Our first guest blogger is Simon Catterall from Syracuse university, who graciously agreed to write a post about the ongoing research attempting to formulate supersymmetric theories on the lattice. Enjoy!
The problem of formulating supersymmetric theories on lattices has a long history going back to the earliest days of lattice gauge theory. However, after initial efforts failed to produce useful supersymmetric lattice actions the topic languished for many years. Indeed a folklore developed that supersymmetry and the lattice were mutually incompatible. However, recently, the problem has been re-examined using new tools and ideas such as topological twisting, orbifold projection and deconstruction and a class of lattice models have been constructed which maintain one or more supersymmetries exactly at non-zero lattice spacing.
While in low dimensions there are many continuum supersymmetric theories that can be discretized this way, in four dimensions there appears to a unique solution to the constraints — super Yang-Mills. The availability of a supersymmetric lattice construction for this theory is clearly very exciting from the point of view of exploring the connection between gauge theories and string/gravitational theories. In this posting I will outline some of the key ingredients that go into these constructions, the kinds of applications that have been considered so far and highlight the remaining difficulties.
First, let me explain why discretization of supersymmetric theories resisted solution for so long. The central problem is that naive discretizations of continuum supersymmetric theories break supersymmetry completely and radiative effects lead to a profusion of relevant supersymmetry breaking counter terms in the renormalized lattice action. The coefficients to these counter terms must then be carefully fine tuned as the lattice spacing is sent to zero in order to arrive at a supersymmetric theory in the continuum limit.
In most cases this is both unnatural and practically impossible — particularly if the theory contains scalar fields. Of course, one might have expected problems — the supersymmetry algebra is an extension of the Poincare algebra which is explicitly broken on the lattice. Specifically, there are no infinitesimal translation generators on a discrete spacetime so that the algebra is broken at the classical level.
One option is arrange for supersymmetry to emerge as an accidental symmetry in the continuum limit in a manner analogous to the way Poincare symmetry is restored in lattice QCD for vanishing lattice spacing. In the latter case the existence of a subgroup of discrete translational and rotational symmetries in the lattice action protect the theory from developing relevant counter terms which violate the full continuum symmetry which then re-emerges automatically in the continuum limit. Ideally one would like to arrange for something similar to happen in the case of supersymmetry. In the case of SYM chiral symmetry can enforce just such a condition — the only relevant susy breaking operator is the gaugino mass which is prohibited by chiral symmetry. If one employs a lattice action which respects chiral symmetry such as domain wall fermions one can hope that supersymmetry, while broken at non-zero lattice spacing, should be recovered without fine tuning in the continuum limit. Numerical work exploring this conclusion has started (1,2). Unfortunately, this nice state of affairs does not persist to more interesting models such as super QCD or models with extended supersymmetry.
The new idea
In the last five years or so this problem has been revisited using new theoretical tools and ideas and a set of lattice models have been constructed which retain exactly some of the continuum supersymmetry at non-zero lattice spacing. The basic idea is to use this subalgbra to constrain the effective lattice action and protect the theory from the dangerous susy violating counter terms. The class of theories which can be treated this way is quite restrictive — the continuum theories need to possess multiples of supercharges — but nevertheless this set includes many interesting theories.
The simplest way to understand how this subalgebra emerges is to reformulate the target theory in terms of “twisted fields”. The basic idea of twisting goes back to Witten (3) in his seminal paper on topological field theory. In our context the idea is decompose the fields of the theory in terms of representations not of the original (Euclidean) rotational symmetry but a twisted rotational symmetry which is the diagonal subgroup of this symmetry and an subgroup of the R-symmetry of the theory.
To be explicit consider the case where the total number of supersymmetries is . In this case I can treat the supercharges of the twisted theory as a matrix . This matrix can be expanded on products of gamma matrices It is easy to show from the original supersymmetry algebra that the scalar supercharge appearing in this expression is nilpotent and furthermore that the supersymmetric action can generically be written in -exact form (actually for SYM one needs an additional -closed term also).
It is important to recognize that the transformation to twisted variables corresponds to a simple change of variables in flat space — one more suitable to discretization. A true topological field theory only results when the scalar charge is treated as a BRST charge and attention is restricted to states annihilated by this charge. In the language of the supersymmetric parent theory such a restriction corresponds to a projection to the vacua of the theory. It is not employed in these lattice constructions.
When one discretizes the theory one tries to preserve just the subalgebra which then guarantees that the lattice theory is invariant under . Of course one must simultaneously require that the theory not suffer from “doubling problems” and, in the case of Yang-Mills theories retains exact gauge invariance. Remarkably, both of these additional constraints may be satisfied together with -symmetry provided the discretization is chosen carefully.
The absence of fermion doubling is intimately linked to the fact that the fermions, like the supercharges, are treated as a set of antisymmetric tensors in the twisted reformulation of the supersymmetric theory. For the case of supersymmetries it is possible to show that the original action describing degenerate Majorana fermions, can be re-written in an entirely geometrical way requiring only the (gauged) exterior derivative and its adjoint.
