D-brane Instantons in Type II String Theory Page: 3 of 70
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functions. Therefore, they are more strongly suppressed than any perturbative
These a priori subleading non-perturbative corrections can however become
very important when all potentially larger corrections are known to be absent
due to non-renormalization theorems. Such situations can be realized in super-
symmetric quantum field theories. For instance, in the context of N = 1 super-
symmetric four-dimensional theories, there exist holomorphic quantities such as
the superpotential W,
Sw = Jd 4xd20 W(), (1)
and the gauge kinetic function f,
SGauge = J 4 2xd20 f (02) tr (W W) , (2)
which are only integrated over half of the superspace and depend holomorphi-
cally on the chiral superfields #i. At the perturbative level, the superpotential
and gauge kinetic function respectively receive only tree-level and up to one-loop
level contributions [3, 4]. As a consequence, non-perturbative corrections can be-
come very important for the dynamics of the system, in particular if for instance
the tree-level superpotential coupling vanishes. Since these non-perturbative con-
tributions are exponentially suppressed in the weak-coupling regime, when they
are the leading effect they may provide a dynamical explanation of some of the
hierarchy problems of fundamental physics.
In gauge theories such non-perturbative corrections arise from so-called gauge
instantons. These are solutions to the Euclidean self-duality equation
F = *F (3)
for the Yang-Mills gauge field. Such solutions can be explicitly constructed as
local minima of the action and are classified by the instanton number N =
fR4 trF A F. Around each instanton saddle point, one can again perform per-
turbation theory and compute the contributions to certain correlation functions.
The final result will then involve summation over all topologically non-trivial
sectors. The prescription to carry out these computations is determined by the
so-called instanton calculus. As a main ingredient it involves integration over the
collective coordinates, also known as the moduli space of the instanton solution.
In string theory the situation is very similar. Also here one can compute
perturbative corrections to tree-level correlation functions.' These are given by
'Here we are speaking loosely. In asymptotically AdS solutions, one is computing correlation
functions (of a dual field theory). In asymptotically Minkowski backgrounds, one computes an
S-matrix, and infers an effective action indirectly. Then the corrections we discuss are really
to terms in this effective action. In a non-gravitational theory, this action would give rise to
meaningful off-shell correlation functions.
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Blumenhagen, Ralph; /Munich, Max Planck Inst.; Cvetic, Mirjam; U., /Pennsylvania; Kachru, Shamit; /Stanford U., Phys. Dept. /SLAC et al. D-brane Instantons in Type II String Theory, article, June 19, 2009; United States. (digital.library.unt.edu/ark:/67531/metadc926940/m1/3/: accessed January 23, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.