Gauge-fixing on the Lattice via Orbifolding
Dhagash Mehta, Noah S Daleo, Jonathan D Hauenstein, Christopher Seaton
GGauge-fixing on the Lattice via Orbifolding
Dhagash Mehta ∗ Dept. of Mathematics, North Carolina State University, Raleigh, NC 27695, USA, andDept. of Chemistry, The University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK.
Noah S. Daleo † and Jonathan D. Hauenstein ‡ Dept. of Mathematics, North Carolina State University, Raleigh, NC 27695, USA.
Christopher Seaton § Dept. of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway Memphis, TN 38112, USA.
When fixing a covariant gauge, most popularly the Landau gauge, on the lattice one encountersthe Neuberger 0/0 problem which prevents one from formulating a Becchi–Rouet–Stora–Tyutinsymmetry on the lattice. Following the interpretation of this problem in terms of Witten-typetopological field theory and using the recently developed Morse theory for orbifolds, we propose amodification of the lattice Landau gauge via orbifolding of the gauge-fixing group manifold and showthat this modification circumvents the orbit-dependence issue and hence can be a viable candidate forevading the Neuberger problem. Using algebraic geometry, we also show that though the previouslyproposed modification of the lattice Landau gauge via stereographic projection relies on delicatedeparture from the standard Morse theory due to the non-compactness of the underlying manifold,the corresponding gauge-fixing partition function turns out to be orbit independent for all the orbitsexcept in a region of measure zero.
I. INTRODUCTION
Lattice field theories have proved to be a very successfulway of exploring the nonperturbative regime of quantumfield theories. They also provide valuable insight and in-put to the nonperturbative approaches in the continuumsuch as the Dyson-Schwinger equations (DSEs), functionalrenormalization group studies (FRGs), etc. [1]. Since eachgauge configuration comes with infinitely many equivalentphysical copies, the set of which is called a gauge-orbit, toremove such redundant degrees of freedom from the gen-erating functional, one must fix a gauge in the continuumapproaches. Hence, to have a direct comparison betweenthe continuum approaches with the corresponding resultsfrom the lattice field theories, one also needs to fix a gaugeon the lattice, even though in general gauge-fixing is not re-quired on the lattice due to the manifest gauge invarianceof the lattice field theories. For this reason, gauge-fixedsimulations have recently attracted a considerable amountof interest.In the perturbative limit, the standard approach of fixinga gauge is the Faddeev-Popov (FP) procedure [2]. In thisprocedure, a gauge-fixing device called the gauge-fixing par-tition function, Z GF , is formulated out of the gauge-fixingcondition. For an ideal gauge-fixing condition, Z GF = 1 .The unity is then inserted in the measure of the generatingfunctional, so that the redundant degrees of freedom areremoved after appropriate integration. This procedure wasgeneralized in [3] and is called Becchi–Rouet–Stora–Tyutin(BRST) formulation. Gribov showed that in non-Abeliangauge theories a generalized Landau gauge-fixing condi-tion, if treated non-perturbatively, would have multiple ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] solutions, called Gribov or Gribov–Singer copies [1, 4, 5].Hence, the effects of Gribov copies should be properly takeninto account within the Faddeev-Popov procedure. In fact,on the lattice, for any Standard Model groups, the cor-responding Z G F turns out to be zero [6, 7] due to a per-fect cancelation among Gribov copies. Thus, when insertedinto the generating functional, the expectation value of agauge-fixed observable turns out to be of the indeterminateform / , called the Neuberger / problem. The problemyields that a BRST formulation on the lattice can not beconstructed and it is argued this may also hamper com-parisons of the results from the lattice with the continuumapproaches [8–10].In theory, to fix a gauge, one must solve the gauge-fixingcondition, a task that could turn out to be extremely diffi-cult in the nonperturbative regime due to the nonlinearityof the equations. Hence, gauge-fixing is currently formu-lated as a functional minimization problem in the latticefield theory simulations because, generally speaking, nu-merical minimization is a less difficult task than findingsolutions of a system of nonlinear equations.Let us consider an action that is invariant under the gaugetransformation U j ,µ → g † j U j ,µ g j +ˆ µ , where U j ,µ ∈ SU( N C )are the gauge-fields, g j ∈ SU( N C ) are the gauge transfor-mations, j is the lattice-site index, and µ is the directionalindex. Then, the standard choice (using the Wilson formu-lation of gauge field theories on the lattice) of the latticeLandau gauge-fixing functional, which we call the naïve lat-tice Landau gauge functional, to be minimized with respectto g j , is F U ( g ) = (cid:88) j ,µ (cid:18) − N c Re Tr g † j U j ,µ g j +ˆ µ (cid:19) , (1)for SU( N c ) groups. Points which are roots of the firstderivatives f j ( g ) := ∂F U ( g ) ∂g j = 0 for each lattice site j yield the lattice divergence of the lattice gauge fields and a r X i v : . [ h e p - l a t ] J un in the naïve continuum limit recovers the Landau gauge ∂ µ A µ = 0 . The matrix M F P is the Hessian matrix of F U ( g ) with respect to the gauge transformations. Z GF is then thesum of the signs of the determinants of M F P computed atthe Gribov copies.The minima of F U ( g ) are by definition solutions of thegauge-fixing conditions, but the minima only form a sub-set of the set of all Gribov copies, since the latter includessaddles and maxima in addition