aa r X i v : . [ h e p - l a t ] N ov Witten index from lattice simulation
Issaku Kanamori ∗ † INFN sezione di Torino, and Dipartimento di Fisica Teorica, Università di Torino,Via P. Giuria 1, 10125 Torino, ItalyE-mail: [email protected]
I propose a method for measuring the Witten index using a lattice simulation. The index is usefulto discuss spontaneous breaking of supersymmetry. As a test of the method, I also report somenumerical results for the supersymmetric quantum mechanics, for which the index is known.
The XXVIII International Symposium on Lattice Filed TheoryJune 14-19,2010Villasimius, Sardinia Italy ∗ Speaker. † The current affiliation: Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ itten index from lattice simulation
Issaku Kanamori
1. Introduction
The supersymmetry (SUSY) is believed as a symmetry of the unification theory such as su-perstring theory and supersymmetric gauge theory is a candidate of a theory beyond the Standardmodel. However, it is broken in our current universe anyway. Since it cannot be broken by higherloop effects in perturbation, it is important to study the breaking nonperturbatively.Witten index [1] is a useful index related to the spontaneous SUSY breaking, which is definednonperturbatively. Using the fermion number operator F , it is given by the following trace, w = tr ( − ) F e − b H = ( N B − N F ) (cid:12)(cid:12) E = , (1.1)where H is the Hamiltonian of the system and E is its eigenvalue. As long as the spectrum isdiscrete, the index does not depend on a parameter b . It is simply a difference of numbers ofbosonic supersymmetric vacua and fermionic vacua. If the index is not zero, there exists at leastone supersymmetric vacuum so SUSY is not broken. But if the index is zero, SUSY may or may notbe broken, since it can be a result of cancellation between bosonic and fermion vacua, or a resultof no supersymmetric vacua at all. The purpose of this talk is to propose a method to measurethe Witten index using lattice simulation based on Ref. [2]. For a different approach from latticesimulation, see Ref. [3].In terms of the path integral, the index becomes a partition function with periodic boundarycondition [4, 5] w = Z P = Z D f D y D y exp ( − S P ) , (1.2)where f is boson, y and y are fermion, and subscript P stands for periodic boundary conditionsfor all the fields in the temporal direction. It seems difficult to measure this quantity using latticesimulation, since what we usually measure is an expectation value normalized by the partitionfunction but we need the normalization factor here. The normalization of the path integral measureis relevant as well.In the following section, we will discuss how to obtain the correct normalization of the par-tition function and thus the Witten index. And then in section 3 we confirm that it in fact worksin supersymmetric quantum mechanics of which the Witten index is well known using a latticesimulation. We also test a method which would improve the efficiency of the measuring the Wittenindex.
2. Idea
We have to determine two normalizations: one for the path integral measure and the other isfor the partition function (from the lattice data).The path integral measure with a correct normalization is easy to obtain. We only have tofollow a standard derivation of the path integral from the operator formalism, where we insert normalized complete sets at each of discretized time slices ( Fig. 1 ). Regarding the discretization If the spectrum is continuum, one has to take a limit b → ¥ . itten index from lattice simulation Issaku Kanamori insert complete sets a T = aN Figure 1:
