Lattice Formulation of ${\cal N} = 2^*$ Yang-Mills

We formulate ${\cal N} = 2^*$ supersymmetric Yang-Mills theory on a Euclidean spacetime lattice using the method of topological twisting. The lattice formulation preserves one scalar supersymmetry charge at finite lattice spacing. The lattice theory is also local, gauge invariant and free from doublers. We can use the lattice formulation of ${\cal N} = 2^*$ supersymmetric Yang-Mills to study finite temperature nonperturbative sectors of the theory and thus validate the gauge-gravity duality conjecture in a nonconformal theory.


Introduction
Supersymmetric quantum field theories are interesting classes of theories by themselves. They can also be used to construct many phenomenologically relevant models such as Minimal Supersymmetric Standard Model (MSSM). Supersymmetric quantum field theories exhibit many interesting features when they are strongly coupled. It is in general difficult to study analytically the strong coupling regimes of supersymmetric quantum field theories. If we could formulate such theories on a spacetime lattice, in a consistent manner, we would have a first principles definition of the theory that can be used to study their nonperturbative sectors. Certain classes of supersymmetric field theories can be formulated on a spacetime lattice by preserving a subset of the supersymmetry charges. These approaches are based on the methods of topological twisting [1] and orbifolding [2] and they can be used to formulate lattice theories with extended supersymmetries.
In this work we provide the lattice construction of a very interesting theory, which is known as the four-dimensional N = 2 * super Yang-Mills theory [20]. This non-conformal field theory is obtained by giving mass to two of the chiral multiplets of four-dimensional N = 4 SYM theory. The N = 2 * SYM theory also takes part in the AdS/CFT correspondence. Its gravitational dual has been constructed by Pilch and Warner [21].
We use the method of topological twisting to construct N = 2 * SYM on a Euclidean spacetime lattice. The continuum twisted N = 2 * SYM theory is obtained by introducing mass deformation terms to the Vafa-Witten twisted N = 4 SYM theory [36]. Once we have a twisted version of N = 2 * theory in the continuum it is straightforward to implement the theory on the lattice. We use the discretization prescription provided by Sugino [3]. The lattice formulation preserves one supersymmetry charge at finite lattice spacing. The lattice construction is also local, gauge invariant and free from the problem of fermion doublers.
One could use the lattice construction of N = 2 * SYM to explore the nonperturbative regimes of the theory, including its thermodynamic properties, and compare with the results obtained from the dual gravitational theory.

Four-dimensional N = 4 Yang-Mills
We consider N = 4 supersymmetric Yang-Mills theory (SYM) on flat R 4 . In the language of N = 1 superfields N = 4 SYM theory contains one vector multiplet and three adjoint chiral multiplets. We denote them as superfields V and Φ s , with s = 1, 2, 3.
The scalars can be packaged into an antisymmetric and self-conjugate tensor φ uv with u, v = 1, 2, 3, 4 representing the indices of the fundamental representation of SU (4). In this notation the gauginos of vector and chiral multiplets can be combined: λ uα , λ uα . All fields of the theory take values in the adjoint representation of the gauge group. Here we take the gauge group to be SU (N ). We use anti-hermitian basis for the generators of the gauge group, with the normalization Tr (T a T b ) = −δ ab .

