on the lattice. Thus, a naive 2 m 2 = P ], it has been found that some nilpotent supercharges } ˙ β ¯ Q 117 , , α 72 ], Seiberg-Witten theory[ Q , 4 { 61 , 17 ]. Furthermore, many exact results and predictions have been obtained in SUSY ]. SUSY theory has been studied as a strong candidate for physics beyond the 3 ]. Thus, lattice SUSY has become a feasible and powerful tool for revealing the 2 98 ]. In addition, SUSY is also included in superstring theory which may produce a “theory ], [ 1 107 2 Wess-Zumino (WZ) model and gauge theories with extended SUSY. These lattice formula- ]. Therefore, a natural extension is to apply the lattice theory to SUSY, in order to study the A large number of attempts have been made to solve this problem. One possible solution is a In this paper, the current status of research into lattice SUSY is reviewed. First, an overview The first paper on lattice SUSY was published in 1976, and many studies related to this topic Lattice field theory has been used to determine the dynamics of field theory via Supersymmetry (SUSY) has been intensively studied in theoretical physics for a long period Although broken SUSY is, of course, restored at the classical continuum limit, such restora- ]-[ 8 = , 100 7 partial realization of SUSY that yields afine supersymmetric tuning. continuum In limit previous without studies (or [ with reduced) tions have been constructed with aused small to number study of exact interesting supersymmetries phenomena and such have already as been SUSY breaking [ calculations and has achieved great success in[ the form of lattice quantum chromodynamics (QCD) approach is ineffective for general SUSYtionally theories expensive. in Therefore, practice, it because is this difficult fine to tuning apply is lattice computa- theory to SUSY theory. can be realized on theN lattice for theories with extended SUSY; for instance, the two-dimensional [ non-perturbative physics of SUSY theories. of the history of latticerealized SUSY research on is the provided lattice and innumerical the simple manner tests models in of is which gauge/gravity explained partialSection duality SUSY in 4 can in section contains be 2. maximally a summary SYM Then, and theories a recent discussion are lattice of discussed results the future for in2. outlook section for Lattice 3. this supersymmetry field. insights into various aspects ofdynamics of quantum SUSY field theories in theory. more detail, Nevertheless, as it they is are not important yet to fully reveal understood. the non-perturbative physics of the latter. However, SUSY algebra has the form [ theories, for instance, Seiberg duality[ of everything” [ standard model [ of time, motivated by alocal variety symmetry of and reasons. yields a Forbehavior [ field example, theoretical supergravity description includes of this gravity concept with as the a improved ultraviolet tion does not occur atbreaking the operators. quantum level, The because supersymmetric ultraviolet continuum divergencetuning limit yields the is the relevant achieved relevant SUSY at SUSY breaking the operators, quantum level and by fine Recent progress in lattice supersymmetry: from lattice gauge theory to black holes 1. Introduction contains the infinitesimal translationlattice operator discretization breaks the SUSY at the cutoff scale. already been studied by using fine tuning the gluino mass [ PoS(LATTICE 2015)017 ]- 34 (2.1) (2.2) ], for ]. The 17 -exact: 7 ]. Q 18 Daisuke Kadoh In this section, 1 ]. The only relevant ], lattice field theory ], this group has ob- 41 10 45 ]. Montvay’s approach was , and the action is This approach was based on ]. Q 2 43 40 , , is replaced with a local difference 42 39 µ ]. Lattice formulation with Nicolai , ∂ ) 22 g , . µ 0 ∂ 21 ( -exact action in SUSY quantum mechanics, f = ]. In recent papers [ Q 2 ]. 