Holography on the lattice: the string worldsheet perspective

We review the study, on the lattice, of the Green–Schwarz gauge-fixed string action describing worldsheet fluctuations about the minimal surface holographically dual to the null cusp Wilson loop, useful to evaluate the cusp anomaly of N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {N}}=4$$\end{document} super Yang–Mills (sYM). We comment on discretization, numerical explorations and challenges for the nonperturbative study of this benchmark model of gauge-fixed worldsheet actions.

of the trace of implicit matrices [12], which was so far used only in the machine learning community and may prove useful in other high-energy-physics areas.However, the Wilsonlike fermionic discretization adopted in [7,10] breaks part of the global symmetry of the model.The resulting divergences observed e.g. in simulations for fermionic correlators [10] urged for further investigation.More recently, a new discretization has been worked out for the same model, which is invariant under the full group of global symmetries -and therefore it is the best premise for investigations [11].As we review below, however, a lattice perturbation theory analysis of one loop renormalizability reveals that the situation is much more complicated than in dimensional regularization, and it is related to the presence of power UV divergences.In order to remove them at one loop, it is necessary to introduce two extra parameters in the action, which need to be either fine-tuned at tree level or renormalized at one loop.These do not seem to have any deep meaning, besides the fact that they make the bare propagators particularly simple.
That the one-loop fine-tuning of [11] is enough to make all physical observables finite at all orders in perturbation theory seems unlikely.It would be still important to understand whether the number of parameters needed to achieve finiteness of all correlators -via finetuning or renormalization -is finite.If not, the discretized model has no predictivity.A complete one-loop analysis of the divergences of n-point functions, e.g. on the general lines of [13] and relying on the technology developed in [11], would be necessary here.This calculation would be easily generalisable to a number of observables and backgrounds.Obviously, a positive answer to this question would not be a definite statement on the nonperturbative renormalizability of the model, but a strong indication of the action of symmetries.In addition, notice that the divergences requiring the introduction and tuning of extra-parameters in [11] are of linear type and very specific of the lattice discretization.One may try to find a general mechanism that prevents linear divergences in the first place, trying to exploit a spurionic symmetry involving the replacement m → −m, the reflection of both worldsheet coordinates and an SO(5) rotation, which is enjoyed by the continuous action.Preliminary explorations seem to indicate that it is not completely trivial to preserve this symmetry on the lattice while avoiding the doubling problem.A crucial point is that the symmetries preserved in [11] are only a small subset of the set of symmetries initially present of the AdS 5 × S 5 superstring sigma-model: target space supersymmetry -the P SU (2, 2|4) supergroup -is broken and local symmetries (diffeomorphism and fermionic kappa-symmetry) are all fixed.This holds for the large majority of the studied string configurations dual to interesting gauge theory observables, but not much has been said, in the continuum, on the "remnants" of these symmetries in terms of a nonlinear realization, or of "BRST"-like mechanism (for the local ones) and related Ward identities.An explicit investigation of this kind, in particular for the fermionic κ-symmetry -which plays a central role in the formal argument for UV finiteness of the GS string -appears important in the continuum in the first place (in a related setup, some discussion on κ-symmetry ghost contributions appears in [14]).Then, one would have to search for a discretization able to preserve (at least a subgroup of) such remnant symmetries [15], and study the corresponding continuum limit.
We conclude by remarking that, in many known cases, symmetries play a crucial role in proving renormalizability of theories that seem to be non-renormalizable on the basis of power counting alone (as it happens for the discretized cusp model of [11], as we see below).This is what happens for the well-known 2d O(N ) non-linear sigma-model both in dimensional regularization and on the lattice [16], and it would be pedagogical -to go closer to the superstring worldsheet setup -to conduct numerical explorations in the setup obtained adding target space supersymmetry 1 .This review proceeds as follows.In Section 2 we sketch general aspects of the GS action in AdS and introduce the model in the continuum (some details of which are in Appendix A), in Section 3 we review some of the findings of [7,8,9,10,11] and in Section 4 we report the more recent analysis of [11], via which we have a better picture of the challenges underlying this program.