Mass Deformation and N = 2 * Yang-Mills
We can combine the superfield V and one of the adjoint chiral superfields say, Φ 3 to form an N = 2 vector multiplet. The chiral superfields Φ 1 and Φ 2 can be combined to form an N = 2 hypermultiplet. The N = 2 * SYM theory is a one-parameter (real) mass deformation of N = 4 SYM obtained by giving mass to the fields of N = 2 hypermultiplet. The mass terms softly break supersymmetry from N = 4 to N = 2. The N = 2 * SYM theory has a fixed point in the far UV, which is the conformal N = 4 SYM theory. The mass deformation is relevant and it induces running in the coupling, so that the theory becomes pure N = 2 SYM theory in the deep IR. On flat R 4 the mass deformation takes the following form in terms of the component fields [37,38] where m is the mass parameter and g the coupling constant of the theory. The mass deformation gives conventional mass terms for two Weyl fermions and two complex scalars and also tri-linear couplings between the N = 2 hypermultiplet scalars and the vector multiplet scalar φ 3 . Motivated by the supergravity dual geometry of N = 2 * Yang-Mills theory, it is convenient to write the mass deformation in terms of relevant operators in irreducible representations of the N = 4 R-symmetry group, SO(6) SU (4). There are two contributions that correspond to scalars in the bulk dual.
First, there is a dimension two bosonic operator O 2 : This operator is an element of the 20 of SO (6). It contributes the usual positive bosonic mass terms for the hypermultiplet scalars φ 1 and φ 2 . But it also destabilizes the scalar φ 3 belonging to the vector multiplet.
The second contribution O 3 is a dimension three fermionic operator. It introduces mass terms for the Weyl fermions in the hypermultiplet, in addition to tri-linear scalar terms and scalar mass terms. The operator O 3 is We also note that O 3 contains the Konishi operator, which is an SO(6) singlet This term is crucial since it cancels the negative potential energy for φ 3 introduced by operator O 2 .
Thus the action of the N = 2 * SYM theory can be expressed as [38,39] In general one could consider the case where the mass parameter is unequal for the two operators. In Ref. [40] the authors have explored the thermodynamics of the N = 2 * Yang-Mills plasma for a wide range of temperatures and for different mass deformations (m bosonic , m fermionic ). Supersymmetry is softly broken by the temperature and the unequal values of mass parameters in such cases.
The N = 2 * SYM theory has an SU (2) × U (1) R-symmetry. The symmetry breaking gives equal masses to two of the four Weyl fermions. The SU (2) acts on the two massless fermions, and the U (1) SO(2) mixes the two massive fermions. As m → 0, we recover N = 4 SYM theory. When m → ∞, the massive fields decouple from the theory and we end up with four-dimensional Yang-Mills with N = 2 supersymmetry.

Vafa-Witten Twist of N = SYM
We are interested in formulating N = 2 * Yang-Mills theory on a Euclidean spacetime lattice. We begin with N = 4 Yang-Mills theory on flat R 4 . We are interested in discretizing a twisted version of N = 2 * Yang-Mills theory.
Four-dimensional N = 4 Yang-Mills theory can be twisted in three inequivalent ways, giving rise to (i) half-twisted theory [41], (ii) Vafa-Witten theory (gauged four-dimensional A model) [36] and (iii) geometric Langlands twisted theory (gauged four-dimensional B model or Marcus twisted theory) [41,42]. When the theory is formulated on a flat manifold or in general on a hyper-Kahler manifold, the twisted theories coincide with the untwisted N = 4 SYM theory [36].
We are interested in the Vafa-Witten twist of the N = 4 Yang-Mills theory. For this particular twist, the internal symmetry group SU (4) is decomposed as SU (2) F × SU (2) I such that the twisted global symmetry group is where and SU (2) F remains as a residual internal symmetry group. The fields and supercharges of the untwisted theory are rewritten in terms of the twisted fields. After performing the twist, the fields of N = 4 SYM decompose in the following way [43] with i, j = 1, 2 representing the indices of the residual internal rotation group SU (2) F ; ϕ ij is a symmetric tensor. The fields χ i µν and B µν are self-dual with respect to the Euclidean Lorentz indices.
The theory exhibits flat directions along which the fields φ, φ, C commute with each other. Such configurations are given by diagonal matrices up to gauge transformations, in general.
The subset of the twisted fields (A µ , φ, φ, η, ψ µ , χ µν ) can be readily recognized as the twisted vector multiplet of the four-dimensional N = 2 SYM theory (Donaldson-Witten theory) [1]. The twisted theory contains an N = 2 hypermultiplet with the field content (C, B µν , ζ, χ µ , ψ µν ). We make this hypermultiplet massive when we construct the twisted N = 2 * SYM theory. A mass deformed version of Vafa-Witten twisted theory was constructed in Ref. [44] but this does not correspond to the N = 2 * SYM theory.

