Fermionic Zero Modes Research Articles

Discrete gauge anomalies and instantons

We revisit anomalous phases related to large gauge transformations, such as the Witten anomaly. The latter, known to plague d = 4 Sp(k) theories, is well-understood in terms of π4(Sp(k)) = ℤ2, but it also has an oblique relation to the instantons, labeled by π3(G) = ℤ, via the fermion zero mode counting. We revisit this relation and point out how SU(N) theories escape an anomalous sign of the latter type, only thanks to the perturbative anomaly cancelation condition that restricts the chiral fermion spectrum. This leads to the question of what happens if the latter, more mundane anomaly is canceled by an inflow instead. After raising an open question about fractional D3 probe theories, we explore the simplest bottom-up model of such a kind, due to Witten and Yonekura, from which we find the relevant chiral theories to be free of such a disease despite the unrestricted chiral spectra. We close with a simple but often-overlooked observation about how fermionic zero modes enter physics differently between Euclidean and Lorentzian descriptions and point out a related issue in d = 3.

Spectral flow of fermions in the ℂP2 (anti-)instanton, and the sphaleron with vanishing topological charge

The spectral flow is ubiquitous in the physics of soliton-fermion interacting systems. We study the spectral flows related to a continuous deformation of background soliton solutions, which enable us to develop insight into the emergence of fermionic zero modes and the localization mechanism of fermion densities. We investigate a ℂP2 nonlinear sigma model in which there are the (anti-) instantons and also the sphalerons with vanishing topological charge. The standard Yukawa coupling of the fermion successfully generates infinite towers of the spectra and the spectral flow is observed when increasing the size of such solitons. At that moment, the localization of the fermions on the solitons emerges. The avoided crossings are also observed in several stages of the exchange of the flows, they are indicating a manifestation of the fermion exchange of the localizing nature.

Supersymmetry and trace formulas. Part I. Compact Lie groups

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula.

Modified supersymmetric indices in AdS3/CFT2

We consider the \U0001d4a9 = (2, 2) AdS3/CFT2 dualities proposed by Eberhardt, where the bulk geometry is AdS3 × (S3 × T4)/ℤk, and the CFT is a deformation of the symmetric orbifold of the supersymmetric sigma model T4/ℤk (with k = 2, 3, 4, 6). The elliptic genera of the two sides vanish due to fermionic zero modes, so for microstate counting applications one must consider modified supersymmetric indices. In an analysis similar to that of Maldacena, Moore, and Strominger for the standard \U0001d4a9 = (4, 4) case of T4, we study the appropriate helicity-trace index of the boundary CFTs. We encounter a strange phenomenon where a saddle-point analysis of our indices reproduces only a fraction (respectively 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}, 23\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{2}{3} $$\\end{document}, 34\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{3}{4} $$\\end{document}, 56\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{5}{6} $$\\end{document}) of the Bekenstein-Hawking entropy of the associated macroscopic black branes.

A nontopological soliton in an [formula omitted] supersymmetric Abelian gauge model

A version of N=1 supersymmetric scalar electrodynamics is considered here, and it is shown that an electrically charged nontopological soliton exists in this model. In addition to the long-range electric field, the soliton also possesses a long-range scalar field, which leads to a modification of the intersoliton interaction potential at large distances. The supersymmetry of the model makes it possible to express fermionic zero modes of the soliton in terms of bosonic fields. The properties of the nontopological soliton are investigated using analytical and numerical methods.

New conjecture on exact Dirac zero-modes of lattice fermions

Abstract We propose a new conjecture on the relation between the exact Dirac zero-modes of free and massless lattice fermions and the topology of the manifold on which the fermion action is defined. Our conjecture claims that the maximal number of exact Dirac zero-modes of fermions on finite-volume and finite-spacing lattices defined by a discretizing torus, hyperball, their direct-product space, and hypersphere is equal to the summation of the Betti numbers of their manifolds if several specific conditions on lattice formulations are satisfied. We start with reconsidering exact Dirac zero-modes of naive fermions on the lattices whose topologies are a torus, hyperball, and their direct-product space (TD × Bd). We find that the maximal number of exact zero-modes of free Dirac fermions is in exact agreement with the sum of Betti numbers $\sum ^{D}_{r=0} \beta _{r}$ for these manifolds. Indeed, the 4D lattice fermion on a torus has up to 16 zero-modes while the sum of Betti numbers of T4 is 16. This coincidence holds also for the D-dimensional hyperball and their direct-product space TD × Bd. We study several examples of lattice fermions defined on a certain discretized hypersphere (SD), and find that it has up to two exact zero-modes, which is the same number as the sum of Betti numbers of SD. From these facts, we conjecture the equivalence of the maximal number of exact Dirac zero-modes and the summation of Betti numbers under specific conditions. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory.