1 SYM in four dimensions, which consists of + Q 44 g 16 = ) 3 f ]-[ ] (so-called “Nicolai mapping”), through which the N µ ∂ , -transformation without the Leibniz rule. In section 12 20 , Q QV 19 ) = ( = f g S ( µ ∂ ] noted that this breaking prevents the realization of SUSY on the 11 ]. ], and lattice WZ models have also been numerically examined [ 46 33 SYM with the Wilson fermion in the mid 1990s [ 2 WZ model and using the Wilson fermion. ]-[ ) 2 = ( 23 N 150 proceedings. SU ∼ ]. To obtain SUSY invariance of the action, the Leibniz rule 1 11 = N Approximately 1,000 papers can be found in INSPIRE by searching for two keywords: “lattice” and “supersym- Cecotti and Girardello independently discovered the same method using a Hamiltonian formulation[ In the 1990s, lattice SUSY progressed to SUSY was introduced into field theory at the beginning of the 1970s [ The first lattice action with exact SUSY was reported by Sakai and Sakamoto in 1983 [ 1 2 ]. The sigma model has also been formulated on the lattice [ 38 is required. However, this rule isoperator. broken Dondi when and the Nicolai derivative [ metry”. However, 600 of those papers250, are irrelevant excluding to 130 this topic. In fact, the number of relevant papers is approximately Thus, the lattice action is invariant under mapping, refinements to thisstudied approach, intensively and [ the other[ formulations of the WZ model have been SUSY breaking operator iscorresponds the to gluino the mass SUSY term, limit by and fine-tuning one the can gluino construct mass the [ massless limit that served the mass degeneracy between bosonictheoretically and predicted fermionic in bound [ states at low energy, which was lattice and defined the latticeformulations WZ with exact model SUSY without and locality. locality [ No other early studies obtained lattice boson action becomes a simple Gaussianfermion integral determinant. and The the theory Jacobian has of an the exact nilpotent transformation supercharge gives the 2.2, this method is explained from the perspective of a gauge field (gluon) and aulation massless of Majorana fermion (gluino). Montvay began a numerical sim- then employed by the DESY-Munster collaboration,in and the the field resultant series of of lattice work SUSY is from the largest 1996 onwards [ which is another example involving Nicolai mapping [ Recent progress in lattice supersymmetry: from lattice gauge theory to black holes have since been conducted.lattice field Approximately theory, 24,000 and approximately papers 400the have papers history been of of published which lattice SUSY discuss onon is lattice the a summarized SUSY. topic lattice and via the of super manner quantum in mechanics which and partial SYM SUSY is2.1 can reviewed. be History realized of lattice supersymmetry research was first proposed infirst the paper 1960s, on and lattice SUSY lattice wasframeworks gauge published [ in theory 1976 was by Dondi first and formulated Nicolai, in who combined 1974 the [ above a two-dimensional a Nicolai-Parisi-Sourlas transformation [ PoS(LATTICE 2015)017 ]. 75 ]-[ ]. The ]. The ]. Using 72 79 51 57 ]. Moreover, ]-[ Daisuke Kadoh ]. In addition, 73 54 ], and Giedt has ]. ], and by Sugino, 101 69 ], and Catterall has 60 78 ], [ 83 ]-[ ]. The original Cohen- 82 58 ], while the same theory ] for fundamental matter. 