Fermionic and bosonic mass deformations of N=4 SYM and their bulk supergravity dual

We examine the AdS-CFT dual of arbitrary (non)supersymmetric fermionic mass deformations of N=4 SYM, and investigate how the backreaction of the RR and NS-NS two-form potentials dual to the fermion masses contribute to Coulomb-branch potential of D3 branes, which we interpret as the bulk boson mass matrix. Using representation-theory and supergravity arguments we show that the fermion masses completely determine the trace of this matrix, and that on the other hand its traceless components have to be turned on as non-normalizable modes. Our result resolves the tension between the belief that the AdS bulk dual of the trace of the boson mass matrix (which is not a chiral operator) is a stringy excitation with dimension of order $(g_s N)^{1/4}$ and the existence of non-stringy supergravity flows describing theories where this trace is nonzero, by showing that the stringy mode does not parameterize the sum of the squares of the boson masses but rather its departure from the trace of the square of the fermion mass matrix. Hence, asymptotically-AdS flows can only describe holographically theories where the sums of the squares of the bosonic and fermionic masses are equal, which is consistent with the weakly-coupled result that only such theories can have a conformal UV fixed point.

term was guessed by using supersymmetry to complete the squares in the polarization potential. When the masses of the fermions in all the three chiral multiplets are equal, the value of this term was computed directly in supergravity by Freedman and Minahan [4] and found to be exactly the one guessed in [1].
Our main goal is to study the non-supersymmetric version of the Polchinski-Strassler story, and in particular to spell out a method to determine completely the D3-brane Coulomb branch potential (or the quadratic term in the polarization potential) for the N = 4 SYM theory deformed with a generic supersymmetry-breaking combination of fermion and boson masses. Many of the issues in the problem we are adressing have been touched upon in previous explorations, but when one tries to bring these pieces of the puzzle together one seems to run into contradictions. We will try to explain how these contradictions are resolved, and give a clear picture of what happens in the supergravity dual of the mass-deformed N = 4 theory.
As explained in [1], a fermion mass deformation of the N = 4 SYM field theory, λ i M ij λ j , corresponds in the bulk to a combination of RR and NSNS three-form field strengths with legs orthogonal to the directions of the field theory, that transforms in the 10 of the SU (4) Rsymmetry group. The complex conjugate of the fermion mass, M † , corresponds to the complex conjugate combination transforming in the 10. Since the dimension of these fields is 3, the normalizable and non-normalizable modes dual to them behave asymptotically as r −3 and r −1 .
The boson mass deformation in the field theory, φ a M ab φ b , can be decomposed into a term proportional to the trace of M, which is a singlet under the SU (4) ≃ SO(6) R-symmetry, and a symmetric traceless mass operator, which has dimension 2 and transforms in the 20 ′ of SO (6). The traceless mass operator in the 20 ′ corresponds in the AdS 5 × S 5 bulk dual to a deformation of the metric, dilaton and the RR four-form potential that is an L = 2 mode on the five-sphere, and whose normalizable and non-normalizable asymptotic behaviors are r −2 and r −2 log r [5]. On the other hand, the dimension of the trace operator is not protected, and hence, according to the standard lore, turning on this operator in the boundary theory does not correspond to deforming AdS 5 × S 5 with a supergravity field 1 , but rather with a stringy operator [6]. The anomalous dimension of this operator at strong coupling has consequently been argued to be of order (g s N ) 1/4 .
On the other hand, there exist quite a few supergravity flows dual to field theories in which the sum of the squares of the masses of the bosons are not zero [7][8][9][10][11][12][13][14][15], and none of these solutions has any stringy mode turned on, which seems to contradict the standard lore above. In this paper we would like to argue that the solution to this puzzle comes from the fact that the backreaction of the bulk fields dual to the fermions determines completely the singlet piece in the quadratic term of the Coulomb branch potential of a probe D3-brane. Therefore, the trace of the boson mass matrix that one reads off from the bulk will always be equal to the trace of the square of the fermion mass matrix.
This, in turn, indicates that in the presence of fermion masses, the stringy operator is not dual to the sum of the squares of the boson masses, but to the difference between it and the sum of the squares of the fermion masses. Mass deformations of the N = 4 theory where the supertrace of the square of the masses is zero can therefore be described holographically by asymptotically-AdS supergravity solutions [7][8][9][10][11][12][13][14][15]. However, to describe theories where this supertrace is nonzero, one has to turn on "stringy" non-normalizable modes that correspond to dimension-(g s N ) 1/4 operators, which will destroy the AdS asymptotics.
