Yang-Mills mass gap, Floer homology, glueball spectrum, and conformal window in large-
Roughly speaking Morse-Smale-Floer homology associates the critical points of the action functional of a classical field theory over a manifold to its homology. We associate to the homology of two punctured Lagrangian submanifolds of intersecting at the cusps the critical points of a quantum effective action of large- , thus realizing a quantum field-theoretical version of Lagrangian intersection Floer homology. For this purpose we construct in a trivial Topological Field Theory at , defined by twistor Wilson loops whose v.e.v. is 1 in the large- limit for any shape of the loops supported on certain punctured Lagrangian submanifolds. We derive a new version of Makeenko-Migdal loop equation for the topological twistor Wilson loops, the holomorphic loop equation, that involves the change of variables in the functional integral from the connection to the anti-selfdual part of the curvature and the choice of a holomorphic gauge. Employing the holomorphic loop equation at , and viewing Floer homology the other way around, we associate to arcs asymptotic in both directions to the cusps of the Lagrangian submanifolds the critical points of an effective action implied by the holomorphic loop equation. The critical points, being associated to the homology of the punctured Lagrangian submanifolds, consist of (magnetic) surface operators supported on the punctures, thus realizing constructively by magnetic condensation a version of ’t Hooft duality. At the next-to-leading order a certain correlator of surface operators is non-topological and non-trivial, controls the mass gap of theory, and is saturated by an infinite sum of pure poles of scalar and pseudoscalar glueballs with positive charge conjugation. Besides, it satisfies asymptotically for large momentum fundamental universal constraints arising from the asymptotic freedom and the renormalization group. We predict at large- the ratio of the masses of the two lower-mass scalar glueballs , to be compared with the measure in lattice by Meyer-Teper , and with the value implied by the Particle Data Group (2014) .The construction extends to massless Veneziano large- limit of , for which we determine the lower edge of the conformal window and the corresponding quark-mass anomalous dimension .
1 Introduction and Conclusions
The aim of this paper is fourfold.
Firstly, we show that in four-dimensional large- pure , that in the pure-glue sector coincides with in ’t Hooft large- H1 () limit, there exist some gauge-invariant observables, a special kind of Wilson loops, that we call twistor Wilson loops for geometrical reasons explained later (see section 4), whose vacuum expectation value (v.e.v.) is in the large- limit for any shape of the loops supported on a certain Lagrangian submanifold of Euclidean (complexified) space-time. They define a trivial Topological Field Theory () at underlying . We show that twistor Wilson loops can be localized at , i.e. at the leading order, on a set of critical points of a certain quantum effective action, that turn out to be a special kind of surface operators, i.e. singular instantons, indexed by the integers labeling the elements of the center of the gauge group. In fact, the localization on surface operators of holonomy realizes in the by magnetic condensation a version of ’t Hooft duality (see section 2).
Secondly, we show that a certain two-point connected correlator of these surface operators supported on the aforementioned Lagrangian submanifold analytically continued to Minkowski or to ultra-hyperbolic signature, at the next-to-leading order, controls the mass gap of theory at large , in fact the joint spectrum of an infinite tower of scalar and pseudoscalar glueballs with positive charge conjugation. The spectrum of the glueballs that couple to our surface operators turns out to be:
where is the renormalization-group () invariant scale (see section 3) in the scheme in which it coincides with the mass gap. This statement about the exact linearity, as opposed to asymptotic linearity, of the large- joint scalar and pseudoscalar glueball spectrum of the masses squared is as strong as it sounds very unlikely, and easy to falsify by numerical lattice gauge theories computations. Therefore, despite the aim of this paper is to present Eq.(1) as a deduction from first principles, we mention that Eq.(1) agrees accurately with the first low-lying points of the glueball spectrum computed by Meyer-Teper T0 (); T (), employing a supercomputer, for lattice on the presently largest lattice () with the smallest value 111This is the smallest value of coupling constant for ever reached to date in lattice computations for glueballs and perhaps in general. For a critical discussion see MBN (). of coupling constant (): In loose words, in the lattice computation for that is presently closest MBN () to the continuum limit. Indeed, Meyer-Teper result T () for the ratios , , , agrees sharply MBN () with Eq.(1), that implies . Moreover, there is a numerical evidence MBS () that also the masses squared of glueballs of higher spin are integer valued in the large- limit in units of as summarized in the plot in Fig.(1), a fact that may be suggestive of the existence MBS () of a Topological String Theory dual to the of large- (see appendix A).
Thirdly, the physical identification of the spectrum in Eq.(1) is based on our interpretation of a certain two-point correlator of surface operators 222The overall normalization of our surface operators is chosen in order to match the normalization of the corresponding correlators in perturbation theory. (see section 11) on the Lagrangian submanifold, actually isomorphic to in the infinite volume limit, as the two-point correlator of the Euclidean local operator , with the anti-selfdual () part of the curvature of the gauge connection and the Hodge dual (see section 3), analytically continued to ultra-hyperbolic signature:
and to Minkowski signature 333In Minkowski signature the correlator is proportional to the difference of the scalar and pseudoscalar correlators because of the factor of in the Hodge dual (see section 3). Nevertheless, according to Veltman conventions Velt2 (), we maintain Euclidean notation for the momenta . The analytic continuation is understood but not explicitly displayed. ():
both restricted to momentum dual in Fourier sense to the aforementioned . is ’t Hooft (see section 3) canonical running coupling constant at the scale of the -th pole (in ultra-hyperbolic or in Minkowski space-time) in the scheme defined in Eq.(19), i.e. . Since the lowest-mass glueball is believed to be a scalar (see section 3) and the correlators in Eq.(2) and Eq.(3) couple to scalars and pseudoscalars (see section 3), Eq.(2) or Eq.(3) suffice to imply the mass gap in the large- limit. We would like to make it clear that in this paper we take the point of view that functional integrals are defined by the rules by which they are computed, and therefore we refer to all the results of this paper as computations rather than mathematical proofs. Nevertheless, we solve explicitly the , in such a way that its existence in mathematical sense is a matter of definitions.