64 ]. ]-[ , 77 ], respectively, after which ] indicates that the Sugino 68 84 81 63 62 ]. The corresponding actions -invariance of the action comes 101 2 supersymmetric QCD (SQCD) 72 ]. , Q 1 SYM in four dimensions without = 71 81 that is nilpotent up to a gauge trans- ] and [ N = Q 61 N ] beyond the constraint of the Nielsen- ], and the 52 76 4 ] obeying the Ginsparg-Wilson relation [ ]. The lattice gauge theories using orbifolding are 50 65 ] for adjoint matter and by Matsuura [ -exactness of the SYM action, for several SYM theories [ ], and independently by Sugino [ 1 SYM has been numerically investigated [ 67 Q 61 ], and defines the massless fermions on the lattice. If a massless gauge = 53 N ] and the overlap fermion [ ]. Moreover, SUSY gauge theories with matter fields have been formulated on the 49 66 ]-[ 47 ]. The Sugino model has already been studied numerically [ 80 Cohen et al. have constructed lattice SYM theories using orbifolding from the matrix models Sugino has defined a lattice action with a partial SUSY Other significant progress in the 1990s is directly related to the discovery of the domain wall As regards developments in the 21st century, significant progress was made in relation to ]. The CKKU model has already been studied numerically [ 70 Kaplan-Katz-Unsal (CKKU) method uses athe non-compact compact gauge gauge field, field and can Unsal also has be shown used that [ In addition, Joseph has utilizeddiscussed the the CKKU positivity method or to[ non-positivity define of several theories the [ fermion determinant for several related theories This method is based on the topological field theory [ was also formulated on thefor flavor lattice symmetry by to Kikukawa avoid the andSuzuki, extra Sugino and restriction using the on the present the author Ginsparg-Wilson numberimportance relation of using of fermions the a [ admissible different gauge typeMatsuura action of and in twist Sugino Sugino’s have and method proposed the was anothertion first Wilson type [ noted fermion of in [ gauge [ actionnumerical without an evaluation admissibility of condi- themodel SUSY reproduces a Ward-Takahashi identity desirable continuum [ theory in two dimensions, beyond the perturbation theory. classified in [ lattice by Endres and Kaplan [ obtained by reducing the dimensionscommute with of all the of target SUSY, and theories.for the SYM resultant The with lattice actions orbifolding four have and procedure partial eight does SUSY. The supercharges not lattice were actions given in [ Ninomiya no-go theorem [ defined lattice actions with partial SUSY (the geometrical approach) [ Kaplan and Unsal reported a lattice SYM with sixteen supercharges [ formation, by focusing on the from its algebraic property and the symmetry2.3. can Later, be Sugino kept defined even on a the lattice lattice, action as of explained two-dimensional in section field and a massless adjointexact chiral fermion symmetry exist, forbids they the gluino form mass a and realizes supersymmetric pair. In other words, the fermion [ SUSY gauge theory indiscovered 2003. by Cohen Lattice et formulations al. of [ the extended SUSY gauge theories were Recent progress in lattice supersymmetry: from lattice gauge theory to black holes overlap fermion realizes the exact chiral symmetry [ fine tuning. In this sense, latticethe formulations domain without wall fine-tuning fermion, are discussed in [ have exact partial SUSY onthat the lattice. exact SUSY In perturbation yields theory,with the the less correct power fine continuum counting tuning theorem limit in states discussed with three to no and date four fine and dimensions. other tuning formulationsconstructed in As have lattice explained two also actions below, been dimensions, by many proposed. respecting and variants For SUSY have instance, algebra been Dadda (the et link al. approach) have [ with an equal number of fundamental and anti-fundamental fermions [ PoS(LATTICE 2015)017 2 = (2.3) N -exact form 2 WZ model ]. In two di- Q Daisuke Kadoh = , 98 N ) ]. ], and topological ψ 96 ]. 112 ], it is shown that the ) , 95 φ 86 ( 111 4 SYM from a lattice SYM ∂φ W ∂ = ], which will be introduced in Numerical simulation with the + N 102 4 d dt , ]. ¯ 101 ]. 