Obtaining the correct measure: Derivation of the path integral is exactly the lattice regularization. is the lattice discretization, we obtain the following measures for bosons and fermions:Bosons: Z D f = Z ¥ − ¥ (cid:213) i √ p d f ( lat ) i , (2.1)Fermions: Z D y D y = Z (cid:213) i d y ( lat ) i d y ( lat ) i . (2.2)The correct normalization of the partition function is non-trivial. Let us start with a 1-dimensionalbosonic system with N lattice sites and consider the following quantity: * e + S exp (cid:20) − (cid:229) i m ( f lat i ) (cid:21)| {z } regularization functional + ≡ C R D f e − S , (2.3)where m is an arbitrary (positive) real number which should be tuned later and C = Z D f exp " − (cid:229) i m ( f lat i ) = m − N . (2.4)Here, we have used eq. (2.1). Combining eq. (2.3) and (2.4), we obtain Z = Z D f e − S = C (cid:28) exp (cid:20) + S − (cid:229) i m ( f lat i ) (cid:21)(cid:29) . (2.5)Since we can calculate the value of C analytically, and the denominator in the r.h.s is an observablein the lattice simulation, we can measure the partition function Z . Notice that though we have useda gaussian functional as a regularization functional in eq. (2.3), one can use any functional as longas it gives a calculable and convergent value like in eq. (2.4).In the r.h.s. of eq. (2.5), the action S appears with a “wrong sign” which cancels the originaldistribution. That is, the partition function is calculated using an extreme reweighting. To obtain abetter efficiency, we have to tune the value of m .Next let us introduce fermions. After integrating out the fermions, we obtain the effectiveaction as usual: S ′ = S B − ln | det ( D ) | , (2.6)where S B is the bosonic part of the action and D is the the fermion bilinear operator (i.e., theDirac operator plus the Yukawa interactions) . The phase factor of det ( D ) should be reweighted If the fermion is Majorana, the determinant should be replaced with a Pfaffian itten index from lattice simulation Issaku Kanamori afterwards which gives for arbitrary expectation values h A i = R D f A s [ D ] e − S ′ R D fs [ D ] e − S ′ = h A s [ D ] i h s [ D ] i , (2.7)where s [ D ] is the phase factor and the subscript 0 stands for a phase quenched average. Thistime we have to cancel e a factor s [ D ] e − S ′ to obtain the partition function. Therefore, measuring h s [ D ] − e + S ′ exp ( − (cid:229) i m ( f lat i ) ) i , we obtain the Witten index as w = Z P = C h s [ D P ] i , P (cid:10) exp (cid:2) + S ′ P − (cid:229) i m f i (cid:3)(cid:11) , P , (2.8)where C is given in eq. (2.4).The r.h.s of eq. (2.7) implies the phase quenched average of the phase factor s [ D ] is almostthe partition function. This observation is correct, and eq. (2.8) provides the correct normalizationto the partition function.
3. Numerical Test: Supersymmetric Quantum Mechanics
We test our method using supersymmetric quantum mechanics (of N = Q L with anexact supertransformation which satisfies Q =
0, we can repeat a similar argument to the contin-uum case. As a result, the index is well defined even at finite lattice spacing in such lattice models.In particular, the index from a finite lattice spacing should be an integer.A Q -exact lattice action for the supersymmetric quantum mechenics is given as [7] S = N − (cid:229) k = h ( f k + − f k ) + W ′ ( f k ) + ( f k + − f k ) W ′ ( f k ) − F k + y k ( y k + − y k ) + W ′′ ( f k ) y k y k i , (3.1)where f k is a real boson, y k and y k are fermions, and F k is a real bosonic auxiliary field. Thepotential W is a function of f and the prime ( ′ ) indicates a derivative. If the asymptotic behavioris W (+ ¥ ) W ( − ¥ ) > W (+ ¥ ) W ( − ¥ ) < • n = W = l f + l f SUSY, w = • n = W = l f + l f ✏✏✏✏ SUSY, w = l i are parameters of the potential. We use the Hybrid Monte Carlo algorithm. See [8] forthe implementation for this system.The results are plotted in Figs. 2 and 3. With a suitable choice of m , the known indexesare reproduced. There is almost no dependence on the lattice spacing, as expected from the exact Q -symmetry of the action.Next, we consider a possible way to improve the efficiency. Because of the factor e S ′ P ineq. (2.8), the efficiency is poor and we need large statistics. This factor cancels the weight from the4 itten index from lattice simulation Issaku Kanamori m set 4a, N=21set 4b, N=21set 4c, N=21 0.9 1 1.1 1.5 2 2.5 3 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 0 0.02 0.04 0.06 0.08 0.1 0.121/N set 4b, m =2.0 set 4a ( L l = , L l =
1) : m = . , w = . ( ) set 4b ( L l = , L l =
1) : m = . , w = . ( ) set 4c ( L l = , L l =