Moreover, we believe that the results of our computations are correct. Indeed, Eq.(2) and Eq.(3) agree asymptotically MBN () for large momentum (see section 3), up to physically-irrelevant contact terms, with the universal, i.e. the scheme-independent, leading and next-to-leading logarithms of the first-two coefficient functions of the -improved operator product expansion (), according to fundamental principles of the and the asymptotic freedom of (see section 3), in Euclidean (and ultra-hyperbolic) signature:
and in Minkowski signature:
with the coefficients denoted by vanishing up to scheme-dependent terms on the order of , and (see section 3):
The vanishing of the universal part of either the second or the first coefficient function in the of the correlator, in Euclidean (and ultra-hyperbolic) signature and in Minkowski signature respectively, is due to peculiar partial cancellations (see section 3) that occur taking the appropriate linear combination of the scalar and pseudoscalar correlators: They are implied both in the (see section 11), and in massless on the basis of remarkable recent computations by Chetyrkin-Zoller Che3 () and Zoller Zo () (see section 3).
In particular, in the a powerful non-trivial check of the correct asymptotic behavior SM (); MBN () of the correlators in Eq.(2) and Eq.(3) occurs as the consequence of the conspiracy SM () between the fourth power of the running coupling evaluated on shell, that arises by the anomalous dimension of our surface operators, and the linearity of the spectrum of the masses squared (see section 10), that arises by the localization on surface operators of holonomy (see section 2).
We mention that in the aforementioned cancellations are complete due to space-time supersymmetry and the correlator actually vanishes Shif () in Minkowski space-time up to physically-irrelevant contact terms, as it has been proved by Shifman Shif (), by means of methods inspired by some of the techniques employed in this paper (i.e. the change to the variables, known also as Nicolai map (see section 5)).
The aforementioned agreement with fundamental principles of the and the asymptotic freedom is presently a unique feature of Eq.(2) and Eq.(3), not shared by any other proposal, in particular based on the String/ Large- Gauge Theories correspondence (see MBN (); SM () and references therein for a critical discussion), for the scalar or pseudoscalar glueball propagators.
Fourthly, we extend the construction of the and the results for the correlator of surface operators to massless Veneziano limit Veneziano () of large- , , with quarks in the fundamental representation (see section 12). In the flavor-singlet sector, where the propagator of the lives, pure-glue operators mix with flavor-singlet quark operators in such a way that in general there is no clear distinction between glueballs and flavor-singlet mesons in Veneziano limit (see Veneziano () for a neat review and detailed estimates of the couplings of glueballs and mesons both in ’t Hooft and Veneziano limits).
In particular the operator in is a linear combination of the scalar and pseudoscalar pure-glue operators (see section 3), and mixes with the divergence of the flavor-singlet axial current because of the chiral anomaly Che3 (); Zo (). But the mixing with the axial current affects the terms that involve the anomalous dimension in the Callan-Symansik equation only at order of Zo (), and therefore it does not affect the leading universal asymptotic behavior of the correlator but for the change of the coefficients of the beta function in Eq.(6) and Eq(7). Moreover, in the we compute only the correlator of surface operators, that field-theoretically are pure-glue operators.
Accordingly, we compute the correlator in massless Veneziano limit of in the local approximation (see section 12) of the effective action in the , from which we extract the critical value of at which the lower edge of the conformal window occurs, and the corresponding value of the quark-mass anomalous dimension in the massless limit (see section 12).
The lower edge of the conformal window occurs at the value of for which the kinetic term in the correlator in the changes sign due to the quark contribution to the effective action, i.e. the kinetic part of the Hessian of the effective action at the critical points changes sign, signaling an instability of fluctuations around the magnetic condensate of surface operators, i.e. a phase transition from confinement to another phase, identified with a conformal Coulomb phase. We show that at this value the canonical beta function in our family of schemes develops an infrared zero that occurs independently on the scheme (see section 12). The argument mimics Seiberg argument for the lower edge of the conformal window in gauge theories (see section 12). In the last case the lower edge is determined as the point at which the anomalous dimension of operators in the holomorphic ring reaches the unitarity limit, and correspondingly a zero of the beta-function NSVZ () occurs.
Remarkably, the canonical beta function in the exactly reproduces by first principles the universal part of the perturbative large- beta function in Veneziano limit (see section 12).
Moreover, we compute the value of the ratio of the two lower-mass glueballs 444Strictly speaking we cannot determine the parity of the glueball states, but only the joint spectrum of any parity, because the correlator couples to scalar and pseudoscalars. at the leading order, that turns out to be the same both in ’t Hooft and in massless Veneziano large- limits: , the only difference between the local parts of the effective action in the two limits being the change of the beta function that defines the -invariant scale , but the subtle difference being that in ’t Hooft limit the effective action of the is large- exact (see section 2), while in massless Veneziano limit only the local part of the effective action of the is in fact large- exact (see section 12). In fact, the two-point correlators in Veneziano limit cannot be saturated only by single-particle states, because the widths are on the order of , as opposed to ’t Hooft limit. Therefore, the fact that the spectrum is of pure poles also in Veneziano limit of the is an artifact of the local approximation for the effective action.
