Comments on the NSVZ $\beta$ Functions in Two-dimensional $\mathcal N=(0,2)$ Supersymmetric Models

The NSVZ $\beta$ functions in two-dimensional $\mathcal N=(0,2)$ supersymmetric models are revisited. We construct and discuss a broad class of such models using the gauge formulation. All of them represent direct analogs of four-dimensional ${\mathcal N} =1$ Yang-Mills theories and are free of anomalies. Following the same line of reasoning as in four dimensions we distinguish between the holomorphic and canonical coupling constants. This allows us to derive the exact two-dimensional $\beta$ functions in all models from the above class. We then compare our results with a few examples which have been studied previously.

The goal of this paper is to derive NSVZ-like β functions [25][26][27][28][29] in general two-dimensional N = (0, 2) supersymmetric gauge theories adding new evidence for the 2d/4d correspondence. A number of 2d analogs of the NSVZ β functions were obtained in the past via both perturbative methods and instanton calculus in the N = (0, 2) CP 1 model [24] and in a large class of heterotically deformed non-linear sigma models (NLSMs) which are deformations of their N = (2, 2) cousins [6]. Here we focus on another general class of N = (0, 2) gauged linear sigma models (GLSMs) and obtain the general form of the corresponding β functions. They have the same structure as the NSVZ β function in 4d. In those cases where comparison with the previous results is possible our newly derived GLSM β functions are identical to those of NLSMs. This is not surprising since the NLSMs studied previously can be embedded in GLSMs.
We want to emphasize not only the ubiquity of 2d/4d correspondence, but also the conspiracy of methodologies applicable to both 2d and 4d theories. Historically, 2d sigma models were considered as simplified toy models useful for understanding real world physics in 4d. Instead, in this paper, we follow the opposite direction, from 4d to 2d, establishing and using the 2d analog of the Konishi anomaly [30] and scaling anomalies in 2d N = (0, 2) gauge theories,à la Arkani-Hamed and Murayama in 4d N = 1 case [31]. This observation helps us relate holomorphic coupling constants to canonic ones in 2d GLSMs thus trivializing derivation of their β functions. The general master formula obtained in this paper is in the case of 2d N = (0, 2) gauge theories with a single FI coupling where q i 's are the U(1) gauge charges of the bosonic matter fields,q a and γ a 's are the U(1) gauge charges and anomalous dimensions of the fermionic matter fields.
The paper is organized as follows: We will briefly review the building blocks of 2d N = (0, 2) supersymmetric GLSMs in section 2 and a non-renormalization theorem for the FI coupling constants in section 3. We then explain the difference between holomorphic and canonical coupling constants both from the perspectives of the Konishi anomaly and the scaling anomalies of matter fields, and derive the master equation (1.1) in section 4. Finally, we apply the formula in several examples.
Gauge multiplets: is real and adjoint-valued in superfield formalism. Here are the 2d gauge fields, λ − andλ − are the gaugino fields, and the real field D is auxiliary. The field A ++ is an N = (0, 2) singlet. Next, we can promote superderivatives to be covariant, namely The superfield strength of the gauge multiplet is given by is the field strength of the A µ field. The conjugated superfieldῩ − is defined accordingly. The action of the gauge multiplet is as follows: Here e 2 is the gauge coupling. The corresponding NLSM can be obtained in the limit e 2 → ∞.

Chiral multiplets:
The N = (0, 2) chiral multiplet Φ i = (φ i , ψ i + ) satisfies the usual chiral constraint In the superfield formalism it is written as Moreover, q i is the charge of the field Φ i with respect to the U(1) gauge field. The action of the chiral multiplets can be written as (2.10)

Fermi multiplets:
Another important matter superfield consists of a fermion χ a − and an auxiliary field It is not necessary chiral, but, instead, satisfies the constraint where E(Φ) is an arbitrary holomorphic function with respect to chiral boson fields Φ's. In the superfield formalism, it can be expanded as The action for the fermi multiplet reduces to (2.14) Note that the gauge field strength Υ − is a particular case of the fermi multiplets in the adjoint representation of the gauge group, satisfyinḡ

Superpotentials:
Last but not least, we need to introduce superpotentials J a (Φ) as holomorphic functions of chiral superfields, whose action reduces to a half of the superspace (accompanied by fermi multiplets Γ a − ), Of the utmost interest is the Fayet-Iliopoulos (FI) term as a superpotential given by the gauge field strength, if it admits U(1) factors, where for simplicity we only consider theories with a single FI term, and is the complexitied FI coupling constant.

