Nicola Maggiore
email: nicola.maggiore@ge.infn.it2mpicollo@selex-si.com

Marco Picollo

Let us twist again: N = 2 super-YangMills theory coupled to matter

Keywords: Topological Quantum Field Theories, Extended Supersymmetry, BRST Renormalization PACS Nos: 03.70.+k (theory of quantized fields), 11.10.Gh (renormalization), 11.30.Pb (supersymmetry), 12.60.Jv (supersymmetric models)

We give the twisted version of N=2 Super Yang-Mills theory coupled to matter, including quantum fields, supersymmetry transformations, action and algebraic structure. We show that the whole action, coupled to matter, can be written as the variation of a nilpotent operator, modulo field equations. An extended Slavnov-Taylor identity , collecting gauge symmetry and supersymmetry, is written, which allows to define the web of algebraic constraints, in view of the algebraic renormalization and of the extension of the algebraic proof of the non-renormalization theorems holding for N=2 SYM theory without matter.

Introduction

Before 1994, the reasons to discard N=2 Super Yang-Mills (SYM) theory prevailed over those for studying it. This is testified by the weak occurrence in the literature before then, of theories with extended supersymmetries.

Concerning phenomenology, the presence of the so called "mirror particles" eliminates every possible physical interest: fermions of opposite chirality, but in the same representation of the gauge group, unavoidably appear in the theory, which hence is not chiral, and consequently not realistic, if one wants to include the particles of the Standard Model [START_REF] West | Introduction to supersymmetry and supergravity[END_REF].

From the Quantum Field Theory (QFT) point of view, on the other hand, theories with extended supersymmetry represent a real challenge, as explained in [START_REF] Breitenlohner | Renormalization of supersymmetric Yang-Mills theories[END_REF].

In fact, while for N=1 SYM the superspace formalism based on unconstrained superfields allowed to perform the algebraic quantum extension of the theory [START_REF] Piguet | Renormalized supersymmetry. The perturbation theory of N=1 supersymmetric theories in flat space-time[END_REF], the superfield approach to theories with extended supersymmetry is troublesome for several reasons. N=2 supersymmetry can be realized by means of of N=1 superfields, but the necessary additional symmetry involving N=1 superfields in non-polynomial. On the other hand, the harmonic superspace approach [START_REF] Galperin | Harmonic Superspace[END_REF] is possible, but a regularization scheme preserving both supersymmetry and gauge invariance, to all orders of perturbation theory, is still lacking. Despite this, the most celebrated results concerning the good renormalization properties of theories with extended supersymmetry, in particular the vanishing of the βfunction above one loop, have been obtained in a superspace (N=1 and/or N=2) framework [START_REF] Grisaru | Supergraphity. 2. Manifestly covariant rules and higher loop finiteness[END_REF][START_REF] Howe | Miraculous ultraviolet cancellations in supersymmetry made manifest[END_REF][START_REF] Howe | A class of finite four-dimensional supersymmetric field theories[END_REF]. A review of these results, of the ways employed to get them and also of as the weaknesses of each of them, can be found in Chapter 18 of [START_REF] West | Introduction to supersymmetry and supergravity[END_REF] 1 .

The situation doesn't sound much better in components. The drawback of adopting the WZ gauge, is that the supersymmetry transformations are nonlinear, and the supersymmetry algebra does not close on translations, but two kinds of obstructions occur: field dependent gauge transformations and field equations of motion. This fact has two consequences: the difficulty of defining a gauge fixing term, which is invariant under both supersymmetry and BRS symmetry, and the need of an infinite number of external sources, with increasing negative mass dimensions, in order to control the algebra [START_REF] Breitenlohner | Renormalization of supersymmetric Yang-Mills theories[END_REF].

After the appearance of the celebrated Seiberg-Witten papers [START_REF] Seiberg | Monopole condensation and confinement In N=2 supersymmetric Yang-Mills theory[END_REF][START_REF] Seiberg | Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD[END_REF] on the electric-magnetic duality in N=2 SYM theory, which relates the weak and strong coupling regimes of that theory, the N=2 susy theories faced a kind of second youth, becoming extremely popular, and were massively reconsidered by the community.

Most of the problems described in [START_REF] Breitenlohner | Renormalization of supersymmetric Yang-Mills theories[END_REF] were solved, and the renormalizability of N=2 coupled to matter, by means of a non-anomalous Slavnov-Taylor identity, was rigorously established [START_REF] Maggiore | Algebraic renormalization of N=2 super Yang-Mills theories coupled to matter[END_REF][START_REF] Maggiore | Off-shell formulation of N=2 super Yang-Mills theories coupled to matter without auxiliary fields[END_REF], using a technique which has been successfully repeated since, and which we are adopting also in this paper for the classical definition of the theory (see Section 5).

More recently, the method of "shadow fields" has been introduced, which allowed to write a system of Slavnov-Taylor identities by means of which supersymmetric gauge field theories can be renormalized in a regularization independent way, permitting also to study the observables which are not scalar under supersymmetric transformations [START_REF] Baulieu | Shadow fields and local supersymmetric gauges[END_REF].

Other important goals have been reached exploiting the twist [START_REF] Witten | Topological quantum field theory[END_REF][START_REF] Witten | Supersymmetric Yang-Mills theory on a four manifold[END_REF]. Indeed, it was a known fact that N=2 SYM is related to topological field theories, in particular Topological Yang-Mills (TYM) theory, by means of a twist, which ultimately reduces to a linear redefinition of the quantum fields, to which the path integral defining the generating functionals is insensitive (we shall be more precise in Section 3). Consequently, the twisted-related theories are completely equivalent.

The bad news driven by this, is that the results concerning N=2,4 SYM theories can hardly be extended to the more realistic N=1 SYM theories, which are definitely not topological QFTs, having local degrees of freedom. This drawback is partially compensated by the fact that some important facts concerning N=2 SYM can be proved through their equivalent twisted version: TYM. This has been the case, for instance, for the theorem concerning the N=2 SYM β-function, whose finiteness above one loop has been algebraically demonstrated in [START_REF] Blasi | Perturbative beta function of N = 2 super Yang-Mills theories[END_REF] exploiting the existence of the twist.

Later, it has also been algebraically proved by means of the shadow technique that, chosen the matter hypermultiplet in order to have a vanishing β-function at one loop, it vanishes at all orders of perturbation theory [START_REF] Baulieu | Superconformal invariance from N=2 supersymmetry Ward identities[END_REF].

Moreover, a central role is played by the operator Tr φ 2 , which, in the Seiberg-Witten supersymmetric theory, is the gauge invariant quantity parametrizing the space of vacua of the theory, and, in the twisted topological theory, is the finite operator [START_REF] Maggiore | Protected operators in N = 2,4 supersymmetric theories[END_REF] by means of which the pure gauge theory can be defined [START_REF] Blasi | Perturbative beta function of N = 2 super Yang-Mills theories[END_REF][START_REF] Fucito | Algebraic renormalization: perturbative twisted considerations on topological Yang-Mills theory and on N = 2 supersymmetric gauge theories[END_REF]. So far the state of the art on which this paper stands. The program is not yet completely carried out: the non-renormalization theorems concerning the β-function are algebraically proved for the N=2 SYM, in absence of matter. We recall that matter is coupled to the pure gauge theory by means of the hypermultiplet [START_REF] West | Introduction to supersymmetry and supergravity[END_REF], in a generic representation of the gauge group. As a consequence of the second supersymmetry, the theory, even in presence of matter, has only one coupling constant. It is natural to ask which is the fate of the non-renormalization theorem concerning the unique β-function in presence of matter. Related to this, it is interesting to know if, as in the pure gauge case, the whole theory can be written in terms of a single operator, which is finite to all orders of perturbation theory, and, if the answer is positive, which this operator is. Finally, the inclusion of matter allows also for taking into account N=4 SYM, which can be reached from N=2 in the particular case of matter in the adjoint, rather than generic, representation of the gauge group.