Here we outline the mathematical features of our approach. Our surface operators belong to a new underlying pure . For clarity we recall briefly topological field theories in the supersymmetric () setting, because of the loose connection with our non- . The basic idea of topological field theories W1 () is to view certain functional integrals as cohomology classes associated to a nilpotent differential defined by the charge obtained by a twist of the supersymmetry, with the property for any . Thus these physical theories contain a very special topological subsector defined by closed forms modulo total differentials , the cohomology class of . Moreover, the topological field theory is often solvable, since the cohomology classes are localized on critical points by means of deformations trivial in cohomology:
because the saddle-point approximation for large , being independent for the class , is in fact exact DE (); A (); Bis1 (); W (). The aforementioned localization extends to the whole cohomology ring generated by the closed forms. Hence a small subset of the observables of the theory is localized on critical points. For a review of cohomological localization see Szabo (). The canonical example from the physics point of view is the gluino condensate NSVZ (), for which cohomological localization on critical points (see section 3.2 in GGI ()), in this case on gauge connections with vanishing curvature, leads to the exact beta function NSVZ () of .
The canonical example from the mathematical point of view leads to a twisting of the supersymmetry of and to a field theoretical interpretation W1 () of Donaldson invariants by localization of the functional integral in the topological sector of on the very same configurations with vanishing curvature.
This cannot work in pure , because of the lack of the differential of the would-be cohomology, i.e. the lack of .
However, in the finite dimensional setting, Morse-Smale theory Morse () (and references therein) associates to the critical points of a function on a compact manifold the homology groups of the manifold. We recall briefly Morse theory. Given a non-degenerate Morse function on a manifold, Morse inequality bounds from below the number of critical points of by the sum of the dimension of the homology groups of the manifold: . Moreover, Morse-Smale homology allows us to reconstruct the homology groups directly from the knowledge of the critical points.
The basic idea of Smale is to connect two critical points, whose Morse index differs by , by arcs associated to the gradient flow of on the manifold. Then to the arcs it is associated a complex, that turns out to be isomorphic to the usual singular homology of the manifold. Morse index is the number of negative eigenvalues of the Hessian of at the critical points. Hence Morse-Smale homology involves necessarily both the stable and the unstable manifold of the critical points, since the arcs of Smale flow connect necessarily the unstable with the stable manifolds.
Besides, in the infinite dimensional setting, Floer theory Floer () (and references therein) associates to the critical points of a classical field theory on a manifold certain Floer homology groups, that are topological invariants of the underlying manifold.
The basic idea of Floer homology is to extend Smale construction to infinite dimension, i.e. to a field theory over a manifold, for which the function is now the classical action functional of the theory, by allowing Morse index to be infinite. Floer preserves the construction of the gradient flow associated to the functional , by requiring that the flow connects two critical points whose relative Morse index is finite and equal to , despite each Morse index may actually be infinite. Floer shows that in certain cases this construction defines new homology groups that are topological invariants of the manifold, that may, or may not in the infinite dimensional setting, be isomorphic to the ordinary singular homology of .
2 A synopsis of the main argument and plan of the paper
To say it in a nutshell, we replace cohomology in function space (i.e. ) with homology in submanifolds of space-time (see section 1), in order to localize on critical points of a quantum effective action in large- .
Indeed, our construction associates critical points of a quantum effective action of large- to the intersection homology of certain Lagrangian submanifolds of (complexified) space-time (see Fig.2 and Fig.3): The critical points occur at the cusps that lie at the intersection of two Lagrangian submanifolds. Conceptually, our construction is a quantum field-theoretical version of Lagrangian intersection Floer homology, that instead, going in the opposite direction, associates the intersection homology of the Lagrangian submanifolds to the critical points of certain functionals Wiki (); Floer ().
Thus, roughly speaking, our strategy consists in constructing a very special topological subsector of pure , that represents by gauge-invariant observables numerical topological invariants defined on homology classes in the loop algebra of closed curves based over a point on a complex submanifold of space-time (or of its complexification ). Then we associate, by means of new field theoretical methods, to the aforementioned homology classes the critical points of a quantum effective action of restricted 555This restriction is the counterpart in homology of the restriction to the cohomology ring in cohomological localization. to the subalgebra of gauge-invariant observables in the topological subsector. Therefore, our construction can be regarded as Morse-Smale-Floer homology (see section 1) seen the other way around: From the homology classes to the critical points.
The natural candidates to represent homology classes of closed curves in gauge theories are the Wilson loops. However, in pure in general they represent the loop algebra over the manifold (see MBT () for a short summary of the properties of the representation by Wilson loop operators of the loop algebra at large-), but this representation does not lift to homology classes. Indeed, the v.e.v. of Wilson loops in gauge theories is not in general homotopy invariant, as opposed to the homology groups. This is due to the fact that Wilson loops in general capture detailed information about the physics of pure , that certainly is not a topological field theory.
Therefore, we define in section 4 special Wilson loops constructed by means of a modified gauge connection , that we call twistor Wilson loops for geometrical reasons explained in section 4, whose v.e.v. is a numerical homotopy and homology invariant. The twistor connection depends on a complex non-vanishing fixed parameter . is a functional of the ordinary connection . Thus we introduce new gauge-invariant observables, but we do not change the action of theory, that defines the v.e.v. through the functional integral. In fact, twistor Wilson loops in the adjoint representation are completely trivial homotopy and homology invariants, but only in the large- limit of theory: Their v.e.v. is in the large- limit (, see section 4) for any shape of loops supported on certain Lagrangian submanifolds immersed in or in its complexification . Yet, the very existence of twistor Wilson loops is non-trivial.