To see this we begin by considering the backreaction of the three-form field strengths corresponding to fermion mass deformations on the metric, the dilaton and the four-form potential, which has been done explicitly for several particular choices of masses [4,16]. This backreaction can give several terms that modify the action of a probe D3 brane, giving rise to a Coulombbranch potential that is quadratic in the fermion masses and that transforms either in the 1 or in the 20 ′ of SO (6). Furthermore, one can independently turn on non-normalizable modes in the 20 ′ of SO(6) that correspond to deforming the Lagrangian with traceless boson bilinears, and that can also give rise to a Coulomb-branch potential. Since all these terms behave asymptotically as r −2 and transform in the same SO(6) representation, disentangling the contributions of the non-normalizable modes from the terms coming from the backreaction of the three-forms can be quite nontrivial. For example, in equation (62) in [1], the Coulomb-branch potential appears to contain both contributions in the 1 and in the 20 ′ of SO(6) coming from the backreaction of the fermion mass tensor T ijk , and to have no non-normalizable contribution.
We will show that the backreaction of the modes dual to the fermion masses can only source terms in the D3 brane Coulomb-branch potential that are singlets under SO (6), and hence the Coulomb-branch potential terms that transform in the 20 ′ of SO(6) can only come from nonnormalizable L = 2 (traceless) modes that one has to turn on separately from the fermion masses. Since the singlet term in the Coulomb-branch potential is the supergravity incarnation of the trace of the boson mass matrix, our result implies that in the bulk this boson mass trace is completely determined by the fermion masses: the sum of the squares of the boson masses will always be equal to the sum of the squares of the fermion masses.
Our calculation establishes that asymptotically-AdS 5 solutions can only be dual to theories in which the sum of the squares of the boson masses is the same as the sum of the squares of the fermion masses. Theories where these quantities are not equal cannot by described holographically by such solutions.
From a field theory perspective this interpretation is very natural: the solutions that are asymptotically AdS 5 can only be dual to field theories that have a UV conformal fixed point, and therefore their masses and coupling constants should not run logarithmically in the UV (their beta-functions should be zero). At one loop this cannot happen unless the sum of the squares of the boson masses is equal to the sum of the squares of the fermion masses [17], which reduces the degree of divergence in the corresponding Feynman diagram and makes the beta-functions vanish. 2 Thus in perturbative field theory one inputs boson and fermion masses, and one cannot obtain a UV conformal fixed point unless the sums of their squares are equal; in contrast, in holography one inputs an asymptotically-AdS solution (dual to a conformal fixed point) and the non-normalizable modes corresponding to fermion masses, and obtains automatically the sum of the squares of the boson masses.
This understanding of how the sum of the squares of the boson masses appears in AdS-CFT also clarifies some hitherto unexplained miraculous cancellations. In the Pilch-Warner dual of the N = 2 ⋆ theory [9], which from the N = 1 perspective has a massless chiral multiplet and two chiral multiplets with equal masses, the only non-normalizable modes that were turned on in the UV were those corresponding to the fermion masses M = diag(m, m, 0, 0) and to a traceless (L = 2) boson bilinear of the form m 2 3 (|φ 1 | 2 + |φ 2 | 2 − 2|φ 3 | 2 ). Since the latter contains some tachyonic pieces one could have expected the potential for the field φ 3 to be negative, but in the full solution this potential came out to be exactly zero. Using the new understanding developed in this paper it is clear that this "miraculous cancellation" happens because the backreaction of the fields dual to fermion masses gives a non-trivial contribution to the trace of the boson mass, of the form 2m 2 3 (|φ 1 | 2 + |φ 2 | 2 + |φ 3 | 2 ), and as a result the potential for φ 3 exactly cancels. One of the motivations for our work is the realization that the near-horizon regions of anti-branes in backgrounds with charges dissolved in fluxes have tachyonic instabilities [19,20]. From the point of view of the AdS throat sourced by the anti-branes, this tachyon comes from a particular L = 2 bosonic mass term that is determined by the gluing of this throat to the surrounding region. Understanding the interplay between this mass mode and the fluxes of the near-brane region is crucial if one is to determine whether the tachyonic throat has any chance of supporting metastable polarized brane configurations of the type considered in the KPV probe analysis [21]. Preliminary results of this investigation have already appeared in [22].