Q Supersymmetry Transformations
The twisting procedure gives rise to the following twisted supercharges: two scalars (Q, Q), two vectors (Q µ , Q µ ) and two self-dual tensors (Q µν , Q µν ). All twisted supercharges leave the twisted N = 4 SYM action invariant.
We are interested in the scalar supercharges Q and Q. The twisted theory is invariant under the Cartan subgroup of SU (2) F . We can define a conserved charge in the theory. We call it the U (1) R charge. In the topological field theory language it is known as the ghost number.
field U (1) R charge dimension nature The scalar supercharges Q and Q have opposite U (1) R charges. In Table. 1 we provide the U (1) R charges, canonical dimensions and nature (even or odd) of the twisted fields of the N = 4 Yang Mills.
Introducing two auxiliary fields, a vector field H µ and a self-dual tensor H µν , the offshell action of the Q supercharge on the twisted fields takes the following form We define the field strength F µν and covariant derivative D µ the following way We note that the Q supercharge satisfies the following algebra for a generic field X.

Q Supersymmetry Transformations
We can easily obtain the Q transformations by exchanging the two scalar supercharges Q and Q, and exchanging the fields The off-shell Q transformations on the twisted fields are The Q supercharge satisfies the following algebra for a generic field X.

Twisted Action
We can obtain the twisted action of the N = 4 theory through successive variations of Q and Q on a functional F known as the action potential [37,43]. We have the twisted action where the action potential The Vafa-Witten twisted action can be written as the Q variation of a gauge fermion Ψ (which in turn is the Q variation of F) with Ψ taking the form In order to obtain the required gauge fermion from the action potential F we need to redefine the auxiliary fields [36] while maintaining the relation Eq. (5.11). This leads to the following off-shell Q transformations These transformations indeed respect the algebra given in Eq. (5.11).
Applying Q variation on the gauge fermion we obtain the twisted N = 4 SYM action The twisted action given above has net U (1) R charge zero. We also note that the action is invariant under the discrete field transformations given in Eq. (4.14).

N = 2 * SYM Using Twisted Fields
Once we know the transformations from the untwisted fields to twisted fields it is easy to write down the action of the N = 2 * SYM theory in the twisted language. The N = 2 * SYM theory is obtained by giving masses to the N = 2 hypermultiplet fields (C, B µν , ζ, χ µ , ψ µν ).
We can rewrite the mass terms given in Eq. (3.1) using the twisted fields. From Eq. (3.1) we have In the bosonic sector, the components of the untwisted fields are related to that of the twisted fields the following way [37,43] (Note that we use anti-hermitian basis for SU (N ) generators.) Substituting the twisted field variables we obtain the bosonic mass terms and the trilinear coupling terms In the fermionic sector we have the following relations between the twisted and untwisted field variables [37,43]. (We use the conventions given in Ref. [45] for Euclidean spinors.) From the above relations we can write down the fermionic mass terms in the language of twisted fields. (We derive this in Appendix B.) We have the mass terms Having expressed the mass deformation terms using twisted variables it is now straightforward to write down the twisted action of the N = 2 * SYM. We have 6) where S N =4 is given in Eq. (4.22) and S m has the form From the above form of the twisted N = 2 * SYM action we note the following: There are mass terms with U (1) R charge −2, 0 and +2.
(ii.) There are mass terms that are not invariant under twisted Lorentz symmetry.
(iii.) The piece S m breaks the exchange symmetry under Q ↔ Q. This is expected since we have given mass to only one of the N = 2 hypermultiplets.