By a completely different argument, Witten argued 666Talk at the Simons Center workshop Mathematical Foundations of Quantum Field Theories, Jan 2012. See in this respect MBT (); MB1 (). that every theory with a mass gap should contain a possibly trivial topological field theory in the infrared. But what makes the existence of twistor Wilson loops a powerful tool is their triviality at all scales, in particular in the ultraviolet. Indeed, it is precisely the large- ultraviolet triviality of twistor Wilson loops, that implies the absence of cusp anomalies (see below), that allows us to localize them on critical points: A feature much more specific than the triviality of the topological theory in the infrared. Besides, the adjoint twistor Wilson loops factorize in the large- limit into twistor Wilson loops in the fundamental and conjugate representation, that turn out to be homology invariants valued in the center of the gauge group . Twistor Wilson loops in the fundamental representation are the building blocks of our new underlying the large- limit of . The main conceptual point of this paper is that twistor Wilson loops satisfy a new kind of loop equation, the holomorphic loop equation 3MB () derived in section 6:
that involves a change of variables in the functional integral that defines theory. One of these new variables is the curvature of the connection that occurs in twistor Wilson loops. turns out to be a non-Hermitian field of type: .
In fact, the aforementioned change of variables is the map, defined in section 5, from the independent components of the connection in any fixed gauge to the independent components of the field , composed with the change to in a holomorphic gauge (see section 6). This map is implemented employing the resolution of identity in the functional integral in the original variables 3MB ():
that allows us to compute the Jacobian of the change to the variables (see section 5), by integrating exactly in any gauge on the gauge connection in the functional integral in the original variables, because of the delta functional (see section 5). Since the change to the variables is already a non-standard tool in quantum theories, with A. Pilloni we have proved BP () the identity of the one-loop perturbative one-particle irreducible () effective action of , and , and more generally of any gauge theory that extends pure , in the original and in the variables. Therefore, we think that the change to the variables is sufficiently tested at perturbative level to be employed in our non-perturbative approach.
Coming back to our , we choose as support of twistor Wilson loops a Lagrangian surface with the topology of a high-genus Riemann surface immersed in space-time with a canonical basis of cycles and certain cycles Fig.2. We make this choice in order to get a large homology group and potentially a large number of critical points in our , according to Morse-Smale-Floer theory. We specify later in this section the concrete realization of the Lagrangian surface. For our initial considerations only topology matters. Then we use the freedom to deform the loops of our trivial . We deform the Lagrangian surface in such a way that the cycles degenerate to nodal points Fig.3. Each node is then the intersection point of a pair of cusps belonging to the deformed cycles of the Lagrangian nodal surface Fig.3. The first reason for introducing cycles intersecting at cusps is that loop equations carry quantum information precisely at the intersection points. The second reason will become apparent momentarily. Thus pairs of deformed cycles form a loop of Makeenko-Migdal () type MM (); MM1 (), with the shape of the symbol , that decomposes into two petals and .
We need a gauge-invariant regularization of the right-hand side () of the holomorphic loop equation 3MB (), that is obtained introducing a real structure on our Riemann surface (see real () and references therein), that therefore is chosen to be the topological double of a sphere with boundary circles, all of which but one degenerate to cusps Fig.3. Thus the cusps are now real points of the Riemann surface. For real points the result of the regularization of the Cauchy kernel 777Alternatively a real Cauchy kernel is obtained by analytic continuation to Minkowski space-time 3MB (). is the sum of two distributions, the principal part of the real Cauchy kernel and a one-dimensional delta function:
Hence for real points , the holomorphic loop equation reduces to:
because by gauge invariance the principal part does not contribute 3MB (), being supported on open loops for which, by (non-)gauge invariance Mak2 (), . As a consequence the of the holomorphic loop equation is supported on closed loops only, as it must be by gauge invariance, and it is non-vanishing in general, as it is the of the loop equation MM (); MM1 (): It represents the quantum contribution, that is the obstruction to localize the loop equation on the critical points defined by the equation of motion that occurs in the . But the remarkable fact about the holomorphic loop equation, as opposed to the equation, is that for nodal points the of the holomorphic loop equation vanishes, provided the cycles on the Riemann surface are oriented in the same way. Indeed, in this case pairwise intersecting cycles are asymptotic to each cusp in both directions Fig.3. It follows that the contribution in the of the holomorphic loop equation of each backtracking cusp intersecting at a node is exactly :
because of the opposite orientation of the arcs asymptotic to the cusps. Thus the holomorphic loop equation associates to each node of the Lagrangian surface with a real structure a critical equation for the quantum effective action :
that states that all the matrix elements of the critical equation vanish between states created by operators in the subalgebra generated by (cusped) twistor Wilson loops. Hence restricted to this subalgebra:
This is Morse-Smale-Floer homology seen the other way around, from the non-trivial homology to the critical points. In particular, in our inverse construction, the arcs interpolating critical points associated to Smale gradient flow (see section 1) are replaced by arcs belonging to the arc complex 3MB () of the double of our punctured Riemann surface, asymptotic in both directions to the pairwise identified punctures Fig.3. Moreover, despite the of the holomorphic loop equation vanishes also for simple backtracking cusps, and thus this vanishing does not require necessarily nodes, Eq.(15) needs cusps that are double points in order to imply the vanishing of matrix elements between states created by the vacuum by independent petals of the twistor Wilson loops operators. For a single cusp one of these petals would degenerate to the identity operator. Thus the localization on critical points is associated really to the intersection homology.