GLSM action:
Overall we assemble all the above ingredients and arrive at the action of N = (0, 2) supersymmetric GLSM, Here and below, without loss of generality, we will consider theories in which the superpotentials are limited to FI terms. Importantly, for such theories to be consistent at the quantum level (i.e. free of internal anomalies), we need to impose constraints on the representations of the chiral and fermi multiplets to get rid of the gauge anomalies, see also in [33], where q i andq a are U(1) gauge charges of chiral and fermi multiplets, t 2 is the dual Coxeter number, and "i", "a" and "A" denote the Reps. of chiral, fermi and gauge multiplets.

A non-renormalization theorem for the holomorphic coupling τ
In 2d gauge theories, the gauge coupling e has dimension of mass, and is thus superrenormalizable. For energy scale µ ≪ e, the gauge multiplets will be non-dynamical and we arrive at NLSMs. Therefore the only sensible parameter in the theory is its FI coupling constant τ , which is marginal and runs at the quantum level. In much the same way as with the gauge couplings in 4d N = 1 gauge theories, the 2d FI parameter τ , as the coupling of the N = (0, 2) superpotential, is subject to a non-renormalization theorem and receives at most one-loop correction (see e.g. [28]) . We will follow [28,31] in reviewing the relevant argument.
From eq.(2.19), we see that the action S depends on τ holomorphically. It is convenient to use the notation Let us ask ourselves: when we change the cutoff from M 0 to µ, how the coupling 2πiτ (µ) (in the Wilsonian sense) changes to keep the low-energy physics intact. To answer this question, let us examine an ansatz It is worth noting that a 2π shift of the θ angle leads no change of physics, therefore at most, can only take integer values. Furthermore because F (0) = 0, by continuity we conclude that function f is periodic respect to the θ angle. Therefore the β function for 2πiτ , is periodic with respect to θ and admits a Fourier expansion, It is clear that in perturbation theory we can only have non-negative integer values of n appearing in the expansion (3.5). Also, in the perturbative regime we at most It perturbation theory it is obvious that all b n 's with n = 1, 2, 3, ... vanish. Hence the non-renormalization theorem of the absence of higher loops is proven for the holomorphic coupling. Non-perturbatively, one needs to apply the anomalous R-symmetry of N = (0, 2) , which guarantees that the θ angle receives no quantum corrections at all.
Before proceeding to the discussion of the canonical coupling τ c in next sections, let us first calculate b 0 that would be used latter. It can be easily obtained by inspecting the D term of the action (2.19), From (3.7) we see that the real part of τ receives a tadpole one-loop correction. 1 The tadpole graph emerges through contracting φ andφ. As a result,

From the holomorphic to canonic coupling
As known from [28], all higher order loops in the gauge coupling renormalization appear in passing from the holomorphic to canonic coupling from the Z factors of the matter fields (which are converted into the anomalous dimensions in the β functions). To see how this happens we must convert the kinetic terms of the matter fields into (2.17) by virtue of anomalies. In other words, we must take into account a subtle difference between the Wilsonian Lagrangian and 1PI irreducible functional (see [25][26][27][28]). Below we will discuss two alternative (but related) derivations, through the Konishi anomaly [30] and through the scale anomaly [31].

The Konishi anomaly in N = (0, 2) GLSM
It is not difficult to establish the 2d analog of the Konishi anomaly. To this end, as an example, we will consider the operator a Γ − a Γ a − appearing in (2.14) (assuming that E a = 0). Classically, the equation of motion for this operator is This follows, e.g. from inspection of theθ + component. However, at the quantum level this particular component contains a well-known anomaly in the derivative of the χ − current, see more details in appendix B and also [34], analogous to the triangle anomaly in the axial current in 4d, 2 where B is defined in (2.6). Note that the relative coefficient between D and B in (2.5) is rigidly fixed by N = (0, 2) supersymmetries. Needless to say, that the full derivative in the U(1) part does not appear in the action classically (it can be dropped). However, at the quantum level we can establish the following relations (after evolving the action from M 0 down to µ), where in the last step, we uplifted the equation to the level of superspace, cf. (2.17). The Υ − part gives the evolution of the wave function renormalization of fermion Γ a − to the FI-coupling constant τ , see also eq.(4.7). Adding the one-loop tadpole graph and differentiating over µ/∂µ we arrive at theq a γ a term in (1.1).