to the minima. The set ofminima of F U ( g ) is called the first Gribov region. Thereis no cancelation among these Gribov copies, so the Neu-berger / problem does not appear if one restricts thegauge-fixing to the space of minima instead of all solutionsof the gauge-fixing condition. This restricted gauge-fixing iscalled the minimal Landau gauge [11] and can be written interms of a renormalizable action with auxiliary fields (see,e.g., [12] for a review). However, the number of minimamay turn out to be different for different gauge-orbits andincreases exponentially with increasing lattice size, as wasshown for the compact U( ) case in Refs. [13–18]. Thus,the corresponding Z GF , which counts the number of min-ima for each gauge-orbit in the minimal Landau gauge, isorbit-dependent, and inserting Z GF in the generating func-tional becomes a difficult task.To resolve the gauge-dependence issue, one may furtherrestrict the gauge-fixing to the space of global minima,called the fundamental modular region (FMR). In thisgauge, known as the absolute Landau gauge, again the Neu-berger / problem is avoided as in the minimal Landaugauge case. However, the corresponding Z GF may be orbit-independent since the number of global minima is thoughtto be constant for any gauge-orbit (it is also anticipatedthat the set of configurations with degenerate global min-ima is a set of measure zero which forms the boundary of theFMR). Thus, the FMR is expected to not have any Gribovcopies within it [19, 20]. This claim was verified to be truefor the compact U (1) case for the one- and two-dimensionallattice in Refs. [13, 14]. The problem with the absoluteLandau gauge is that one must find the global minimum of F U ( g ) for sampled orbits, which corresponds to finding theglobal minimum of spin glass model Hamiltonians, a taskin most cases known to be an NP hard problem.In the past few years, a few further suggestions to evadethe Neuberger problem and restore BRST formulations onthe lattice have been put forward in Refs. [21–25], whichare reviewed in Ref. [26]. In the current paper, we con-centrate on the stereographic lattice Landau gauge whichwas proposed in Refs. [8, 9, 13]. In Section II, we firstreview this proposed modification of lattice Landau gauge-fixing via stereographic projection of the gauge-fixing man-ifold. We also give a plausible topological argument on whythe proposal might fail. In particular, the orbit indepen-dence of the corresponding Z GF is crucial to ensure thatthe stereographic lattice Landau gauge is a viable candi-date to evade the Neuberger / problem. We also showwhy topologically the stereographic projection might turnout to be orbit dependent. In Refs. [13, 15, 27], the prob-lem of finding all Gribov copies on the lattice was trans-formed into a problem in algebraic geometry. However, forthe stereographic lattice Landau gauge, it was not possibleto solve the corresponding equations using the then avail- able algebraic geometry methods. In Appendix A, withthe improved algorithms, we give explicit calculations ofthe number of Gribov copies using an algebraic geometrybased method which guarantees to find all isolated solu-tions for the simplest non-trivial case of the stereographiclattice Landau gauge, i.e., × lattice with periodic bound-ary conditions. With these stronger results, we show that Z GF for the stereographic lattice Landau gauge is orbit in-dependent over the orbit space except for a region of zeromeasure.In Section III, we propose a novel modification via orb-ifolding of the gauge-fixing manifold that is topologicallyvalid unlike the stereographic case, and show that Z GF isorbit-independent for this gauge-fixing. Though the idea ofan orbifold lattice Landau gauge was conceived in 2009 inRef. [13], the necessary mathematical framework, namely,Morse theory for orbifolds, was published later in thatyear [28]. We briefly review the definition of an orbifoldand Morse theory for orbifolds. Then, we apply these con-cepts to propose a modified lattice Landau gauge based onorbifolding of the gauge-fixing group manifold. We showhow the modification evades the Neuberger / problem forcompact U( ) while maintaining orbit-independence. Wethen conclude the paper in Section IV. II. STEREOGRAPHIC LATTICE LANDAUGAUGE
The following is a review of the stereographic lattice Lan-dau gauge. We start by noting that a major breakthroughto resolve the Neuberger / problem came from Schaden,who in Ref. [29] interpreted the Neuberger / problem interms of Morse theory. It can be shown that the corre-sponding Z GF for Landau gauge on the lattice calculatesthe Euler characteristic χ of the group manifold G at eachsite of the lattice, i.e., for a lattice with N lattice-sites, Z GF = (cid:88) j sign (det M F P ( g )) = ( χ ( G )) N , (2)where the sum runs over all the Gribov copies. This result isbased on the Poincaŕe–Hopf theorem, which states that theEuler characteristic χ ( M ) of a compact, orientable, smoothmanifold M is equal to the sum of indices of the zeros of asmooth vector field on M . In the case that the vector fieldis the gradient of a non-degenerate height function, a dif-ferentiable function from the manifold M to R with isolatedcritical points, the index at a critical point is ± dependingon the sign of the Hessian determinant at the critical point From Eq. (2), we identify F U ( g ) as a height function of thegauge-fixing manifold, Gribov copies as the critical points, It should be emphasised that in Refs. [13, 14, 30], it was shownthat the naïve lattice Landau gauge is not a Morse function at afew special orbits, such as the trivial orbit, due to the existenceof isolated and continuous singular critical points. However, fora generic random