The aim of this paper, is to contribute to answer these questions. The preliminary and necessary step is to give the complete twisted version of N=2 SYM coupled to matter, and to achieve the whole set up for its quantum extension (gauge fixing, BRS symmetry, Slavnov-Taylor identity, algebraic structure, etc.) [START_REF] Piguet | Algebraic renormalization: perturbative renormalization, symmetries and anomalies[END_REF].

The paper is organized as follows. In Section 2 we recall the basics of N=2 SYM theories, with and without matter. In Section 3 we introduce the twist for the pure gauge case. The main results of this paper are contained in Sections 4 and 5, where the twisted version of the whole theory, including matter, is given, as well as the basis for the quantum implementation, which relies on the extended Slavnov-Taylor identity and on the off shell closed algebra. Conclusions and perspectives are summarized in Section 6.

2 The untwisted theory: N=2 SYM coupled to matter

Pure N=2 SYM

The N=2 susy algebra reads

{Q i α , Q j α} = δ i j (σ µ ) α α∂ µ {Q i α , Q j β } = {Q i α, Q j β } = 0 , (2.1)
where (Q i α , Q j α) are the supersymmetry charges, indexed by i = 1, 2 and Weyl spinor indices α, α = 1, 2. The total number of supercharges is therefore eight.

The pure N=2 SYM theory is based on the Yang-Mills (YM) multiplet [START_REF] West | Introduction to supersymmetry and supergravity[END_REF], which belongs to the adjoint representation of the gauge group, and whose field components are (A µ , λ iα , λ i α, φ, φ), where A µ (x) is the gauge field, λ iα (x), λ i α(x) are two pairs of Weyl spinors, and φ(x), φ(x) are two scalars.

The corresponding pure N=2 SYM action reads

S Y M = 1 g 2 Tr d 4 x 1 2 F µν F µν -4λ iα σ µ α αD µ λ α i - 1 2 φD µ D µ φ -φ λ i α, λ i α + φ λ iα , λ iα - 1 32 φ, φ φ, φ , (2.2)
where the Trace Tr is done over the adjoint representation group.

The global symmetry group of the theory is

H = SU (2) L × SU (2) R × SU (2) I × U (1) , (2.3)
where SU (2) L × SU (2) R represents the Lorentz group, SU (2) I × U (1) is the internal symmetry group, SU (2) I referring to the supersymmetry index i = 1, 2 and U (1) being the rigid R-symmetry.

Correspondingly, the fields belonging to the N=2 YM multiplet are assigned the following H-group quantum numbers:

A µ : 1 2 , 1 2 , 0 0 λ iα : 1 2 , 0, 1 2 -1 λ i α : 0, 1 2 , 1 2 +1 (2.4)
φ : (0, 0, 0) +2 φ : (0, 0, 0) -2 ,

where we adopted the notation

(SU (2) L , SU (2) R , SU (2) I ) U (1) . (2.5)
For what concerns the supersymmetry generators, the quantum numbers are:

Q iα = 1 2 , 0, 1 2
+1 ; Q i α = 0, 1 2 , 1 2 -1
.

(2.6)

The supersymmetry transformations of the pure N=2 SYM fields are:

δA µ = - √ 2ξ αj (σ µ ) α αλ α j - √ 2ξ αj (σ µ ) α αλ α i δφ = -4 √ 2ξ αj λ j α δφ = -4 √ 2ξ αj λ jα (2.7)
δλ iκ = √ 2 8 ξ αj ε ακ ε ij φ, φ + 1 √ 2 ξ αj ε ij (σ µν ) κα F µν + 1 √ 2 ξ αj ε ij (σ µ ) κ αD µ φ δλ i κ = 1 √ 2 ξ αj ε ij (σ µ ) α κD µ φ + √ 2 8 ξ αj ε κ αε ji φ, φ + 1 √ 2 ξ αj ε ji (σ µν ) κ αF µν ,
where the operator δ collects the supercharges Q iα and

Q i α through δ = ξ αj Q jα + ξ αj Q j α.
Notice that in the Wess-Zumino gauge the supersymmetry transformations (2.7) are nonlinear.

The action (2.2) is susy invariant :

δS Y M = 0 . (2.8)

N=2 SYM coupled to matter

To couple pure N=2 SYM to matter, we need the matter hypermultiplet (q i , qi , ψ q , ψ q , ψ q, ψ q) [START_REF] West | Introduction to supersymmetry and supergravity[END_REF], formed by two pairs of scalar fields q i (x) and qi (x), two Weyl fermions ψ q (x) and ψ q(x) and their complex conjugates, all in a generic complex representation of the gauge group. The matter H-quantum numbers (2.5) are:

q i : (0, 0, 1 2 ) 0 qi : (0, 0, 1 2 ) 0 (ψ q ) α : ( 1 2 , 0, 0) +1 (2.9) (ψ q ) α : (0, 1 2 , 0) -1 (ψ q) α : ( 1 2 , 0, 0) +1 (ψ q) α : (0, 1 2 , 0) -1 .
The complete N=2 SYM action is:

S = S Y M + S matter , (2.10)
where S Y M is given by (2.2), and

S matter = 1 g 2 Tr m d 4 x 1 2 qi D µ D µ q i + 2q i λ i α(ψ q ) α -2q i λ i α(ψ q) α - 1 2 qi λ iα (ψ q ) α - 1 2 q i λ iα (ψ q) α + (ψ q) α (σ µ ) α αD µ (ψ q ) α -(ψ q) α(σ µ ) αα D µ (ψ q ) α + 1 8 (ψ q) α φ(ψ q ) α -2(ψ q) αφ(ψ q ) α + 1 16 qi φ, φ q i - 1 32 qi q i qj q j .
(2.11)

In the previous expression, Tr m is the Trace over the matter representation of the gauge group.

