We discuss the connection between supersymmetric field theories and
topological field theories and show how this connection may be used to
construct local lattice field theories which maintain an exact supersymmetry.
It is shown how metric independence of the continuum topological field theory
allows us to derive the lattice theory by blocking out of the continuum in a
deformed geometry. This, in turn, allows us to prove the cut-off independence
of certain supersymmetric Ward identities.Comment: Lattice2003(theory

Catterall, Simon

English

arXiv

We discuss the connection between supersymmetric field theories and topological field theories and show how this connection may be used to construct local lattice field theories which maintain an exact supersymmetry. It is shown how metric independence of the continuum topological field theory allows us to derive the lattice theory by blocking out of the continuum in a deformed geometry. This, in turn, allows us to prove the cut-off independence of certain supersymmetric Ward identities

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Syracuse University SURFACE Physics College of Arts and Sciences 9-10-2003 Exact Lattice Supersymmetry from Topological Field Theory Simon Catterall Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Catterall, Simon, "Exact Lattice Supersymmetry from Topological Field Theory" (2003). Physics. 462. https://surface.syr.edu/phy/462 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-lat/0309040v1  10 Sep 20031Lattice Supersymmetry and Topological Field TheorySimon CatterallaaDepartment of Physics, Syracuse University,Syracuse, NY 13244, USAWe discuss the connection between supersymmetric field theories and topological field theories and show howthis connection may be used to construct local lattice field theories which maintain an exact supersymmetry. It isshown how metric independence of the continuum topological field theory allows us to derive the lattice theory byblocking out of the continuum in a deformed geometry. This, in turn allows us to prove the cut-off independenceof certain supersymmetric Ward identities.1. MotivationThere have been many attempts to formulatesupersymmetric theories on lattices. [1]. But,unfortunately, generic lattice models break super-symmetry explicitly allowing relevant SUSY vio-lating operators to appear in the effective action.The couplings of such operators must then be finetuned to approach a supersymmetric fixed pointas the lattice spacing a→ 0.Because of these problems one is motivated totry to preserve an element of SUSY on the latticein the hope that this residual supersymmetry willprotect the lattice theory from dangerous radia-tive corrections. In this talk I will discuss oneway to achieve this – by exploiting a well-knownconnection between theories with extended su-persymmetry and topological field theories [2].The prototype example of this is supersymmetricquantum mechanics viewed as (0+1) dimensionallattice field theory. Another approach which alsoattempts to find lattice actions exhibiting an ex-act supersymmetry can be found in [3].2. SUSY QMConsider a model built from a discrete set ofscalar fields φi with classical action Scl(φ) = 0.Such a model is trivially invariant under the shift(topological) symmetryφi → φi + ǫiTo quantize this theory we need to pick a gaugecondition eg. Ni(φ) = 0. The partition functionof the gauge-fixed theory then takes the formZ =∫Dφe−1αN2i (φ)det(∂Ni∂φj)If the determinant is represented using anticom-muting ghosts ψi, ψi and an auxiliary field Bi isintroduced we can rewrite this asZ =∫DφDψDψDBe−SqwhereSq = −12αB2i +NiBi + ψi∂Ni∂φjψjSq is then invariant under the BRST symme-try:δφi = ψiξδψi = Biξδψi = 0δBi = 0We see that indeed that δ is a nilpotent operatoron these fields. Indeed, the action may be writtenSq = δ(ψi(Ni −α2Bi))We will show in the next section that this struc-ture is rather special and allows us to write downexpectation values in the continuum version ofthis model whose values are independent of themetric. These theories are thus termed topologi-cal quantum field theories TQFT.2Returning to our discrete model we can nowmake a specific choice of gauge function and anassociated physical interpretation of the field in-dices. If the latter are now chosen to label a seriesof lattice sites on a circle and the field Ni chosenasNi = D+ijφj + P′i (φ)whereD+ is the usual forward difference operatoron the lattice and P ′ is some arbitrary polynomialfunction of the field φ, we can reinterpret the re-sulting theory as a lattice regulated version of su-persymmetric quantum mechanics in Euclideanspace [4]. Notice that this requires us also toreinterpret the ghosts of the topological theoryas the physical fermions of the supersymmetrictheory. This is one aspect of the way in whichthe TQFT and its SUSY parent differ. The phys-ical state conditions arising from the gauge fixingof the shift symmetry ensure, in addition, thatthe true TQFT corresponds to the projection tothe vacuum states of the parent SUSY theory. Inour construction we drop this projection and treatthe discrete TQFT action as a bona fide latticeaction of the associated