97 ψ i 107 ) + φ ]-[ ]. Unfortunately, introducing and evaluating ( ]. If fact, a certain equivalence and similarity 103 113 5 85 iFW + ], the cyclic Leibniz rule [ 2 2 superconformal field theory [ F -symmetry; hence they examined whether or not SUSY 110 2 1 = ]. Giguere and the present author have employed the Sug- Q , N ]. + ]. For non-gauge theory, SUSY breaking has been studied in 2 89 100 109 91 φ dt 2 SUSY quantum mechanics (QM) is reviewed, which provides a d = 2 1 ]. Further, Catterall and Veernala studied the two-dimensional N 1 SYM [ ( 90 = dt 2 SYM using the Sugino action and measured the vacuum expectation value -exact formulation of lattice SUSY. N Z Q = = ]. N 88 SQM S ]. In addition, Kikukawa and Kawai have examined the two-dimensional , and have also shown that the lattice action given by the link approach is equivalent to the 94 3 Hanada et al. have studied the same theory numerically using a sharp momentum cut-off [ Such equivalence is also discussed in [ There are many interesting works that should also be introduced here; for instance, the energy- In this section, lattice The continuum euclidean action of SUSY QM is given by Since the discovery of lattice SUSY actions with a few exact supersymmetries, many numer- The lattice formulations discussed above are similar in the sense that the remnant charges are Remarkable applications include numerical verifications of gauge gravity duality in maximally 4 3 ]-[ ] 92 87 SQCD from a similar perspective [ Sugino lattice action has alsosystem been at performed next-to-leading and order the validity in of a the low gauge/gravity temperature duality expansion in has this been shown [ section 3. Further, Catterall et al. have examined four-dimensional twisted SYM on a discretized Riemann surface [ momentum tensor in ino lattice action to examine the duality in two dimensions [ these studies would extend thisleave them review to beyond another the reviewer. page limit, therefore the author2.2 would like Supersymmetric to quantum mechanics is broken in that theory [ [ and shown that this theory reproduces mensions, Catterall, Wiseman and Joseph havehole employed the - CKKU black lattice string action phase to transition study [ a black and its simulations, and tried to verify AdS/CFT [ clear concept of the ical applications have been conducted.two-dimensional For example, Kanamori, Suzukiof and the Sugino Hamiltonian have defined studied from exact between these formulations has been discussed in several papers. In [ CKKU action [ nilpotent at least up to gauge transformations and the lattice actions are written in SYM. In one dimension, Catterall and WisemanSUSY have to employed a study naive the lattice action thermodynamics without of exact a dual black hole [ Recent progress in lattice supersymmetry: from lattice gauge theory to black holes although they are based on different methods [ Sugino model can be reproduced from theof Catterall geometrical freedom approach, by of truncating the the degrees complexifiedMatsuura fields have shown while that preserving the the[ Catterall supercharge. geometrical approach Further, is Damgaard equivalent and to the CKKU method PoS(LATTICE 2015)017 is + ¯ Q (2.4) (2.7) (2.8) (2.9) (2.5) (2.6) 0. In ) also (2.10) W ) states = and + ) for the 2.9 2 2.1 Q iF Q 2.8 , and satisfy ]. − Q Daisuke Kadoh 22 ¯ ) is manifestly ε φ , dt d , + 2.9 ) via ( 21 ψ ¯ are one-component Q ¯ ∆ ψ ε ) i 2.3 ε t ]. ( − = ¯ = ψ , δ = 114 X . , F and , QF 11 ) + t . with ( ), with . The lattice Q-transformation , as the Leibniz rule ( invariance to be realized on the ) φ ) ) )) d t ψ dt φ ¯ dt ε ( d φ dt i φ ( 2.6 , i φ ( ( / 2 φ − or − ) )-( a iW W − ∆ a iW 2 i dX ε + 2 ¯ = t ε 2.4 ( + − = + } φ ) with a difference operator and summation = ¯ F φ L Q φ dt d ¯ , δ ψ ∆ i = 2.3 i ) = Q δ t ¯ { + ψ ( + 6 φ Q F -invariant without the Leibniz rule, as F -exact form as , ∆ , ( Q Q 0 , ψ ¯ ¯ ψ dt ψ d = 1 2 2 1 ¯ ) is , ε ) for any function i 2 F on the lattice. 