One-loop Determinants Research Articles

Computing black hole partition functions from quasinormal modes

We propose a method of computing one-loop determinants in black hole spacetimes (with emphasis on asymptotically anti-de Sitter black holes) that may be used for numerics when completely-analytic results are unattainable. The method utilizes the expression for one-loop determinants in terms of quasinormal frequencies determined by Denef, Hartnoll and Sachdev in \cite{Denef:2009kn}. A numerical evaluation must face the fact that the sum over the quasinormal modes, indexed by momentum and overtone numbers, is divergent. A necessary ingredient is then a regularization scheme to handle the divergent contributions of individual fixed-momentum sectors to the partition function. To this end, we formulate an effective two-dimensional problem in which a natural refinement of standard heat kernel techniques can be used to account for contributions to the partition function at fixed momentum. We test our method in a concrete case by reproducing the scalar one-loop determinant in the BTZ black hole background. We then discuss the application of such techniques to more complicated spacetimes.

Localizing gauge theories on S d

We conjecture the form of the one-loop determinants for localized gauge theories with eight supersymmetries on $d$-dimensional spheres. Combining this with results for the localized action, we investigate the strong coupling behavior in the large $N$ limit for a continuous range of $d$. In particular, we find the $N$ dependence of the free energy for supersymmetric Yang-Mills with only a vector multiplet in $3<d<4$ and for maximally supersymmetric Yang-Mills in $3< d<6$. We also argue that this gives an effective way to regularize divergences after localization in $d=4$ for ${\mathcal N}=2$ gauge theories and $d=6$ for the maximally supersymmetric case.

New supersymmetric index of heterotic compactifications with torsion

We compute the new supersymmetric index of a large class of N=2 heterotic compactifications with torsion, corresponding to principal two-torus bundles over warped K3 surfaces with H-flux. Starting from a UV description as a (0,2) gauged linear sigma-model with torsion, we use supersymmetric localization techniques to provide an explicit expression of the index as a sum over the Jeffrey-Kirwan residues of the one-loop determinant. We finally propose a geometrical formula that gives the new supersymmetric index in terms of bundle data, regardless of any particular choice of underlying two-dimensional theory.

High-temperature expansion of supersymmetric partition functions

Di Pietro and Komargodski have recently demonstrated a four-dimensional counterpart of Cardy's formula, which gives the leading high-temperature ($\beta\rightarrow{0}$) behavior of supersymmetric partition functions $Z^{SUSY}(\beta)$. Focusing on superconformal theories, we elaborate on the subleading contributions to their formula when applied to free chiral and U(1) vector multiplets. In particular, we see that the high-temperature expansion of $\ln Z^{SUSY}(\beta)$ terminates at order $\beta^0$. We also demonstrate how their formula must be modified when applied to SU($N$) toric quiver gauge theories in the planar ($N\rightarrow\infty$) limit. Our method for regularizing the one-loop determinants of chiral and vector multiplets helps to clarify the relation between the 4d $\mathcal{N} = 1$ superconformal index and its corresponding supersymmetric partition function obtained by path-integration.

Elliptic Genera of 2d $${\mathcal{N}}$$ N = 2 Gauge Theories

We compute the elliptic genera of general two-dimensional N=(2,2) and N=(0,2) gauge theories. We find that the elliptic genus is given by the sum of Jeffrey-Kirwan residues of a meromorphic form, representing the one-loop determinant of fields, on the moduli space of flat connections on T^2. We give several examples illustrating our formula, with both Abelian and non-Abelian gauge groups, and discuss some dualities for U(k) and SU(k) theories. This paper is a sequel to the authors' previous paper arXiv:1305.0533.