Scaling anomalies
Now we would like to discuss the 2d N = (0, 2) β function along the the lines of [31]. It is true that the holomorphic τ only receives one-loop correction, however, because of the normalization point running down from M 0 to µ, the kinetic terms of the matter fields will receive a wave function renormalization, To keep all matter fields canonically normalized, we need to change field variables, i.e. redefine Φ i ≡ 1 However, such rescaling will result in anomalous Jacobians from the functional measure. Formally we have where "sDet" and "sTr" denote the super-determinant and super-trace, respectively. The super-trace is superficially vanishing due to supersymmetries. Nevertheless, in a non-trivial gauge field background, we can show that they give rise to terms proportional to the U(1) field strength Υ − . More specifically, The derivation of this formula is presented in appendix B. Therefore, the holomorphic τ will receive non-holomorphic corrections from wave function renormalizations, The anomalous dimensions of Φ i and Γ a − are given by and they are non-holomorphic. This statement is in one-to-one correspondence with the NSVZ β function in four dimensions. Differentiating log µ on both sides of eq.(4.8) and using eq.(3.9), we have (4.10) In terms of coupling constant Furthermore, from eq.(3.7), the β function of g 2 c , or say, ξ, is nothing other than the wave function renormalization of chiral multiplets, i.e. (4.13) Using it, we arrive at the master formula, (4.14) Remark: The gauge multiplets have no contribution to the β function, because τ c is associated with the U(1) factor gauge group, with respect to which the gauge multiplet is U(1) neutral.

Examples
In this section, we will apply eq.(1.1) in various examples.
Therefore the holomorphic τ and canonical τ c coincide, and the β-function terminates at one-loop, in terms of g 2 Especially, for a U(1) gauge theory with all q i = 1, we have the standard N = (2, 2) CP N −1 sigma model, and its β-function is We can deform the previous N = (2, 2) CP N −1 model by deleting part of N = (2, 2) U(1) field strength, considered in [16]. In the language N = (0, 2) supersymmetries, the N = (2, 2) U(1) field strength Σ (2,2) can be decomposed as, where the Σ (0,2) is a N = (0, 2) chiral superfield and Υ − is the N = (0, 2) fermi multiplet as the field strength of U(1) gauge multiplet. N = (2, 2) chiral multiplet Φ i (2,2) also admits a decomposition as (5.5) and the N = (0, 2) fermi multiplet Γ i − satisfy the constraint Now, if we delete Σ (0,2) , the deformed theory will have only N = (0, 2) supersymmetry, and the fermi multiplets satisfyD Its β function turns out to be where γ denotes the anomalous dimension of Fermi multiplet Γ i − . We want to further comment that, in [24], the authors also considered a type of deformed N = (0, 2) CP 1 model at the level of NLSM, which is different from ours. However, we do see that the β functions of the two models are similar. To compare the difference between our model and that in [24], we discuss its non-linear formalism in appendix A.

Heterotically deformed N = (0, 2) CP N −1 model
We can also consider a further deformation from the N = (0, 2) CP N −1 model discussed above, by adding an additional gauge singlet N = (0, 2) fermi multiplet, (5.9) to the N = (0, 2) model, with the corresponding deformed term in the action, where κ is an additional coupling. It is crucial to note that, since we start from the N = (0, 2) model, all fermi multiplets satisfȳ This constraint turns out to be important, because it guarantees that the interaction term can be recast in half superspace as, It was argued in [23] that this type of interaction is subject to a "D-term" nonrenormalization theorem in 2d, see also [6]. Therefore, the holomorphic coupling constant κ is not renormalized. Here we pause and remark that, if one tries to perform the heterotic deformation from N = (2, 2) CP N −1 GLSM, there would be no non-renormalization theorem to protect the coupling κ, because in the N = (2, 2) case,D + Γ i − ∝ Σ (0,2) Φ i , see eq.(5.6). This differs from the situation in [6], where the heterotic deformation is indeed performed on N = (2, 2) CP N −1 NLSM, because the superderivative acting on the fermi multiplet in NLSM automatically vanishes.
Since the coupling κ receives no renormalization, we thereby will focus on the β function of ξ, or say g −2 c , in the presence of the coupling constant κ. Let us first write down the action in components, The key observation, see also [6], is that the evolution of the interaction term iκ∇ ++φi χ i − η − and its Hermitian conjugate will give a finite shift to the kinetic term of φ i , i.e.
14) where we take fermions as quantum fluctuations and bosons as a background. We write the wave function renormalizations of χ − and η − explicitly. It was argued in [6] that this |κ| 2 iteration is limited to one-loop in the computation of the quantum correction in the instanton background. Here we have a similar situation -our 2d GLSM admits an (anti-)vortex background, say, where ∇ z is the Euclidean continuation of ∇ ++ . In this background, the iteration of |κ| 2 will not enter higher loops. Nevertheless, the wave function renormalization of the fields ψ i − and η − will still enter higher loops evaluation. Therefore, we define a new coupling, whose β function is given by where are the anomalous dimension of the fields χ i − and η − . Now we assemble this addition contribution to the one-loop correction of the holomorphic coupling ξ. The imaginary part of eq.(4.8) is thus modified as Differentiating with respect to the running scale µ, and using eqs.(4.13) and (5.17), we arrive at the β function for g 2 c in the heterotically deformed N = (0, 2) CP N −1 GLSM, (5.20) Finally, we can to compare eq.(5.23) to the master formula in [6]. In [6], the kinetic term of the fermion χ i − (in their notation, it was ψ i R ) is non-linearly coupled to the bosonic field φ i . It makes the definition of the wave function renormalizations of the two theories different up to a scale factor g 2 c , i.e.
Therefore it leads us to define (