Mass-dependent Q and Q Transformations
We would like to write down the N = 2 * SYM action in a Q-exact form, with an appropriate gauge fermion. In order to achieve this we need to modify the Q and Q transformations on the twisted fields in a mass dependent way. Let us define a modified supercharge Q (m) . It acts on the twisted fields the following way These transformations respect the following modified algebra for a generic field X; with α = 1 for the fields (η, ψ µ , C, H µ ), α = −1 for the fields (χ µν , B µν ) and α = 0 for the rest of the fields. Similarly, we modify the off-shell Q transformations in a mass dependent way. Defining the modified supercharge Q (m) we have the following transformations The Q (m) supercharge satisfies the following algebra for a generic field X; with α = 1 for the fields ζ, χ µ , ψ µν , C, H µ , B µν and α = 0 for the rest of the fields. It would be interesting to see if the deformation part of the algebra represents rotation by an R-symmetry generator. Similar topics were considered in Ref. [46] by Hanada, Matsuura and Sugino and they were extended to various cases by Kato, Kondo and Miyake in Ref. [44]. It would be interesting to find the structure of the mass deformed supersymmetry algebra in this case.
In order to derive the twisted Lorentz non-invariant part of the N = 2 * SYM action let us consider linear combinations of the massive fields. The Q (m) transformations give We can now obtain the N = 2 * SYM action as a Q (m) variation of the following modified gauge fermion where 14) We derive the Q (m) transformations of the gauge fermion components V, W, Y and T in Appendix C. We also note that the terms W, Y and T contain the rotated fields that give appropriate twisted Lorentz non-invariant mass terms of the theory.
It is straightforward to show that the Q (m) variation of Ψ (m) will produce the twisted action of N = 2 * SYM S N =2 * = 1 g 2ˆd 4 x Q (m) Ψ (m) . 6 Lattice Formulation

Balanced Topological Field Theory Form
We can rewrite the Vafa-Witten twisted N = 4 SYM theory in a form known as the balanced topological field theory (BTFT) form. The existence of two scalar supercharges Q and Q would allow us to express the N = 4 theory in this form. In Ref. [47] Dijkgraf and Moore wrote down the BTFT form of the Vafa-Witten twisted theory. Sugino has used this approach to formulate four-dimensional N = 4 and N = 2 SYM theories on the lattice [3]. We can define a three component vector Φ, which is a function of the field strength. The components of this vector take the form The action potential takes the following form in the BTFT notation and the Q (m) transformations take the form The action of twisted N = 2 * SYM can again be written as Q (m) variation of gauge fermion