Partition functions in even dimensional AdS via quasinormal mode methods

In this note, we calculate the one-loop determinant for a massive scalar (with conformal dimension $\Delta$) in even-dimensional AdS$_{d+1}$ space, using the quasinormal mode method developed in arXiv:0908.2657 by Denef, Hartnoll, and Sachdev. Working first in two dimensions on the related Euclidean hyperbolic plane $H_2$, we find a series of zero modes for negative real values of $\Delta$ whose presence indicates a series of poles in the one-loop partition function $Z(\Delta)$ in the $\Delta$ complex plane; these poles contribute temperature-independent terms to the thermal AdS partition function computed in arXiv:0908.2657. Our results match those in a series of papers by Camporesi and Higuchi, as well as Gopakumar et al. in arXiv:1103.3627 and Banerjee et al. in arXiv:1005.3044. We additionally examine the meaning of these zero modes, finding that they Wick-rotate to quasinormal modes of the AdS$_2$ black hole. They are also interpretable as matrix elements of the discrete series representations of $SO(2,1)$ in the space of smooth functions on $S^1$. We generalize our results to general even dimensional AdS$_{2n}$, again finding a series of zero modes which are related to discrete series representations of $SO(2n,1)$, the motion group of $H_{2n}$.

Rényi entropies of free bosons on the torus and holography

We analytically evaluate the Renyi entropies for the two dimensional free boson CFT. The CFT is considered to be compactified on a circle and at finite temperature. The Renyi entropies S_n are evaluated for a single interval using the two point function of bosonic twist fields on a torus. For the case of the compact boson, the sum over the classical saddle points results in the Riemann-Siegel theta function associated with the A_{n-1} lattice. We then study the Renyi entropies in the decompactification regime. We show that in the limit when the size of the interval becomes the size of the spatial circle, the entanglement entropy reduces to the thermal entropy of free bosons on a circle. We then set up a systematic high temperature expansion of the Renyi entropies and evaluate the finite size corrections for free bosons. Finally we compare these finite size corrections both for the free boson CFT and the free fermion CFT with the one-loop corrections obtained from bulk three dimensional handlebody spacetimes which have higher genus Riemann surfaces as its boundary. One-loop corrections in these geometries are entirely determined by quantum numbers of the excitations present in the bulk. This implies that the leading finite size corrections contributions from one-loop determinants of the Chern-Simons gauge field and the Dirac field in the dual geometry should reproduce that of the free boson and the free fermion CFT respectively. By evaluating these corrections both in the bulk and in the CFT explicitly we show that this expectation is indeed true.

Continuity, deconfinement, and (super) Yang-Mills theory

We study the phase diagram of SU(2) Yang-Mills theory with one adjoint Weyl fermion on R^3xS^1 as a function of the fermion mass m and the compactification scale L. This theory reduces to thermal pure gauge theory as m->infinity and to circle-compactified (non-thermal) supersymmetric gluodynamics in the limit m->0. In the m-L plane, there is a line of center symmetry changing phase transitions. In the limit m->infinity, this transition takes place at L_c=1/T_c, where T_c is the critical temperature of the deconfinement transition in pure Yang-Mills theory. We show that near m=0, the critical compactification scale L_c can be computed using semi-classical methods and that the transition is of second order. This suggests that the deconfining phase transition in pure Yang-Mills theory is continuously connected to a transition that can be studied at weak coupling. The center symmetry changing phase transition arises from the competition of perturbative contributions and monopole-instantons that destabilize the center, and topological molecules (neutral bions) that stabilize the center. The contribution of molecules can be computed using supersymmetry in the limit m=0, and via the Bogomolnyi--Zinn-Justin (BZJ) prescription in the non-supersymmetric gauge theory. Finally, we also give a detailed discussion of an issue that has not received proper attention in the context of N=1 theories---the non-cancellation of nonzero-mode determinants around supersymmetric BPS and KK monopole-instanton backgrounds on R^3xS^1. We explain why the non-cancellation is required for consistency with holomorphy and supersymmetry and perform an explicit calculation of the one-loop determinant ratio.

