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Extended sigma-model in nontrivially deformed field-antifield formalism

We propose an action for the extended sigma - models in the most general setting of the kinetic term allowed in the nontrivially deformed field - antifield formalism. We show that the classical motion equations do naturally take their desired canonical form.


Introduction and summary
The field-antifield formalism [1,2] is known as the most powerful method for covariant quantization of general gauge theories. The method operates with two fundamental objects: the antibracket and the odd nilpotent second-order Delta operator. In the standard version of the method, it is formulated in the Darboux coordinates. Various local deformations of the Delta operator [3] and antibracket [4,5,6,7] elucidate the geometry underlying the fieldantifield formalism [8]. The deformed Delta operator and antibracket do play the central role in formulating the path integral version of the nontrivially deformed field-antifield formalism [9]. It has been shown recently [10] that the nontrivially deformed formalism does exist, as a consistent approach, at the classical level. In the present article, we confirm the latter result as applied to the important case of the topological sigma-models [8,11,12,13,14,15,16,17]. We propose an action of the deformed sigma-models in the most general setting for the kinetic term allowed in the formalism, and show that the motion equations do naturally take their desired canonical form. We define the kinetic part of the action directly in terms of the zero modes for the τ -extended Euler operator N τ , that generalizes naturally the standard power counting operator. We show that these zero modes, although they are nontrivial functions of the original phase variables, do satisfy very simple antibracket algebra, so that they generate effectively their own antisymplectic structure, very similar to the original one. The latter circumstance makes it possible to deduce the desired canonical form of the motion equations in a very simple and natural way. Moreover, we have deduced the so -called canonical "Ward Identities" the complete action does satisfy to at the general (arbitrary) trajectory in the antisymplectic phase space. These identities, in their own turn, are necessary when deducing the functional master equation for the complete action . Thus, we have got to a considerable new progress in our study of the canonical structure in topological field theories.

Trivially deformed sigma-model
The extended trivially deformed sigma -model is formulated via the action where the Lagrangian L is defined via S satisfying the extended trivially deformed master equation, or equivalently, One should seek for a solution to the equation where the nontrivially deformed antibracket is given by Let us consider our new Lagrangian for the deformed sigma-model in the most general setting, is the zero mode for the N τ operator. By making a variation δZ C in (3.8), we find By multiplying the equation (3.10) by the coefficients N C from the left, we get On the other hand, by making a variation δ ln t in (3.8), we have It follows from (3.11), (3.12) that Dθ + N τ (T S) = 0, (3.13) or, equivalently, in terms of the extended τ -antibracket, That is exactly the desired canonical motion equation for the variable θ. The two other canonical motion equations do follow from (3.8) in the usual way, as well, 15) or in its canonical form, together with the canonical form of the equation (3.10), The complete set of the canonical equations (3.14), (3.16), (3.17) tells us that the action (3.8) yields the correct general canonical description to the deformed sigma -model at the classical level.
For the sake of completeness, an explicit derivation of the canonical form (3.17) directly from the original one (3.10) is given below. As the operator (N τ − 2) does differentiate the antibracket (due to the second in (2.8)), we have It follows from (3.18) that, within the class of regular functions of ln t, The latter rewrites in the explicit form, which implies in turn, Also, we have where we have used (3.15). By inserting (3.22) into (3.10), and using (3.21), we get In terms of the extended antibracket (2.6), the equation (3.23) takes immediately its desired canonical form (3.17).
Although we have shown that all the motion equations generated by the action (3.1)/(3.8) have the canonical form, that is not the case for functional derivatives of the action taken at the general trajectory in the phase space. Instead, the functional derivatives do satisfy the so-called canonical "Ward Identities", In the right-hand sides in these relations, one can recognize the canonical form of the lefthand sides of the motion equations (3.14), (3.16), (3.17). Now, let [F, G] τ be the τ -extended antibracket in the space of functionals, generated by the ultralocal τ -extended antibracket (F, G) τ , The first equality in (3.28) holds due to the following integral identity, where while ω αβ is an inverse to (3.32).
Notice that all the above reasoning did use no further restrictions to the Euler operator N. However, it follows from the second in (2.8), that The most general form allowed for F is Notice that, in the case (3.38), one can deduce (3.17) from (3.10) even more explicitly by making use of the representation (3.40) for the zero modes.