The paper is organized as follows. In Sections 2 and 3 we use group theory to find the bosonic potential, both the singlet and the 20 ′ pieces, arising from the square of the fermionic masses living in the 10 of SU (4). Although the group theory is well-known and most of Section 2 is a review, our final formulas in Section 3 are new, as only their supersymmetric versions have so far appeared in the literature. In Section 4 we explain how the bosonic masses appear in supergravity. This section contains the main observations of the paper. In Section 5 we recapitulate the main conclusions of our analysis and their relation to perturbative gauge theories. The appendix includes a summary of useful formulas for intertwining between SO(6) and SU (4) representations.

The Group Theory of the Mass Deformations
The goal of this section is to identify the SO(6) representation of the fermionic and bosonic mass deformations. We begin by reviewing in detail the group theory behind the mass deformations because this will play an important role in our discussion.

Fermionic masses
The most general non-supersymmetric fermionic mass deformation of N = 4 SYM is given by the operator: where λ i , i = 1, ..., 4 are the 4 Weyl fermions of the N = 4 theory, that in N = 1 language are the three fermions in chiral multiplets plus the gaugino. The mass matrix M is in the 10 of SU(4), which is the symmetric part of 4 × 4: As noted in [1], this matrix in the 10 of SU (4) ∼ = SO(6) can equivalently be encoded in an imaginary anti-self dual 3-form 4 T ABC . The map between them will be given in the next section.
In the language of N = 1, one distinguishes a U (1) R ⊂ SU (4) R that singles out the gaugino within the 4 fermions, or in other words the SU (4) R-symmetry group is broken as: corresponding to the splitting of the fundamental index 4 = 3 + 1 (i = {I, 4}). In this breaking, the fermionic mass matrix in the 10 decomposes as This corresponds to the breaking of M into the following pieces where m IJ ,m I andm are respectively in the 6, 3 and 1.

Bosonic Masses
A generic 6 × 6 bosonic mass matrix M 2 AB has 21 components, coming from the symmetric piece in If bosonic masses come from the backreaction of the fermion masses on the supergravity fields, M 2 should be of order M 2 . The most naive guess is that they are related to the hermitian matrix M M † , which involves the following SU(4) representations: From these very simple group-theory arguments one can immediately conclude that either our naive guess was too simple, or that the backreaction of the fermionic masses only generates the singlet (the trace) in the bosonic masses. However, since this goes against most people's intuition, particularly when there is some supersymmetry preserved, let us then push a bit further the possibility that our naive guess was wrong, or in other words that the bosonic masses are determined by fermionic ones, and see where it takes us.
The 20 ′ representation in (2.6), which is not in the product (2.7), appears instead in In terms of SU (4), the 20 ′ s is one of the three 20-dimensional representations whose Young tableau and Dynkin label are: There is an important caveat here: this representation is complex, and we therefore have to project out half of the components in order to get a real representation for the bosonic masses. As we will see in the next section this projection is directly related to the map between SU (4) and SO (6). A straightforward check that this representation is the one describing bosonic masses is to see what happens when N = 1 supersymmetry is preserved (m I =m = 0 in (2.5)). The bosonic mass matrix should then be proportional to mm † in (2.10) The 1 representation is the one we discussed above, while the 8 representation indeed appears in 20 ′ , with the right U (1) R charge, since for the breaking (2.3), we have [23]: From these group-theory arguments we conclude that if boson masses are generated by fermion masses at second order, then while the other 20 components of M 2 come from the product M M . Anticipating, we will see this map explicitly in the next section, from which we will conclude that only the former is true.

The explicit map between bosonic and fermionic mass matrices
In this section we will construct explicitly the maps (2.7) and (2.8), and the relationship between SU (4) and SO (6) representations. This will give the form of the possible terms in the supergravity fields that depend quadratically on fermion masses, which come from the backreaction of the fields dual to these masses. As shown in the previous section, the backreaction splits into two parts, corresponding to the 20 ′ and 1 representations.