0 ¯ , Q − 2.8 + Q 2.3 t = dt ∑ ¯ ¯ is a real auxiliary field, and εψ φ ψ , ψ dt dt ) d Z d 0 t − i Qa Q ( ε Q ¯ i = ψ F = ) and the integral in ( ε ε − . The off-shell SUSY transformation with two Grassmann parame- = 2 t Q ) leads to the lattice action, 2.6 = = = , LAT ) coincides with the continuum action given in ( S ψ SQM )-( F 2.8 S -exact action ( − δφ 2.9 δ ) is then expressed in δψ 2.3 Q = 2.3 φ -exact form of the action allows either invariance requires the Leibniz rule because the anti-commutator of -invariance does not require the Leibniz rule and can be realized on the lattice. Q Q ¯ 0. Q Q are supertransformations extracted from ( 0, even for finite lattice spacing. Therefore, the lattice action ( ¯ = Q S = , is a real scalar field, δ -exact action ( ¯ 2 ε ) and Q t is a forward difference operator, with Q ( -derivative in ( Q t ∆ φ and , and It is obvious that ( Instead, the The One can easily define a naive lattice action by placing the fields on the lattice sites and replac- ε ). Although W 2.8 ¯ classical continuum limit. Numerical simulationsindicate with that the this lattice formalism SUSY reproduces action SUSY QM given beyond in perturbation ( theory [ ters fermions with euclidean time The action given in ( We can show that the satisfies invariant under partial supersymmetry that the Lagrangian varies up to a total divergence, where over the sites, respectively.because the However, Leibniz this rule does naive not action hold breaks for any SUSY difference operators at [ a finite lattice spacing Recent progress in lattice supersymmetry: from lattice gauge theory to black holes where indicates the invariance of the action ( εψ ing the other words, we have already employed( the Leibniz rule to expressa the derivative, action the in the form given in lattice. PoS(LATTICE 2015)017 , φ The (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.11) (2.12) (2.13) (2.14) 5 are origi- ]. , a . 76 ) ω ] L Daisuke Kadoh ¯ R ψ ψ + , R φ [ ψ , ( are defined by L 2 ¯ , a complex scalar ψ η ¯ √ µ φ , , 2 i ) µ χ A − , D D µ ] + , = µ ψ L . Although the 2 ψ η ψ ψ , i D ω , . φ ¯ φ − C + as in the case of SUSY QM. To [ , ] , ] 2 , = R ¯ ] ∈ ) φ ν ¯ φ Q f ¯ L ψ , -exactness of the action. The same φ a A µ ¯ , , ( 2 ψ , , φ ] ] -exact form in twisted fields: Q [ ω ¯ µ φ iD − φ χ + Q [ η , , A R , as field variables. Its euclidean action is L 4 1 [ ] 1 4 = φ i φ ψ ψ ( ω D . µ ) + 2 + ] , 2 1 = [ ¯ = [ f ψ ) + φ φ √ µ [ , µ i Q iD η 12 A µ = ν D − Q iF QH + A ∂ χ 7 [ φ 2 1 i = µ − D , − ]. f , + D ) ( ν µ 2 and the twisted fermion fields R L ω H A φ 72 , ¯ , ¯ + ψ δ ψ ( √ , ψ µ µ / H 0 η 4 χ ∂ ∂ − ) µν being nilpotent and = R L = F = = ¯ + µ Q ψ = = tr Q ( ¯ R χ φ φ , µν + 2 x φ ψ i F L QA Q Q Q 2 µν ω ) √ µ can be maintained on the lattice as an exact symmetry was ex- 1 2 d Q F 2 µ and an auxiliary field -exact form using a nilpotent ( D ) invariant for any gauge parameter = D Q Z T Q iD 2 ) tr − 2 ψ ≡ − R − g x 1 2.11 1 Q 2 = ψ 2 ) in , , d ) D µ L Q ( R A ¯ Z R ψ ψ 2.11 = ¯ ω 2 ψ + 1 ) is expressed as the following simple δ g 4 ] for more details. L = ( ψ cont + 77 = ( S ψ 2.11 2 and 2 , 1 √ 1 ], the Lorentz group is redefined by a topological twist and the fields transform in a new manner under SYM is identified with = 76 = S 1 Q ν acts on the fields as ψ , µ Q We can express ( In TFT [ A definition of lattice SUSY QM was presented in the previous section and the manner in In two dimensions, SYM with four supercharges has a gauge field 5 See the appendix of [ achieve this easily, topological field theory (TFT) field variables can be employed [ the twisted Lorentz group; for instance,supercharge the fermion transforms like a vector or scalar. In this notation, a nilpotent scalar defined by nally real, the invariance holds for complexified parameters action given in ( where which partial supersymmetry Recent progress in lattice supersymmetry: from lattice gauge theory to black holes 2.3 Supersymmetric Yang-Mills theory plained. This approach ismethod based is on applicable to theSYM with gauge four theory. supercharges is In explained [ this section, the Suginoa action Dirac of fermion two-dimensional where The infinitesimal gauge transformation, makes the action given in ( PoS(LATTICE 2015)017 1 ). = a 2.12 (2.23) (2.22) (2.25) (2.21) (2.26) (2.24) ). It is ). The leads to ) 2.20 -symmetry † 12 2.14 P Q ), that is, the )-( ) by replacing Daisuke Kadoh , and set − a ]. 