Indeed, the twisted fermions satisfy an equation — the Kaehler-Dirac equation of the form where is the collection of antisymmetric tensor fermion fields. This equation, unlike its spinorial cousin , can be discretized in such a way that the discrete solutions map one-to-one with those of the continuum equation, as was shown many years ago by Rabin, Becher and Joos (4,5) and thereby evades fermion doubling problems.
The basic idea is to place p-form fermions on p-cells in the lattice and replace by a forward difference operator and by a backward difference. The well-known staggered fermion formulation of lattice QCD can be related to this prescription for discretization of the Kaehler-Dirac equation. In practice the p-form field is placed on a link running through from the origin of the unit hypercube through the center of the p-cell.
This structure for the fermionic action when combined with the exact susy and gauge invariance will then dictate where the bosons of the lattice theory are located. To understand how this works in more detail let me first describe the bosonic sector of the continuum twisted Yang-Mills theory.
The gauge fields are singlets under the R-symmetry and so transform as vectors again under twisted rotations. The scalars however typically transform as scalars under the original rotations but vectors under the R-symmetry. Thus they also transform as vectors under twisted rotations. In practice they may be combined with the gauge fields to yield complexified connections in the twisted theory — the scalar fields being encoded in the imaginary parts of the gauge field. Notice that while the bosonic degrees of freedom are composed of complex connections the final action is only invariant under the usual gauge symmetry corresponding to the real part of this complex connection. The imaginary part of the connection yields the scalar fields of the usual theory. In the lattice theory the complex gauge fields are then exponentiated into complexified Wilson gauge variables and placed on corresponding links in the lattice.
Finally a generic lattice field living on the link is taken to transform under gauge transformations as . A set of rules for implementing gauge covariant lattice difference operators exists which respects these gauge invariance properties (6). This mapping of continuum to lattice fields, the corresponding gauge transformation properties and the mapping of and into difference operators then allows lattice actions to be derived from their continuum cousins with all the desired properties.
Rather remarkably the lattice actions I have described were first constructed using a completely different method by David B. Kaplan Mithat Unsal and collaborators (7,8). Their idea is to start from a zero dimensional supersymmetric matrix theory and carry out an orbifold projection in such a way as to zero out all but a few blocks of the matrices. The surviving blocks can be labelled by a set of charges which originate in the original global symmetries of the matrix theory.
Furthermore, these charges can be interpreted as vectors in a lattice and the non-zero blocks as fields on that lattice. The projected matrix action then inherits a Yang-Mills lattice gauge symmetry. If this projection is chosen carefully a subset of the supersymmetries of the continuum theory can be retained. Finally, a deconstruction step is performed in which the bosonic link matrices are expanded about points in the moduli space of the lattice theory to induce kinetic terms controlling a continuum limit. The precise connection between orbifolding and twisted constructions has now been explored in detail and the precise mapping between the two constructions is explicitly known (9,10).
In four dimensions the constraint that the target theory possess 16 supercharges singles out a single theory — SYM. Furthermore, both orbifold and twisting methods yield a unique lattice action which is invariant under a single real supersymmetry.
The lattice theory is most easily discussed in the language of twisting and so I will employ that here. The continuum twist of that is the starting point of the construction was first written down by Marcus in 1995 although it now plays a important role in the Geometric-Langlands program and is hence sometimes called the GL-twist. This four dimensional twisted theory is most compactly expressed as the dimensional reduction of a five dimensional theory in which the ten (one gauge field and six scalars) bosonic fields are realized as the components of a complexified five dimensional gauge field while the 16 twisted fermions naturally span one of the two Kaehler-Dirac fields needed in five dimensions.
The lattice that emerges is called and is constructed from the set of five basis vectors pointing out from the center of a four dimensional equilateral simplex out to its vertices together with their inverses . It is the four dimensional analog of the 3D bcc lattice. Complexified Wilson gauge link variables are placed on these links together with their -superpartners . Another 10 fermions are associated with the diagonal links with . Finally, the exact scalar supersymmetry implies the existence of a single fermion for every lattice site.
The lattice action corresponds to a discretization of the Marcus twist on this lattice and can be represented as a set of traced closed bosonic and fermionic loops. It is invariant under the exact scalar susy, lattice gauge transformations and a global permutation symmetry and can be proven free of fermion doubling problems as discussed before.
The lattice theory may be simulated using the standard algorithms of lattice QCD and work in this direction has already started (11).
The lattice constructions I have described play a role similar to lattice QCD in providing a non-perturbative definition of the corresponding continuum supersymmetric field theory. But at least in the case of sixteen supercharges they also have potential value as tools to explore the connection between gauge theory and string and supergravity models. Of course the most obvious example of the latter is the AdS-CFT correspondence which in its weakest form tell us that type IIB string theory should be dual to maximally supersymmetric large N Yang-Mills theory at strong coupling. The constructions described here may provide a new window on this strongly coupled theory both conceptually and practically through the use of Monte Carlo simulation.