To build a map between SU (4) and SO(6) one identifies the 6 a representation of SU (4) we have encountered above in (2.2) with the fundamental representation of SO (6). The former is given by a 4 × 4 antisymmetric matrix, ϕ T = −ϕ, that transforms as ϕ → U ϕU T under U ∈ SU (4). The complex 6 can be further decomposed into two real representations, 6 = 6 + + 6 − , by imposing the duality condition: where (⋆ϕ) ij = 1 2 ǫ ijkl ϕ kl . In what follows we will use the following parametrization of 6 + : where the Φ 1,2,3 are complex combinations of the six real scalars φ A=1,...,6 in the fundamental representation of SO(6). We choose conventions such that Φ I = φ I + i φ I+3 for I = 1, 2, 3. This parametrization is convenient as it makes explicit the 6 → 3 + 3 decomposition and the relation with the three chiral multiplets of N = 4. From (3.2) we find: where the six matrices G A are antisymmetric self-dual matrices (sometimes referred as 't Hooft symbols, or generalized Weyl matrices) which intertwine between SO(6) and SU (4), and whose form and explicit properties we give in Appendix A, and G A ij ≡ G A ji . An SU (4) rotation given by a matrix U is related to an SO(6) rotation by a matrix O via: 6 Note that the action of SO (6) is the same when U → −U , and so, as expected, SO(6) = SU (4)/Z 2 . With the help of t'Hooft matrices, we can work out the explicit map between the fermion mass matrix M ij and an anti-self dual 3-form T ABC . We get 3) to verify that T ABC is indeed an anti-self-dual three-form.
In terms of the 3-form T , the different representations correspond to the following components: 7 where J IJ is the symplectic structure associated to the SU (3) group. In our conventions it is just J 11 = J 22 = J 33 = i. Let us now discuss the bosonic masses, in the 20 ′ + 1 representations of SO(6). In terms of SU (4), the 20 ′ representation is labelled by four indices and from its Young tableau (2.9) we learn that: Furthermore, the zero-trace condition ǫ ijkl B ij,kl = 0 (3.8) eliminates the singlet leaving only 20 ′ from 20 ′ ⊕ 1. Following our discussion we can decompose this complex SU (4) representation into two real SO(6) representations, 20 ′ C = 20 ′ + + 20 ′ − . This is achieved by requiring: and we will use in this paper the choice 20 ′ + . The explicit map between the 20 ′ representations of SU (4) and SO(6) then works very similarly to (3.3): (3.10) 6 The SO(6) indices are raised with δ AB . 7 The primitive 6 and non-primitive3 pieces of a 3-form G are obtained as follows It is straightforward to verify that V AB 20 ′ is symmetric and real when B ij,kl satisfies (3.7) and (3.9) with the upper sign. Moreover, by using the fact that the 't Hooft matrices satisfy (A.2), one can see that the tracelessness of V AB 20 ′ is guaranteed by (3.8). Now, given a fermionic mass matrix M , one can build the following matrix in the 20 ′ + : Here the first term is dictated by the Young tableau (2.9) and the second guarantees (3.9) with the 20 ′ + choice. Furthermore, it is by construction traceless. One can add a trace to this, which, as discussed, should be built from M M † . We define which in turn, using the properties listed in the Appendix, implies that: To summarize, the most general bosonic mass matrix produced by the backreaction of the fermionic masses is V AB quad. , given by some linear combination of the 20 ′ and 1 contributions, V AB 20 ′ and V AB 1 . The latter is related to the fermion masses as in (3.13), while the former is determined by (3.10) with (3.11). Out of this we can build a scalar φ A V AB quad. φ B , or identifying the scalars φ A with some local coordinates on the six-dimensional space x A we get the "potentials" (3.14) Let us now examine the form of these potentials for the simple example of a diagonal fermionic mass matrix: which yields It is not hard to see that when the fourth fermionic mass is zero, and hence N = 1 supersymmetry is preserved, there is no combination of these two terms that can yield the N = 1 ⋆ supersymmetric bosonic mass potential Hence, the bosonic mass matrix cannot be fully determined by the fermion mass matrix.