4) : m = . , w = . ( ) Figure 2: n = L is the physical size of the system. (left panel) m dependence. (right panel) Lattice spacing a = / N dependence. (bottom) values of m and the measuredindex w , which minimize the error. -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0.5 1 1.5 2 2.5 3 3.5 4 4.5 m set 3a, N=21set 3b, N=21set 3c, N=21set 3d, N=21 -0.003-0.002-0.001 0 0.001 0 0.02 0.04 0.06 0.08 0.1 0.121/N set 3d, m =3.0 set 3a ( L l = , L / l = m = . , − . ( ) set 3b ( L l = , L / l =
16) : m = . , . ( ) set 3c ( L l = , L / l =
32) : m = . , − . ( ) set 3d ( L l = , L / l =
16) : m = . , − . ( ) Figure 3: n = L is the physical size of the system. (left panel) m dependence. (right panel) Lattice spacing a = / N dependence. (bottom) values of m and the measuredindex w , which minimize the error. action so we do not have to use importance sampling with respect to a weight factor e − S . Therefore,we can also use configurations generated with less importance sampling. Decomposing the weightfactor as e − S = e − rS e − ( − r ) S , we rewrite a general expectation value as h A i = R D f Ae − rS e − ( − r ) S R D f e − rS e − ( − r ) S = h Ae − rS i r h e − rS i r , (3.2)where h · i r is an expectation value with a weight factor e ( − r ) S . Therefore, preparing configurations5 itten index from lattice simulation Issaku Kanamori using e ( − r ) S ′ P , we can obtain the Witten index as follow: w = C h s [ D P ] e − rS ′ P i r , P h exp (cid:2) ( − r ) S ′ P − (cid:229) i m ( f lat i ) (cid:3) i r , P . (3.3)Note that r = m . On the right panel, we plot thebehavior of the errors versus number of the configurations used in the measurements. Contrary tothe naive expectation, the magnitudes of the error are the same for large statistics in both r = r > r > ( num. of confs. ) − / faster than r = m number of configurationsr=0, m =2.0 0.01 0.1 1 r=0.25, m =1.5 0.01 0.1 1 r=0.5, m =1.0 Figure 4:
Results from the less importance sampling. (left panel) The obtained Witten index. Set la-bels are the same in Figs. 2 and 3. (right panel) Behavior of the errors, for set 4b. The dotted line is ( num. of conf. ) − / .
4. Conclusion and Discussion
We proposed a method for measuring the Witten index, which is a useful index to detect aspontaneous supersymmetry breaking. Since the index is given as a partition function under theperiodic boundary condition, it is important to use the correct normalization of the path integralmeasure. We also normalized overall factor of the partition function measuring a special regular-ization functional. As a test of the method, we measured the index of supersymmetric quantummechanics. The results reproduced the known values of the index. A disadvantage of the methodis its poor efficiency. A less importance sampling method may improve it to some extend.Finally, we mention possible applications of the method, which may or may not be practical.It is straightforward to use the method in higher dimensional systems. Within one-dimensional6 itten index from lattice simulation
Issaku Kanamori systems, the most interesting one is supersymmetric Yang-Mills quantum mechanics with 16 su-percharges. This model is one of the candidates of M(atrix)-theory, and assumes the Witten indexshould be 1 to obtain a suitable supergravity limit.
Acknowledgements
I. K. was financially supported by Nishina Memorial Foundation. He also thanks InsitutoNazionale di Fisica Nucleare (INFN).
References [1] E. Witten,
Constraints On Supersymmetry Breaking , Nucl. Phys. B (1982) 253.[2] I. Kanamori,
A Method for Measuring the Witten Index Using Lattice Simulation , Nucl. Phys. B (2010) 426 [ arXiv:1006.2468 [hep-lat] ].[3] H. Kawai and Y. Kikukawa,
A lattice study of N=2 Landau-Ginzburg model using a Nicolai map , arXiv:1005.4671 [hep-lat] .[4] S. Cecotti and L. Girardello, Functional Measure, Topology And Dynamical Supersymmetry Breaking ,Phys. Lett. B (1982) 39.[5] K. Fujikawa,
Comment On The Supersymmetry At Finite Temperatures , Z. Phys. C (1982) 275.[6] E. Witten, Dynamical Breaking Of Supersymmetry , Nucl. Phys. B (1981) 513.[7] S. Catterall and E. Gregory,
A lattice path integral for supersymmetric quantum mechanics , Phys.Lett. B (2000) 349 [ hep-lat/0006013 ].[8] I. Kanamori, F. Sugino and H. Suzuki,
Observing dynamical supersymmetry breaking with euclideanlattice simulations , Prog. Theor. Phys. (2008) 797 [ arXiv:0711.2132 [hep-lat] ].].