The model in the continuum
The Type IIB string background emerging in the near-horizon limit of D3-branes -the setup at the heart of the AdS/CFT correspondence -is the 10-dimensional spacetime product of the five-dimensional Anti-de-Sitter space AdS 5 and the five-sphere S 5 , and supported by a self-dual five-form Ramond-Ramond flux [19].Because of the presence of the latter and its nonlocal features when coupled with worldsheet fields, the standard worldsheet supersymmetric formalism of Ramond, Neveu and Schwarz for the string action appears of problematic use.More adequate is the Green-Schwarz (GS) approach, which is supersymmetric only in target space, and whose fundamental fields, both bosonic and fermionic, are worldsheet scalars.The corresponding TypeIIB GS action in this background can be usefully written as a Wess-Zumino-Witten-type nonlinear sigma-model 2 on the target coset superspace P SU (2,2|4)  SO(4,2)×SO(5) [1], for a pedagogical review see [20].The global symmetry of the action is the supergroup P SU (2, 2|4) 3 , which is the N = 4 super-extension of the conformal group in four dimensions SO (4,2).The superstring action has two additional local symmetries: a bosonic one, the invariance under reparametrization of the worldsheet or diffeomorphism, and a fermionic one, the κ-symmetry [21], which is crucial to remove unphysical degrees of freedom and ensure space-time supersymmetry of the spectrum.This action is classically integrable, as its equations of motion are equivalent to the flatness condition for a Lax pair and infinitely many non-local conserved charges can be derived [22].About the quantum integrability, for 1 Preliminary investigations, conducted together with B. Hoare, A. Patella, T. Meier and J. Weber, on the "supersphere" setup of Ref. [17] show that both in the continuum (in dim reg) and on the lattice renormalizability at all loop orders can be proven on the same lines, i.e. exploiting the constraints of the nonlinear symmetry on the possible counterterms.See also the talk of I. Costa at Lattice 2022 [18]. 2 The action is namely the sum of a "kinetic term", namely a 2d integral quadratic in invariant Cartan 1-forms on the superspace, and a Wess-Zumino type term, that is an integral of a closed 3-form over a 3d space which has the world-sheet as its boundary.Because the 3-form is closed, in a local coordinate system, namely choosing a specific parametrization of the Cartan 1-forms in terms of target superspace coordinates, the action takes the usual 2d sigma-model form.the string worldsheet action this is a solid statement only at low perturbative orders in sigmamodel loop expansion, verified through the factorization of the worldsheet S-matrix [23] or because of the fact that 1-loop [24] and 2-loop [4] corrections to energies of certain string configurations match the strong-coupling predictions of the N = 4 sYM Bethe Ansatz.At finite coupling, integrability remains a conjecture for the whole AdS/CFT system.
The AdS 5 × S5 GS sigma-model is expected to be UV finite.The argument [1] 4 for this is however very formal, and based on the assumption that both global symmetries and the local fermionic κ-symmetry are actually preserved at the quantum level.In practice, UV finiteness is not manifest.A direct, canonical quantization of the AdS string -namely, deriving the quantum S-matrix from the classically integrable structure -is beyond current methods, and one proceeds in a semiclassical fashion which begins, usually, by choosing a physical, lightcone gauge-fixing for both diffeomorphism and κ-symmetry invariance.The model is formally power-counting non-renormalizable beyond one loop [25,2,14] and verifying the cancellation of UV divergences may be non-trivial.
The original action has no scale -natural scales are the inverse string tension α and the radius R common to AdS 5 and S 5 , but they combine in a dimensionless combination, the effective string tension g = √ λ 4π , equated via AdS/CFT to the t'Hooft parameter λ of the N = 4 super Yang-Mills theory.The bosonic light-cone gauge fixing X + = P + τ 5 where P + is the (dimensionful) light-cone momentum, introduces however a "fiducial" mass scale which below we call m.The fermionic κ-symmetry light-cone gauge reduces the 32 fermionic coordinates Θ I α of the two (I = 1, 2) Type IIB left Majorana-Weyl 10-d spinors to 16 physical Graßmann variables, θ i and η i (i = 1, 2, 3, 4) below (the action is linear in the first and quartic in the second), which transform according to the fundamental representations of SU (4).