The mass deformation from supergravity
In this section, we will discuss how to get the bulk boson masses from the dual supergravity solution given by the full backreaction of the dual of the fermion masses on AdS 5 × S 5 . The fully backreacted ten-dimensional (Einstein frame) metric is generically of the form with the RR four-form potential along space-time a dilaton φ and some internal 3-form fluxes that are usually combined into the complex form where τ = C − ie −φ is the combination of RR axion and dilaton. As explained in the Introduction, and as can be seen from the explicit flow solutions corresponding to mass deformations of N = 4 theory that have been constructed explicitly [7][8][9] the boson masses can be read off from the quadratic terms in the D3 Coulomb-branch potential, given by: where the warp factor and four-form potential are those of the fully backreacted solution. This computation is quite complicated for generic fermion masses, and was only obtained for some special choices, corresponding to the equal-mass N = 1 ⋆ theory ( M = diag(m, m, m, 0)) [4] and the supersymmetry-breaking-SO(4)-invariant N = 0 ⋆ theory (M = diag(m, m, m, m)) [16]. We will see how much of the quadratic term of V we can infer from these examples and from our group-theoretic arguments in the previous sections. On the gravity side the fermionic mass deformation corresponds to the non-normalizable modes of the complex 3-form flux G 3 [8], [1]. As we argued in the previous sections, the 10 representation of the SU(4) fermion mass matrix M ij is equivalent to the 10 of SO(6) corresponding to imaginary anti-self-dual 3-forms. At first order in the mass perturbation the supergravity equations of motion are satisfied if the imaginary anti-self-dual 3-form e 4A (⋆ 6 G 3 − iG 3 ) is closed and co-closed. One option is to set this to zero, i.e. to have G 3 be purely in the 10 (imaginary self-dual), but this solution does not correspond to the dual of the N = 1 ⋆ gauge theory. 8 The three form flux has therefore both 10 and 10 components, and has the r −1 behavior of a nonnormalizable mode dual to the ∆ = 3 operator corresponding to the fermion masses. It is given by: where c is a constant, T 3 is the imaginary anti-self-dual 3-form corresponding to the fermion masses, Eq. (3.5), and V 3 is constructed from T 3 and combinations of the vector x A , and it has both 10 and 10 components: At second order (quadratic in the fermionic masses) one has to solve for the dilaton, the metric and the 4-form potential, whose equations of motion depend quadratically on G 3 , and this was only done for the special mass deformations discussed above [4,16]; for supersymmetric unequal masses only the solution for the dilaton-axion is known [24]. Here we will not need the details of these solutions, but we note a few key points from which we will draw our conclusions.
The EOMs for the dilaton, warp factor and four-form potential have schematically the following structure: ∇ · ∇ (Bosonic fields) = (3-form Fluxes) 2 , (4.7) Since the fluxes are known, a general solution for the bosonic fields has inhomogeneous and homogeneous parts. For fluctuations around AdS 5 × S 5 , the homogenous part is a combination of harmonics of the sphere with different fall-offs in r. The quadratic term in (4.4) comes from modes with a r −2 fall off (the background warp factor e 4A 0 ∼ r 4 ), or in other words from modes which are dual to an operator of dimension ∆ = 2. Only the 20 ′ representation in the combination of metric and four-form potential that is relevant to compute (4.4) has this behavior [5]. It corresponds to the second harmonic on the five-sphere, and was referred in [1] as the L = 2 mode.
The inhomogeneous piece is sourced by quadratic combinations of the three-form fluxes, which transform in (2.7) and (2.8). Out of these, only the 1 and 20 ′ contribute to the masses of the bosons. The corresponding pieces in the fields that give rise to these masses can then be schematically represented as: where the first two terms in each line correspond to inhomogeneous solutions, whose dependence on the fermionic mass we computed in the previous section (equation (3.16) for a diagonal mass matrix), and the last term is the contribution from the homogeneous solution whose angular dependence, is determined by 20 free parameters µ 20 ′ AB , that have the dimension of mass squared. 9 It is important to note that, unlike the components of V 20 ′ , the components of U 20 ′ are not related in any direct way to the fermionic masses M ij , but are determined in a given configuration by IR and UV boundary conditions.
With the solution for the metric and the 4-form potential at hand, one can compute the boson masses directly in supergravity, through (4.4). If one works in Einstein frame, this requires only the combination of warp factor and four-form potential Φ − = e 4A − α, whose equation of motion has a right-hand side of the form (see (2.30) of [26]): where the . . . stand for the terms that are higher order in the mass deformation, and in the last step we have used (4.5) together with the duality properties ⋆ 6 T 3 = −iT 3 and The crucial observation is that V 3 drops out of the equation. The remaining piece, |T 3 | 2 , has no x-dependence and as a result is proportional to the singlet of the 10 × 10 product. We see that out of the 20 ′ and the 1 parts in the inhomogeneous solution (4.8), only the latter contributes to the Φ − equation 10 . Furthermore, as we already mentioned, Φ − unambiguously determines the r 2 part of the potential. We therefore conclude that the quadratic piece in the bosonic potential is necessarily of the form: V quad.