Indeed, while deforming an ordinary Wilson loop by a backtracking cusp leaves the loop invariant at classical level because of the zig-zag symmetry Gr () of Wilson loops, at quantum level it introduces an additional logarithmic divergence known as cusp anomaly Po2 (); Kr (). The cusp anomaly is reflected in an extra divergence Gr () in loop equation at the cusp, as opposed to the holomorphic loop equation, in addition to the usual perimeter divergence:
where is the perimeter of the loop and the cusp angle at the cusp. For our conventions for no cusp, while for a backtracking cusp. Thus in no way the quantum contribution in the of the equation vanishes for ordinary Wilson loops, as opposed to the holomorphic loop equation for (cusped) twistor Wilson loops. It is therefore clear that our localization depends crucially on the large- triviality of our at all scales and in particular in the ultraviolet, that implies the absence of cusp anomalies.
Correspondingly to the localization, the effective action does not get quantum corrections at and therefore it must contain information, as we check a posteriori by direct computation in the following sections, about the beta function through the Jacobian of the change to the variables, and about the mass gap through the Jacobian to the holomorphic gauge. We prove for consistency in this section, after some preparation, that does not get quantum corrections at around the critical points, as an independent check of the localization of the holomorphic loop equation (see section 11 for the direct computation).
Indeed, in a certain technical sense Eq.(16), that for the moment is restricted to the subalgebra generated by adjoint twistor Wilson loops in Eq.(14), can be extended in the large- limit to a set dense in the whole algebra. In fact, the algebra generated by adjoint twistor Wilson loops for any fixed contains only the local algebra generated by and because of the large- factorization into the fundamental and conjugate representation, but we will show for consistency that fluctuations around the critical points of Eq.(16) in the whole local algebra generated by , our new independent variables in force of Eq.(10), are suppressed in the large- limit, provided that a certain basis dense in function space is chosen to compute the functional integral in the . This justifies interpreting Eq.(14) strongly, as Eq.(16) extended to the whole local algebra.
We describe now how the choice of this basis arises. Since our considerations are of topological nature, we can deform our nodal surface arbitrarily, to the extent that we do not change its topology. Topologically to the nodal surface it is associated its normalization, that is a punctured sphere with certain pairwise identifications of the punctures Fig.4. The normalizing surface can also be seen as two topological disks glued at the boundary, with certain identifications of the fibers of the vector bundles involved over the punctures Fig.4. The local picture obtained restricting to each punctured disk Fig.5 is the most convenient for the following field-theoretical considerations, and it is our actual choice in the functional integral.
In the the functional integral on a punctured disk is fixed specifying as boundary condition that the connection carries a polar singularity at the punctures (see section 7 and 8). Correspondingly, the curvature is a distribution supported on the punctures. But while in the the number of punctures depends on our choice and it is arbitrary, field theoretically we have the freedom to construct the vacuum of the in such a way that it is translational invariant, and therefore there must be a critical point for each point of space-time. Since in the critical points arise by double points, translational invariance cannot be achieved globally because of the opposite orientations of the cycles along which the holonomy of twistor Wilson loops is computed Fig.4, but only locally on each Lagrangian disk. However, it turns out that the critical points at the punctures on the two disks are charge conjugate, in such a way that both the action and the fluctuations are the same, being the theory invariant for charge conjugation.
Besides, we introduce a lattice of punctures of not-necessarily-equal uniform spacings and , as in lattice field theory, and then we take either the continuum limit keeping fixed the areas for largely separated ultraviolet () and infrared () divisors, or a scaling limit (see section 3) for largely separated divisors, with the number of punctures going to infinity Fig.5. Locally, to the extent that the areas of the and Lagrangian disks are much larger than the -invariant scale, translational invariance is achieved for all the physical purposes. In both the continuum limit and the scaling limit the lattice spacing plays the role of the (inverse of the) cutoff , while an intermediate infrared subtraction scale , not necessarily equal at the and divisors, is introduced in the renormalized theory. Taking the continuum limit or the large-area scaling limit must be compatible with the renormalization of implied by ordinary perturbation theory, as in lattice gauge theories: The universal scheme-independent part of the beta function implied by the effective action of the must coincide with the universal part of the perturbative beta function, that is fixed by its first and second coefficient. Most remarkably, this turns out to be the case 3MB (), as it is checked by direct computation in section 9.
In fact, as in theory, two different renormalization schemes can be defined for the gauge coupling of the underlying large- , the Wilsonian and the canonical scheme (see section 9). In the Wilsonian scheme the gauge coupling turns out to be one-loop exact, as in (for a compact exposition of the supersymmetric case see section 3.2 in GGI ()). The Wilsonian scheme is implemented in the for the Wilsonian normalization of the action (see section 9) if the same choice is made for both the cutoff and the subtraction scales on the two disks. If different subtraction scales are instead chosen, one large at the divisor and one small at the divisor, a canonical coupling in the can be defined (see section 9), that correctly reproduces in the the universal part of the perturbative beta function for ’t Hooft coupling (see section 3) :
with and a scheme-dependent constant. can be fixed imposing a physical condition on the scheme that occurs in the correlators (see section 11). It is easy to check that the canonical beta function reproduces 3MB () the correct universal one- and two-loop coefficients of the perturbative AF () beta function and , by noticing that for small , within the leading logarithmic accuracy, and expanding in powers of (see section 9). Remarkably, the occurrence of the correct canonical beta function in the 3MB () is somehow linked to the entanglement of the and degrees of freedom due to the pairwise identification of the fibers over the punctures on the normalization of the nodal surface, a direct consequence of the homological nature of our localization. Indeed, the canonical coupling of the gets also contributions from the infrared divisor, as opposed to the Wilsonian coupling, as somehow expected by the analogy with the case (see Sh () p.89). In fact, the homology interpretation associated to the nodal points allow us a more precise and rigid realization, with respect to the original computation of the beta function in 3MB (), of the Wilsonian and canonical flow in the .