The AdS light-cone gauge-fixed action6 is presented in [4] 7 .Here we are interested in the action S cusp -equation (2.2) below -for the quantum fluctuations about one of its classical solutions, the one representing an open-string euclidean world surface ending on a closed contour with a light-like cusp on the boundary of AdS.This is because, according to AdS/CFT [28,29], the euclidean path integral weighted by the action for the quantum fluctuations about this classical solution evaluates the vacuum expectation value of the Wilson loop with the null cusp contour The effective action Γ eff to which the path integral is equivalent is then the holographic definition of the cusp anomalous dimension f (g) governing the renormalization of the Wilson loop [30].The cusp anomaly f (g) can be evaluated perturbatively in gauge theory [31] (g 1) and in sigma-model loop expansion [3,24,4] (g 1).Considering its equivalent definition as leading coefficient in the anomalous dimension of twist operators at large spin 8 and assuming all-order integrability, an integral equation [32] can be derived which gives f (g) exactly at each value of the coupling g.The action S cusp for the bosonic and fermionic fluctuations X(t, s), Ψ(t, s) over the cusp minimal surface -with t, s the temporal and spatial coordinate spanning the string worldsheet -has constant coefficients, hence the worldsheet volume V 2 = dtds factorizes out (the normalization factor 1/8 takes also into account the conventions of [4]).The m-dependence, in (2.1) and below, is introduced by dimensional analysis for convenience.
The explicit form for S cusp is [4] Above, x is a complex bosonic field whose real and imaginary part are the AdS 5 (coordinate) fields transverse to the AdS 3 subspace spanned by the classical solution (A.8).
is the radial coordinate of the AdS 5 space with z M /z = u M identifying points on S 5 , and together with x they are the bosonic coordinates of the AdS 5 × S 5 background in Poincaré parametrization remaining after fixing the light-cone gauge.As mentioned above, θ i , η i , i = 1, 2, 3, 4 are 4+4 complex anticommuting variables for which θ i = (θ i ) † , η i = (η i ) † .They transform in the fundamental representation of the SU (4) R-symmetry and do not carry (Lorentz) spinor indices.The matrices ρ M ij are the off-diagonal blocks of SO(6) Dirac matrices γ M in the chiral representation The two off-diagonal blocks, carrying upper and lower indices respectively, are related by 6) generators.Some details are collected in Appendix A. The action (2.2) is invariant under a U (1) × SU (4) global symmetry acting on the fields as x → e iα x , where U is an element of SU (4) and its representative in the antisymmetric, denoted here as as(U ), is an element of SO (6).The U (1) is the rotational symmetry in the two AdS 5 directions orthogonal to the classical solution, while the SU (4) ∼ SO(6) symmetry originates from the isometries of S 5 , unaffected by the (purely) AdS light-cone gauge fixing.Notice that the euclidean action (2.2) is not invariant under worldsheet rotation, parity (s → −s), time reversal (t → −t).
The UV finiteness of the model has been verified up to two loops in sigma-model perturbation theory via the evaluation at this order of the cusp anomaly in (2.1) [4] and of the "generalized scaling function" of [33], both reproducing the result predicted by integrability [34].Also in agreement with the exact result obtained via Bethe Ansatz is the one-loop calculation of the dispersion relation of excitations around the long spinning string with one angular momentum in AdS 5 and one in S 5 [35].The regularization adopted in all these cases is a version of dimensional regularization9 introduced in [4, 2, 14].

Explorations in the broken symmetry setup
In this Section we summarise some theoretical and numerical aspects of the simulations addressed in [7,8,9,10], where -oriented to simulations -the model was first linearized (the quartic fermionic interaction being reduced to quadratic, so to be formally integrated out) then discretized with a Wilson-like treatment of the fermionic sector which would break some of the global symmetries of the problem.Simulations were then performed employing a Rational Hybrid Monte Carlo (RHMC) algorithm [37,38].While repeating the kind of analysis in the setup of [11] or similar gauge-fixed ones is first subject to clarifying the question of its finiteness on the lattice, it remains an interesting fact that on these models algorithms and techniques employed in unrelated models with fermionic interactions as well as in lattice QCD could be tested, implemented and shown to improve the analysis.Among these, the procedure of eliminating a complex phase in the fermionic Pfaffian and the attempts to reduce numerical instabilities.We do not report on this here, it is still interesting to recall that in this setup we have tested [12] a stochastic trace estimator algorithm introduced in [39], revealing advantages in terms of variance with respect to traditional methods, such e.g. the Gaussian estimator, and as such of potential benefit in the area of lattice field theory.
The AdS light-cone gauge-fixed action is at most quartic in fermions.Having numerical simulations as a goal, the common way to proceed is to linearize the action thus making it quadratic in fermions, then formally integrate them out and re-exponentiate the determinant (here a Pfaffian) obtained in terms of a set of pseudo-fermions so to enter the Boltzmann weight of configurations in the statistical ensemble.