12 2.16 2.18 , P 2.16 ) ( x i 2 101 , ( , µ − ¯ ψ 81 φ , ) = µ 1 x ( ∇ , − ). lat is a lattice version of ( 12 ) µ µ , , φ x F ) ) ψ ψ ( || i lat i ) reproduces ( x defined by 12 ˆ † ) ν µ ( F -exact action ( x ) − U φ ( x + ) ] ) + Q ( ¯ 2.24 ˆ x x 12 ν φ − ( ( ]. The numerical result for the SUSY , P ) µν + φ ) and x ω φ P [ − ( 72 µ ) x 1 ( x is an exact symmetry of the lattice action 1 η ∇ ( † − µ i µ || 4 1 2.22 µ Q 2 U U 1 ) ) ε . Under the lattice gauge transformations, the iU ˆ ˆ ) = µ ) + µ G x − − ( + lat + ) ∈ 1 12 are µ ]. Moreover, instead of the admissible field tensor, 8 x x x ) , but a simple choice ( ( iF µ ( ψ x 73 ) ν 2 µ ( φ ψ [ Q lat 12 ) † µ U U 12 − ]. ) ε F x ) P U ( x x and the plaquette ( H ( and 80 µ − ( µ µ ω U µ , χ i 12 ) U ∇ U x P ( ( ) = tr i µ 2 ) = ) = x x ( x U x ( ( ) ∑ φ µ x µ 2 ( ≡ − µν U even on the lattice. Thus, g 1 µ P ∇ ) ω 2 φ x ψ δ i ( δ Q i lat 12 − = (the gauge transformation with the parameter F ) = ). In two dimensions, perturbative power counting indicates that the only x φ = ( δ 2 i LAT µ S Q -transformations of − 2.11 Q QU = 2 ). symmetry of the Sugino action forbid the mass term, and the SUSY is restored in the Q -type tensor can also be used [ is the covariant forward difference operator ( ) R 2 µ ) 2.24 / 1 ∇ ( θ ( We can formally define the Sugino lattice action from the The lattice In the naive continuum limit, the Sugino action given in ( We define the lattice theory on a two dimensional lattice with a lattice spacing U are gauge covariant. degenerate vacua that are irrelevant to the correct continuum limit. The admissible field tensor, avoids the extra vacua for sufficiently small Ward-Takahashi identity indicates that the Sugino action reproducesbeyond the desirable the continuum perturbation theory theory at the continuum limit in two dimensions [ link fields transform as for simplicity. The fermion,fields scalar are defined and on auxiliary the fields links as are link defined fields on the sites, while the gauge and the fermion, scalar and auxiliary fields transform like the second equation of ( There are a number of possible choices for relevant SUSY breaking operator is the mass term for the scalar field. The exact a tan covariant forward difference operator the integral over the space time withderivative summation and over the the lattice field sites, strength and with by their replacing lattice the covariant respective versions, such that while the other transformations are identicaleasy to to show the that continuum transformations, ( give in ( action given in ( and quantum continuum limit (in perturbation theory, at least) [ Recent progress in lattice supersymmetry: from lattice gauge theory to black holes and satisfy where PoS(LATTICE 2015)017 ] ]. ], 99 102 (3.1) (3.3) (3.2) 100 , ]-[ 96 101 -exactness Q Daisuke Kadoh ]. . The ] in order to verify 0 and in [ 102 T , / = 1 107 coefficient can be esti- p 1 = 101 , ]-[ c , β -branes at low temperature ··· 98 p 103 ] for ) in the zero temperature limit + 6 98 . 4 3.3 .... 3 407 / . 1 T ]: 7 λ = 108 1 5 2 c , i + S 14 h 8 . 2 3 limit. The value of the β 9 7 π − N 3 / 2 1 = T λ 15 E 13 4 1 c 6 = 9 14 3 = 1 for the large / 1 1 E c λ corrections in string theory, and its value is currently unknown. 3 0 / 2 1 α 1 λ N / T ]. 