orbit, it is indeed a Morse function and it isthis property that saves the topological interpretation of the gauge-fixing procedure [29]. and M F P as the corresponding Hessian matrix. This in-terpretation establishes the fact that the gauge-fixing onthe lattice can be viewed as a Witten-type topological fieldtheory [31].For compact U( ), for which the group manifold is S , thelink variables and gauge transformations in terms of angles φ j ,µ , θ j ∈ ( − π, π ] mod 2 π are U j ,µ = e iφ j ,µ and g j = e iθ j ,respectively. Thus, the naïve gauge fixing functional inEq. (1) is reduced to F φ ( θ ) = (cid:88) j ,µ (cid:0) − cos( φ j ,µ + θ j +ˆ µ − θ j ) (cid:1) ≡ (cid:88) j ,µ (1 − cos φ θ j ,µ ) , (3)and the corresponding gauge-fixing conditions are: f j ( θ ) = − d (cid:88) µ =1 (cid:16) sin φ θ j ,µ − sin φ θ j − ˆ µ,µ (cid:17) = 0 , (4)where φ θ j ,µ := φ j ,µ + θ j +ˆ µ − θ j . A given random set of φ j ,µ is called a random orbit . Moreover, when all φ j ,µ are zero,it is called the trivial orbit . We choose periodic bound-ary conditions (PBC) which are given by θ j + N ˆ µ = θ j and φ j + N ˆ µ,µ = φ j ,µ , where N is the total number of lattice sitesin the µ -direction. With PBC, there is a global degree offreedom leading to a one-parameter family of solutions with θ j → θ j + ϑ, ∀ j where ϑ is an arbitrary constant angle. Weremove this degree of freedom by fixing one of the variablesto be zero, i.e., θ ( N,...,N ) = 0 . Then, { φ j ,µ } take randomvalues independent of the action, i.e., the strong couplinglimit β = 0 , which is sufficient to answer the questions weare interested in this paper.We can view Eq. (3) as a height function from S × · · · × S to R . Since χ ( S ) = 0 , Z GF = 0 . In fact, for any compact,connected Lie group G that is not -dimensional, it is wellknown that χ ( G ) is zero .To evade the Neuberger / problem, Schaden proposedto construct a BRST formulation for the coset spaceSU( ) / U( ) of a SU( ) theory. For this coset space, χ is non-zero. The proposal was generalized to fix gaugeof an SU( N c ) gauge theory to the maximal Abelian sub-group ( U(1) ) N c − in Refs. [32, 33]. In short, the Neuberger / problem for an SU( N c ) lattice gauge theory lies in ( U(1) ) N c − , and can be avoided if the problem for com-pact U( ) is avoided.Following this interpretation, a promising proposal to evadethe Neuberger / problem via a modification of the gauge-fixing group manifold (i.e., the manifold of the combina-tion g † j U j ,µ g j +ˆ µ ) of compact U( ) developed using stere-ographic projection at each lattice site was presented inRefs. [8, 9, 13]. The stereographic gauge fixing functional To see this, note that if t (cid:55)→ g ( t ) is a one-parameter group in G and L g ( t ) denotes left-multiplication by g ( t ) , then the derivative of L g ( t ) at t = 0 produces a vector field on G which never vanishes.Then χ ( G ) = 0 follows from the Poincaré–Hopf theorem. was proposed as: F sφ ( θ ) = − (cid:88) j ,µ ln(cos( φ θ j ,µ / , (5)and the corresponding gauge-fixing conditions are: f s j ( θ ) = − d (cid:88) µ =1 (cid:16) tan( φ θ j ,µ / − tan( φ θ j − ˆ µ,µ / (cid:17) = 0 (6)for all lattice sites j .Here, the Euler characteristic of the modified manifold isnon-zero, so the Neuberger / problem is avoided. Ap-plying the same approach to the maximal Abelian sub-group (U(1) ) N c − , as mentioned above, the generalizationas stereographic projection for SU( N c ) lattice gauge the-ories is also possible when the odd-dimensional spheres S k +1 , k = 1 , . . . , N c − , are stereographically projectedto the real projective space R P (2 k ) . In those references,using topological arguments the number of Gribov copieswas shown to be exponentially suppressed for the stereo-graphic lattice Landau gauge compared to the naïve gaugeand the corresponding Z GF for the stereographic latticeLandau gauge was shown to be orbit-independent for com-pact U( ) in one dimension. Since it can be shown thatthe FP operator for the stereographic lattice Landau gaugeis generically positive (semi-)definite, Z GF counts the totalnumber of local and global minima. The stereographic lat-tice Landau gauge is thought to be a promising alternativeto the naïve lattice Landau gauge, except that the orbit-independence of Z GF was yet to be confirmed for latticesin more than one dimension.It is interesting to point out that in supersymmetric Yang–Mills theories on the lattice, non-compact parameteriza-tions of the gauge fields similar to the stereographic pro-jection have been used [34], independently of the develop-ment of the stereographic lattice Landau gauge (see, e.g.,[35, 36] for earlier accounts on non-compact gauge-fieldson the lattice). The non-compact parameterization in thesupersymmetric lattice field theories, unlike the compact(group based) parameterization, surprisingly avoids thewell-known sign problem in these lattice theories [37, 38].Recently, a more direct connection between the sign prob-lem in lattice supersymmetric theories and the Neuberger / problem has been established [39] by noticing that thecomplete action of, for example, the N = 2 supersymmet-ric Yang-Mills theories in two dimensions can be shown tobe a gauge-fixing action via Faddeev-Popov procedure tofix a topological gauge symmetry in this case. A. Orbit-dependence of the Stereographic LatticeLandau Gauge