In the case at hand, one starts with that part of the cusp fluctuation lagrangian La- and notices a "repulsive" potential, i.e. the plus sign in front of the second term in (3.1), that squares an hermitian bilinear (i . This has the effect of introducing a complex phase in the auxiliary, linearized Lagrangian.Indeed, a standard Hubbard-Stratonovich transformation [6] exp which introduces 7 auxiliary fields (one scalar φ and a SO (6) vector field φ M ), will generate a non-hermitian term (the last one above) and lead to a complex-valued Pfaffian [7].To eliminate the complex phase, an alternative procedure -inspired by the analysis of [40] for the much simpler case of a SO(4) four-fermion interaction -was considered in [10], consisting in an algebraic manipulation of the original four-fermion Lagrangian which would essentially change the problematic sign in (3.1).After some ρ-matrices algebra, equation (3.1) can be written as Now one can define the fermionic bilinear Σ i j ≡ η i η j and introduce a Hodge-duality-like transformation to obtain its dual Σj i as well as its self-and antiself-dual part 10 In terms of these, one can write two equivalent forms of the same action, differing by a sign Above, the trace is over SU (4) fundamental indices 11 , and one uses that TrΣ ± Σ ± = 2 Tr(ΣΣ± Σ Σ).Notice that there is no ambiguity in the double sign of equation (3.5): Writing the Lagrangian in terms of the self-dual part of Σ requires the minus sign, writing it in terms of the antiself-dual part requires the plus sign.It is only in the second case that a complex phase appears, proving that the latter is an artefact of our naive linearization.Choosing the part involving the Σ + , one obtains (3.6) 10 Notice that Σ = Σ and Σ i j ≡ (Σi j ) † = Σj i . 11Notice that Tr Σ Σ = TrΣΣ.
This introduces a total of 17 auxiliary fields (φ is real, φ i j is a 4 × 4 complex hermitian matrix with 16 real degrees of freedom), and leads to a Lagrangian now quadratic in fermions where In simpler cases of models with four-fermion interactions [40,41] a similar algebraic manipulation ensures a positive-definite Pfaffian 12 .In [7,10] a (U (1)-breaking) discretization was adopted, to avoid fermion doublers a Wilson-like term was added in the main diagonal of the fermionic operator.The resulting discretized fermionic operator ÔF is antisymmetric and γ 5 -hermitian" Ô † F = Γ 5 ÔF Γ 5 (with Γ 5 a unitary, antihermitian matrix, Γ † 5 Γ 5 = 1, Γ † 5 = −Γ 5 ) leading to a real and non-negative det ÔF .A study of the fermionic spectrum [10] confirms that the Yukawa-like terms (given in terms of the auxiliary fields resulting from the linearization but also present in the original lagrangian) are responsible for the occurrence of sign flips in the Pfaffian 13 .
The absence of a complex phase is a manifest improvement (which e.g.allows to eliminate systematic errors), but it is not enough to make the Pfaffian positive-definite.Indeed a sign problem has been detected in the simulations of [10] for small values of the effective string tension g -this, in a large tension expansion like the one of sigma-model perturbation theory, identifies the strong coupling regime of the model.We have not repeated a similar analysis of the Pfaffian in the symmetry-preserving discretized setup of [11], see Section 4.However, for this and similar uses of the light-cone gauge fixed action of [26] (namely, considering the action expanded around string configurations holographically dual to other interesting observables) it is very likely algebraic manipulations of the four-fermion terms as the one described above would be necessary to eliminate complex phases -it is also rather obvious that a sign problem will similarly appear.
Together with the sign problem, for lower values of g (and thus when the string sigma-model is strongly coupled) simulations in the setup described here run into numerical instabilities which can be associated with zero eigenvalues in the fermionic operator and cause the nonconvergence of the inverter for the fermionic matrix.This may be cured by a "twistedmass" IR regularization [42], adding a massive term to the fermionic matrix (µ being a real parameter) so that the µ 2 1 term above shifts the eigenvalues of ÔF Ô † F apart from zero.Concretely, then, simulations are not done with the string worldsheet action in the chosen discretization, but differ due to both the replacement of the Pfaffian by its absolute value and the addition of the "twisted mass" in (3.9).The sign of the Pfaffian and the low modes of O F are then taken into account by a standard reweighting procedure, for which the expectation values O of observables in the underlying, target theory are obtained from the expectation values O m in the theory with the modified, positive-definite fermionic determinant and the total reweighting factor W in this case reads In the setup of [10] -where two different values for µ were chosen -this procedure lead to stability of the simulations in a very large region of the parameter space.We refer the reader to the detailed analysis there included, but report that in the case of numerical values of the ensemble averages for the two-point functions of bosonic and fermionic fluctuations over the cusp (the configurations being generated by the standard RHMC algorithm) the sign-reweighting seemed not to have effect on the measured observables.