SUSY model possessingan exact supersymmetry. Our numerical resultsindicate that this is fine at least in low dimensions[5].3. Continuum TQFTThe ingredients of a topological quantum fieldtheory are a Riemannian manifold gµν , a setof fields Φ and an action S(Φ) together with aspecial class of so-called topological observablesO (Φ) which have the propertyδδgµν〈O〉 = 0This property can be guaranteed if the action isthe variation of some other function with respectto a nilpotent symmetry S = δΛ. The proof re-quires that both the measure and the operatorO be invariant under the symmetry (and further-more the latter should not contain the metric ex-plicitly) It is important for our purposes to noticethat any such theory always contains a trivial setof topological observables – namely those func-tions which arise from the symmetry variation ofsome other function O = δO′. In the language ofsupersymmetry these operators generate the su-persymmetric Ward identities of the parent SUSYtheory4. Lattice Action as a Perfect ActionThis metric independence of the continuumtheory may be exploited so as to derive the latticemodel directly from the continuum. The latticetheory can be obtained by a process of blockingout of the continuum in a deformed metric. We il-lustrate this in the 1D model we have introduced.In the continuum the bosonic action takes theformSB =∫dt e(t)[1e(t)dφdt+ P ′(φ)]2where e(t) is the einbein Change variables to blockfields:φB(t) =∫dt′ e(t′)B−β (t− t′)φ(t′)whereB−β (t) =12a[Lβ (t+ δ)− Lβ (t− a+ δ)]withlimβ→∞Lβ(t) = θ(t)Consider also an associated deformed metriceβ(t) =N∑n=1aA(β)L′β(t− na)We can easily show that in limβ→∞φB(t) =N∑n=1φ(na)12AL[θ(t− na)− θ(t− (n+ 1)a)]Thus the field φB approaches a constant valuewithin cells of a 1D lattice – changing only onmoving from one cell to another. Furthermore,its derivative takes the form1eβ(t)dφBdt=N∑n=112aALD−nmφm×[θ(t− (n− 1/2)a)− θ(t− (n+ 1/2)a)]3which is constant now within cells of a dual lat-tice. Furthermore, the continuum bosonic actionevaluated on such a continuum field resembles thelattice bosonic actionlimβ→∞SB(φB) =N∑n=1a[12ALaD−nmφBm + P′n(φB)]2Similar arguments apply to the full action evalu-ated on both scalar and fermion block fields. Asβ → ∞ one can show that Z is dominated bysuch block fields (the form of the scalar kineticterm ensures this) Thus the lattice theory, fromthe point of view of topological observables, ismerely the continuum theory in the limit of asingular background and evaluated using a newset of block variables. We would then expect allthe BRST symmetry of continuum theory to bemanifested at the quantum level on the lattice.Our simple model affords an example of the lat-ter feature.SUSY QM actually has 2 topological symme-tries in the continuum which correspond to theflip ψ → ψ – the original BRST symmetry (nowcalled δ1) and a second BRST symmetry δ2. Nowthe lattice action is, by construction, classicallyinvariant under δ1 but not under the second sym-metry δ2S 6= 0. The breaking term would bea total derivative in the continuum but is non-vanishing in general on the lattice. Neverthe-less, as our previous argument would have usbelieve, the Ward identities following from δ2are accurately satisfied on the lattice. The ta-bles below show numerical data for the bosonicand fermionic contributions to Ward identities forboth δ1 and δ2 for a simple model with P′ = gφ3on a N = 4 site lattice with large lattice spacing.Table 1δ(1)(ψtx0)t < x(0)N (1)(t) > < ψ(0)ψ(t) >0 0.8895(11) -0.8898(3)1 0.6152(10) -0.6155(3)2 0.4294(11) -0.4295(3)3 0.3024(11) -0.3028(3)Table 2δ(2) (ψtx0)t < x(0)N (2)(t) > < ψ(0)ψ(t) >0 -0.8895(11) 0.8898(3)1 -0.3016(11) 0.3028(3)2 -0.4294(11) 0.4295(3)3 -0.6160(10) 0.6155(3)5. ConclusionsCertain models with extended SUSY can be re-lated to topological quantum field theories. TheBRST charge(s) that appear in the latter theo-ries are formed from linear combinations of super-charges (a procedure termed twisting in the liter-ature) The topological symmetries may often beextended to the lattice and lead to models whichmaintain some exact residual supersymmetry. In-deed, the algebra Q2 = 0 evades all the usual no-go theorems associated with lattice SUSY. Fur-thermore, we have showed how the lattice theoriesmay be obtained by blocking out of continuum incarefully chosen background geometry. This pro-cedure naturally generates a r = 1 Wilson massterm for the fermions. Finally these ideas maybe extended to 2D sigma models and Yang-Millsmodels in four dimensions which is work currentlyunderway.REFERENCES1. A. Feo, Supersymmetry on the lattice,hep-lat/0210015.2. S. Catterall, JHEP 0305 (2003) 038.3. A. Cohen, D. Kaplan, E. Katz and M. Unsal,hep-lat/0307012A. Cohen, D. Kaplan, E. Katz and M. Unsal,JHEP 0308 (2003) 024.D. Kaplan, E. Katz and M. Unsal, JHEP 0305(2003) 037.4. S. Catterall and E. Gregory, Phys. Lett. B487(2000) 349.5. S. Catterall and S. Karamov, Phys. Rev. D65(2002) 094501.

Exact Lattice Supersymmetry from Topological Field Theory

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Exact Lattice Supersymmetry from Topological Field Theory

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