Lattice Regularized Theory
We formulate the theory on a four-dimensional hypercubic lattice following the discretization prescription given by Sugino [3]. In the lattice theory the gauge fields A µ are promoted to compact unitary variables living on the links (n, n + µ) and (n, n − µ), respectively. All other variables are distributed on the sites. Upon using the language of BTFT form we have the Q (m) transformations on the lattice These transformations were originally proposed by Sugino [3], for the m = 0 case, while formulating the N = 4 and N = 2 SYM theories on the lattice.
In the above transformations, D (+) µ is the forward covariant difference operator and D (−) µ represents the backward difference operator The Q (m) transformations reduce to their continuum counterparts in the limit of vanishing lattice spacing. The term quadratic in ψ µ is suppressed by additional power of the lattice spacing. Q (m) 2 on the lattice obeys a relation similar to the one given in the continuum.
Once we have the Q (m) transformation rule closed among lattice variables, it is almost straightforward to construct the lattice action.
The functional Φ A takes the following form on the lattice [3] The plaquette variables U µν (x) are defined as We can integrate out the auxiliary field H(n) so that the Φ(n) 2 term gives the gauge kinetic term on the lattice (6.12) We note that there are also additional terms appearing in Φ(n) 2 as cross terms. They become topological (total derivative) terms in the continuum limit however, we should keep them at the lattice level. The gauge terms in the continuum are F µν + F µν 2 rather than conventional F 2 µν . The vacua in the continuum theory are instanton solutions (anti-self-dual field strengths) corresponding to Φ A = 0.
We note that the above term Eq. (6.12) contains double winding plaquette terms. On the other hand, the standard Wilson action has the form 1 2g 2 0 n µ<ν Tr 2 − U µν (n) − U νµ (n) , (6.13) which has a unique minimum U µν = I. The action obtained through discretizing the twisted theory this way has many classical vacua U µν = diag(±1, · · · , ±1), (6.14) up to gauge transformations, where any combinations of ±1, with '−1' appearing even times are allowed in the diagonal entries. We also note that in the case of G = SU (N ), in addition to Eq. (6.14) there also appear the center elements U µν = z k I N = exp(2πik/N ) diag(1, 1, · · · , 1), (k = 1, 2, · · · , N − 1), (6.15) as the minima. The existence of many classical vacua has some serious consequences. Since the diagonal entries can be taken freely for each plaquette, it results in a huge degeneracy of vacua with the number growing as exponential of the number of plaquettes. We need to add up contributions from all the minima in order to see the dynamics of the model. In this case, the ordinary weak field expansion around a single vacuum U µν = I cannot be justified. That is, we are unable to say anything about the continuum limit of the lattice theory without its nonperturbative investigations.
We could add a term proportional to the standard Wilson action to the lattice action in order to resolve the degeneracy where ρ is a parameter to be tuned. This term resolves the degeneracy with the split 4ρ/g 2 0 [3]. We note that this breaks the supersymmetry Q (m) , even though it justifies the expansion around the vacuum U µν = I.
On the lattice we have a lattice version of the anti-self-dual equations for the minima. A discussion about lattice anti-self-dual equations is lacking in the literature. Thus we are not completely sure about the vacuum structure of the theory. In particular we note that the answer to the following question has not been established: Is it enough to remove the unwanted vacua in Eqs. (6.14) and (6.15) in the four-dimensional theory? For additional degeneracy, due to the instantons that is already in the continuum, we do not have to remove such degeneracy on the lattice because it is physical. If any degeneracy of Φ A = 0 that has no counterpart in the continuum other than the type of Eqs. (6.14) and (6.15) we should care about that.
In any event, if we introduce the supersymmetry breaking term Eq. (6.16), the trivial vacuum is singled out and we can proceed.
The N = 2 * SYM action takes the following Q (m) -exact form on the lattice (6.17) with β L denoting the lattice coupling. It is possible to show that the lattice theory has no fermion doubling problem by following an analysis similar to the one given in Ref. [3].
We note that the lattice action of N = 2 * SYM formulated here is We also note that it would be possible to impose the admissibility condition [48] ||1 − U µν || < , (6.18) on each plaquette variable in order to solve the issues with vacuum degeneracy. We note that Eq. (6.18) resolves the degeneracy Eqs. (6.14) and (6.15) with keeping supersymmetry because the admissibility condition is imposed on the gauge fermion Ψ (m) of the Q (m) -exact action and it does not affect the Q (m) -exact structure. Ref. [49] discusses another method to avoid the vacuum degeneracy while keeping supersymmetry.
Although yet another vacua appear as discussed in Ref. [48], it is irrelevant to the discussion for the admissibility condition Eq. (6.18).
We do not know which value should be chosen for in Eq. (6.18) because we do not know the vacuum structure of Φ A = 0. The value of in the admissibility condition should be determined so as to exclude the unphysical vacua from in Eqs. (6.14) and (6.15).
For the case that the gauge field has no topologically nontrivial structure (zero Pontryagin index), we think that it would be enough to remove the degeneracy Eqs. (6.14) and (6.15) even in four dimensions and Eq. (6.18) would be available since the nontriviality from the lattice version of the instantons could be irrelevant. In the numerical simulation it would be a good starting point to try to simulate the lattice action with the boundary conditions of topologically trivial gauge fields and with the use of Eq. (6.16) or the supersymmetry preserving Eq. (6.18).