Symmetry-preserving discretization and fine-tuning
We review here the analysis of [11], in which for the null cusp effective worldsheet action a new discretization has been worked out, which is invariant under the full U (1) × SU (4) group of internal symmetries.
The proposed action where a is the lattice spacing and b ± are two auxiliary parameters whose role will become clear below.The action is written, for both the bosonic and the fermionic sector, in terms of the forward and backward discrete derivatives with e µ is the unit vector in the direction µ = 0, 1, and σ is a shorthand notation for (s, t), the worldsheet coordinates.Using forward and backward discrete derivatives breaks parity and time-reversal, which is the reason why this is normally avoided for fields satisfying first-order equations of motion -which is usually the case for fermions.As noticed above, however, parity and time-reversal are not symmetries of the action even in the continuum, so that their use is harmless here and -in fact -is the key fact allowing the preservation of the full group of internal symmetries.
Notice that in the naive continuum limit a → 0, setting b ± = 1 in (4.1) gives back the continuum action S cont cusp in equation ( 2.2) 14 .The one loop analysis of [11], summarized below, shows however that with this choice of the parameters no cancellation of UV divergences occurs.For this to happen, they must be set as in (4.14), so that they effectively play the role of fine-tuning parameters.
Choosing a flat measure for the fields 15 , the expectation values of a generic observable A in the lattice discretized theory is then defined by The partition function Z cusp is as usually fixed by the requirement 1 = 1.
In fact, for an analysis in lattice perturbation theory one proceeds reparametrizing the bosonic fluctuations as in continuum perturbation theory [4] -see Appendix A, equations (A.9)-(A.10)-so that the corresponding Jacobian determinant contributes to an effective action and the path integral (4.3) is reformulated as The sigma-model perturbative expansion goes in inverse powers of the effective string tension g and it effectively splits the action in the sum of its free (quadratic in fluctuations) and 14 Notice that in the continuum action (2.2) the term 1/z 4 (z M ∂sz M ) is actually a vanishing boundary term (this can be seen recalling that z M z M = z 2 ).Omitting it from the discretized version of the action is then possible.This choice, together with the introduction of the b± parameters in the way specified by (4.1) is important to get the propagators as below and -with the fine-tuning discussed -is responsible for canceling divergences and reproducing the continuum results for observables. 15Notice that this choice is rather arbitrary, as it is not invariant under reparametrization of the target AdS5 × S 5 target space.
interacting part, S eff = S 0 + S int , with S 0 proportional to g (the g-independent quadratic terms coming from the expansion of the Jacobian determinant belong to S int ).The quadratic action can be conveniently written in momentum space as where Φ = (Re x, Im x, y 1 , . . ., y 5 , φ) t (4.8) Ψ = (θ 1 , . . ., θ 4 , θ 1 , . . ., θ 4 , η 1 , . . ., η 4 , η 1 , . . ., η 4 ) t (4.9) collect the bosonic and fermionic fields respectively, and K B , K F are the corresponding quadratic operators.K B (p) is a 8 × 8 diagonal matrix with non-vanishing components given by For the two matrices it holds that K t B (p) = K B (−p) and K t F (p) = −K F (−p).Propagators in momentum space are defined by the entries of the inverted matrices up to prefactors 17 and read explicitly (σ ≡ (s, t)) , and all other 2-point functions vanish.Notice that the choice c ± = 1, at which the auxiliary parameters in the discretized action read b± = 1 + am 4 corresponds to a particularly simple form of the propagators.In fact, it is also the choice which reproduces the continuum results for the observables calculated in [11], see below.
The interaction Lagrangian is obtained expanding further S eff in powers of the fields, taking into account the explicit expressions of the terms involving the forward derivatives By expanding the exponentials in the first line and denominators in the second and third line in powers of y one obtain terms with an arbitrary number of powers of ∂k φ(σ) and ∂k y c (σ), respectively, multiplied by explicit powers of a.The number of derivatives and the number of factors of a are related by dimensional analysis.