The (nonlinear) supersymmetry transformations of the matter fields are:

δq i = √ 2ε ji ξ αj (ψ q ) α + √ 2ε ji ξ αj (ψ q ) α δ qi = √ 2ε ji ξ αj (ψ q) α + √ 2ε ji ξ αj (ψ q) α δ(ψ q ) γ = √ 2ε γα ξ αj φq j + 1 √ 2 ξ αj (σ ν ) γ αD ν q j δ(ψ q ) γ = - 1 √ 2 ξ αj (σ ν ) α γ D ν q j - √ 2 16
ε γ αξ αj φq j (2.12)

δ(ψ q) γ = - √ 2ε γα ξ αj φq j + 1 √ 2 ξ αj (σ ν ) γ αD ν qj δ(ψ q) γ = - 1 √ 2 ξ αj (σ ν ) α γ D ν qj + √ 2 16
ε γ αξ αj φq j , and the matter action (2.11) is susy invariant:

δS matter = 0 , (2.13)
so that, finally, one has

δS = δ(S Y M + S matter ) = 0 . (2.14)
3 Introducing the twist: the pure N=2 SYM theory

As we said, the global symmetry group for N=2 SYM in four dimensions is given by H (2.3), and the total number of generators, including supersymmetry, are:

SU (2) L × SU (2) R Susy SU (2) I U (1) generators P µ (4) , M µν (6) Q iα (4) , Q i α(4) T i j (3) R(1)
The nonvanishing algebraic relations are

[M µν , M ρσ ] = -i(η µρ M νσ -η µσ M νρ -η νρ M µσ + η µσ M µρ ) [M µν , P ρ ] = i(η νρ P µ -η µρ P ν ) M µν , Q i α = -(σ µν ) β α Q i β {Q i α , Q j β } = 2σ µ α β P µ δ ij (3.1)
T j i , Q kα = - 1 2 (δ j k Q iα - 1 2 δ j i Q kα ) T j i , T l k = 1 2 (δ l i T j k -δ j k T l i ) [R, Q iα ] = Q iα ,
and their hermitian conjugates. Remember that the SU (2) I generators are traceless: T i i = 0, hence only three of them are independent. It is now convenient to rearrange the Lorentz and translations generators M µν and P µ as follows:

J αβ := 1 2 (σ µν ) αβ M µν ; J α β := 1 2 (σ µν ) α β M µν ; P α β := (σ µ ) α β P µ , (3.2)
exploiting the isomorphism ρ between the Minkowski space M 4 and the space H(2, C) of 2 × 2 hermitian matrices:

ρ : M 4 → H , ρ(x µ ) = x µ σ µ (3.3) ρ -1 : H → M 4 , ρ -1 (h) = 1 2 Tr [hσ µ ] . (3.4)
The twisting procedure, introduced by Witten in [START_REF] Witten | Topological quantum field theory[END_REF][START_REF] Witten | Supersymmetric Yang-Mills theory on a four manifold[END_REF], simply consists into a redefinition of the internal group indices i as lefthanded spinorial indices α:

i twist -→ α . (3.5)
This is possible thanks to the fact that both the spinorial indices {α, α} and the susy index i run from 1 to 2. The Lorentz group generators J αβ are correspondingly redefined through a linear combination J αβ with the SU (2) I internal group generators, which, after the twist, are written as T β α :

J αβ := J αβ + kT αβ , (3.6)
where k is a constant to be fixed by requiring that [J , J ] = [J, J]. Since both J and T are symmetric in (α, β), the same holds also for J . Notice that lefthandedness is a possibility, the twist defined through the identification of i and α being equally legitimate.

If SU (2) L is the group associated to the generators J αβ , the redefinition (3.6) corresponds to twisting the Lorentz group

SU (2) L × SU (2) R into SU (2) L × SU (2) R , where SU (2) L is the diagonal sum of SU (2) L and SU (2) I .
The new, twisted, global symmetry group H is

H twist -→ H = SU (2) L × SU (2) R × U (1) . (3.7)
The supersymmetry charges become:

Q iα twist -→ Q βα and Q i α twist -→ Q β α . (3.8)
The twisted supercharges under H transform as

Q βα = (0, 0) +1 ⊕ (1, 0) +1 and Q β α = ( 1 2 , 1 2 ) -1
, or, more explicitly, the four supercharges Q βα under the twist can be rearranged into a scalar δ W and an anti-selfdual tensor δ µν , while the other four Q β α become a vector operator δ µ :

Q βα twist -→ δ W := 1 √ 2 ε αβ Q βα ⊕ δ µν := 1 √ 2 (σ µν ) αβ Q βα Q β α twist -→ δ µ := 1 √ 2 Q β α(σ µ ) α β , (3.9)
and δ µν is selfdual

δ µν = δµν = 1 2 ε µνρσ δ ρσ . (3.10)
The subalgebra formed by the eight twisted supercharges δ W , δ µ , δ µν and the R symmetry, reads:

{δ W , δ W } = 2δ 2 W = 0 {δ W , δ µ } = ∂ µ {δ µ , δ ν } = 0 (3.11) {δ µ , δ ρσ } = -(ε µρσν ∂ ν + g µρ ∂ σ -g µσ ∂ ρ ) {δ W , δ µν } = 0 [R, δ W ] = +δ W [R, δ µ ] = -δ µ [R, δ µν ] = +δ µν ,
where g µν = diag(+, +, +, +) is the euclidean flat space metric.

A few remarks are in order:

1. The operator δ W , which coincides with the "fermionic symmetry" introduced by Witten in [START_REF] Witten | Topological quantum field theory[END_REF][START_REF] Witten | Supersymmetric Yang-Mills theory on a four manifold[END_REF], is nilpotent

δ 2 W = 0 . (3.12)
In the Wess Zumino gauge its realization is nonlinear, as we shall see, and δ W will turn out to be nilpotent modulo (field-dependent) gauge transformations and field equations, as usual in supersymmetry algebras.

2. The operators δ W and δ µ form a subalgebra which closes on translations. This is a common, and somehow defining, feature of topological models [START_REF] Birmingham | Topological field theory[END_REF], and remarkably suggests that the twist has deeply to do with topological quantum field theories and their algebraic structure.

In fact the common feature of all topological field theories, is the existence of three operators δ, δ µ , ∂ µ satisfying the following algebra [START_REF] Piguet | Algebraic renormalization: perturbative renormalization, symmetries and anomalies[END_REF]]

δ 2 = 0 , {δ, δ µ } = ∂ µ , {δ µ , δ ν } = 0 . (3.13)
In other words, it is not surprising at all that, twisting N=2 SYM, a topological quantum field theory is recovered.

3. The twist does not change the mass dimensions of the supersymmetry charges, which is 1 2 . The R-charge is +1 for δ W and δ µν , and -1 for the vector symmetry δ µ .

The following table summarizes the effect of the twist on the global

group, its generators and on the supersymmetry charges:

UNTWISTED TWISTED group SU (2) L SU (2) R SU (2) I SU (2) L SU (2) R generators J αβ (3) J α β (3) T ij (3) J αβ (3) J α β (3) Q iα 1/2 0 1/2 δ W 0 0 δ µν 1 0 Q i α 0 1/2 1/2 δ µ 1/2 1/2

Twisted fields

The fields belonging to the YM multiplet concerned by the twist are the fermionic fields λ iα (x) and λ i α(x), i.e. those carrying the internal supersymmetry index i, which, like the supercharges Q iα and Q i α, are twisted as follows

λ iα → λ βα ( 1 2 , 0, 1 2 ) -1 → η(0, 0) -1 ⊕ χ µν (1, 0) -1 (3.14) λ i α → λ β α(0, 1 2 , 1 2 ) +1 → ψ µ ( 1 2 , 1 2 ) +1 . (3.15)
The field λ(x) is twisted into a scalar field η(x) and an antiselfdual antisymmetric tensor χ µν (x), while λ(x) yields a vector field ψ µ (x):

λ βα → η := ε αβ λ [βα] ⊕ χ µν := 1 4 (σ µν ) αβ λ (βα) λ β α → ψ µ := λ β α(σ µ ) α β , (3.16)
with

χ µν = χµν = 1 2 ε µνρσ χ ρσ . (3.17)
The bi-spinor λ βα (x) is thus decomposed into its symmetric and antisymmetric part:

λ βα = 1 2 (λ [βα] + λ (βα)
). Summarizing, the effect of the twist on the fields of the YM multiplet is

(A µ , λ iα , λ i α, φ, φ) twist -→ (A µ , ψ µ , χ µν , η, φ, φ) , (3.18)
which, not by chance, coincides with the field content of the Donaldson-Witten topological QFT [START_REF] Witten | Topological quantum field theory[END_REF][START_REF] Donaldson | An Application of gauge theory to four-dimensional topology[END_REF].

