One dimensional supersymmetric Yang-Mills theory with 16 supercharges
We report on numerical simulations of one dimensional maximally supersymmetric SU(N) Yang-Mills theory, by using the lattice action with two exact supercharges. Based on the gauge/gravity duality, the gauge theory corresponds to N D0-branes system in type IIA superstring theory at finite temperature. We aim to verify the gauge/gravity duality numerically by comparing our results of the gauge side with analytic solutions of the gravity side. First of all, by examining the supersymmetric Ward-Takahashi relation, we show that supersymmetry breaking effects from the cut-off vanish in the continuum limit and our lattice theory has the desired continuum limit. Then, we find that, at low temperature, the black hole internal energy obtained from our data is close to the analytic solution of the gravity side. It suggests the validity of the duality.
One dimensional supersymmetric Yang-Mills theory with 16 supercharges
Graduate School of Science, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
Gauge/gravity duality asserts an equivalence between strongly coupled gauge theory and the classical gravity on curved space, which was originally stated as AdS/CFT correspondence which includes the supersymmetry by Maldacena. From the duality, we expect that strongly coupled gauge theories, which are usually difficult to calculate by hand, can be analytically solved via the gravity side. So, there are many applications from the context, getting over the barrier among fields (for example, elementary particle physics, cosmology, condensed matter physics and so on). However, it is a conjecture and therefore verifying the duality in some way is desirable.
We aim to verify the gauge/gravity duality from lattice simulations in one dimensional supersymmetric Yang-Mills theory with sixteen supercharges. The theory is obtained by dimensional reduction from 10d SYM (or 4d SYM). Actually, we introduce temperature into the theory by imposing the anti-periodic boundary conditions on fermions. Based on the duality, the gauge theory corresponds to N D0-branes system in type IIA superstring theory at finite temperature. In particular, at low temperature, the gauge theory becomes strong coupling and using analytic techniques to examine the dual black hole physics from gauge theory become difficult. So, we use the lattice gauge theory to analize the gauge theory. From comparisons between lattice results and analytic solutions of the gravity side, we discuss the validity of the gauge/gravity duality.
There are two previous works about numerical simulations of the 1d maximally supersymmetric Yang-Mills theory: non-lattice simulations done by Nishimura et al.[2, 3, 4, 5] and lattice simulations done by Catterall and Wiseman. Supersymmetry is broken by the cut-off effects in their regularized theories. Nevertheless, both results are consistent with the gravity side from UV-finiteness of 1d gauge theories. In contrast, we employ the lattice formulations with a few supersymmetric charges on the lattice [7, 8], which have been recently developing, in particular, our lattice theory has two exact supercharges even on the lattice. We expect that the lattice theory has some advantages, for example, clear signals etc., thanks to the exact charges, in high accuracy verifications of the duality.
In section 2 we explain our lattice theory and then in section 3 we see some details of simulation techniques. In section 4 we show that the lattice theory has the correct continuum limit by computing the supersymmetric Ward-Takahashi relation. In section 5 we show the internal energy of the dual black hole obtained from gauge theory side as an evidence of the gauge/gravity duality.
2 1d SYM with 16 supercharges
The supersymmetric Yang-Mills theory with sixteen supercharges is a gauge theory in which a gauge field of the temporal direction interacts with nine scalar fields and real sixteen fermions . The continuum action is given by
where is the ’t Hooft coupling constant. Here, all fields are expanded as by gauge group generators of the group 111 The generators satisfy the normalization condition, . . Also, the covariant differential operator is defined through . The are real symmetric matrices which satisfy the nine dimensional Euclidean Clifford algebra.
The realization of supersymmetry on the lattice has been a difficult issue due to the lack of Leibniz rule on the lattice for a long time. However, recently, Sugino proposed a lattice formulation of maximally supersymmetric Yang-Mills theory with two exact supercharges from the topological twisted version . In the twisted theory, the original action eq.(2.0) can be rewritten as a closed form, , using two supercharges where are gauge transformations. From the nilpotency of up to gauge transformations, we can see that the action is invariant under -transformations, without the obvious use of Leibniz rule.
