One-loop Determinants Research Articles

Topological gauge theories on local spaces and black hole entropy countings

We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by $U(1)$-equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov–Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuation determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi–Yau local surfaces, describing the quantum foam for the $A$-model, relevant to the calculation of Donaldson–Thomas invariants.

Instantons and holomorphic couplings in intersecting D-brane models

We clarify certain aspects and discuss extensions of the recently introduced string D-instanton calculus (hep-th/0609191). The one-loop determinants are related to one-loop open string threshold corrections in intersecting D6-brane models. Utilising a non-renormalisation theorem for the holomorphic Wilsonian gauge kinetic functions, we derive a number of constraints for the moduli dependence of the matter field Kaehler potentials of intersecting D6-brane models on the torus. Moreover, we compute string one-loop corrections to the Fayet-Iliopoulos terms on the D6-branes finding that they are proportional to the gauge threshold corrections. Employing these results, we discuss the issue of holomorphy for E2-instanton corrections to the superpotential. Eventually, we discuss E2-instanton corrections to the gauge kinetic functions and the FI-terms.

Generalized proper-time approach for the case of broken isospin symmetry

We present a derivation of the low-energy effective meson Lagrangian of the Nambu -- Jona-Lasinio (NJL) model on the basis of Schwinger's proper-time regularization of the one-loop fermion determinant. We consider the case in which the $SU(2)\times SU(2)$ chiral symmetry of the NJL Lagrangian is broken by the current quark mass matrix with $\hat{m}_u\ne\hat{m}_d$. The non-degeneracy of $d$ and $u$ masses destroys one of the most crucial features of the proper-time expansion -- the chiral-invariant structure of Seeley -- DeWitt coefficients. We show however that systematic resummations inside the proper-time expansion are still possible and derive a result which is in full agreement with the chiral Ward -- Takahashi identities.

One-loop fermion determinant with explicit chiral symmetry breaking

We use a proper-time regularization to define the real part of the one-loop fermion determinant for the case in which explicit chiral symmetry breaking takes place. We demonstrate that beyond the chiral limit the standard definition of Re[lndet D] by ln|det D| deforms the symmetry breaking pattern of the basic fermionic Lagrangian. We show how to obtain the correcting functional in order to arrive at the fermion determinant whose transformation properties are consistent with the general symmetry requirements. As an example it is shown how the fundamental symmetries and the explicit chiral symmetry breaking pattern associated with the ENJL model are consistently preserved in this way.

Black hole pair creation in de Sitter space: a complete one-loop analysis

We present an exact one-loop calculation of the tunneling process in Euclidean quantum gravity describing creation of black hole pairs in a de Sitter universe. Such processes are mediated by $S^2\times S^2$ gravitational instantons giving an imaginary contribution to the partition function. The required energy is provided by the expansion of the universe. We utilize the thermal properties of de Sitter space to describe the process as the decay of a metastable thermal state. Within the Euclidean path integral approach to gravity, we explicitly determine the spectra of the fluctuation operators, exactly calculate the one-loop fluctuation determinants in the $\zeta$-function regularization scheme, and check the agreement with the expected scaling behaviour. Our results show a constant volume density of created black holes at late times, and a very strong suppression of the nucleation rate for small values of $\Lambda$.

One-loop determinant of Dirac operator in non-renormalizable models

We use proper-time regularizations to define the one-loop fermion determinant in the form suggested by Gasser and Leutwyler some years ago. We show how to obtain the polynomial by which this definition of ln detD needs to be modified in order to arrive at the fermion determinant whose modulus is invariant under chiral transformations. As an example it is shown how the fundamental symmetries associated with the NJL model are preserved in a consistent way.

Supersymmetry and the multi-instanton measure II. From N = 4 to N = 0

Extending recent N = 1 and N = 2 results, we propose an explicit formula for the integration measure on the moduli space of charge- n ADHM multi-instantons in N = 4 supersymmetric SU(2) gauge theory. As a consistency check, we derive a renormalization group relation between the N = 4, N = 2, and N = 1 measures. We then use this relation to construct the purely bosonic (“ N = 0”) measure as well, in the classical approximation in which the one-loop small-fluctuations determinants is not included.

The 2D effective field theory of interfaces derived from 3D field theory

The one-loop determinant computed around the kink solution in the 3D φ 4 theory, in cylindrical geometry, allows one to obtain the partition function of the interface separating coexisting phases. The quantum fluctuations of the interface around its equilibrium position are described by a c = 1 two-dimensional conformal field theory, namely a 2D free massless scalar field living on the interface. In this way the capillary wave model conjecture for the interface free energy in its gaussian approximation is proved.