Twisted action

The twisting procedure changes the action (2.2) accordingly. It is important to stress that the twist, as far as quantum fields are concerned, is simply a linear rearrangement, which does not modify the path integral defining the functional generators. The partition function is not affected by the twist, hence the two theories, the untwisted and the twisted one, are completely equivalent. This means, in particular, that the physical observables should be the same in the two theories, that the finiteness properties must be preserved, and that the susy invariance should reflect into invariance under the twisted operators δ W , δ µ and δ µν .

In order to verify this latter property, let us twist the pure N=2 SYM action (2.2):

S Y M twist -→ S T Y M = 1 g 2 Tr d 4 x 1 2 F + µν F +µν -χ µν (D µ ψ ν -D ν ψ µ ) + + ηD µ ψ µ - 1 2 φD µ D µ φ + 1 2 φ{ψ µ , ψ µ } (3.19) - 1 2 φ{χ µν , χ µν } - 1 8 [φ, η] η - 1 32 φ, φ φ, φ .
It is evident that the twisted N=2 SYM theory coincides with the Topological Yang-Mills (TYM) theory, as expected [START_REF] Witten | Topological quantum field theory[END_REF][START_REF] Witten | Supersymmetric Yang-Mills theory on a four manifold[END_REF].

Of course, under the gauge transformations As we already said, we expect that S T Y M keeps memory of susy invariance through its invariance under the twisted operators δ W , δ µ and δ µν . Notice that it would have been a very difficult task to identify a priori the symmetries of the TYM action (3.19). This, on the contrary, turns out to be quite natural thanks to the twisting procedure. Before verifying that the twisted operators are indeed symmetries of the TYM action, we have to write down their action on the twisted fields.

δ g A µ = -D µ (3.20) δ g λ = [ , λ] , λ = χ, ψ, η, φ, φ ,

Twisted supersymmetry transformations on twisted fields

Recalling the definition of the twisted operators (3.9) and of the twisted fields (3.16), after a little algebra, one gets δ W transformations on twisted fields

δ W A µ = ψ µ δ W ψ µ = -D µ φ δ W φ = 0 δ W χ µν = F + µν (3.22) δ W φ = 2η δ W η = 1 2 φ, φ
δ µ transformations on twisted fields

δ µ A ν = 1 2 χ µν + 1 8 g µν η δ µ ψ ν = F µν - 1 2 F + µν - 1 16 g µν φ, φ δ µ η = 1 2 D µ φ (3.23)
δ µ χ στ = 1 8 ε µστ ν D ν φ + g µσ D τ φ -g µτ D σ φ δ µ φ = -ψ µ δ µ φ = 0
δ µν transformations on twisted fields

δ µν A σ = -(ε µνστ ψ τ + g µσ ψ ν -g νσ ψ µ ) δ µν ψ σ = -(ε µνστ D τ φ + g µσ D ν φ -g νσ D µ φ) δ µν φ = 0 δ µν φ = 8χ µν (3.24) δ µν η = -4F + µν δ µν χ στ = 1 8 (ε µνστ + g µσ g ντ -g µτ g νσ ) φ, φ + F + µσ g ντ -F + νσ g µτ -F + µτ g νσ + F + ντ g µσ + ε α µνσ F + τ α -ε α µντ F + σα + ε α στ µ F + να -ε α στ ν F + µα .
The following tables summarize the quantum numbers of the twisted fields twisted fields

A µ χ µν ψ µ η φ φ dim. 1 3/2 3/2 3/2 1 1 R -charge. 0 -1 1 -1 2 -2 statistics
comm ant ant ant comm comm and of the twisted operators

twisted operators δ W δ µ δ µν dim. 1/2 1/2 1/2 R -charge. 1 1 -1 statistics ant ant ant
where comm and ant stand for commuting and anticommuting respectively.

Long but straightforward calculations confirm that, indeed, the twisted operators are symmetries of the twisted action:

δ W S T Y M = δ µ S T Y M = δ µν S T Y M = 0 . (3.25)
It is important to stress that the fermionic, nilpotent, Witten's δ W symmetry does not completely fix the coefficients of every term appearing in S T Y M . In other words, S T Y M is not the most general action invariant under δ W . In order to fix completely all the terms by means of a unique coupling constant, the role of the vector δ µ symmetry is crucial. On the other hand, the three δ µν twisted symmetries are automatically satisfied, therefore, under this respect, they seem to be redundant.

Twisted algebra

Let us see what becomes the twisted supersymmetry algebra in the Wess-Zumino gauge, where the symmetries are nonlinearly realized. The following algebraic relations hold:

δ 2 W = δ g φ + (field equations) , (3.26)
where δ g φ is a gauge transformation whose gauge parameter is the field φ(x). The operator δ W is therefore on shell nilpotent in the space of gauge invariant local functionals. The cohomology in this constrained functional space defines the so called Witten observables [START_REF] Witten | Topological quantum field theory[END_REF][START_REF] Witten | Supersymmetric Yang-Mills theory on a four manifold[END_REF];

{δ µ , δ ν } = - 1 8 g µν δ g φ + (field equations), (3.27)
where δ g φ is a field dependent gauge transformation, with the field φ(x) as gauge parameter;

{δ W , δ µ } = ∂ µ + δ g Aµ + (field equations), (3.28)
where δ g Aµ is a field dependent gauge transformation, with the field A µ (x) as gauge parameter.

Finally, the algebraic relations involving δ µν are: The above algebraic structure is typical of the supersymmetry in the Wess-Zumino gauge. Two kind of obstructions to the closure of the algebra on translations occur: field equations and field dependent gauge transformations. The canonical way to proceed (see, for instance, [START_REF] West | Introduction to supersymmetry and supergravity[END_REF]), is to take care of the first type of obstructions, namely the field equations, introducing auxiliary fields, whose transformations coincide with the field equations. Still, the other kind of obstruction, namely the field dependent gauge transformations, remains, and the algebra is open, needing an infinite number of external fields. This problem has been exhaustively treated in [START_REF] Breitenlohner | Renormalization of supersymmetric Yang-Mills theories[END_REF], where the non-renormalizability of theories with extended supersymmetry is discussed. The problem has nonetheless been solved, turning the situation the other way around [START_REF] Maggiore | Algebraic renormalization of N=2 super Yang-Mills theories coupled to matter[END_REF][START_REF] Maggiore | Off-shell formulation of N=2 super Yang-Mills theories coupled to matter without auxiliary fields[END_REF], as we shall see.

The twisted theory: N=2 SYM coupled to matter

Let us now apply the twisting procedure, described in the previous section, to the complete N=2 SYM theory, coupled to matter. Besides the pure YM multiplet, belonging to the adjoint representation of the gauge group, the field content of the theory is completed by the hypermultiplet, in a generic representation of the gauge group. The global symmetry group does not change, and the twist goes the same way:

H twist -→ H , (4.1)
where H and H are defined in (2.3) and (3.7) respectively.