Conclusions
We have provided a Euclidean lattice construction of four-dimensional N = 2 * SYM that respects gauge invariance, locality, and supersymmetry invariance under one supercharge. The lattice formulation is also free from the fermion doubling problem. We have also provided the continuum twisted formulation of four-dimensional N = 2 * SYM starting from the Vafa-Witten twist of the N = 4 SYM theory. The lattice theory is obtained by transporting the twisted N = 2 * SYM theory on to the lattice. The advantage of twisting is that we can preserve a part of the supersymmetry algebra involving one of the scalar supercharges that results from twisting, on the lattice.
The nonperturbative construction of four-dimensional N = 2 * SYM discussed here can be used to simulate the theory at any finite value of the gauge coupling, mass parameter and number of colors. It would be interesting to simulate the lattice N = 2 * SYM theory and study the observables related to the AdS/CFT correspondence.
We note that there are many aspects of N = 2 * SYM which would be interesting to study on the lattice. In Ref. [39] it was discussed that the N → ∞ theory, which has a holographic dual, evidently has no thermal phase transition at any non-zero temperature. But for finite values of N , there should be a distinct low temperature phase. Seeing evidence of this from lattice gauge theory simulations, and gaining information about the N dependence of the transition would be interesting 1 .
A Gravitational Dual of N = 2 * SYM In this section we give a brief review of the gravitational dual of N = 2 * SYM theory, at zero and finite temperatures.

A.1 Zero Temperature
The holographic dual of N = 2 * SYM theory at zero temperature was constructed by Pilch and Warner [21]. The dual geometry is a warped product of a deformed AdS 5 and a deformed five-sphere. The deformed five-sphere is foliated by elongated three-spheres, whose SU (2) × U (1) isometry realizes geometrically the R-symmetry of the dual N = 2 * SYM theory. The "uplift" of the five-dimensional supergravity to ten dimensions was also successfully constructed by Pilch and warner. In the full ten-dimensional type IIB supergravity the two scalars are Kaluza-Klein modes, which deform the AdS 5 ×S 5 geometry dual to the N = 4 SYM theory. We can consider the dual theory as Einstein gravity coupled to two real supergravity scalars, which we denote as α and χ, in five dimensions. The holographic dual of N = 2 * SYM theory was well explored in Refs. [21,[50][51][52].
We can also interpret the above mentioned gravitational background as a dual description of N = 4 SYM theory perturbed by two relevant operators -a bosonic operator O 2 and a fermionic operator O 3 . The supergravity scalars can be interpreted as bosonic and fermionic deformations of the D3-brane geometry. According to the general framework of holographic renormalization group flows [53,54] the asymptotic boundary behavior of scalars α and χ contains information about the couplings and expectation values of the dual operators O 2 and O 3 in the boundary gauge theory.
The appropriate terms in the five-dimensional supergravity action, including the scalars α and χ, can be written as where the matter Lagrangian is with the potential P = g 2 1 16 determined by the superpotential The dimensionful gauged supergravity coupling is where L is the radius of the five-sphere, and the five-dimensional Newton's constant is From the action Eq. (A.1) we have the Einstein's equations and the equations for the scalars

A.2 Finite Temperature
The supergravity background geometry dual to finite temperature N = 2 * SYM theory was constructed by Buchel and Liu in Ref. [55]. When the temperature goes to zero this geometry becomes the Pilch-Warner geometry [21]. One can construct a map between finite temperature N = 2 * SYM theory parameters and the parameters of the dual non-extremal geometry [51,55]. There are three supergravity parameters uniquely determining a non-singular RG flow in the dual non-conformal gauge theory [40]. They are unambiguously related to the three physical parameters of the N = 2 * SYM theory: the temperature T , the bosonic mass m b and the fermionic mass m f . (Note that for the case of N = 2 * SYM theory we have strictly m ≡ m b = m f . It is still possible to consider the theory with unequal m b and m f . The resulting theory will of course break supersymmetry further.) In Ref. [40] the thermal Pilch-Warner flow was investigated near the boundary of the supergravity geometry. From the asymptotic expansions near the boundary it is possible to identify the conformal weight two supergravity scalar, defined as α ≡ log ρ, as dual to turning on a mass for the bosonic components of the N = 2 * hypermultiplet. The asymptotic expansion of ρ contains parameters ρ 11 and ρ 10 [40], which can be interpreted as the coefficients of its non-normalizable and normalizable modes, respectively. The conformal weight one supergravity scalar χ can be identified as dual to turning on a mass for the fermionic components of the N = 2 * hypermultiplet [40].
Once the potential P and the superpotential W are given, it is possible to consistently truncate the finite temperature supergravity system to a purely bosonic deformation. This corresponds to the choice χ = 0. However, it is inconsistent, beyond the linear approximation, to set the bosonic deformation to zero. That is, setting α = 0 while keeping a fermionic deformation.