Let us consider one dimensional lattice of the size with the periodic boundary condition. Scalars and fermions are defined on sites labeled by while a gauge field is defined on links through the link field to realize the exact gauge invariance. Our lattice action is defined by
where is a dimensionless ’t Hooft coupling constant defined by with the lattice spacing . Here, and are some combinations of original scalars and those of fermions, respectively (our notation follows , or see ). Lattice -transformations are defined over the new variables 222The transformations here are those for and (for the others, see  ).,
From the definitions above, are gauge transformations with gauge parameters ,
where is a gauge transformation with the parameter . As a result, -invariance is realized even on the lattice, because of eq.(2.0) and the exact gauge invariance of the lattice theory.
The continuum limit is realized by taking to zero while keeping a typical scale of this system (e.g., the dimensionful ’t Hooft coupling ). Also, the lattice action has no doublers because where is the free limit of the lattice Dirac operator.
To introduce temperature, we change the boundary condition on fermions from periodic one to anti-periodic one, while keeping that on bosons,
where temperature is defined by . Temperature breaks all supersymmetries explicitly and, of course, -invariance. Hereafter, denotes a dimensionless temperature .
3 Simulation details
We use the standard Hybrid Monte Carlo method. But, there are two additional difficulties in our fermion sector: the 4-fermi interaction and the pfaffian, as explained below.
After -transformations, the action eq.(2.0) includes a cut-off order 4-fermi interaction as
For the 4-fermi interaction, we introduce an auxiliary field to write it as the third term in eq.(3.0). We treat as a part of the boson action and as a part of the fermion bilinear , without integrating . So, we regard 10+1 bosonic fields, , as configurations generated by HMC method.
The integration of fermions becomes the pfaffian, , which generally takes complex values. We treat the absolute value and the complex phase of the pfaffian, individually, to avoid the sign problem. The absolute value can be given as an integral by pseudo fermion since , and the 4th root is approximately by the rational expansion,
where the order and the coefficients of the approximation are determined from the range of ’s eigenvalues measured in the simulations. We compute inversions of with shifts in eq.(3.0) by using multi-mass shifted solver. Also, for the phase of pfaffian, we use the phase quench, or use the phase reweighting method if we want to include the effect in the results.
For , HMC method stably works and we can obtain sufficient statistics. However, as temperature decreases, at some temperature (which actually depend on ), the magnitudes of scalar fields monotonically increase against Monte Carlo trajectories and therefore the thermalization does not occur. This instability is related to the classical flat direction of the boson action. To avoid the instability, in parameter regions where run-away modes to the flat direction appear, we introduce a mass term of scalar fields,
where is a dimensionless mass. Hereafter denotes a dimensionless mass .
4 Supersymmetric Ward-Takahashi relation
Supersymmetry is broken by the lattice cut-off. In the classical continuum limit, the breaking effects by the cut-off identically vanish. By contrast, in the quantum theory, it is not so clear whether the breaking effects generally vanish in the continuum limit because of ultra violet divergences and non-perturbative effects. Fortunately, the 1d gauge theory is UV-finite and any operators which break supersymmetry are not generated radiatively, so our lattice theory has the correct continuum limit, at least, in the perturbation theory. However, we do not know a-priori whether the similar argument is possible beyond the perturbation theory. Also, we must divide the cut-off effects from the other breaking sources, the temperature and the mass term, eq.(3.0).
For the issue, Kanamori and Suzuki used a method which can extract only the cut-off effects in lattice simulations of 2d SU(2) SYM. The method is a simple one measuring of the partially conserved SUSY current on the lattice. We use the same method and check whether the cut-off effects vanish in the continuum limit without relying on the perturbation theory.
For the continuum theory with mass term, we have the partially conserved supersymmetric Ward-Takahashi relation,
where is an arbitrary operator and are supercharges which generate supersymmetric transformations. Here and are the supercurrent and the breaking term from mass term, respectively, which are defined as
In the continuum theory, the supersymmetric Ward-Takahashi relation, eq.(4.0), holds even at finite temperature. It means that we can find the cut-off effects by measuring a lattice counterpart of the relation. Actually, we compute the following ratio,
where is the symmetric covariant difference operator. Here we only use the forward covariant difference operator in the lattice definition of the supercurrent . Also, for fixed , is uniquely determined because correlators in the denominator with other are nearly zero, that is, the ratio is meaningless. If the cut-off effects vanish in the continuum limit, the ratio must be in the limit, from eq.(4.0).
In Figure 1, we plot the ratio for and with , and the lattice size . The horizontal axis represents the temporal direction. The corresponding lattice spacing is in the unit of . For both cases, clear plateaus are observed within statistical errors. We perform the constant fit for the plateaus. The obtained values are consistent with within the statistical errors. The result suggests that cut-off effects mostly vanish near the continuum limit and our lattice theory has the correct continuum limit beyond the perturbation theory for, at least, of and .