Operator regularization and the phase of one-loop determinants

Operator regularization is a symmetry preserving regularization procedure to all orders of perturbation theory that avoids explicit divergences. At one-loop order it is det H which is regularized, where H is an operator appearing in the theory. For some theories it is not possible to treat det H directly; in previous implementations det H has simply been replaced by det 1 2 H 2 in such cases. However, if H has negative or imaginary eigenvalues then information about the phase of det H may be lost in this replacement. We discuss this general problem for an arbitrary operator H and we give a prescription for finding the phase of det H that reduces, if H is Hermitian, to the η-function analysis introduced by Gilkey and further exploited by Witten and others. We consider explicitly one example of a non-Hermitian operator and three examples of a Hermitian operator. To illustrate how we treat the phase of the determinant of a non-Hermitian operator, we consider the determinant associated with a spinor coupled to an external axial vector field and show how the phase of this determinant is associated with the axial anomaly, using a technique introduced by Bukhbinder, Gusynin, and Fomin. In three-dimensional quantum electrodynamics (or, more generally, its nonabelian analogue) the η-function can be computed exactly (following Birmingham, Cho, Kantowski, and Rakowski) when there is an external vector field, showing that an effective Chern-Simons action is generated, resulting in a quantization of the gauge coupling constant. In the final two examples H is a super-operator, spanning a space containing both bosons and fermions. We compute the contribution of the two-point spinor function to the η-function when the vector field to which the spinor couples has a Chern-Simons action; it simply serves to renormalize the coefficient of the kinetic term for the spinor field. Finally, we consider the self-energy in a two-dimensional model in which a Majorana spinor couples to a vector field which has a topological action. In this case the η-function vanishes, but the ζ-function indicates that chiral symmetry is broken despite the fact that all the unregulated Feynman diagrams in this model vanish.

Operator regularization and the phase of one-loop determinants: D. G. C. McKeon. Department of Mathematical Physics, University College Galway, Galway, Ireland; and T. N. Sherry. School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

Operator regularization and the phase of one-loop determinants: D. G. C. McKeon. Department of Mathematical Physics, University College Galway, Galway, Ireland; and T. N. Sherry. School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

Instantons in conformal gravity

We study extrema of the general conformally invariant action: S c = ∫ 1 α 2 C abcdC abcd+γR abcd ∗R abcd ∗+iθR abcd∗R abcd . We find the first examples in four dimensions of asymptotically euclidean gravitational instantons. These have arbitrary Euler number and Hirzebruch signature. Some of these instantons represent tunneling between zero-curvature vacua that are not related by small gauge transformations. Others represent tunneling between flat space and topologically non-trivial zero-energy initial data. A general formula for the one-loop determinant is derived in terms of the renormalization group invariant masses, the volume of space-time, the Euler number and the Hirzebruch signature.

Phase transitions in De Sitter space

This paper uses zeta-function regularisation to calculate the one-loop functional determinants for fields of any spin in De Sitter space. As an example, we investigate the Coleman-Weinberg spontaneous symmetry breaking mechanism in massless scalar electrodynamics. The effective potential is calculated in Landau gauge. It depends upon the curvature, and upon the renormalised value of ζ (in ζRφ 2). The phase transition will be first or second order, and the critical curvature and mass are found. The methods can be applied to any gauge theory.

Calculation of one-loop instanton determinants using propagators with space-time dependent mass

One-loop quantum tunneling amplitude due to a single (anti-) instanton is evaluated analytically by exploiting explicitly constructed particle propagators with a space-time dependent mass. The O(5) formulation is not used. Our method is more elegant than previous ones, and great care is taken to ensure that renormalization is done correctly. The results agree with the original computation of 't Hooft. A brief discussion is also given to explain, with some rigor, the method of using particle propagators to calculate induced one-loop effective actions in a given background gauge field (and particularly in a self-dual gauge field).

Instanton interactions at the one-loop level

We discuss the interactions between widely separated instantons (but not instantons and anti-instantons) due to the one-loop fluctuations in a non-Abelian gauge theory. A slightly generalized version of the recent calculation of Brown and Creamer for the one-loop determinant about an exact self-dual solution is used as a starting point. An apparently dangerous 1/r/sup 2/ interaction is found to disappear with the proper parametrization of the exact solution. As a by-product of our procedure, we develop a more precise understanding of the relation between exact solutions of topological charge k and the superposition of k instantons. In particular, the meaning of the parameters of the Jackiw-Nohl-Rebbi k = 2 solution is understood in these terms.

Givental $J$-functions, quantum integrable systems, AGT relation with surface operator

We study 4d $\mathcal{N}=2$ gauge theories with a co-dimension two full surface operator, which exhibit a fascinating interplay of supersymmetric gauge theories, equivariant Gromov-Witten theory and geometric representation theory. For pure Yang-Mills and $\mathcal{N}=2^*$ theory, we describe a full surface operator as the 4d gauge theory coupled to a 2d $\mathcal{N}=(2,2)$ gauge theory. By supersymmetric localizations, we present the exact partition functions of both 4d and 2d theories which satisfy integrable equations. In addition, the form of the structure constants with a semi-degenerate field in SL(N,R) WZNW model is predicted from one-loop determinants of 4d gauge theories with a full surface operator via the AGT relation.