Multi-instanton Measure from Recursion Relations in N=2 Supersymmetric Yang-mills Theory

By using the recursion relations found in the framework of N = 2 Super Yang-Mills theory with gauge group SU(2), we reconstruct the structure of the instanton moduli space and its volume form for all winding numbers. In [1] the entire nonperturbative contribution to the holomorphic part of the Wilsonian effective action was computed for N = 2 globally supersymmetric (SUSY) theories with gauge group SU(2), using ansätze dictated by physical intuitions. There are several aspects of the Seiberg-Witten (SW) model [1] which are related to the theory of moduli spaces of Riemann surfaces. In particular, here, we will consider the recursion relations for nonperturbative (instanton) contributions to the N = 2 Super Yang-Mills (SYM) effective prepotential [2] and will compare them with the recursion relations for the Weil-Petersson volumes of punctured Riemann spheres. In the Seiberg-Witten model there exists a relation the modulus u = 〈Trφ〉 and the effective prepotential [2] (see also [3]). This allowed to prove the SW conjecture by using the reflection symmetry of vacua [4]. On the other hand, it is rather surprising that, while on one side all the instanton coefficients have been computed in [2], explicit calculations have been performed only in the one and two-instanton background [5, 6, 7], while the above mentioned relation has been shown to hold to all instanton orders [8, 9]. The problem for instanton number k ≥ 3 seems extremely difficult to solve. Indeed, the ADHM constraint equations become nonlinear and have not been explicitly solved up to now. Moreover, neither the structure of the moduli space, nor the volume form are known. The instanton measure for all winding numbers has been written in [10], but only in an implicit form (i.e. by implementing the bosonic and fermionic ADHM constraints through the use of Dirac delta functions), which in some special cases allows to extract information on the instanton moduli space [11]. However, the mathematical challenging problem of finding the explicit structure of the instanton moduli space for generic winding numbers still remains unsolved. On the other hand, the simple way in which the recursion relations have been derived, strongly suggests that there may be some mechanism which should make the explicit calculations possible. The investigation of such mechanism would provide important information on the structure of the instanton moduli space (of which only the boundary à la Donaldson-Uhlenbeck is known for generic winding number [12, 13, 14]) and of the associated volume form. In particular, even if the integrals seem impossible to compute, (actually, as we stated before we know neither the structure of the space nor the volume form), the existence of recursion relations and the simple way in which they arise, seem to suggest that these integrals could be easy to compute because of some underlying geometrical recursive structure. It has been claimed for some time, but only recently proven [15], that the nonperturbative contributions to u actually can be written as total derivatives, i.e. as pure boundary terms, on the moduli space. If the boundary is composed by moduli spaces of instantons of lower winding number times zero-size instantons moduli spaces, as it happens in the Donaldson-Uhlenbeck compactification, this would immediately provide, in the case of a suitable volume form, a recursion relation. We will now see how the similar problem one finds in computing the Weil-Petersson (WP) volumes