5 Internal energy
The internal energy of the black hole, associated with the black hole thermodynamics, is one of simple examples to test the gauge/gravity duality. In the gravity side, the internal energy is related to temperature through an analytic formula,
We compute , where is the inverse temperature, from our data and compare it with the above analytic formula.
In Figure 2, we show the internal energy versus temperature for and . We used two different lattice sizes, for and for . For high temperature, we see that our data and the result obtained by high temperature expansion at next to leading order agree as expected. However, as temperature decreases, our data departs from the curve of the high temperature expansion around and it is smoothly close to the analytic curve of the gravity side, eq.(5.0). This suggests the validity of the gauge/gravity duality in this system.
In , HMC-runs are stable for without the mass term, however, for , the instability from the flat direction occurs. So, we must use to explore possible further lower temperature. Also, simulations with different lattice spacings at same temperature are necessary to take the continuum limit. Such simulations to obtain high accuracy results are in progress.
-  J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998) [hep-th/9711200].
-  K. N. Anagnostopoulos, M. Hanada, J. Nishimura and S. Takeuchi, “Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature,” Phys. Rev. Lett. 100, 021601 (2008) [arXiv:0707.4454 [hep-th]].
-  M. Hanada, A. Miwa, J. Nishimura and S. Takeuchi, “Schwarzschild radius from Monte Carlo calculation of the Wilson loop in supersymmetric matrix quantum mechanics,” Phys. Rev. Lett. 102, 181602 (2009) [arXiv:0811.2081 [hep-th]].
-  M. Hanada, Y. Hyakutake, J. Nishimura and S. Takeuchi, “Higher derivative corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics,” Phys. Rev. Lett. 102, 191602 (2009) [arXiv:0811.3102 [hep-th]].
-  M. Hanada, J. Nishimura, Y. Sekino and T. Yoneya, “Monte Carlo studies of Matrix theory correlation functions,” Phys. Rev. Lett. 104, 151601 (2010) [arXiv:0911.1623 [hep-th]].
-  S. Catterall and T. Wiseman, “Towards lattice simulation of the gauge theory duals to black holes and hot strings,” JHEP 0712, 104 (2007) [arXiv:0706.3518 [hep-lat]]. “Black hole thermodynamics from simulations of lattice Yang-Mills theory,” Phys. Rev. D 78, 041502 (2008) [arXiv:0803.4273 [hep-th]]. “Extracting black hole physics from the lattice,” JHEP 1004, 077 (2010) [arXiv:0909.4947 [hep-th]].
-  F. Sugino, “A Lattice formulation of super Yang-Mills theories with exact supersymmetry,” Nucl. Phys. Proc. Suppl. 140, 763 (2005) [Prog. Theor. Phys. Suppl. 164, 138 (2007)] [hep-lat/0409036]. “Various super Yang-Mills theories with exact supersymmetry on the lattice,” JHEP 0501, 016 (2005) [hep-lat/0410035].
-  D. B. Kaplan and M. Unsal, “A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges,” JHEP 0509, 042 (2005) [hep-lat/0503039].
-  C. Vafa and E. Witten, “A Strong coupling test of S duality,” Nucl. Phys. B 431, 3 (1994) [hep-th/9408074].
-  I. Kanamori and H. Suzuki, “Restoration of supersymmetry on the lattice: Two-dimensional N = (2,2) supersymmetric Yang-Mills theory,” Nucl. Phys. B 811, 420 (2009) [arXiv:0809.2856 [hep-lat]].
-  I. R. Klebanov and A. A. Tseytlin, “Entropy of near extremal black p-branes,” Nucl. Phys. B 475, 164 (1996) [hep-th/9604089].
-  N. Kawahara, J. Nishimura and S. Takeuchi, “High temperature expansion in supersymmetric matrix quantum mechanics,” JHEP 0712, 103 (2007) [arXiv:0710.2188 [hep-th]].
-  D.Kadoh and S.Kamata, "Toward numerical verification of the gauge/gravity duality in 1d SYM with 16 supercharges.", in preparation.
-  D.Kadoh and S.Kamata, "Black hole thermodynamics from lattice simulations of 1d SYM with 16 supercharges.", in preparation.