The following provides an explanation of toopologicallysubtleties of the stereographic gauge (see [40, 41] for furtherbackground). Let M be a closed manifold (i.e., compactand without boundary). A smooth function f : M → R has a critical point at x if df x is nonsingular; a criticalpoint x is degenerate if the Hessian Hf ( x ) of f at x is sin-gular and non-degenerate otherwise. A Morse function isa smooth function whose critical points are isolated andnon-degenerate. Given such a Morse function of f , thegradient ∇ f is a tangent vector field to M that vanishesat exactly the critical points x ∈ M for f . As f is Morse,it has isolated critical points, which must then be finite as M is closed. The requirement that a critical point x of f be nondegenerate implies that the index ind x ( ∇ f ) of thevector field ∇ f at x is ± , depending only on the sign ofthe determinant of the Hessian Hf ( x ) of f at x . Therefore,letting C denote the set of critical points in M , we have (cid:88) x ∈ C sign ( det Hf ( x )) = (cid:88) x ∈ C ind x ( ∇ f ) (7) = χ ( M ) , where the last equality follows from the Poincaré–Hopf the-orem. Hence, in the case where M = (cid:81) j S is the productof circles parameterized by the { φ θ j ,µ } at each lattice site,the partition function Z GF in fact depends only on M , andcomputes χ ( M ) for any collection of { φ j ,µ } or any choice ofMorse function F φ .In the case that M is not closed but rather an open manifoldwithout boundary, the sum in Eq. (7) depends on f , and notsimply on M . This can be seen, for instance, by choosinga Morse function on the circle S with at least two criticalpoints (whose indices must sum to ) and then by defining M to be an open subset of S . Then, M can be chosen tobe an interval in S which contains a single critical point x ,in which case the sum is ± depending on ind f ( x ) . Also,one can choose M to be an open interval in S containingno critical points, in which case the sum is . Note that ineach of these cases, the manifold M is diffeomorphic to anopen interval. In short, when M is not closed, the sum ofthe indices depends on the height function.Using the stereographic gauge fixing functional Eq. (5), itcan be shown that the Hessian is generically positive [15],so that Z GF is strictly positive and counts the number ofcritical points. For a -dimensional lattice, there are only N critical points [13, 42], so the corresponding Z GF = N ,which is independent of orbits, and thus Z GF does not de-pend on the choice of { φ j ,µ } . In higher dimensions, how-ever, the above phenomenon may occur, and Z GF may varywith the choice of { φ j ,µ } since the stereographic gauge isoutside the applicability of Morse theory.Appendix A demonstrates that, for the stereographic lat-tice Landau gauge for a -dimensional lattice, the numberof Gribov copies and hence Z GF indeed are orbit indepen-dent quantities except in a region of orbit space with mea-sure zero, via explicit calculations. Specifically, we use analgebraic geometry based method which guarantees to findall isolated solutions of a given nonlinear system of equa-tions with polynomial-like nonlinearity to show that thoughthe number of Gribov copies for the × lattice for the com-pact U( ) case is constant, , for most of the randomorbits { φ j ,µ } , there are regions in the orbit space for whichthe numbers of Gribov copies differ from this number. III. ORBIFOLDING
The following uses orbifolding to develop a modification oflattice Landau gauge which is topologically rigorous unlikethe steregraphic gauge. We start by reviewing some of thebasic concepts about orbifolds. We give the definition of aorbifold and then describe Morse theory for orbifolds. Wethen apply Morse theory for orbifolds to propose a modifiedlattice Landau gauge via orbifolding the gauge-group man-ifold that evades the Neuberger / problem while beingorbit-independent.Let M be a manifold and G a finite group of diffeomor-phisms of M . Then the quotient G \ M is an example ofa global quotient orbifold or simply orbifold . Note that ingeneral, orbifolds are required to be only locally of the form G \ M , but we restrict our attention here to global quotientorbifolds; e.g., see [43]. A point in G \ M corresponds to the G -orbit Gx = { gx : g ∈ G } of x ∈ M .There are several Euler characteristics for orbifolds, andeach can be computed using a Morse function with modi-fications to the method of Eq. (7). The reader is warnedthat the term “orbifold Euler characteristic” can refer todifferent Euler characteristics in the literature. The mostprimitive Euler-characteristic, in the sense that other Eulercharacteristics can be defined in terms of it, is the so-called Euler–Satake characteristic χ ES ( M , G ) , which is given by χ ES ( M , G ) = χ ( M ) / | G | , (8)where | G | denotes the order, or number of elements, of G .It was defined for general orbifolds in [44]; see also [45, 46].Note that in general, χ ES is a rational number. One mayalso consider the usual Euler characteristic (of the under-lying topological space) χ ( G \ M ) , which is related to theEuler–Satake characteristic via χ ( G \ M ) = 1 | G | (cid:88) g ∈ G χ ( M g )= (cid:88) ( g ) ∈ G ∗ χ ( M g ) / | Z ( g ) | (9) = (cid:88) ( g ) ∈ G ∗ χ ES ( M g , Z ( g )) , where Z ( g ) = { h ∈ G : gh = hg } , M g = { x ∈ M : gx = x } is the set of points in M fixed by g , ( g ) = { hgh − : h ∈ G } is the conjugacy class of g in G , and G ∗ the set of con-jugacy classes in G . Note that χ ES ( M g , Z ( g )) coincidesfor each element of a conjugacy class, so that the last twosums are well-defined. In particular, χ ( G \ M ) is the sum ofthe Euler–Satake characteristics of the orbifolds Z ( g ) \ M g ,which for g (cid:54) = 1 are called twisted sectors . The nontwistedsector corresponding to g = 1 coincides with G \ M . Thecollection (cid:116) ( g ) ∈ G ∗ Z ( g ) \ M g is called the inertia orbifold , de-noted Λ( G \ M ) , (see e.g. [43]) so that succinctly, the usualEuler characteristic χ ( G \ M ) is the Euler–Satake character-istic of the inertia orbifold.The stringy orbifold Euler characteristic χ str ( M , G ) , intro-duced in [47, 48] for global quotients and [49] for generalorbifolds, see also [50], is defined analogously as χ str ( M , G ) = 1 | G | (cid:88) ( g,h ) ∈ G com χ ( M (cid:104) g,h (cid:105) ) , (10)where G com denotes the set of ( g, h ) ∈ G = G × G suchthat gh = hg and M (cid:104) g,h (cid:105) = { x ∈ M : gx = hx = x } denotes the set of points fixed by both g and h . This Eu-ler characteristic is related to the others as follows. Fora pair of commuting elements ( g, h ) ∈ G com , let [ g, h ] = { ( kgk − , khk − ) : k ∈ G } (the orbit of ( g, h ) under theaction of G on G com by simultaneous conjugation), let G com ∗ = { [ g, h ] : ( g, h ) ∈ G com } (the set of orbits), andlet Z ( g, h ) = Z ( g ) ∩ Z ( h ) denote the subgroup of G con-sisting of elements that commute with both g and h . Thencomputations similar to those in Eq. (9) demonstrate that χ str ( M , G ) = (cid:88) ( g ) ∈ G ∗ χ ( Z ( g ) \ M g ) (11) = (cid:88) [ g,h ] ∈ G com ∗ χ ES ( M (cid:104) g,h (cid:105) , Z ( g, h )) . In other words, χ str ( M , G ) is the usual Euler char-acteristic of the inertia orbifold, and as well coin-cides with the Euler–Satake characteristic of the orbifold (cid:116) [ g,h ] ∈ G com ∗ Z ( g, h ) \ M (cid:104) g,h (cid:105) . Observe that this latter dis-joint union is in fact the inertia orbifold of the inertiaorbifold, which we refer to as the double-inertia orbifold Λ ( G \ M ) . The orbifold corresponding to [ g, h ] = [1 , isthe nontwisted double-sector , while the other orbifolds arereferred to as twisted double-sectors . The reader is warnedthat double-sectors do not coincide with -multi-sectors de-fined in [43] unless G is abelian .A Morse function on a global quotient orbifold G \ M is de-fined to be a Morse function f : M → R that is G -invariant,i.e. f ( gx ) = x for each g ∈ G and x ∈ M . The latter condi-tion implies that f yields a continuous function ˜ f : G \ M → R on the topological space G \ M given by ˜ f ( Gx ) = f ( x ) .Morse theory has recently been developed for orbifolds inthe general context of Deligne-Mumford stacks [28] which,in particular, demonstrates that orbifolds always admitMorse functions, and establishes Morse inequalities for anorbifold and the corresponding inertia orbifold.To compute the Euler characteristic χ ES using a Morsefunction , one can apply the Poincaré–Hopf theorem fororbifolds as demonstrated in Ref. [44]. The reader may have noticed that the three Euler characteristics χ ES , χ , and χ str form the th, st, and nd elements of a sequenceof Euler characteristics for orbifolds, so that others can be defined.This was observed in [51], and this sequence was defined and studiedfor global quotients in [52]. More generally, an Euler characteristiccorresponding to each finitely generated discrete group (with theabove sequence corresponding to the groups Z m for m = 0 , , , . . . )was assigned to a global quotient an orbifold in [53, 54], and theseEuler characteristics were defined for arbitrary orbifolds in [55]. Satake worked with
V-manifolds , orbifolds where each group ele-ment is assumed to fix a subset of codimension at least . However,this result can be extended to general orbifolds by applying it tothe orientable double-cover, which always satisfies this hypothesis,and can be proved directly for global quotient orbifolds using thePoincaré–Hopf theorem for manifolds. For a global quotient orbifold G \ M , a point Gx is a criticalpoint of ˜ f if x is a critical point of f , and Gx is said to be degenerate (respectively non-degenerate ) if x is degenerate(respectively non-degenerate) for f . Note that the require-ment that f is G -invariant implies that these notions donot depend on the choice of x from the orbit Gx .Similarly, the gradient ∇ f (depending on a choice of Rie-mannian metric) defines a G -equivariant vector field on M ,which induces a vector field denoted ∇ ˜ f on the orbifold G \ M . If Gx is a zero of ∇ ˜ f (equivalently, a critical pointof ˜ f ), then the index of ∇ ˜ f at Gx is defined to beind orbGx ( ∇ ˜ f ) = 1 | G x | ind x ( f ) where G x = { g ∈ G : gx = x } is the subgroup of G thatfixes x . That is, the index of a critical point on an orbifoldis the index of a corresponding critical point on the man-ifold divided by | G x | . Again, the (manifold) index can becomputed as the sign of the determinant of the Hessian.If C denotes the set of critical points of ˜ f on G \ M , thenSatake’s Poincaré–Hopf theorem for orbifolds implies (cid:88) Gx ∈ C | G x | sign ( det Hf ( x )) = (cid:88) Gx ∈ C ind orbGx ( ˜ f )= χ ES ( M , G ) . Therefore, the sum of the (orbifold) indices of the criticalpoints computes the Euler–Satake characteristic. In thecontext of global quotients, it is not hard to show that aMorse function ˜ f on G \ M defines a Morse function Λ ˜ f onthe inertia orbifold Λ( G \ M ) as well as a Morse function Λ ˜ f on the double-inertia orbifold Λ ( G \ M ) by restricting ˜ f to the appropriate fixed-point submanifolds. By Eq. (9)and (11), we obtain that applying the procedure above to Λ ˜ f or Λ ˜ f yields χ ( G \ M ) and χ str ( M , G ) , respectively. A. A simple example
To illustrate this procedure, consider the orbifold given by M = S and G = Z , where the nontrivial element a of Z acts via e iθ (cid:55)→ e − iθ . The resulting orbifold can be identifiedwith { e iθ : 0 ≤ θ ≤ π } , as each e iθ with π < θ < π is inthe orbit of e i (2 π − θ ) . It is therefore homeomorphic to aclosed interval, where the endpoints are the images of thetwo points fixed by Z . Then we have that χ ES ( M , G ) = 0 ,as χ ( S ) = 0 , and χ ( G \ M ) = 1 , the Euler characteristicof a closed interval. To compute χ str ( M , G ) , note that allelements of G = { (1 , , (1 , a ) , ( a, , ( a, a ) } are mutuallycommuting, and the common fixed-point set of each is twopoints except for the trivial pair (1 , which fixes all of S .Hence, applying Eq. (10) yields χ str ( M , G ) = 3 .To compute these Euler characteristics using a Morse func-tion, we choose f ( θ ) = cos( θ ) . The corresponding ˜ f hascritical points at the orbits of θ = 0 and θ = π . TheHessians of f at these two critical points are − and , re-spectively, and the isotropy groups are both Z , so that wecompute χ ES ( M , G ) = ind orbG ( ∇ ˜ f ) + ind orbGπ ( ∇ ˜ f )= − | Z | + 1 | Z | = −
