Singular Instanton Research Articles

The D-instanton partition function

The D-instanton partition function is a fascinating quantity because in the presence of N D3-branes, and in a certain decoupling limit, it reduces to the functional integral of = 4 U(N) supersymmetric gauge theory for multi-instanton solutions. We study this quantity as a function of non-commutativity in the D3-brane theory, VEVs corresponding to separating the D3-branes and α'. Explicit calculations are presented in the one-instanton sector with arbitrary N, and in the large-N limit for all instanton charge. We find that for general instanton charge, the matrix theory admits a nilpotent fermionic symmetry and that the action is -exact. Consequently the partition function localizes on the minima of the matrix theory action. This allows us to prove some general properties of these integrals. In the non-commutative theory, the contributions come from the ``Higgs Branch'' and are equal to the Gauss-Bonnet-Chern integral of the resolved instanton moduli space. Separating the D3-branes leads to additional localizations on products of abelian instanton moduli spaces. In the commutative theory, there are additional contributions from the ``Coulomb Branch'' associated to the small instanton singularities of the instanton moduli space. We also argue that both non-commutativity and α'-corrections play a similar rôle in suppressing the contributions from these singularities. Finally we elucidate the relation between the partition function and the Euler characteristic of the instanton moduli space.

ON HYPER-KAHLER SINGULARITIES

Using a geometric realization of the SU (2)R symmetry and a procedure of factorization of the gauge and SU (2)R charges, we study the small instanton singularities of the Higgs branch of supersymmetric U (1)r gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preserving manifestly the eight supercharges. In particular, we obtain an extension of the ordinary ADE singularities for hyper-Kahler manifolds and show that the classical moduli space of vacua is in general given by cotangent bundles of compact weighted projective spaces.

Open inflation and the singular boundary

The singularity in the Hawking-Turok model of open inflation has some appealing properties, such as the fact that its action is integrable. Also, if one thinks of the singularity as the boundary of spacetime, then the Gibbons-Hawking term is nonvanishing and finite. Here, we consider a model where the gravitational and scalar fields are coupled to a dynamical membrane. The singular instanton can then be obtained as the limit of a family of ``no-boundary'' solutions where both the geometry and the scalar field are regular. Using this procedure, the contribution of the singularity to the Euclidean action is just 1/3 of the Gibbons-Hawking term. Unrelated to this issue, we also point out that the singularity acts as a reflecting boundary for scalar perturbations and gravity waves. Therefore, the quantization of cosmological perturbations seems to be well posed in this background.

High orders of perturbation theory. Are renormalons significant?

According to Lipatov, the high orders of perturbation theory are determined by saddle-point configurations (instantons) of the corresponding functional integrals. According to t'Hooft, some individual large diagrams, renormalons, are also significant and they are not contained in the Lipatov contribution. The history of the conception of renormalons is presented, and the arguments in favor of and against their significance are discussed. The analytic properties of the Borel transforms of functional integrals, Green functions, vertex parts, and scaling functions are investigated in the case of \phi^4 theory. Their analyticity in a complex plane with a cut from the first instanton singularity to infinity (the Le Guillou - Zinn-Justin hypothesis) is proved. It rules out the existence of the renormalon singularities pointed out by t'Hooft and demonstrates the nonconstructiveness of the conception of renormalons as a whole. The results can be interpreted as an indication of the internal consistency of \phi^4 theory.

Resolving the singularity of the Hawking–Turok type instanton

We point out that the singular instanton of Hawking–Turok type, in which the singularity occurs due to the divergence of the massless scalar field, can be generated by Euclideanized regular p-brane solutions in string (or M-) theory upon compactification to four dimensions.

Instantons from low energy string actions

We look for instanton solutions in a class of two scalar field gravity models, which includes the low energy string action in four dimensions. In models where the matter field has a potential with a false vacuum, we find that non-singular instantons exist as long as the Dilaton field found in string theory has a potential with a minimum, and provide an example of such an instanton. The class of singular instanton solutions are also examined, and we find that depending on the parameter values, the volume factor of the Euclidean region does not always vanish fast enough at the singularity to make the action finite.

Higgs Branch, Hyper-Kähler Quotient and Duality in SUSY N = 2 Yang–Mills Theories

Low-energy limits of N = 2 supersymmetric field theories in the Higgs branch are described in terms of a nonlinear four-dimensional σ-model on a hyper-Kähler target space, classically obtained as a hyper-Kähler quotient of the original flat hypermultiplet space by the gauge group. We review in a pedagogical way this construction, and illustrate it in various examples, with special attention given to the singularities emerging in the low-energy theory. In particular, we thoroughly study the Higgs branch singularity of Seiberg–Witten SU(2) theory with Nf flavors, interpreted by Witten as a small instanton singularity in the moduli space of one instanton on ℝ4. By explicitly evaluating the metric, we show that this Higgs branch coincides with the Higgs branch of a U(1) N = 2 SUSY theory with the number of flavors predicted by the singularity structure of Seiberg–Witten's theory in the Coulomb phase. We find another example of Higgs phase duality, namely between the Higgs phases of U(Nc)Nf flavors and U(Nf-Nc)Nf flavors theories, by using a geometric interpretation due to Biquard et al. This duality may be relevant for understanding Seiberg's conjectured duality Nc ↔ Nf-Nc in N = 1 SUSY SU(Nc) gauge theories.

PHYSICAL INTERPRETATION OF CERTAIN STRONG COUPLING SINGULARITIES

We interpret certain strong coupling singularities of the E8×E8 heterotic string on K3 in terms of exotic six-dimensional theories in which E8 is a gauge symmetry. These theories are closely related to theories obtained at small instanton singularities, which have E8 as a global symmetry.

QCD perturbative expansions in large orders

We consider implications of divergencies of QCD perturbative expansions in large orders for the total cross sections of e +e − annihilation into hadrons at high energies. The divergencies bring in uncertainties which are suppressed as powers of the total energy √ s and we present numerical estimates of these uncertainties. The leading uncertainty is associated with the so-called ultraviolet renormalon. It scales with energy like s −1 ln −2 s and at the energies of order Z-boson mass does not exceed 10 −6. The next-to-leading uncertainty is due to the infrared renormalon and scales like s −2 ln −2 s. This contribution is not Borel summable and its treatment calls for some input on non-perturbative physics. We consider then the status of the instanton singularity. Although it produces only a subleading, Borel non-summable contribution it reflects a specific high-energy phenomenon, that is the growth of multi-jet production with the multiplicity. Phenomenologically, the corresponding events would look as anomalous isotropic multi-jet events. The mechanisms of the termination of this growth lie outside the perturbation theory itself. We observe that even if such events dominate the total cross section asymptotically, the onset of this asymptotics could be delayed until the (1–10) TeV region.

Asymptotics of the cut discontinuity and large order behaviour from the instanton singularity: The case of lattice Schr�dinger operators with exponential disorder

We discuss the relation between the singularity structure of the Borel transform, the asymptotics of the cut discontinuity and the large order behaviour of perturbation theory. In an explicit example — a tight binding model with exponential disorder — we show how to obtain the first instanton singularity from a cluster expansion in the Borel variable, and as an application we determine the exact decay of the density of states asE → ∞. The method opens some perspectives for similar problems arising from different models of Mathematical Physics.

Equivariant rho-invariants and instanton homology of torus knots

The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the generators and Floer gradings in the singular instanton chain complex of (p,q)-torus knots with odd p and q.