Analytically, in the functional integral we associate to the lattice of punctures a special basis in the resolution of identity that defines the change to the variables 3MB () in Eq.(10). Indeed, we assume the further resolution (see section 7) 3MB ():
parametrized by the lattice field , that is dense in function space in the sense of distributions in the gauge theory on non-commutative space-time in which twistor loops are in fact defined (see section 4). The non-commutativity of space-time arises here just as a device to define the large- limit, since it is known that the large- limit and the large- limit on commutative space-time coincide, by the modern version Mak2 (); Szabo2 () of the non-commutative Twc (); Twl1 () Eguchi-Kawai reduction EK (), both at perturbative level Seiberg () and at level of loop equations loop (). This special choice of a dense basis of cylindrical functions in the resolution of identity has crucial properties that we summarize as follows, first on the mathematics side and then on the physics side.
Firstly, it allows us to actually expand the around its critical points only, by restricting to the fluctuations of the lattice field , since critical points of topological origin occur only at the singular fibers over the punctures: Any other choice of basis, say the ordinary Fourier transform, would imply that fluctuations are computed outside the singular support of the critical points, and therefore it would be meaningless in the framework of the .
Secondly, it implies that the functional integral of the , despite being over a dense set in function space in the curvature, is in fact restricted to connections flat away from the punctures, because of Eq.(22) and Eq.(10). The equations reduce in theory on non-commutative space-time (see section 6) to (an infinite dimensional version of) Hitchin equations H2 () (see section 7 and 8) for a flat (away from the punctures) non-Hermitian two-dimensional connection, precisely our twistor connection . Therefore, as in Floer theory Floer (), our construction involves the fundamental group of the underlying Riemann surface rather than the homology group, and a field theoretical representation of it by twistor Wilson loops evaluated on the flat connection . Besides, as in Floer theory, to the representation of the fundamental group it is associated a homology theory in the moduli space of the flat connection Floer () over the Riemann surface, that reflects the topology of the underlying manifold via its fundamental group.
Thirdly, the choice of this basis guarantees us that the critical points of the effective action carry homological information about the Lagrangian surface, since they are associated, in the spirit of Floer theory, to a representation of the fundamental group of the Lagrangian surface, whose Abelianization is the middle homology group of the surface. In fact, we can determine a priori what the critical points are: Under the assumption that the global gauge group of , and of the as well, be unbroken, the critical points of the must be the twistor connections whose holonomy in the fundamental representation is valued in the center of . In section 9 we check by direct computation that the twistor connections with holonomy are in fact degenerate minima of the effective action of the . The point-like singularities on the Lagrangian disks lift to surface-like singularities MB2 (); MB3 () known as surface operators W2 () because of the immersion in . They are singular instantons, i.e. singular connections satisfying equations away from the punctures, carrying, at the critical points of the effective action, magnetic charge valued in on each point of the lattice of punctures. Therefore, since the theory must be translational invariant on the local wedge at the or at the , the translational invariant critical points belong to sectors labelled by the integer that determines the element of .
Moreover, the degeneracy of the eigenvalues of the curvature for surface operators of holonomy is and , and the glueball masses squared, in the spectral formula Eq.(1) derived in section 10, turn out to be precisely proportional to these degeneracies, thus making the statement of the integral nature (in units of ) of the glueball masses squared that couple to our surface operators a very rigid prediction of the .
Finally, the most remarkable self-consistency check of our localization by the holomorphic loop equation Eq.(16) evaluated in our special basis arises by the following property of the equations around the local singularities: The lattice fields at each puncture must commute S5 (); S6 () (see also W3 () p.6 and p.31 for a proof of this statement in the physics style) in order for the equations in the resolution of identity to admit a solution.
This is most relevant to compute two-point connected correlators of surface operators at the next-to-leading order, by computing non-topological and non-trivial fluctuations around the critical points of the , that is trivial at the leading order. Indeed, as a consequence of the aforementioned commutativity there exists a gauge, on the complement of the set of measure zero in the functional integral for which no solution of the equations in the resolution of identity Eq.(22) and Eq.(10) exists, in which the local magnetic fluctuations of the field around the critical points can be diagonalized simultaneously at each puncture, implying the local Abelianization of the lattice field.
Hence by the standard large- argument in the large- solution of vector-like models Zinn (); Mak1 () and of one-matrix models Par (), the quantum corrections to the effective action in Eq.(16) are suppressed in the large- limit, since only the eigenvalues fluctuate around each singularity, as opposed to the matrix elements of the field around smooth points, that are not critical points. This argument applies also to fluctuations outside the subalgebra generated by twistor Wilson loops, provided they are computed around the critical points in the topological sector, i.e. around the punctures. This furnishes the promised extension of Eq.(16) outside the subalgebra generated by twistor Wilson loops, that for the adjoint representation contains only the holomorphic/anti-holomorphic sector generated by and in the fundamental representation.
Thus, on the mathematics side, the special choice of variables obtained restricting to the makes the non-topological fluctuations around the weakly coupled at large- and explicitly computable by saddle-point methods in the large- limit. In loose words the fluctuations around the condensate of surface operators are self-consistently suppressed in the large- limit, in such a way that the aforementioned formal localization is in fact checked by direct computation.
This is perhaps the deepest result of this paper, because it allows us to get a real control of the fluctuations around the condensate of critical points in the large- limit, and because of its importance we rephrase it in another way, that does not make use of restricting a priori to the fluctuations of the lattice field.