In this section, we shall find out the twisted matter fields, the complete twisted action, the twisted operators and the corresponding twisted algebra. We shall moreover verify that the twisted operators are still symmetries of the twisted theory. The result should not be taken for granted, since the topological character of the twist is spoiled by the introduction of matter, and therefore we do not expect that the twisted theory is topological. Hence, the algebraic topological structure, which we shall find for a non-topological field theory, comes somehow as a surprise.

Twisted hypermultiplet

The matter hypermultiplet is (q i , qi , ψ q , ψ q , ψ q, ψ q). Only the bosonic fields q i (x) and qi (x), which have a nonvanishing SU (2) I quantum number, will be twisted, the other fields remaining unchanged. The action of the twist is as follows, and we rename the fields in order to simplify notations:

q i twist -→ q α (0, 0, 1 2 ) 0 → H α ( 1 2 , 0) 0 qi twist -→ qα (0, 0, 1 2 ) 0 → H α ( 1 2 , 0) 0 (ψ q ) α ( 1 2 , 0, 0) +1 → u α ( 1 2 , 0) +1 (4.2) (ψ q ) α(0, 1 2 , 0) -1 → v α(0, 1 2 ) -1 (ψ q) α ( 1 2 , 0, 0) +1 → u α ( 1 2 , 0) +1 (ψ q) α(0, 1 2 , 0) -1 → v α(0, 1 2 ) -1 .
Notice that, while in the pure N=2 SYM the twist gets rid of the spinorial fields, this does not happen for the hypermultiplet, whose twisted version, on the contrary, is entirely formed by spinors.

Twisted matter action

Twisting the matter N=2 SYM action (2.11), we get

S matter twist -→ S T matter , (4.3)
with

S T matter = 1 g 2 Tr m d 4 x 1 2 H γ D µ D µ H γ + H γ (σ µ ) γ γ ψ µ v γ (4.4) -v γ (σ µ ) γγ ψ µ H γ + 1 8 H γ ηu γ + 1 8 H γ (σ µν ) γβ χ µν u β + 1 8 u γ ηH γ - 1 8 u γ (σ µν ) γβ χ µν H β + u γ (σ µ ) γ γ D µ v γ -v γ (σ µ ) γγ D µ u γ + 1 8 u γ φu γ -2v γ φv γ + 1 16 H γ {φ, φ}H γ - 1 32 H γ H γ H δ H δ

Twisted supersymmetry transformations on twisted fields

The action of the twisted operators (δ W , δ µ , δ µν ) on the twisted matter fields, is as follows δ W transformations on twisted matter fields

δ W H γ = 1 √ 2 ε αβ Q βα H γ = 1 √ 2 ε αβ ( √ 2u α ε βγ ) = u γ δ W H γ = u γ δ W u γ = 1 √ 2 ε αβ ( √ 2ε γα φH β ) = +φH γ δ W u γ = -φH γ (4.5) δ W v γ = 1 √ 2 ε αβ - 1 √ 2 (σ ν ) α γ D ν H β = - 1 2 (σ ν ) α γ D ν H α δ W v γ = - 1 2 (σ ν ) α γ D ν H α δ µ transformations on twisted matter fields δ µ H γ = 1 √ 2 (σ µ ) αβ √ 2v αε βα = (σ µ ) γ αv α δ µ H γ = (σ µ ) γ αv α δ µ u γ = 1 √ 2 (σ µ ) αβ 1 √ 2 (σ ν ) γ αD ν H β = 1 2 D µ H γ - 1 2 (σ µν ) β γ D ν H β δ µ u γ = 1 2 D µ H γ - 1 2 (σ µν ) β γ D ν H β (4.6) δ µ v γ = 1 √ 2 (σ µ ) αβ √ 2 16 ε γ αφH β = - 1 16 (σ µ ) γβ φH β δ µ v γ = 1 16 (σ µ ) γβ φH β
δ µν transformations on twisted matter fields

δ µν H γ = 1 √ 2 (σ µν ) αβ √ 2u α ε βγ = -(σ µν ) α γ u α δ µν H γ = -(σ µν ) α γ u α δ µν u γ = 1 √ 2 (σ µν ) αβ √ 2ε γα φH β = (σ µν ) β γ φH β δ µν u γ = -(σ µν ) β γ φH β (4.7)
δ µν v γ = 1 √ 2 (σ µν ) αβ - 1 √ 2 (σ λ ) α γ D λ H β = 1 2 (σ µν ) α β (σ λ ) α γ D λ H β = 1 2 (σ µ ) β γ D ν H β - 1 2 (σ ν ) β γ D µ H β - 1 2 ε µνλτ (σ τ ) β γ D λ H β = 1 2 (σ µ ) β γ D ν H β -(σ ν ) β γ D µ H β + δ µν v γ = 1 2 (σ µ ) β γ D ν H β -(σ ν ) β γ D µ H β + .
The following table summarizes the quantum number and the statistics of the twisted matter fields:

twisted hypermultiplet H H u u v v dim. 1 1 3/2 3/2 3/2 3/2 R -charge. 0 0 +1 +1 -1 -1 statistics
comm. comm. ant. ant. ant. ant.

Twisted algebra

Once we have the twisted action (4.4) and the twisted field transformations (4.5) and (4.6), we can verify that the algebra formed by the twisted operators (δ W , δ µ , δ µν ) is the same of the pure gauge case, i.e. it is a topological, supersymmetric algebra which closes on translations, modulo gauge dependent field transformations and equations of motion. Notice that the fields on which the gauge transformations depend, are the same as in the pure gauge case.

The complete N=2 SYM theory contains interactions terms between the two supermultiplets, the pure gauge and the matter one. Hence, the field equations of motion for the vector supermultiplet change. In order to preserve the algebra, which depends on the field equations, we must modify the transformations of the gauge multiplet. Let us see how this can be done. The field equations appearing in the algebra as obstructions, are those concerning the fields η(x), χ(x) and ψ(x), which appear in the interaction terms of the complete action. Their transformations under the twisted operators are those to be changed. Let us see in detail how, for example, the transformation δ W χ µν must be modified.

Since

δ 2 W χ µν = [φ, χ µν ] -g 2 δS T Y M δχ µν = [φ, χ µν ] -[φ, χ µν ] + (D µ ψ ν -D ν ψ µ ) + (4.8) - 1 8 H γ (σ µν ) γβ u β - 1 8 u γ (σ µν ) γβ H β , it must be δ W χ µν = F + µν - 1 8 H γ (σ µν ) γβ H γ . (4.9)
Analogously, by analyzing the whole set of algebraic relations, from (3.26) to (3.31), we can infer the modified transformations of the fields belonging to the YM multiplet, when coupled to the matter hypermultiplet:

δ W χ µν = F + µν - 1 8 H γ (σ µν ) γβ H γ δ µ ψ ν = F - µν - 1 16 g µν φ, φ + 1 16 H α (σ µν ) αβ H β δ µν η = -4F + µν + 1 2 H γ (σ µν ) γβ H β (4.10)
δ µν χ ρσ = 1 8 (ε µνστ + g µσ g ντ -g µτ g νσ ) φ, φ + F + µσ g ντ -F + νσ g µτ -F + µτ g νσ + F + ντ g µσ + ε α µνσ F + τ α -ε α µντ F + σα + ε α στ µ F + να -ε α στ ν F + µα - 1 16 [g µσ (σ ρν ) β γ -g µρ (σ σν ) β γ +g ρν (σ σν ) β γ + g σν (σ µρ ) β γ ] H β H γ + H γ H β ,
all the other transformations remaining unchanged.