12 + 12 = 0 . To compute χ , we note that the inertia orbifold Λ( G \ M ) inthis case has three connected components, the nontwistedsector as well as two points, each equipped with trivial Z -actions. The function f restricted to a point triviallyhas a non-degenerate critical point of index . It followsthat χ ( G \ M ) is given by the sum of χ ES ( M , G ) , computedabove, as well as one term of the form / | Z | = 1 / for eachtwisted sector. That is, χ ( G \ M ) = χ ES ( M , G ) + 1 | Z | + 1 | Z | = 1 . Similarly, as Λ ( G \ M ) consists of G \ M as well as six points,each with isotropy Z , we have χ str ( M , G ) = χ ES ( M , G ) + 6 (cid:18) | Z | (cid:19) = 3 . B. Orbifolding the lattice Landau gauge
To apply the lattice Landau gauge procedure for com-pact
U(1) to orbifolds, we define a Z -action on the spacevariables { φ θ j ,µ } by letting the nontrivial element a ∈ Z act via a : φ θ j ,µ (cid:55)→ − φ θ j ,µ . The choice of group ac-tion is motivated by the fact that the gauge fixing func-tion Eq. (3) is invariant under this action. However,though it is the case that χ ES (( S ) N d − , Z ) = 0 , neither χ ( Z \ ( S ) N d − ) nor χ str (( S ) N d − , Z ) vanish. The in-ertia orbifold Λ( Z \ ( S ) N d − ) consists of the nontwistedsector as well as N d − points with trivial Z -action, eachgiven by the orbit of a point ( φ θ j ,µ ) where each φ θ j ,µ is or π , so that χ (cid:0) Z \ ( S ) N d − (cid:1) = 2 N d − . Similarly, as each of the pairs of group elements (1 , a ) , ( a, , and ( a, a ) fix again N d − points, the double-inertia Λ ( Z \ ( S ) N d − ) consists of the nontwisted sector and · N d − points with trivial Z -action, so that χ str (cid:0) ( S ) N d , Z (cid:1) = 3 · N d − . To apply the procedure, then, given a random choice of { φ j ,µ } , is to use the Morse function ˜ F on Z \ ( S ) N d − induced by F on ( S ) N d − defined in Eq. (3) with nochanges to the gauge-fixing and boundary conditions. Since Λ ( Z \ ( S ) N d − ) consists only of the nontwisted double-sector and -dimensional twisted double-sectors, the re-striction of Λ ˜ F to each connected component of a twisteddouble-sectors trivially has a nondegenerate critical pointwith positive index. Hence, if C denotes the set of criticalpoints on the nontwisted sector, we have Z GF = (cid:88) Z θ ∈ C | ( Z ) θ | sign ( det M F P ) + 3 · N d − = 3 · N d − . Note that the sum vanishes because it computes χ ES (( S ) N d − , Z ) = 0 . Hence the critical points in thenontwisted sectors occur in pairs with positive and neg-ative Hessian determinants. Furthermore, note that thecomputation of the sum differs from the manifold case inthat a pair of stationary points { φ θ j ,µ } and {− φ θ j ,µ } of F are the same stationary point for ˜ F , and hence the sign ofdet M F P is counted only once. This may be accomplishedalgebraically by choosing a single φ θ j ,µ and considering onlycritical points such that ≤ φ θ j ,µ ≤ π ; for critical pointssuch that φ θ j ,µ = 0 or π , we choose another variable andrestrict in the same way.As an example, let N = 3 and d = 1 . We consider thetrivial orbit for simplicity, i.e. each φ i = 0 , and fix θ = 0 to remove the global degree of freedom arising from the pe-riodic boundary condition θ N +3 = θ i ; see Section II. Thenwe have F φ ( θ ) = N (cid:88) i =1 (1 − cos φ θi )= 3 − cos( θ − θ ) − cos( − θ ) − cos( θ ) . Setting ∂∂θ i F φ ( θ ) = 0 for i = 1 , , we find five solutionsfor ( θ , θ ) : (0 , , (0 , π ) , ( π, , ( π, π ) , and (2 π/ , − π/ .Note that we only consider solutions such that ≤ θ ≤ π as above, because the solution ( − π/ , π/ is in the same Z -orbit as (2 π/ , − π/ and hence represents the samepoint on the orbifold. The Hessian determinants of thesecritical points are +3 , − , − , − , and / , respectively,and the first four critical points are fixed by Z while thelast is fixed only by the trivial element. It follows thatthe indices are given by / , − / , − / , − / , and , re-spectively, and their sum computes χ ES (cid:0) ( S ) , Z ) = 0 .To compute χ (cid:0) Z \ ( S ) ) , we consider F φ ( θ ) as a functionon the larger space Λ (cid:0) Z \ ( S ) ) consisting of Z \ ( S ) as well as four isolated points fixed by Z correspondingto the fixed points (0 , , (0 , π ) , ( π, , and ( π, π ) . Eachpoint is isolated and hence trivially a critical point withindex / | Z | , so summing these indices along with thoseon Z \ ( S ) described above yields χ (cid:0) Z \ ( S ) ) = 2 . For Λ (cid:0) Z \ ( S ) ) , we consider instead three copies of each iso-lated fixed point, one for each nontrivial commuting pair (1 , a ) , ( a, , and ( a, a ) , yielding twelve critical points withindex / | Z | and hence χ str (cid:0) ( S ) , Z ) = 6 . C. An Integral Formulation of Z GF for Orbifolding For the sake of completeness, we also provide an expres-sion of Z GF in the usual integral formulation a la Faddeev-Popov procedure, which we plan to further refine to suitthe needs of the lattice simulations. To compute the topo-logical Euler characteristic χ ( Z \ ( S ) N d − ) , we have Z GF = (cid:90) orb Λ( Z \ ( S ) Nd − ) D θ D φ N d − (cid:89) i =1 δ ( f i ) H ( F ) | H ( F ) | (12)where D θ indicates integration over each θ , the f i are thestationary equations, i.e., f i = ∂F∂θ i , and H ( F ) is the hessiandeterminant of F . The integral is computed in the orbifoldsense, see [43, Section 2.1]. If we let X denote the subset of ( S ) N d − × Z consisting of pairs ( θ, g ) such that gθ = θ ,then this orbifold integral can be expressed using the usualintegral as Z GF = 12 (cid:90) X D θ D φ N d − (cid:89) i =1 δ ( f i ) H ( F ) | H ( F ) | , (13)where the prefactor / arises from the order of Z and thedefinition of orbifold integration. D. Summary of the Procedure