A.3 Relating Supergravity and Gauge Theory Parameters
The relation between N = 2 * SYM theory and the supergravity parameters of the thermal Pilch-Warner geometry was established by Buchel, Peet and Polchinski in Ref. [51] and later by Buchel and Liu in Ref. [55].
Finite temperature softly breaks supersymmetry. Thus we could generalize the thermal N = 2 * SYM theory by allowing different masses, m b and m f , for the bosonic and fermionic components of the N = 2 * hypermultiplet. Note that it is only when m b = m f ≡ m and T = 0 we have N = 2 supersymmetry.
Turning on the bosonic and fermionic masses for the components of the N = 2 hypermultiplet sets a strong coupling scale Λ in the theory. In this case, we could expect two qualitatively different thermal phases of the gauge theory. It depends on on whether T Λ or T Λ. When T Λ, we expect that the thermodynamics to be qualitatively (and quantitatively when T /Λ → ∞) similar to that of the N = 4 SYM theory plasma. When T ∼ Λ and m f = 0, we expect an instability in the system. Turning on only the supergravity scalar α, that is, setting m b = 0 and m f = 0, corresponds to giving positive mass squared to four N = 4 SYM scalars (the bosonic components of the N = 2 hypermultiplet). At the same time, the remaining two N = 4 SYM scalars acquire a negative mass squared. That is, they are the tachyons at zero temperature. However, at high enough temperatures, the thermal corrections would come into effect and stabilize these tachyons. As the temperature is lowered, we expect the re-emergence of these tachyons. This is due to the fact that dynamical instabilities in thermal systems can manifest as thermodynamic instabilities. (See Ref. [56] for arguments leading to this.) It was argued in Ref. [57] that in general, thermodynamic instabilities are reflected to developing c 2 s < 0, where c s is the speed of sound waves in the thermal gauge theory plasma.

B Euclidean Spinor Conventions and Mass Terms
Following the conventions given in Ref. [45] we define the Euclidean Dirac spinors λ, λ using Weyl spinors λ iα , λ iα , with i = 1, 2 denoting the internal symmetry index and α,α = 1, 2 denoting the spinor indices We also have λ i α * = −λ iα and λ iα * = λ iα . With these conventions it is straightforward to show that the fermion mass terms Upon using the relations between the twisted and untwisted fermionic field variables we have the mass term −m Tr λ α 1 λ 2α = −mTr λ 1 1 λ 21 + λ 2 1 λ 22 Let us look at the following mass term Upon using with the Euclidean convention σ µ ≡ ( σ, iI), the fermion mass term becomes  The Q (m) variation of the quantity P ≡ Tr (χ 1 − iχ 2 )(H 1 + iH 2 ) will contain the mass term −(im/ √ 2)Tr (χ 1 χ 2 ). We have Similarly the linear combination R ≡ Tr (χ 3 + iχ 4 )(H 3 − iH 4 ) will contain the mass term (im/ Defining T ≡ 1 4 (P + R) we have This reproduces two of the terms in the N = 4 twisted SYM action and also two of the mass terms that appear in N = 2 * SYM action.
Let us now consider the Q (m) variation of the product of the terms A ≡ (ψ 12 − iψ 23 ) and B ≡ (B 12 + iB 23 ).