Taking into account the above transformations, now the whole algebra formed by the twisted operators (δ W , δ µ , δ µν ) is closed, modulo field dependent gauge transformations and field equations, for the whole theory, including matter.

Symmetries of the complete twisted action

Lengthy and uninstructive computations lead us to claim that the complete, twisted action

S T = S T Y M + S T matter , (4.11)
where S T Y M and S T matter are given in (3.19) and (4.4) respectively, is invariant under the twisted supercharges:

δ W S T = δ µ S T = δ µν S T = 0 . (4.12)
As for the twisted pure gauge N=2 SYM action, the total S T action is univocally determined by the two symmetries δ W and δ µ , δ W alone being not sufficient. Only one coupling constant is left, as expected.

Starting from the untwisted N=2 SYM theory coupled to matter, through the twisting procedure we got an action, gauge invariant, and invariant as well under two symmetries δ W and δ µ , remnant of five over the eight supercharges, three of them turning out to be redundant. The resulting action is equivalent to the starting, untwisted supersymmetric action, and the twist revealed an algebraic topological structure.

The twisted N=2 sym action coupled to matter appears to be a Witten-type topological action, since it can be written as the variation of a nilpotent operator (δ W , in our case), modulo field equations:

S T = δ W ∆ + ∆ , (4.13)
where

∆ = Tr d 4 x 1 2 χ µν F + µν - 1 2 φD µ ψ µ + 1 16 η φ, φ + 1 16 H g (σ µν ) β γ χ µν H β + 1 16 u γ φH γ + 1 16 u γ φH γ , (4.14)
and ∆ is a contact term

∆ = 1 2 χ µν δS T δχ µν + v γ δS T δv γ + v γ δS T δv γ . (4.15)
We recall that δ W is nilpotent only on shell, and the above relation is not yet an exact relation, in the cohomological sense. But, still, this last result is quite remarkable. It suggests indeed to consider the operator δ W -which, we stress again, is not sufficient, alone, to completely determine the actionas the starting point towards the identification of a nilpotent operator under which the total action is off shell exact.

Towards quantum extension

In the previous sections, we managed in order to treat a known, though complex, situation. Namely we are now dealing with the gauge invariant action S T (4.11), invariant also under the scalar operator δ W and the vector operator δ µ . The underlying supersymmetry algebra closes on translations, modulo field equations and gauge dependent field transformations. The situation is similar to that encountered in topological field theories (like Chern-Simons theory or BF models) and in supersymmetric field theories (N=1 and N=2 SYM). We shall adopt in this case the same technique successfully used there, to define the classical theory and to proceed towards the algebraic renormalization.

The study of the divergences of a quantum field theory and of the possible quantum extension of its classical symmetries requires the usual renormalizations tools. In the case of supersymmetric field theories, so far it is not known a completely satisfactory regularization scheme which preserves at the same time BRS symmetry and supersymmetry. The algebraic renormalization [START_REF] Piguet | Algebraic renormalization: perturbative renormalization, symmetries and anomalies[END_REF], which does not rely on any regularization scheme, is, hence, a mandatory choice. The first algebraic study of the renormalizability of a supersymmetric QFT, has been completely performed, for the N=1 case, using the superspace formalism (see [START_REF] Piguet | Renormalized supersymmetry. The perturbation theory of N=1 supersymmetric theories in flat space-time[END_REF] and references therein, in particular [START_REF] Piguet | The anomaly in the Slavnov identity for N=1 supersymmetric Yang-Mills theories[END_REF]), and a class of N=1 SYM theories has been shown to have no coupling constant renormalization at all [START_REF] Piguet | Non-renormalization theorems of chiral anomalies and finiteness[END_REF][START_REF] Piguet | Non-renormalization theorems of chiral anomalies and finiteness in supersymmetric Yang-Mills theories[END_REF][START_REF] Lucchesi | Necessary and sufficient conditions for all order vanishing beta functions in supersymmetric Yang-Mills theories[END_REF][START_REF] Parkes | Finiteness in rigid supersymmetric theories[END_REF][START_REF] Parkes | Three loop results in two loop finite supersymmetric gauge theories[END_REF].

For what concerns N=2 SYM, with and without matter, the first algebraic approach to the study of counterterms and anomalies has been given in [START_REF] Maggiore | Algebraic renormalization of N=2 super Yang-Mills theories coupled to matter[END_REF][START_REF] Maggiore | Off-shell formulation of N=2 super Yang-Mills theories coupled to matter without auxiliary fields[END_REF].

In this Section, we set the standard for the quantum extension of the twisted N=2 TYM, all the results obtained previously thanks to the twist being valid at the classical level only. The basic steps of the procedure are the construction of an invariant gauge fixing term, the definition of a classical action, including gauge fixing and source-dependent terms, which satisfies all the symmetries of the theory through an extended Slavnov-Taylor identity, which resumes both gauge symmetries and supersymmetries. The key point in our reasoning is the closure of the algebra off shell.

TYM and the extended BRS operator

Our starting point is the classical action S T (4.11), equivalent to the classical N=2 SYM coupled to matter.

Besides being gauge invariant, the action S T is invariant also under a set of global transformations, whose generators δ W , δ µ , δ µν commute with the gauge transformations δ g (3.20), and satisfy the following algebra:

[δ W , δ g ] = [δ µ , δ g ] = [δ µν , δ g ] = 0 δ 2 W = δ g φ + (field eq.) {δ µ , δ ν } = - 1 8 g µν δ g φ + (field eq.) {δ W , δ µ } = ∂ µ + δ g
Aµ + (field eq.) (5.1) {δ W , δ µν } = (gauge transf.) + (field eq.) {δ µν , δ ρσ } = (gauge transf.) + (field eq.) {δ µ , δ ρσ } = -(ε µρσν ∂ ν + g µρ ∂ σ -g µσ ∂ ρ ) + (gauge transf.) + (field eq.) .

The quantum extension of susy, or susy-like, theories presents some serious difficulties, as explained in [START_REF] Breitenlohner | Renormalization of supersymmetric Yang-Mills theories[END_REF]: gauge fixing term: In absence of supersymmetry, the gauge fixing term is a BRS variation, hence it is BRS invariant by construction, being the BRS operator nilpotent. In presence of supersymmetry, instead, because of the algebra, which in the WZ gauge does not simply close on translations, such a term is not susy invariant. The usual way to add a gauge fixing term cannot be applied for supersymmetric QFT.

open algebra: In the Wess Zumino gauge, the susy transformations (and hence their twisted versions) are not linear. The algebra closes only on shell and modulo field dependent gauge transformations. The standard way to deal with this kind of algebras is to introduce auxiliary fields in order to get rid of the field equations, but still the algebra does not close, and an infinite number of external fields is needed, which renders the quantum extension of the theory meaningless.

A solution is to define an extended BRS operator which collects all the symmetries of the theory [START_REF] Maggiore | Algebraic renormalization of N=2 super Yang-Mills theories coupled to matter[END_REF][START_REF] Maggiore | Off-shell formulation of N=2 super Yang-Mills theories coupled to matter without auxiliary fields[END_REF], but, before doing that, let us write the usual BRS operator s, promoting the gauge parameter (x) to a ghost field c(x), so that the gauge transformation δ g becomes the BRS operator s:

a (x) → c a (x) , δ g → s .