To summarize, the procedure for computing the topologicaland stringy Euler characteristic from the naive gauge-fixingfunctional can be divided in three steps. In the first step:1. Find all the stationary points of F φ ( θ ) as given inEq. 3 by solving ∂F∂θ i = 0 , i = 1 , ..., N d .2. Call the solution vectors of these equations φ θ . Let’ssay there are M solutions.3. If for two solutions, say φ θ (1) and φ θ (2) , we have φ θ (1) = − φ θ (2) , then discard one of them. Hence, m ≤ M solutions are left in the end.4. Compute the hessian determinant at each of the m solutions.5. For each solution φ θ , the index is ± if φ θ (cid:54) = − φ θ and ± if φ θ = − φ θ , where the sign is that of the hessiandeterminant.6. Compute the sum of the indices for each solution.This sum will be always zero in our case.For the second step (the fixed points):1. The fixed vectors are simply φ θ =(0 , , ..., , (0 , , ..., , π ) , ..., ( π, π, ..., π ) , i.e., allthe N d − combinations of and π . These solutionsalready appeared in the first step, but are now con-sidered as isolated points (twisted sectors) associatedto the nontrivial group element.2. By convention, the ‘hessian determinant’ for each ofthese solutions is positive, and each solution is fixedby construction, so the index of each of these pointsis + .3. The (topological) Euler characteristic χ ( G \ M ) isgiven by the sum of all indices found in the first twosteps, χ ( G \ M ) = 0 + ( ) · N d − = 2 N d − .Finally, the third step (for the fixed points associated tocommuting pairs):1. The fixed vectors are the same as in the secondstep, but we now consider three copies of each forthe three nontrivial commuting pairs of group ele-ments ( ( a, , (1 , a ) , and ( a, a ) where a is the non-trivial element of Z ). 2. We again have that the index of each such point is + .3. The stringy Euler characteristic of the orbifold is thenthe sum of the indices from first and third step, i.e., Z GF = χ str ( M , Z ) = 0 + ( ) · · N d − = 3 · N d − . IV. CONCLUSION
Like many other crucial nonperturbative phenomena,gauge-fixing and the BRST symmetry are yet to be wellunderstood in the nonperturbative regime of gauge fieldtheories. In this paper, we first reviewed and investigateda recently proposed modified Landau gauge on the lattice,known as stereographic lattice Landau gauge. We gaveplausible arguments to demonstrate why this gauge maynot turn out to be a valid topological field theory due tothe fact that the procedure is outside the applicability ofMorse theory. In Appendix A, we use algebraic geometryto show for the simplest non-trivial example of × lat-tice with periodic boundary conditions for compact U( )that though the number of Gribov copies for the stereo-graphic lattice Landau gauge remains constant for almostall the random gauge-orbits, there are certain regions in thegauge-orbit space for which the number of Gribov copiesdiffers from the generic case. Since the corresponding Z GF counts the number of Gribov copies for the stereographiclattice Landau gauge, our results yields that the Z GF isorbit independent over the orbit space except for a regionwith measure zero.We then proposed modified lattice Landau gauge via orb-ifolding of the gauge-fixing manifold which is mathemat-ically more rigorous due to the recently developed Morsetheory for orbifolds. We reviewed the definition and de-scription of Morse theory for an orbifold. We also discussedthree different Euler characteristics of an orbifold. We thendemonstrated how Morse theory for orbifolds can be ap-plied to modify the naïve lattice Landau gauge so that thecorresponding Z GF for the orbifold lattice Landau gauge,which computes the stringy (or the usual) Euler charac-teristic of an orbifold, is orbit-independent and also evadesthe Neuberger / problem since the Euler characteristic isnon-zero. The orbifolds we considered are always compactsince the original manifold is compact. Thus, our modifiedlattice Landau gauge is fundamentally different than thestereographic lattice Landau gauge in that the former re-tains the compactness of the gauge-fixing manifold, and isclose in the spirit of the standard Wilsonian formulation oflattice gauge theories.We anticipate that our modified lattice Landau gauge, com-bined with the coset space gauge-fixing as proposed bySchaden, may turn out to be the most viable candidate toevade the Neuberger / problem which has prohibited re-alizing the BRST symmetry on the lattice for over 25 years. ACKNOWLEDGMENT
The first three authors were supported in part by a DARPAYoung Faculty Award (YFA). NSD and JDH were sup-ported in part by NCSU Faculty Research and Develop-ment Fund, and JDH was supported in part by NSF grantDMS-1262428. CS was supported by a Rhodes College Fac-ulty Development Endowment grant and the Ellett Profes-sorship in Mathematics. A part of this paper is based onDM’s thesis, and he would like to thank Lorenz von Smekalfor numerous discussions related to this work. DM wouldalso like to thank Maarten Golterman, Axel Maas, YigalShamir, Jon-Ivar Skullerud, Andre Sternbeck and AnthonyWilliams for valuable discussions related to this work.