One-loop corrections to a holographic Wilson loop in AdS 4 × ℂℙ3

The evaluation of BPS Wilson loops in N=6, D=3 Chern-Simons matter theory is reduced to ordinary matrix integrals via localization technique. It is easy to check that the vacuum expectation value of 1/2 BPS Wilson loops at leading order in planar limit agrees with the regularized classical string action, via AdS/CFT. Then the subleading terms in principle can be calculated by treating the string theory semi-classically. In this article we calculate the one-loop determinant for fluctuation modes of holographic Wilson loop in the dual geometry AdS4xCP3. The fermionic normal mode frequencies are expressed in terms of the hypergeometric function, and we compute the one-loop effective action numerically. The discrepancy with localization formula is due to the zero mode normalization constant, which is yet to be determined.

Geometric Quantization and the Metric Dependence of the Self-Dual Field Theory

We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. This connection governs metric dependence of the partition function of the self-dual field. We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. As an application, we show that the one-loop determinant of the (2,0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model.

One-loop partition function from normal modes for supergravity in AdS3

With the recently discovered formula, which relates the off-shell Euclidean one-loop determinant to the on-shell quantities such as normal and quasinormal modes in real spacetime, we work out the one-loop partition function for supergravity in AdS3 from scratch. In passing, we also provide the explicit expression for the one-loop determinant of a field with an arbitrary spin in AdS3. To achieve this, we first derive the determinant expression for the one-loop partition function in question using the powerful decomposition technique, and then we construct the normal modes in a purely algebraic way by demonstrating that the space of normal modes falls into the representation of SL(2,R) Lie algebra associated with AdS3. The whole procedure developed here turns out to be much simpler than the previous strategy.

One-loop effective action of the holographic antisymmetric Wilson loop

We systematically study the spectrum of excitations and the one-loop determinant of holographic Wilson loop operators in antisymmetric representations of $\mathcal{N}=4$ supersymmetric Yang-Mills theory. Holographically, these operators are described by D5-branes carrying electric flux and wrapping an ${S}^{4}\ensuremath{\subset}{S}^{5}$ in the ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ bulk background. We derive the dynamics of both bosonic and fermionic excitations for such D5-branes. A particularly important configuration in this class is the D5-brane with ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{S}^{4}$ world volume and $k$ units of electric flux, which is dual to the circular Wilson loop in the totally antisymmetric representation of rank $k$. For this Wilson loop, we obtain the spectrum, show explicitly that it is supersymmetric and calculate the one-loop effective action using heat kernel techniques.

Higher spin quasinormal modes and one-loop determinants in the BTZ black hole

We solve the wave equations of arbitrary integer spin fields in the BTZ black hole background and obtain exact expressions for their quasinormal modes. We show that these quasinormal modes precisely agree with the location of the poles of the corresponding two point function in the dual conformal field theory as predicted by the AdS/CFT correspondence. We then use these quasinormal modes to construct the one-loop determinant of the higher spin field in the thermal BTZ background. This is shown to agree with that obtained from the corresponding heat kernel constructed recently by group theoretic methods.

Effective string theory and nonlinear Lorentz invariance

We study the low-energy effective action governing the transverse fluctuations of a long string, such as a confining flux tube in QCD. We work in the static gauge where this action contains only the transverse excitations of the string. The static gauge action is strongly constrained by the requirement that the Lorentz symmetry, that is spontaneously broken by the long string vacuum, is nonlinearly realized on the Nambu-Goldstone bosons. One solution to the constraints (at the classical level) is the Nambu-Goto action, and the general solution contains higher derivative corrections to this. We show that in 2+1 dimensions, the first allowed correction to the Nambu-Goto action is proportional to the squared curvature of the induced metric on the worldsheet. In higher dimensions, there is a more complicated allowed correction that appears at lower order than the curvature squared. We argue that this leading correction is similar to, but not identical to, the one-loop determinant (\sqrt{-h} R \Box^{-1} R) computed by Polyakov for the bosonic fundamental string.