(5.2)

In addition, we introduce an antighost field c(x) and a Lagrange multiplier b(x), always in the adjoint representation of the gauge group, so that the BRS operator

sA µ = -D µ c sψ µ = {c, ψ µ } sχ µν = {c, χ µν } sη = {c, η} sφ = [c, φ] sφ = c, φ sc = c 2 = 1 2 f abc c b c c sc = b (5.3) sb = 0 sH = [c, H] sH = c, H su = {c, u} su = {c, u} sv = {c, v} sv = {c, v} is nilpotent s 2 = 0 . (5.4)
Let us now introduce global ghosts ω, µ and v µ , coupled respectively to δ W , δ µ and to the translations

∂ µ ω ↔ δ W , ε µ ↔ δ µ , v µ ↔ ∂ µ , (5.5)
in order to define the extended BRS operator as

Q =s + ωδ W + ε µ δ µ + v µ ∂ µ -ωε µ ∂ ∂v µ . (5.6)
The mass dimensions, R-charge, ghost number and the statistics of the ghosts (both global and local), of the antighost and of the Lagrange multiplier, are summarized in the following table

ω ε µ v µ c c b dim -1/2 -1/2 -1 0 2 2 R-charge -1 1 0 0 0 0 ghost number 1 1 1 1 -1 0 statistics comm comm ant ant ant comm
The extended BRS operator Q has ghost number +1, zero R-charge and mass dimensions, is a symmetry of the action S T and it is nilpotent on shell:

Q S T = 0 , (5.7) Q 2 = field equations .
The Q-invariance of the action is obvious, since S T does not depend on the ghosts and it is invariant under translations. On the other hand, the nilpotency of Q is obtained defining the action of the twisted operators δ W , δ µ (and hence of Q) on the ghosts (c(x), ω, ε µ , v µ ) suitably.

It must be

Qc = c 2 -ω 2 φ -ωε µ A µ + ε 2 16 φ + v µ ∂ µ c Qω = 0 (5.8) Qε µ = 0 Qv µ = -ωε µ .
The antighost c(x) and the Lagrange multiplier b(x) form a Q-doublet:

Qc = b + v µ ∂ µ c , (5.9) Qb = ωε µ ∂ µ c + v µ ∂ µ b , with Q 2 c = Q 2 b = 0 . (5.10)
At this point we are able to define a gauge fixing term, as the Q-variation of the usual "gauge fermion":

S gf = Q Tr d 4 x c∂A (5.11) = Tr d 4 x b∂ µ A µ + c∂ µ D µ c -ωc∂ µ ψ µ - ε ν 2 c∂ µ χ νµ - ε µ 8 c∂ µ η .
Since the extended BRS operator Q is strictly nilpotent on the fields appearing in S gf , the gauge fixed action S is Q-invariant by construction:

Q (S) = Q (S T + S gf ) = 0 .
(5.12)

The gauge fixing procedure takes into account not only the local pure gauge symmetry, but also the twisted symmetries δ W and δ µ , as can be seen by the presence in the gauge fixing term (5.11) of the global ghosts ω and µ . The absence of v µ is due to the translation invariance.

Therefore, the action of the extended BRS operator Q on the whole set of fields and ghosts, is:

QA µ = -D µ c + ωψ µ + ε ν 2 χ νµ + ε µ 8 η + v ν ∂ ν A µ Qψ µ = {c, ψ µ } -ωD µ φ + ε ν F νµ - 1 2 F + νµ - ε µ 16 [φ, φ] +v ν ∂ ν ψ µ + 1 16 H γ (σ νµ ) γβ H β ε ν Qχ στ = {c, χ στ } + ωF + στ + ε µ 8 (ε µστ ν + g µσ g ντ -g µτ g νσ )D ν φ +v ν ∂ ν χ στ - ω 8 H γ (σ στ ) γβ H β Qη = {c, η} + ω 2 [φ, φ] + ε µ 2 D µ φ + v ν ∂ ν η Qφ = [c, φ] -ε µ ψ µ + v ν ∂ ν φ Qφ = c, φ + 2ωη + v ν ∂ ν φ Qc = c 2 -ω 2 φ -ωε µ A µ + ε 2 16 φ + v ν ∂ ν c Qω = 0 Qε µ = 0 (5.13) Qv µ = -ωε µ Qc = b + v µ ∂ µ c Qb = ωε µ ∂ µ c + v µ ∂ µ b QH γ = [c, H γ ] + ωu γ + ε µ (σ µ ) γ αv α + v µ ∂ µ H γ QH γ = c, H γ + ωu γ + ε µ (σ µ ) γ αv α + v µ ∂ µ H γ Qu γ = [c, u γ ] + ωφH γ + ε µ 1 2 D µ H γ - 1 2 (σ µν ) β γ D ν H β + v µ ∂ µ u γ Qu γ = [c, u γ ] + ωφH γ + ε µ 1 2 D µ H γ - 1 2 (σ µν ) β γ D ν H β + v µ ∂ µ u γ Qv γ = [c, v γ ] - 1 2 ω(σ ν ) α γ D ν H α - 1 16 ε µ (σ µ ) γβ φH β + v µ ∂ µ v γ Qv γ = [c, v γ ] - 1 2 ω(σ ν ) α γ D ν H α - 1 16 ε µ (σ µ ) γβ φH β + v µ ∂ µ v γ , with Q 2 = 0 on A, φ, φ, η, H, H, c, ω, ε, v, c, b , (5.14)
and

Q 2 ψ σ = g 2 4 ωε µ δS δχ µσ (5.15) + g 2 32 ε µ ε ν g µσ δS δψ ν + g νσ δS δψ µ -2g µν δS δψ σ Q 2 χ στ = - g 2 2 ω 2 δS δχ στ (5.16) + g 2 8 ωε µ ε µστ ν δS δψ ν + g µσ δS δψ τ -g µτ δS δψ σ Q 2 u γ = g 2 2 ωε µ (σ µ ) γ γ δS δv γ + ε 2 δS δu γ (5.17) Q 2 u γ = g 2 2 ωε µ (σ µ ) γ γ δS δv γ + ε 2 δS δu γ (5.18) Q 2 v γ = g 2 2 ω 2 δS δv γ -ωε µ (σ µ ) β γ δS δu β (5.19) Q 2 v γ = g 2 2 ω 2 δS δv γ -ωε µ (σ µ ) β γ δS δu β .
(5.20)

The Slavnov-Taylor identity

For the functional implementation of the extended BRS operator Q, we must couple external sources Φ i (x) to the nonlinear Q-transformations of the fields Φ i (x) (5.13):

L → c , X γ → H γ D → φ , X γ → H γ Ω µ → A µ , U γ → u γ ξ µ → ψ µ , U γ → u γ ρ → φ , V γ → v γ τ → η , V γ → v γ B µν → χ µν ,
so that we can add to S = S T + S gf the "external" term

S ext = Tr d 4 x Φ i QΦ i , (5.21)
where we collectively denoted with Φ i (x) all the fields transforming nonlinearly under Q, and with Φ i (x) the corresponding external sources, whose quantum numbers and statistics are