Appendix A: Results from Homotopy Continuation forthe Stereographic Projection
The following shows that Z GF for the stereographic latticeLandau gauge-fixing functional is orbit independent overthe orbit space except for regions having measure zero. Forthis, we first note that the Hessian matrix of Eq. (5) isgenerically positive-definite [13, 15]. Hence, Z GF in Eq. (2)computes the number of stationary points of Eq. (5) forthe given gauge-orbit. Thus, we need to compute the num-ber of solutions of Eq. (6) for various orbits (i.e., randomvalues of { φ j ,µ } , at the strong coupling limit β = 0 ) and de-termine if they remain constant. Finding all the solutionsof such a nonlinear system of equations is a very difficulttask. In Refs.[13, 15, 27] the problem of solving gauge-fixing conditions on the lattice was translated in terms ofalgebraic geometry in order to be able to use the numeri-cal algebraic geometry methods to find all the solutions ofthese equations. With the improved version of the corre-sponding algorithms, we can now solve the equations for atleast the simplest nontrivial lattices in 2D successfully. Touse this method for our purposes, we begin by transform-ing our system of trigonometric equations into a system ofpolynomial equations by first expanding Eq. (6) using thetrigonometric identity tan x + y + z x + cos z sin y + cos y sin z cos x + cos y cos z − sin y sin z . (A1)Replacing sin θ j and cos θ j with s j and c j , resp., yields f s j ( c, s ) = (cid:88) µ (cid:16) sin φ j ,µ c j − cos φ j ,µ s j + s j +ˆ µ sin φ j ,µ s j + cos φ j ,µ c j + c j +ˆ µ − sin φ j − ˆ µ,µ c j − ˆ µ − cos φ j − ˆ µ,µ s j − ˆ µ + s j sin φ j − ˆ µ,µ s j − ˆ µ + cos φ j − ˆ µ,µ c j − ˆ µ + c j (cid:17) . (A2)Due to the Pythagorean identity, we add the additional con-straint equations c j + s j − for each j . As the simplestnon-trivial case, we take the × lattice. To make sure thatthere are only isolated solutions, we also fix θ , = 0 andthen remove the equation f , = 0 from the system. Sincethe above equations have denominators, we introduce aux-iliary variables to produce polynomial conditions to satisfythe system. For the × lattice, this produces a systemof quadratic polynomial equations in variables thatdepends on 18 parameters { φ j ,µ } . This procedure is a one-to-one transformation so that no solutions of the originalsystem are lost in the transformation.
1. Methods
We solve the system consisting of equations using atwo-phase methodology from numerical algebraic geome-try known as a parameter homotopy which guarantees tofind all the solutions of a given system of multivariate poly-nomial equations for any given parameter points. We givea brief summary; for further details, see Refs. [56, 57] andRefs. [15, 27, 30, 58–67] for the related applications in lat-tice field theories and other particle physics areas.First, in the ab initio phase, we choose a random set of com-plex parameters P := { φ ∗ j ,µ } and numerically compute theset of solutions S to the system using homotopy continu-ation with regeneration [68], implemented in Bertini [69].This phase, which is performed only once, took roughly . hours on a cluster consisting of four AMD 6376 Opteronprocessors, i.e., computing cores running at 2.3 GHz.Subsequent computations make use of these results to sig-nificantly reduce the effort involved in solving the system.In particular, we find that there are nonsingular iso-lated solutions for the random set of parameters P .In the second phase, known as the parameter homotopyphase , we solve the system for various choices of parame-ters. For each set of parameters { φ j ,µ } , we use Bertini to numerically track paths starting at the points in S .We numerically follow paths defined by a continuousdeformation of the parameters from P to { φ j ,µ } , so thatthe endpoints are the solutions we seek. On the samecluster, this phase takes an average of minutes tocompute solutions for a given set of parameters.
2. Results
First, to determine the behavior of the system at generalpoints in the parameter space, we solved the system for random sets of real parameters { φ j ,µ } . In each instance, wefind that there are real solutions. Thus, we conjec-ture that all complex solutions are real for all pointsin the real parameter space except on several regions.Next, we investigate the discriminant locus, which is the seton which the system has nongeneric behavior. We find thatwhen the angles in { φ j ,µ } are deliberately chosen so thatthey adhere to some structure, such as rational multiplesof π , it is quite easy to find a point in the parameter spacesuch that the system has fewer than real solutions.Thus, the number of stationary points of Eq. (5) differsfor various orbits, and Z GF for the stereographic latticeLandau gauge-fixing functional is orbit-dependent.The following figures summarize these results. Figure 1plots Z GF (or, equivalently, the number of real solutions)corresponding to various sets of parameters P , . . . , P . Fig-ure 2 plots a subset of the discriminant locus projected ontothe two parameters φ (1 , , and φ (1 , , in which the rest ofthe parameters are fixed to the angles given in Table I. Tolocate points on the discriminant locus, we used the factthat for parameter values to have fewer than realsolutions, we must have corresponding denominators equalto zero in Eq. (A2). Since we introduced auxiliary vari-ables for denominators when constructing the polynomialsystem, we can perform parameter homotopies in whichthe destination systems have these ‘denominators’ equal tozero. We note that the points shown here are only a subsetof the discriminant locus, which is an algebraic curve in thisprojection. Nevertheless, these computed points illustratethe abundance of parameter choices for which the systemhas nongeneric behavior. Figure 1. Z GF corresponding to various sets of parameters P k which are defined as follows. For P , we set each parameter toa distinct angle via φ j ,µ = π/ ( j + 3( j −
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23 28 π
31 24 π − π j j µ φ ( j ,j ,µ π − π π
13 17 π
19 27 π − π − π
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