97 -exact action of maximally SYM in one dimension and limit. As reported in previous works using a sharp momentum cut-off [ Q N ) from the boson action. 3.3 is the corresponds to 1 are considered. S 2 = c See also Ref. [ The gravitational theory corresponding to one-dimensional maximally SYM predicts that the From the gauge theory perspective, the internal energy can be given by the expectation value In this section, the present author’s lattice studies that showThe the gauge/gravity validity duality, such of as the AdS/CFT, gauge/gravity states that gauge theory is equivalent to the the- Lattice gauge theory enables us to test gauge/gravity duality, as it is applicable to a strongly 1-dimensional maximally SYM is expected to be dual to black 6 p + p Moreover, N=4 SYM has alreadyAdS/CFT. been Hereafter, studied the using present lattice author’s theory latticefor [ results given in [ at low temperature mated from a type IIA ten-dimensional SUGRA action[ where corresponding to the SUSYevaluate limit. ( We integrate the fermions and auxiliary fields by hand and internal energy of the black hole behaves according to of the action itself: the one-dimensional theory reproduces blacktwo-dimensional hole case, thermodynamics the at black low temperatures. hole-black string For the phase transition has been investigated [ Here of the action yields the correct zero point of the internal energy ( duality of maximally SYM in one and two dimensions are presentedory [ of gravity on thestrongly basis of coupled superstring gauge theory. theory Therefore,study the using the duality the quantum can corresponding behavior be of employed gravitationalbility to gravity implies theory, study from that and the a conversely, gauge gaugedescription. to theory theory yields perspective. In a this Moreover, definition sense, this ofduality possi- many superstring is interesting theory accepted. beyond applications its ariseimportant However, perturbative once to this the test duality the concept validity has of of not this gauge/gravity duality. yet been provencoupled mathematically. gauge theory Thus, that it describesthis is dual duality is classical accurate gravity by well. comparingtions the yielded We lattice by can results gravity examine in theory. gauge whether For theory or finite with not the temperature theoretical and predic- low dimensional versions of AdS/CFT, Recent progress in lattice supersymmetry: from lattice gauge theory to black holes 3. Gauge/gravity duality and lattice gauge theory for the large PoS(LATTICE 2015)017 475 . (3.6) (3.7) (3.4) (3.5) 0 ≤ ]. 12 was set T = ≤ 102 Daisuke Kadoh N -axis denotes the 375 x . ]. 0. = 101 16 and the corresponding 2 c = L . ) with 35 . 3.1 0 ]. ± , p 98 , , 74 3 . i Cx ), if the assumption of duality is valid, the 4 S T h 2 + 14 and 32, respectively. The = L 3.1 8 N . 2 0 p 2 β = c x 10 − 3 1 N 41 . = = 7 is pressure. The gravity dual of the target gauge theory , p p 6 p . . The lattice size is fixed at − 2 − ) = 3 / ε x ε 1 ± ( f 0 λ . / 9 T = = C . The dashed blue curve represents the theoretical prediction yielded at low temperature and the large N limit. eff L T ) / 1 ... 6. We performed the fit using the five points within the 0 . = 37 . is consistent with the theoretical prediction within a statistical error of approx- eff 10 p aT . In this case, the expectation value of the action gives are the fitting parameters. From ( (= L 2 p / (right) shows the internal energy minus pressure as a function of temperature, which (left) shows the internal energy of the black hole. In the lattice simulation of SYM in 5 3 π 3 1 1 4 should be 4 2 and p is the internal energy density and = C ε 0 c We consider this theory for finite temperature and on a compactified space on a circle with As an extension to higher dimensions, we are performing a lattice simulation of maximally Figure Figure The