At each order in the perturbative expansion the interaction Lagrangian density in σ is a polynomial of the fields Φ(x), Ψ(x), their first derivatives ∂Φ(x), ∂Ψ(x), ∂Ψ(x), the lattice spacing a, and the mass m.We do not write down explicitly all vertices, but notice that the possible vertices -which must have dimension 2 -are constrained by dimensional analysis (bosonic fields have mass dimension 0, fermions have dimension 1/2, discrete derivatives as well as m have dimension 1, the lattice spacing has dimension -1), exist only with 0, 2, or 4 fermion fields and are proportional to m 0 , m 1 or m 2 , and to a p with p ≥ −2.In particular, terms proportional to a −2 are generated by the Jacobian determinant in eq.(4.4).
An analysis of the superficial degree of divergence of a generic Feynman diagram for the lattice-discretized theory -which can be performed following [43] (see also e.g.[44,45])shows that it is non-renormalizable by power counting.One can indeed calculate the superficial degree of divergence of the generic one-particle irreducible Feynman diagram A contributing to an amputated n-point function in momentum space, which has the general form where q i=1,...,L are the loop momenta, W is the product of all vertices, p i=1,...,E denotes the external momenta and l i=1,...,I denotes the momentum flowing in the i-th internal line, D i is the propagator associated to the i-th internal line.The analysis in [11] 18 leads to the following result for the degree of divergence of A where E F is the number of external fermionic lines, P m is the total number of m factors and D E is the total number of discrete derivatives acting on external lines.The formula shows that the degree of divergence of one-particle irreducible diagrams cannot be larger than 2, but a key fact is that deg A does not depend on the number of external bosonic legs.Therefore at any loop order the number of divergent diagrams is infinite, implying the need of infinitely many counterterms at any loop order to cancel the UV divergences.As mentioned in Section 2, this is not surprising as the same conclusion holds in the continuum 19 , see [25,2,14], and yet -provided dimensional regularization is adopted (see above) -non-trivial cancellations of UV divergences happen.This is not the case for the lattice regularization provided here, as a certain amount of fine-tuning is needed in order to reproduce the continuum results at one loop order in the sigma-model expansion.
A useful way to define the cusp anomaly, free from normalization ambiguities (see discussion in [11]) is through the following derivative of the free energy density in the infinite-volume limit consistently with the definition (2.1).
At quadratic order the relevant calculation is a Gaussian integral and yields where . (4.23) 18 See Section 4 there. 19From the point of view of the analysis in [11], in which the total number Pa of factors of a appears at intermediate steps, this can be seen as the fact that Feynman diagrams appearing in the continuum are the same one as the diagrams with Pa = 0 on the lattice.
A small-a asymptotic expansion of the relevant integral, whose details can be found in [11] 20 , leads to where the convention c ± = 1 + a m δc ± has been used.For the cusp anomaly one then obtains Using the naive choice b ± = 1 for the auxiliary parameters, corresponding to δc ± = ∓1/2, the cusp anomaly contains a linear divergence.On the other hand, with the special choice b ± = b± , see (4.14), which corresponds to c ± = 1 and δc ± = 0, the linear divergence is canceled, and we obtain the same same result as in dimensional regularization [4,24], Turning to the calculation of correlators, it is easy to realize that the symmetry-preserving discretization leads to some vanishing one-point functions: x = 0 because of the U (1) symmetry, and y a = 0 because of the SO(5) ⊂ SO( 6) SU (4) symmetry, the rotations for y a , a = 1, . . ., 5. The fluctuation φ, however, acquires a non-trivial UV-divergent onepoint function at one loop.In fact, since φ appears as a subdiagram in any other n-point function, its UV divergence contributes to any physical observable and its inclusion, both in the continuum (in dimensional regularization [4,35,46]) and on the lattice, is important for the cancellation of divergences -as we see below.