Furthermore these dualities are conjectured to persist for type II string theory containing Dp-branes and -dimensional Yang-Mills with 16 supercharges. The simplest of these dualities corresponds to large N Yang-Mills quantum mechanics compactified on a thermal circle and IIa supergravity at low temperature. Quite a bit of numerical work both on and off the lattice has already been attempted for this system (12,13) yielding encouraging agreement between the thermodynamics of the Yang-Mills system and the corresponding black hole solution in supergravity. With these new formulations a much wider class of these dualities may be subjected to non-perturbative scrutiny. There has also been some work to extend the class of model that can be handled this way to systems with both adjoint and fundamental fermions in two dimensions.
Difficulties and the Future
These mainly relate to the practical problems of simulating these theories. Since we cannot represent anticommuting numbers on a computer one typically integrates out the fermionic degrees of freedom before devising a numerical algorithm to sample the path integral. This procedure generates the Pfaffian or determinant of a large sparse matrix whose size is given by the total number of degrees of freedom. In general, expectation values must be weighted by this Pfaffian in the path integral. However in many cases the Pfaffian is complex and hence carries a non-trivial phase factor. Monte Carlo algorithms require that the weight be be real, positive definite so typically this phase is dropped in the simulation. In principle, this can be corrected for when measuring expectation values by a well known re-weighting procedure. However, this technique will fail if the phase fluctuates too rapidly — this is the well known sign problem of finite density QCD simulations.
A priori it is not known whether this sign problem will make simulation of these supersymmetric lattice systems impossible. Luckily, preliminary numerical work has been rather encouraging in this respect at least for the sixteen supercharge models. However, the lattice sizes used so far have been rather modest while sign problems typically increase with the volume of the lattice. So more work must be done to understand the practical feasibility of using numerical simulation to explore SYM.
The second problem is of a more theoretical nature; the lattice actions preserve only a fraction of the supercharges. How well does this exact symmetry help to reduce the fine tuning problems associated with the restoration of full supersymmetry in the continuum limit ? Afterall this was the original motivation for this work. It turns out that the symmetries of the lattice theory, particularly the implementation of gauge invariance, strongly constrains the possible relevant counter terms that can occur.
Indeed, one can show on quite general grounds that in the case of SYM there only are only 3 relevant couplings in the renormalized action of the lattice theory and that two of these correspond to renormalization of the couplings of existing marginal operators in the bare lattice action. Restoration of full supersymmetry then requires that these couplings receive no renormalizations that diverge as the lattice spacing is sent to zero. A one loop calculation in the lattice theory, which has yet to be done, is clearly of great importance in this respect. This will be particularly true if a running in the lattice theory is found — for then the perturbative calculations would give a guide as to how to fine tune these remaining operators when taking the continuum limit.
In spite of these remaining difficulties it is exciting that we have at hand, for the first time, a set of supersymmetric lattice theories which should allow us to ask and answer a number of questions concerning the non-perturbative structure of at least some supersymmetric theories – for example questions concerning dynamical supersymmetry breaking. The potential connection to string and supergravity theories is also particularly exciting.
Much of the work discussed here should appear shortly in an upcoming Physics Report written jointly with David. B. Kaplan and Mithat Unsal. In addition, I would like to acknowledge many useful conversations over the past few years with Poul Damgaard, Joel Giedt, Noboru Kawamoto, So Matsuura, Fumihiko Sugino and Toby Wiseman, all of whom have contributed greatly to our understanding of these theories.
1. Lattice super-Yang-Mills using domain wall fermions in the chiral limit. Joel Giedt, Richard Brower, Simon Catterall, George T. Fleming, Pavlos Vranas Phys.Rev.D79:025015,2009.
2. Dynamical simulation of N=1 supersymmetric Yang-Mills theory with domain wall fermions. Michael G. Endres, arXiv:0902.4267 [hep-lat]
3. Topological field theory, E. Witten, Comm. Math. Phys. 117 (1988) 353.
4. Homology theory of lattice fermion doubling, J. Rabin, Nucl. Phys. B201 (1982) 315.
5. Dirac-Kaehler equation and fermions on the lattice, P. Becher and H. Joos, Zeit. Phys. C15 (1982) 343.
6. A Geometrical approach to N=2 super Yang-Mills theory on the two dimensional lattice, Simon Catterall, JHEP 0411:006,2004.
7. Supersymmetry on a Euclidean space-time lattice. 1. A Target theory with four supercharges. Andrew G. Cohen, David B. Kaplan, Emanuel Katz, Mithat Unsal, JHEP 0308:024,2003.
8. A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges, David B. Kaplan, Mithat Unsal, JHEP 0509:042,2005.
9. From Twisted Supersymmetry to Orbifold Lattices, Simon Catterall JHEP 0801:048,2008.
10. Geometry of Orbifolded Supersymmetric Lattice Gauge Theories, Poul H. Damgaard, So Matsuura, Phys.Lett.B661:52-56,2008.
11. First results from simulations of supersymmetric lattices, Simon Catterall, JHEP 0901:040,2009.
12. Black hole thermodynamics from simulations of lattice Yang-Mills theory, Simon Catterall, Toby Wiseman 2008, Phys.Rev.D78:041502,2008.
13. Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature. Konstantinos N. Anagnostopoulos, Masanori Hanada, Jun Nishimura, Shingo Takeuchi, Phys.Rev.Lett.100:021601,2008.