5.22)
Under these new definition, we exactly reproduce the master formula in [6], (5.23) A NLSM of N = (0, 2) CP N−1 model In this appendix, we transform the action of the deformed N = (0, 2) CP N −1 model of section 5.2 into the corresponding NLSM version. The NLSM can be obtained by integrating out the gauge multiplet of its GLSM cousin at the energy scale µ ≪ e. Then, one can study the model in the geometric formalism. First, by integrating the D term, eq.(3.7), one finds the potential On the level of NLSM, it constrains all bosonic fields on S 2N −1 , i.e. φ i must satisfy the equation On the other hand, integrating the gaugino fields λ − andλ − in eq.(2.10), we see that the fermion fields ψ i + are subject to constraints To obtain the CP N −1 model, we need to also take account of the U(1) gauge imposed on Φ i 's. We can use this gauge to fix one of the chiral multiplet, say the N-th field Φ N , to have its bosonic field real , where ϕ now is a real boson, and κ + is its superpartner that is still a complex Weyl fermion. Further we define the gauge invariant coordinates, from which we find Now, we can solve for Φ i in terms of Z i . From eq.(A.4), we express Φ N as or, in components, We then solve Next, we integrate out the gauge fields A µ in eq.(2.10) and (2.14), and find where, to distinguish the fermi multiplet Γ a from the bosonic one Φ i , we use the Latin letter "a" to label them, with i = 1, 2, . . . , N − 1 and a = 1, 2, . . . , N . Moreover, is the standard Fubini-Study metric on CP N −1 . The bosonic part of the gauge field is in fact the U(1) piece of the holonomy group U(N − 1) of CP N −1 [34], and couple to the left moving fermion χ a − . It implies that the left mover lives on the tautological line bundle O(−1) of CP N −1 .
Using eqs.(A.9), (A.10) and (A.11), we can recast the eqs.(2.10), (2.14) and (2.17) to obtain the NLSM action One can clearly see that unlike N = (2, 2) CP N −1 case, the deformed model has all its left movers living on O(−1) ⊕N . We remark here that at the level of NLSM, the study of isometry/holonomy anomalies is easy. The N − 1 right movers ζ i + living on tangent bundle of CP N −1 contribute to the anomaly proportional to the first Chern class of T CP N −1 , On the other hand, the N left movers χ a − on O(−1) ⊕N contribute Therefore, the deformed model is anomaly-free as its GLSM cousin, for more details see [34].

B Scaling anomalies: technicalities
In this Appendix we explain the technique to compute the anomalous Jacobian in section 4.2, say sTr Φ i 1 and sTr Γ a − 1 in eq.(4.6). A careless treatment of the chiral multiplet Φ i = (φ i , ψ i + ) seemingly tells us that One has to regularize the above super-trace by introducing regulators. To find a proper regulator, it is sufficient to look at the equation of motion of the superfield Φ i which enters the action S chiral , see eq.(2.10), We need to further act byD + to project the operator equation into the half chiral superspace, i.e.D After some algebra, we find Therefore, the super-trace eq.(B.1) is regularized as For trivial fields D and B, the above trace is surely zero. But now let us turn on a non-zero but constant D and B backgrounds. We have Therefore, putting M 2 → ∞, we arrive at or, in superspace, Similarly, for fermi multiplet Γ a − , we also imposeD + D + D −− upon Γ a − and find, Thus we regularize the super-trace of the fermi multiplet as sTr Γ a − 1 = lim log Z a (µ) (B.11) We further remark that, as a consistency check, given a complexified U(1) rotation of the chiral or fermi matter, e.g. For real α, such as the wave function renormalization or a scale transformation, the anomalous Jacobian only gives a correction to the D term, because Im J (α) cancels with the contribution fromΦ i . It simply signals that fermions do not contribute to the one-loop β-function. On the other hand, for imaginary α, it is equivalent to a chiral rotation. We see that J (α) and its conjugation only contribute to the flux B term, which gives us the correct chiral anomaly from the chiral fermions ψ i + (section 4.1).