Zero modes of fermions trapped by giant vortices

Zero-energy solutions of the Dirac equation for the fermions bound to giant vortices of large winding number n are studied in the abelian Higgs and Chern-Simons Higgs models. The case of Jackiw-Rossi theory of the Majorana states in topological superconductors is discussed in detail. By expanding in inverse powers of n we find an analytic result for asymptotically all n solutions required by the index theorem. In the abelian Higgs model the zero modes fill the vortex core and reveal a universal structure independent of fine details of the gauge and scalar field interactions which, in particular, determines the general properties of the large-n superconducting cosmic strings. On the contrary, for the Chern-Simons Higgs vortices the zero modes are localized on the core boundary and the explicit solution is obtained for the supersymmetric couplings in a self-dual background.

Supersymmetric indices factorize

The extent to which quantum mechanical features of black holes can be understood from the Euclidean gravity path integral has recently received significant attention. In this paper, we examine this question for the calculation of the supersymmetric index. For concreteness, we focus on the case of charged black holes in asymptotically flat four-dimensional mathcal{N} = 2 ungauged supergravity. We show that the gravity path integral with supersymmetric boundary conditions has an infinite family of Kerr-Newman classical saddles with different angular velocities. We argue that fermionic zero-mode fluctuations are present around each of these solutions making their contribution vanish, except for a single saddle that is BPS and gives the expected value of the index. We then turn to non-perturbative corrections involving spacetime wormholes and show that a fermionic zero mode is present in all these geometries, making their contribution vanish once again. This mechanism works for both single- and multi-boundary path integrals. In particular, only disconnected geometries without wormholes contribute to the gravitational path integral which computes the index, and the factorization puzzle that plagues the black hole partition function is resolved for the supersymmetric index. Finally, we classify all other single-centered geometries that yield non-perturbative contributions to the gravitational index of each boundary.

Is the right-handed current contribution to B¯→Xulν decays corrected by the nontrivial vacuum in QCD?

We study violation of quark-hadron duality for $\overline{B}\ensuremath{\rightarrow}{X}_{u}l\ensuremath{\nu}$ and $D\ensuremath{\rightarrow}{X}_{d}l\ensuremath{\nu}$ decays in the presence of the right-handed current operator. For this case, we show that duality violation has some aspects different from the standard model due to the existence of an instanton. In particular, the fermionic zero mode of an intermediate light quark can give a nonvanishing contribution owing to its chirality structure. A duality-violating component that arises from a single instanton is then analytically extracted as a finite-distance singularity, leading to an oscillatory correction to the observable in Minkowski space. Finally, we show that the size of duality violation for the lepton energy distribution in $\overline{B}$ decays can be comparable to the chirally suppressed perturbative contribution, although the absolute size depends on the detail of the QCD vacuum.

A 4D IIB flux vacuum and supersymmetry breaking. Part I. Fermionic spectrum

We consider the type-IIB supergravity vacua that include an internal T5, depend on a single coordinate r and respect a four-dimensional Poincaré symmetry, with the aim of highlighting low-energy spectra with broken supersymmetry and a bounded string coupling. These vacua are characterized by the flux Φ of the self-dual five form in the internal torus, the length ℓ of the interval described by the coordinate r, a dilaton profile that is inevitably constant and a strictly positive dimensionless parameter h. As ℓ → ∞ while retaining finite values for Φ and mathrm{ell}{h}^{-frac{5}{4}} , half of the original ten-dimensional supersymmetry is recovered, while finite values of ℓ break it completely. In the large-ℓ limit one boundary disappears but the other is still present, and is felt as a BPS orientifold by a probe brane. In this paper we focus on the fermionic zero modes and show that, although supersymmetry is broken for finite values of ℓ, they are surprisingly those of four-dimensional N = 4 supergravity coupled to five vector multiplets. Still, the gravitini can acquire masses via radiative corrections, absorbing four of the massless spin- frac{1}{2} modes.