The fluctuations around a few isolated critical points cannot be controlled in the large- limit because in general there are smooth fluctuating fields away from the punctures, say by taking the usual Fourier transform. Thus in this case the is still strongly coupled. But in the we have the freedom to deform in order to localize on the configurations that are more convenient to perform an actual computation. Thus we choose to localize on a lattice of critical points, simply introducing a lattice of nodes. For a lattice of critical points the fluctuations of the field can be decomposed into the sum of the distribution-valued fluctuations supported on the lattice and smooth fluctuations, that, to avoid overcounting, must have Fourier modes orthogonal to the momenta of the lattice field:
In this equation the fluctuations of the lattice field are supported on the two-dimensional commutative and the smooth fluctuations are consistently taken on the same , by means of the modern version of the large- non-commutative Eguchi-Kawai reduction, i.e. the fact that the gauge group of gauge theories on non-commutative space-time contains the translations, that allows us to reabsorb the space-time degrees of freedom in the non-commutative directions into color degrees of freedom by a gauge transformation (see section 4).
But in the continuum limit of the lattice field this orthogonal complement of smooth fluctuations converges to zero, because the lattice field becomes dense everywhere in Fourier space in the sense of the distributions (see section 7). Therefore, in the continuum limit of the critical points only the fluctuations of the lattice field survive. As a consequence, after gauge-fixing to the gauge in which the components of the lattice field are diagonal at the same time, that is an allowed gauge for the lattice field on the complement of the set of measure zero in the functional integral for which the resolution of identity in Eq.(22) and Eq.(10) has no solution, the counting of fluctuating fields in the large- limit becomes on the order of , entirely similar to the counting of eigenvalues in the one-matrix model Par () or in vector-like models Zinn (); Mak1 (), and therefore the saddle-point approximation reliably describes the large- theory for the special observables that are in the subalgebra generated by twistor Wilson loops.
On the physics side, the choice of variables associated to the realizes a new version of ’t Hooft electric/magnetic duality Super1 () (for a neat review see DW ()): If theory has a mass gap, either the electric charge condenses (Higgs phase, broken global gauge group) or the magnetic charge condenses (confinement phase, unbroken global gauge group). In the the condensation of the magnetic singularities follows by the asymptotic freedom AF () of the effective action in Eq.(16), that we establish by checking that the beta function of the reproduces the universal, i.e. the scheme independent, first and second coefficients AF () of the perturbative beta function. In fact, introducing the density of surface operators in the sector of magnetic charge , , we find at the critical points, from the asymptotic freedom of the effective action renormalized at a scale on the order of the classical action density itself (see section 10) 888This is precisely the prescription that occurs in Veneziano-Yankielowicz effective action of (see section 9).
with , in the large- limit, that is equivalent to the condition that all the critical points labelled by the elements of , but for , are in fact degenerate minima of the local part of the effective action (see section 10). Indeed, the condensation of the magnetic is believed to be a necessary and sufficient condition for confinement DW (), according to the aforementioned ’t Hooft duality alternatives.
At this point we can forget the holomorphic loop equation and simply regard the as a convenient choice of variables, that furnishes the effective action for special observables in special variables and in a special basis: Purely field theoretically the is a way of testing concretely long-standing conjectures about confinement and mass gap in , since the functional integral of the is precisely the measure induced by the functional integral on a condensate of surface operators in the variables in the holomorphic gauge.
Indeed, we check by direct computation in section 10 that the effective action is a non-degenerate Morse functional, that has a stable critical manifold in the holomorphic/anti-holomorphic sector analytically continued to Minkowski or ultra-hyperbolic signature, obtained factorizing at large- twistor Wilson loops in the adjoint representation into the fundamental and the conjugate representation: i.e. the of has a mass gap in this sector.
In fact, in section 11 we compute at each critical point the holomorphic/anti-holomorphic two-point correlator of lowest dimension constructed by the local single-trace gauge invariant surface operator, on the (analytically-continued) Lagrangian surface of the , that corresponds to in a canonical scheme in perturbation theory. Besides, according to Eq.(2) and Eq.(3) the fluctuations around each minimum contribute one glueball propagator for each term in the spectral sum that saturates the correlator of surface operators. We anticipate that the factor of that occurs in Eq.(2) and Eq.(3) in the residue of each propagator, that is the square of the on-shell renomalization factor associated to the anomalous dimension of the surface operator that corresponds to , and that it is fundamental to reproduce (see section 3) the correct large-momentum asymptotics of the propagator, arises by the canonical normalization of the surface operator that corresponds to in the two-point correlator. The correlators of our surface operators are related GGI () to the ground state of the large- one-loop integrable sector of Ferretti-Heise-Zarembo F (), that involves scalar composite operators constructed by the curvature.
To summarize, what makes our computation possible in the large- limit is the aforementioned Abelianization of the local distribution-valued lattice field and the scaling with of the density in Eq.(24), because it implies that the loop expansion of the non-local effective action in powers of reduces to a purely local one at the relevant order, quadratic in .
The plan of the paper is as follows.
Before presenting the derivation of our results we clarify in section 3 which are the constraints that any solution of the mass-gap problem in large- or in any large- confining asymptotically-free gauge theory has to satisfy. We think that this clarification is needed, since there is some confusion in the physics literature, but essentially no mathematics literature, on the subject.
In section 4 we introduce twistor Wilson loops in non-commutative gauge theories.
In section 5 we define the change to the variables and we remark that the Jacobian of the change of variables and its zero modes contribute to the beta function.
In section 6 we derive the holomorphic loop equation.
In section 7 we restrict the functional integral to a dense basis of surface operators.
In section 8 we classify the moduli of surface operators, that are associated to the zero modes of the Jacobian.