The gravitational exclusion principle and null states in anti-de Sitter space

The holographic principle implies a vast reduction in the number of degrees of freedom of quantum gravity. This idea can be made precise in AdS3, where the the stringy or gravitational exclusion principle asserts that certain perturbative excitations are not present in the exact quantum spectrum. We show that this effect is visible directly in the bulk gravity theory: the norm of the offending linearized state is zero or negative. When the norm is negative, the theory is signalling its own breakdown as an effective field theory; this provides a perturbative bulk explanation for the stringy exclusion principle. When the norm vanishes the bulk state is null rather than physical. This implies that certain non-trivial diffeomorphisms must be regarded as gauge symmetries rather than spectrum-generating elements of the asymptotic symmetry group. This leads to subtle effects in the computation of one-loop determinants for Einstein gravity, higher spin theories and topologically massive gravity in AdS3. In particular, heat kernel methods do not capture the correct spectrum of a theory with null states.Communicated by S Ross

Quantum black holes, localization, and the topological string

We use localization to evaluate the functional integral of string field theory on AdS 2 × S 2 background corresponding to the near horizon geometry of supersymmetric black holes in 4d compactifications with \( \mathcal{N} = 2 \) supersymmetry. In particular, for a theory containing n v + 1 vector multiplets, we show that the functional integral localizes exactly onto an ordinary integral over a finite-dimensional submanifold in the field space labeling a continuous family of instanton solutions in which auxiliary fields in the vector multiplets are excited with nontrivial dependence on AdS 2 coordinates. These localizing solutions are universal in that they follow from the off-shell supersymmetry transformations and do not depend on the choice of the action. They are parametrized by n v + 1 real parameters {C I; I = 0,…, n v } that correspond to the values of the auxiliary fields at the center of AdS 2. In the Type-IIA frame, assuming D-terms evaluate to zero on the solutions for reasons of supersymmetry, the classical part of the integrand equals the absolute square of the partition function of the topological string as conjectured by Ooguri, Strominger, and Vafa; however evaluated at the off-shell values of scalar fields at the center of AdS 2. In addition, there are contributions from one-loop determinants, brane-instantons, and nonperturbative orbifolds that are in principle computable. These results thus provide a concrete method to compute exact quantum entropy of these black holes including all perturbative and nonperturbative corrections and can be used to establish a precise relation between the quantum degeneracies of black holes and the topological string.

On instanton effects in F-theory

We revisit the issue of M5-brane instanton corrections to the superpotential in F-theory compactifications on elliptically fibered Calabi-Yau fourfolds. Elaborating on concrete geometries, we compare the instanton zero modes for non-perturbative F-theory models with the zero modes in their perturbative Sen limit. The fermionic matter zero modes localized on the intersection of the instanton with the space-time filling D7-branes show up in a geometric way in F-theory. Methods for their computation are developed and, not surprisingly, exceptional gauge group structures do appear. Finally, quite intriguing geometrical aspects of the one-loop determinant are discussed.

World-sheet instanton superpotentials in heterotic string theory and their moduli dependence

To understand in detail the contribution of a world-sheet instanton to the superpotential in a heterotic string compactification, one has to understand the moduli dependence (bundle and complex structure moduli) of the one-loop determinants from the fluctuations, which accompany the classical exponential contribution (involving Kahler moduli) when evaluating the world-volume partition function. Here we use techniques to describe geometrically these Pfaffians for spectral bundles over rational base curves in elliptically fibered Calabi-Yau threefolds, and provide a (partially exhaustive) list of cases involving factorising (or vanishing) superpotential. This gives a conceptual explanation and generalisation of the few previously known cases which were obtained just experimentally by a numerical computation.

Truncated heat kernel and one-loop determinants for the BTZ geometry

There is a special relation between the spectrum and the {\it truncated} heat kernel of the Euclidean BTZ black hole with the Patterson-Selberg zeta function. Using an orbifold description of this relation we calculate the on-shell corrections of the gravitational quantum fluctuations.

Quantum stabilization of Z-strings, a status report on D = 3 + 1 dimensions

We investigate an extension to the phase shift formalism for calculating one-loop determinants. This extension is motivated by requirements of the computation of Z-string quantum energies in D = 3 + 1 dimensions. A subtlety that seems to imply that the vacuum polarization diagram in this formalism is (erroneously) finite is thoroughly investigated.