lattice data approach the theoretical prediction given by the gravitational theory as the The obtained and the phase of thelimit has fermion not pfaffian yet was been reached, quenched. theevidence results of It reproduce the the is prediction duality apparent well. between that, This the figure target although provides gauge strong the theory continuum and the dual gravity theory [ circumference where with was calculated via lattice simulations with the Sugino action. In those simulations, SYM in two dimensions withscalar the field Sugino receives lattice a action. quantum Inexact correction SUSY two and renders dimensions, becomes the the a mass massin term relevant term perturbation irrelevant SUSY for theory, and breaking the at guarantees operator. least. the Thethat supersymmetric The numerical SUSY continuum results is limit for restored the at SUSY the Ward-Takahashi identity continuum show limit beyond perturbation theory [ where range and obtained is a black string for which obtained temperature decreases. However, theficiently temperatures low employed to in explain these(NLO) the simulations term leading from were behavior. these not results suf- by Instead, fitting one them can using the extract formula the next-to-leading order imately 7%. This is the firstof lattice the result for duality the conjecture NLO in term, this which one quantitatively dimensional shows theory the validity [ Recent progress in lattice supersymmetry: from lattice gauge theory to black holes one dimension, we employ the SuginoThe lattice red action and green and points quench denote the the phase results of for the fermion pfaffian. by the gravitational theory at the leading order of expansion, ( dimensionless temperature lattice spacing is PoS(LATTICE 2015)017 , 114 0.7 0.6 ]), and (right) Daisuke Kadoh 98 0.5 0 [ = p 0.4 T 0.3 0.2 12 in the right figure. = 0.1 N Gravity L=8x16 L=8x32 0 1 0 1.2 0.8 0.6 0.4 0.2 11 0.9 0.8 0.7 ]). The dashed blue curves are the theoretical predictions given by -branes. (Left) Internal energy of black hole ( p 102 0.6 1 [ 0.5 = T p 0.4 0.3 0.2 ], and numerical simulations with such formulations will provide us with a deep N=32 N=14 0.1 116 Gravity NLO Fit for black string ( 0 Thermodynamics of black 2 1 0 3 2 0.5 2.5 1.5

E/N ) 2 p I would like to thank Noboru Kawamoto, Hiroshi Suzuki, Yoshio Kikukawa, Fumihiko Sugino, Considerable future progress can be expected for lattice SUSY, because it has a wide range Lattice SUSY is a powerful tool for revealing the non-perturbative physics of SUSY theories, The realization of full SUSY on the lattice remains problematic. Some no-go theorems [ ] suggest that it is difficult to retain full SUSY on a lattice with locality. This situation reminds − ε understanding of the interesting aspectsWitten of theory, and SUSY AdS/CFT. theories, Thus, for further progress instance, in Seiberg lattice duality, SUSY Seiberg- is expected inAcknowledgment the near future. So Matsuura, Naoya Ukita, Issaku Kanamori and Eric Giguere for many discussions about lattice of application. It is importantwithout that fine a tuning method is of developed.lattice obtaining SYM Four four-dimensional [ dimensional lattice SYM SUSY can theories be obtained from two-dimensional us of the Nielsen-Ninomiya no-gothink theorem that and the the important history consequence ofof of any exact the attempt chiral Nielsen-Ninomiya to construct theorem symmetry. chiral would invariant One latticehas be models would been discouragement for achieved QCD. However, because exact of chiral symmetry therelation. discovery of the Therefore overlap efforts fermion satisfyingshould to be the construct continued. Ginsparg-Wilson fully SUSY invariant lattice models for SUSY theories 115 Figure 1: the gravitational theory at leading order. 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