In section 9 we obtain the effective action of the and we use it to evaluate the Wilsonian and canonical beta function of the at the leading large- order.
In section 10 we show that the mass gap of the underlying large- arises by the Jacobian to the holomorphic gauge, that the kinetic term arises by the Jacobian to the variables, and we compute explicitly the Wilsonian effective action to the next-to-leading order.
In section 11 we compute the two-point correlator of the surface operator that corresponds to at the next-to-leading order.
In section 12 we extend our approach to large- with quarks in the fundamental representation in massless Veneziano limit, and we determine the beta function, the lower edge of the conformal window and the quark-mass anomalous dimension. We stress that our technique requires that the quark masses be vanishing in Veneziano limit, and therefore our results hold only in the massless limit .
In appendix A, to make contact with reality, we discuss heuristically the actual experimental glueball spectrum in relation to our large- computation.
3 Constraints on any solution of the problem of Yang-Mills mass gap
In gauge theories the only observables that have a physical meaning are the gauge-invariant ones. For example, the gluon propagator is not gauge invariant, and therefore it is not interesting for the problem of the mass gap. Since the lowest-mass state above the vacuum in pure is believed to be a scalar 999There is an overwhelming numerical evidence from lattice gauge theory computations that sustains this belief, see T0 (); T () and references therein., the scalar gauge-invariant local correlators are relevant for the mass-gap problem. By general arguments the Euclidean two-point correlator of a local gauge-invariant single-trace scalar operator in admits the Kallen-Lehmann representation:
theory has a mass gap if and only if the closure of the support of the spectral distribution does not contain zero. For finite is believed to contain a term that accounts for the mass gap and a term that accounts for the possibly continuous spectrum due to the interaction and the multi-particle states.
The mass-gap problem is very difficult in pure , or in any confining asymptotically-free gauge theory with no mass scale in perturbation theory 101010These theories include and with massless quarks. In the last case the theory is believed to have no mass gap since the pion is massless because of the spontaneous breaking of the chiral symmetry, but a mass gap would still occur in the pure-glue sector in the large- limit. as well, because the renormalization group () together with the asymptotic freedom () require that any mass scale of the theory that has a physical meaning, such as the mass gap, must depend on the canonical coupling constant only through the invariant scale , in such a way that in some renormalization scheme, say in the scheme schema ():
where the scheme is defined by the relation:
with , in order to cancel schema () the term proportional to in the solution for . The constant can be reabsorbed in a redefinition of the scheme, in such a way that there is a scheme in which the mass gap is . Therefore, apart from being a -invariant scale, the mass gap is an arbitrary parameter of the theory, whose value in physical units is determined only experimentally. However, fixed the mass gap, any other physical quantity of the theory is determined uniquely. Alternatively, fixed any other physical quantity, the mass gap is uniquely determined.
Physically, the continuum limit is defined removing the cutoff sending at the same time , in such a way that is kept constant. Alternatively but equivalently, in lattice gauge theory computations it is often defined a scaling limit in which the (lattice) cutoff is kept constant, but the infinite-volume limit is taken, while . In the scaling limit the mass gap becomes exponentially small, but the physics is extracted looking at correlators at distances much larger than the correlation length given by the inverse of the mass gap, in such a way that the (finite) lattice spacing becomes invisible.
In both the continuum and the scaling limit the dependence of on is equivalent to the knowledge of the exact beta function of the theory in some scheme. In Eq.(3) the result implied by the two-loop beta function is explicitly displayed, while the dots refer to the scheme-dependent higher-loop contributions irrelevant in the . Eq.(3) in turn implies that an amazing asymptotic accuracy, as vanishes when the cutoff diverges, is needed to solve the mass-gap problem and that the mass gap is zero to every order of perturbation theory, since the Taylor expansion of Eq.(3) around is identically zero. Besides, Eq.(3) requires by consistency to sum to all orders the perturbative expansion associated to correlators involving the mass gap, since perturbative corrections being polynomial in are much larger for small than the dimensionless function of the coupling in Eq.(3).
Therefore, a finest asymptotic accuracy of non-perturbative type is needed to get control over the mass gap.
This rules out any perturbative method, since the mass gap is identically zero to every order of perturbation theory.
This rules out also strong coupling methods, because the mass gap has nothing to do with the coupling being large, since Eq.(3) implies that the existence of the mass gap in the continuum limit, i.e. for arbitrarily-large cutoff , or in the scaling limit, i.e. at fixed cutoff but for exponentially-small mass gap, requires an estimate uniformly in a neighborhood of zero coupling asymptotic to Eq.(3). The same holds if we substitute in Eq.(3) to the cutoff the renormalization scale and to the bare coupling the renormalized coupling at the scale , since the physics of the mass gap must not depend on how large the renormalization scale is chosen.
In fact, strong coupling methods do not allow us to remove the cutoff, since if the coupling is large the cutoff scale that occurs in Eq.(3) must be finite and cannot be large for the mass gap to stay bounded. But then, in absence of a uniform estimate that extends to any positive arbitrarily-small neighborhood of zero coupling, the continuum limit cannot be taken, and the proof is lacking that the would-be mass gap survives the continuum limit and it is not an artifact of the finite cutoff introduced in the theory at strong (or fixed) coupling.
Another unfit feature of any strong coupling approach (see MBN () for a critical discussion of a strong-coupling approach to the mass gap popular in the physics literature, based on the String/ Large- Gauge Theories correspondence) is that it implicitly assumes that the strong coupling expansion in a neighborhood of , or of for ’t Hooft coupling (see below), is connected to the flow of the theory and that it computes meaningful numbers. This statement has no theoretical foundation and on the contrary there is opposite numerical evidence from finite-couplin