L D Ω µ ξ µ ρ τ B µν dim . 4 3 3 5/2 3 5/2 5/2 R -charge 0 -2 0 -1 2 1 1 ghostnumber -2 -1 -1 -1 -1 -1 -1 statistics comm ant ant comm ant comm comm X X U U V V dim . 3 3 5/2 5/2 5/2 5/2 R -charge 0 0 -1 -1 1 1 ghostnumber -1 -1 -1 -1 -1 -1 statistics ant ant comm comm comm comm
In order to write a Slavnov-Taylor identity, the last step is to add to the action S T + S gf + S ext a fourth term S quad which takes into account the fact the the extended BRS operator Q is nilpotent on shell, according to the Batalin-Vilkokisky procedure [START_REF] Batalin | Gauge algebra and quantization[END_REF][START_REF] Batalin | Quantization of gauge theories with linearly dependent generators[END_REF]. Such a term must be quadratic in the external sources Φ * i (x)

S quad = Tr d 4 x Ω ij Φ * i Φ * j , (5.22)
where Ω ij (x) are coefficients which, in general, may depend on the quantum fields and on the global ghosts. They are determined by imposing the validity of the Slavnov-Taylor identity, which we shall write shortly.

The result is the following: which obviously is a symmetry of the theory.

S quad = g 2 Tr d 4 x 1 8 ω 2 B µν B µν - 1 4 ωB µν ε µ ξ ν - 1 32 ε µ ε ν ξ µ ξ ν + 1 32 ε 2 ξ 2 - 1 2 ε 2 U γ U γ + 1 2 ω 2 V γ V γ - 1 2 ωε µ U α (σ µ ) α γ V γ . ( 5
We observe that, since P µ acts linearly on all the fields, the dependence of the action Σ on the global ghost for translations v µ , is fixed by the following identity ∂Σ ∂v µ = ∆ cl µ ,

where

∆ cl µ = Tr d 4 x( L∂ µ c -D∂ µ φ -Ω ν ∂ µ A ν + ξ ν ∂ µ ψ ν -ρ∂ µ φ + τ ∂ µ η + B νσ ∂ µ χ νσ -X γ ∂ µ H γ -X γ ∂ µ H γ +U γ ∂ µ u γ + U γ ∂ µ u γ + V γ ∂ µ v γ + V γ ∂ µ v γ ) , (5.29)
being linear in the quantum fields, is present only at the classical level [START_REF] Piguet | Algebraic renormalization: perturbative renormalization, symmetries and anomalies[END_REF]. We can therefore get rid of the global ghost v µ , introducing the "reduced" classical action Σ Σ = Σ + v µ ∆ cl µ , (5.30)

Conclusions

In this paper we studied the twisted version of N=2 SYM theory coupled to matter.

The twist, being simply a linear redefinition of the quantum fields, does not affect the partition function, and hence two twisted-related theories are completely equivalent. This means, in particular, that they have the same physical content, the same observables and the same coupling constant(s) β-function(s).

In this paper we included matter into the game. We twisted the hypermultiplet, which became entirely spinorial, we modified the (twisted) supersymmetries in order to have a close off-shell algebra, and we achieved the complete off-shell set up by means of a unique Slavnov-Taylor identity, which collects both BRS symmetry and supersymmetries of the theory.

As it is well known [START_REF] Witten | Topological quantum field theory[END_REF][START_REF] Witten | Supersymmetric Yang-Mills theory on a four manifold[END_REF], pure N=2 SYM is twisted to a topological quantum field theory: TYM. An interesting and new result presented in this article is the fact that TYM theory coupled to matter have the same set of invariances of the same theory without matter. Since the theory with matter is not topological, the presence of these symmetries contradicts the common belief that they are peculiar to topological theories. 2The twisted version of the whole theory, including matter, is the necessary step towards the study of the β-function, for which a well known non-renormalization theorem holds, and which has been algebraically proved only in absence of matter [START_REF] Blasi | Perturbative beta function of N = 2 super Yang-Mills theories[END_REF]. We stress also that we never specified to which representation of the gauge group the matter hypermultiplet belongs: in the particular case of matter in the adjoint representation, N=4 SYM is recovered.

In general, the introduction of matter spoils the topological character of the theory. In our case, the relation (4.13) suggests that matter might enter in the theory simply through an extended BRS variation, and this result strongly induces to suppose that matter does not alter neither the physical sector of observables nor the finite, or protected, operators of the theory [START_REF] Maggiore | Protected operators in N = 2,4 supersymmetric theories[END_REF]. In other terms, the presence of matter should not spoil the AdS/CFT duality between non-conformal N=2 theories and string theories [START_REF] Polchinski | N = 2 gauge-gravity duals[END_REF]. It is also natural to expect that the whole action can be written, as in the pure gauge case, in terms of a unique, and probably finite, operator, which in the pure gauge case is Tr φ 2 , whose relevance for the algebraic proof of the non-renormalization theorem of the β-function has been discussed in [START_REF] Blasi | Perturbative beta function of N = 2 super Yang-Mills theories[END_REF].

  (x) being the local infinitesimal gauge parameter, the action S T Y M is invariant δ g S T Y M = 0. (3.21)

  {δ W , δ µν } = (gauge transformation) + (field equations); (3.29) {δ µν , δ ρσ } = (gauge transformation) + (field equations); (3.30) {δ µ , δ ρσ } = -(ε µρσν ∂ ν + g µρ ∂ σ -g µσ ∂ ρ ) (3.31) +(gauge transformation) + (field equations).

. 23 )

 23 With this choice, the complete classical action Σ = S T + S gf + S ext + S quad(5.24) satisfies the Slavnov-Taylor identityS(Σ) = 0 ,(5.25)whereS(Σ) = Tr d 4 x δΣ δΦ i δΣ δΦ i + (b + v µ ∂ µ c) δΣ δc + (ωε µ ∂ µ c + v µ ∂ µ b) δΣ δb -ωε µ ∂Σ ∂v µ .(5.26)We can go further, introducing the translation operatorP µ Σ = Tr d 4 x ∂ µ Φ i δΣ δΦ i + ∂ µ Φ * i δΣ δΦ * i = 0 ,(5.27)

At pag 194 of[START_REF] West | Introduction to supersymmetry and supergravity[END_REF], it is pointed out: "Here we have stressed these weaknesses not because of a mistrust in the arguments for finiteness, but to show that they are not proofs in a mathematical sense and that there is still room for further work".

We thank one of the referees for this remark.

with ∂ Σ ∂v µ = 0 .

(5.31)

It is easily verified that Σ satisfies the modified ST identity

where

(5.33)

The (classically broken) ST identity (5.32) is the one which must be used to determine the quantum extension of the theory. The corresponding linearized ST operator

is not nilpotent. In fact, it holds

that is, B Σ is nilpotent modulo a total derivative. It follows that the operator B Σ is nilpotent in the space of integrated local functionals, which, actually, is the case we are interested in.

Summarizing, we handled the problem in order to be able to deal with the usual web of symmetries and constraints which constitutes the basis for the quantum extension of the model and for the study of its algebraic renormalizability (determination of local counterterms and study of anomalies) [START_REF] Piguet | Algebraic renormalization: perturbative renormalization, symmetries and anomalies[END_REF]:

• ST identity (5.32) ;

• Landau gauge fixing condition

(5.36)

(5.37)

• Landau gauge ghost equation