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| − | == <strong> Konishi Anomaly </strong> ==
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| − | === Definition ===
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| − | The '''Konishi anomaly''' is a set of exact quantum-mechanical relations
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| − | involving various composite operators in supersymmetric gauge theories. They contain anomalous terms as compared to what is expected from the classical field equation of motion. This anomaly can be traced back to the standard chiral U(1) anomalies and to supersymmetry. The Konishi anomaly yields powerful constraints on the possible dynamical properties of the system, such as the way the global symmetry is realized at low energies, or whether or not supersymmetry is broken. It also provides a strong consistency check for any dynamical calculation
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| − | (e.g., instantons) or approximation scheme.
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| − | === General gauge theories ===
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| − | The Konishi anomaly for a generic gauge theory with gauge superfield W, and matter fields :<math> \Phi_i </math>
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| − | reads
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| − | :<math KanomalyBis>
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| − | - \frac{1]{4} {\bar D}^2 \bar{\Phi}_i e^V \Phi_i =
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| − | \frac{\delta}{\delta \Phi_i} {\cal W}(\Phi) \, \Phi_i + \frac{g^2}{32 \pi} \,C_2(\Phi) \, W^{\alpha}(x) \, W_{\alpha}(x) \;, \qquad {\rm no ~ sum ~ over} \,\, i,
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| − | </math>
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| − | where :<math> {\cal W}(\Phi) </math> is the superpotential and :<math> \,C_2(\Phi) </math> is the Dynkin index for the representation according to which the field :<math> \Phi </math> transforms. The formula holds for each matter field, and by considering the vacuum expectation
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| − | === Supersymmetry breaking ===
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| − | Indeed, in a class of chiral gage theories (i.e., theories in which the left-handed and right-handed matters belong to distinct representations of the gauge group, unlike in QCD) the relations such as equation (<ref> KanomalyBis </ref>) and explicit instanton calculations are inconsistent if supersymmetry is assumed to be unbroken. In other words, in such theories, dynamical effects are
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| − | === Gaugino condensates in supersymmetric Yang-Mills theories ===
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| − | === Supergravity and gravitational anomaly ===
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| − | === Generalized Konishi anomaly ===
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