There are two classes of vertices contributing to the one-point function of φ: single-field vertices coming from the measure S φ ∼ g 0 , which produces a tree-level diagram, and threefield vertices coming from the action S φ•• ∼ g, which produces a one-loop (in the x fields and in the fermions) diagram -for explicit expressions and more details see [11].The result (because of the mismatch in the power of g) is a contribution to the same order in g, yielding With the special choice of the auxiliary parameters b ± = b± in (4.14) (c ± = 1), one can use 20 In particular, see Appendix B of [11].the symmetry p 0 ↔ p 1 of the integrals to obtain the logarithmically divergent integral where the numerical constant I (0,0) 0 0.355 is defined through an integral 21 , see Appendix B in [11].Interestingly, measure, fermion-loop and x-loop contributions develop quadratic divergences, whose cancellation is highly non-trivial.In the more general case c ± = 1+a m δc ± where δc ± = O(a 0 ), the asymptotic expansions of the integrals lead to s so that the naive choice b ± = 1, or δc ± = ∓1/2, yields indeed a linear divergence for φ : The last calculation we sketch evaluates the two-point function of the bosonic excitation x at one loop, which we calculate to extract the corresponding dispersion relation.The two classes of vertices contributing are three-field vertices S xx * • and four-field vertices S xx * •• , with resulting Feynman diagrams of three different topologies illustrated in Fig. 1, where the tadpole contribution is proportional to φ .Explicit expressions and more details are, again, in [11].
The two-point function can be expressed, on general grounds, in the form 21 Explicitely, it is 0.355 with the definition where factor c − |p 1 | 2 + m 2 4 comes from the Fourier transform of the combination b − ∂s x − m 2 x , or its complex conjugate, appearing in all interaction vertices with x.Above, Π a (p) can be represented in terms of amputated Feynman diagrams, and reads explicitly As seen above, the φ terms contains in general a linear divergence, while all remaining integrals are logarithmically divergent.Using c ± = 1, which can be done up to terms that vanish in the a → 0 limit, one obtains the simpler expression The leading divergence of these integrals does not depend on the external momentum, exactly as it happens in the continuum.Then the subtracted ∆Π a (p) = Π a (p) − Π a (0) has the finite a → 0 limit + O(a log a) , (4.34) and all the divergences are contained in Using now the result (4.28) for the tadpole contribution and the choice c ± = 1, it is immediate to see that all divergences cancel and Π 0 (0) = 1/π.The two-point function is finite in the continuum limit and

.36)
The two-point function has poles at p 0 = ±iE(p 1 ) for every value of p 1 , where E(p 1 ) is the energy of a single excitation propagating on the worldsheet with momentum p 1 .In the continuum limit one obtains [11] E(p 1 ) 2 = p reproducing the dispersion relation of [35] 22 .
However in the general case c ± = 1 + (am)δc ± where δc ± = O(a 0 ), both the integral Π a (0) and the dispersion relation E(p 1 ) inherit from φ the linear divergence.For example, the naive choice b ± = 1 leads to As discussed in [11], with the naive choice of auxiliary parameters b ± = 1, the divergence in the dispersion relation cannot be eliminated by renormalizing the remaining available parameters, i.e. the effective string tension g and the massive parameter m.This means that the choice b ± = 1 is not stable under renormalization.On the other hand, allowing the coefficients b ± to be renormalized together with m and g, the divergences in the dispersion relation are eliminated for example choosing The continuum dispersion relation has then the same form as eq.(4.37) but with the renormalized m R replacing m.Similarly, a one-loop renormalization of the coupling constant could be chosen so that the cusp anomaly is finite.Such observation do not imply that the chosen lattice theory is renormalizable, something which we do not know.If, however, the lattice theory is renormalizable, it appears not sufficient to renormalize m and g.One also needs to introduce extra coefficients in the action and either fine-tune their tree-level value, or renormalize them.
We conclude summarizing some features of the divergences emerging in the one loop analysis above.About quadratic divergences, the cancel at one loop in the one-point function of φ and in the two-point function of x (they are instead subtracted by hand in the cusp anomaly).The expectation is that such cancellations will always happen in any reasonable discretization of the action.About linear divergences, they generally do not cancel in all considered observables.They arise from the special choice of forward and backward discrete derivatives and are therefore very specific of the lattice discretization.As mentioned, they are cured via the introduction of two extra parameters b ± in the action, to be either fine-tuned at tree level or renormalized at one-loop.Once the linear divergences are removed this way, the logarithmic ones cancel in the cusp anomaly and in the two-point function of x (while they survive in the one-point function of φ, which is what also happens in the continuum).Then the continuum limit of the observables we calculated is the same as in dimensional regularization.

Figure 1 :
Figure 1: Topologies of diagrams contributing to the two point function at 1-loop.
transpose of the block with lower indices.A possible explicit representation is