We emphasize once more that the 20 coefficients µ 20 ′ AB in U 20 ′ are added "by hand" and are fixed only by the boundary conditions. Furthermore, for the N = 1 ⋆ theory (m 4 = 0) we know that this contribution has to be non-zero when the three masses of the chiral multiplets are different. This is obvious from the form of the N = 1 ⋆ bosonic potential in (3.17), which has terms coming from both the 1 (trace) and the 20 ′ representations. Therefore, the solution dual to this theory must contain non-normalizable L = 2 modes.
We close this section by a short summary: when considering the supergravity dual of the mass-deformed N = 4 theory, the backreaction of the fields dual to the fermion masses gives rise to perturbations in the dilaton, metric and 5-form flux proportional to m 2 f , but these conspire to yield an overall zero contribution to the traceless part of the quadratic term of the polarization potential. That term therefore can arise only from the homogeneous traceless L = 2 modes that we referred to as U 20 ′ . This implies that in order to construct the supergravity dual of, say, N = 1 ⋆ SYM theory one has to add "by hand" proper homogeneous 20 ′ UV modes in order to ensure that the bosonic masses will match the fermionic ones. As explained in the Introduction, this result establishes that only theories where the supertrace of the mass squared is zero can be described holographically by asymptotically-AdS solutions. The sum of the squares of the boson masses, which is an unprotected operator (also known as the Konishi) and has been argued to be dual to a stringy mode of dimension (g s N ) 1/4 , can be in fact turned on without turning on stringy corrections, as one could have anticipated from the solutions of [7][8][9]. In the presence of fermion masses, what is dual to a stringy mode is not therefore the sum of the squares of the boson masses, but rather the mass super-trace (the difference between the sums of the squares of the fermion masses and the boson masses). Theories where this supertrace is zero can be described without stringy modes, but to describe theories where this supertrace is nonzero, one has to turn on "stringy" non-normalizable modes which destroy the AdS asymptotics.

The trace of the bosonic and fermionic mass matrices
One can also see the relation between this zero-supertrace condition and the existence of an asymptotically-AdS holographic dual from the dual gauge theory. Indeed, in a gauge theory where supersymmetry is broken by adding bosonic masses, there are no quadratic divergences, and the explicit breaking of supersymmetry is called soft. There are other soft supersymmetrybreaking terms that one can add to an N = 1 Lagrangian, such as gaugino massesm, and trilinear bosonic couplings of the form Similar to the quadratic terms discussed in the previous section, the bosonic cubic terms can also be read off by considering the action of probe D3 branes. They are proportional to the (3,0) and (2,1) imaginary anti-self-dual piece of the three form flux T 3 [2], which in turn are determined by the supersymmetry breaking fermionic massesm I ,m as in (3.6) [28]. One gets 11 Armed with this knowledge, one can compute the one-loop beta functions for all the coupling constants including the "non-standard soft supersymmetry breaking" termsm [30]. If one uses the relation between the soft trilinear terms and the fermion masses (5.3) we find that all the one loop beta functions except the one for the boson masses vanish exactly [17]. The one-loop beta functions for the boson mass trace vanishes if and only if the trace of the boson masses is equal to that of the fermions at tree level, which is precisely what happens for the N = 0 ⋆ theories that have an asymptotically-AdS supergravity dual (Eq. (5.1)), and also for any gauge theory that has a UV conformal fixed point (such as the ones found on D3 branes at singularities). Since the masses do not run with the scale, this is consistent with the fact that this theory has a UV conformal fixed point. Interestingly enough, the two-loop beta functions [31,32] also vanish under the same condition [17].
Hence, the field theory computation of the one and two-loop beta functions confirms the results of our holographic analysis: Asymptotically-AdS solutions are dual to theories with UV conformal fixed points, and if one turns on the fermion masses, the sum of the squares of the boson masses is automatically determined to be equal to the sum of the squares of the fermion masses. Conversely, in perturbative field theory one can turn on arbitrary boson and fermion masses, but for a generic choice of masses the beta-functions will be non-zero and the theory will not have a UV conformal fixed point. These beta-functions only vanish when the sums of the squares of the fermion and boson masses are equal. We can graphically summarize this as two equivalent statements:

A 't Hooft symbols
The explicit form of the 't Hooft matrices G A ij is Here σ 1,2,3 are the standard Pauli matrices and σ 0 is the 2 × 2 unit matrix. These matrices satisfy the following basis independent properties: and