Scattering of fermionic isodoublets on the sine-Gordon kink

The scattering of Dirac fermions on the sine-Gordon kink is studied both analytically and numerically. To achieve invariance with respect to a discrete symmetry, the sine-Gordon model is treated as a nonlinear sigma -model with a circular target space that interacts with fermionic isodublets through the Yukawa interaction. It is shown that the diagonal and antidiagonal parts of the fermionic wave function interact independently with the external field of the sine-Gordon kink. The wave functions of the fermionic scattering states are expressed in terms of the Heun functions. General expressions for the transmission and reflection coefficients are derived, and their dependences on the fermion momentum and mass are studied numerically. The existence condition is found for two fermionic zero modes, and their analytical expressions are obtained. It is shown that the zero modes do not lead to fragmentation of the fermionic charge, but can lead to polarization of the fermionic vacuum. The scattering of the diagonal and antidiagonal fermionic states is found to be significantly different; this difference is shown to be due to the different dependences of the energy levels of these bound states on the fermion mass, and is in accordance with Levinson’s theorem.

D-instanton induced superpotential

We use string field theory to fix the normalization of the D-instanton corrections to the superpotential involving the moduli fields of type II string theory compactified on an orientifold of a Calabi-Yau threefold in the absence of fluxes. We focus on O(1) instantons whose only zero modes are the four bosonic modes associated with translation of the instanton in non-compact directions and a pair of fermionic zero modes associated with the two supercharges broken by the instanton. We work with a generic superconformal field theory and express our answer in terms of the spectrum of open strings on the instanton. We analyse the contribution of multi-instantons of this kind to the superpotential and argue that it vanishes when background fluxes are absent.

Mechanism of skyrmion condensation and pairing for twisted bilayer graphene

When quantum flavor Hall insulator phases of itinerant fermions are disordered by strong quantum fluctuations, the condensation of skyrmion textures of order parameter fields can lead to superconductivity. In this work, we address the mechanism of skyrmion condensation by considering the scattering between (2+1)-dimensional, Weyl fermions and hedgehog type tunneling configurations of order parameters that violate the skyrmion-number conservation law. We show the quantized, flavor Hall conductivity ($\sigma^f_{xy}$) controls the degeneracy of topologically protected, fermion zero-modes, localized on hedgehogs, and the overlap between zero-mode eigenfunctions or 't Hooft vertex determines the nature of pairing. We demonstrate the quantum-disordered, flavor Hall insulators with $\sigma^f_{xy}= 2 N$ lead to different types of charge $2 N e^-$ superconductivity. Some implications for the competition among flavor Hall insulators, the charge $2e^-$ paired states in BCS and pair-density-wave channels, and the composite, charge $4e^-$ superconductors for twisted bilayer graphene are outlined.

Quantum tunneling dynamics in a complex-valued Sachdev-Ye-Kitaev model quench-coupled to a cool bath

The Sachdev-Ye-Kitaev (SYK) model describes interacting fermionic zero modes in zero spatial dimensions, e.g. quantum dot, with interactions strong enough to completely washout quasiparticle excitations in the infrared. In this paper, we consider the complex-valued SYK model at initial temperature $T$ and chemical potential $\mu$ coupled to a large reservoir by a quench at time $t=0$. The reservoir is kept at zero temperature and charge neutrality. We find that the dynamics of the discharging process of the SYK quantum dot reveals a distinctive characteristic of the SYK non-Fermi liquid (nFl) state. In particular, we focus on the tunneling current induced by the quench. We show that the temperature dependent contribution to the current's half-life scales linearly in $T$ at low temperatures for the SYK nFl state, while for the Fermi liquid it scales as $T^2$.

A comment on instantons and their fermion zero modes in adjoint QCD_2

The adjoint 2-dimensional QCD with the gauge group SU(N)/Z_NSU(N)/ZN admits topologically nontrivial gauge field configurations associated with nontrivial \pi_1[SU(N)/Z_N] = Z_Nπ1[SU(N)/ZN]=ZN. The topological sectors are labelled by an integer k=0,\ldots, N-1k=0,…,N−1. However, in contrast to QED_2QED2 and QCD_4QCD4, this topology is not associated with an integral invariant like the magnetic flux or Pontryagin index. These instantons may admit fermion zero modes, but there is always an equal number of left-handed and right-handed modes, so that the Atiyah-Singer theorem, which determines in other cases the number of the modes, does not apply. The mod. 2 argument [1] suggests that, for a generic gauge field configuration, there is either a single doublet of such zero modes or no modes whatsoever. However, the known solution of the Dirac problem for a wide class of gauge field configurations indicates the presence of k(N-k) zero mode doublets in the topological sector k. In this note, we demonstrate in an explicit way that these modes are not robust under a generic enough deformation of the gauge background and confirm thereby the conjecture of Ref. [1]. The implications for the physics of this theory (screening vs. confinement issue) are briefly discussed.

Localization of Dirac Fermions in Finite-Temperature Gauge Theory

It is by now well established that Dirac fermions coupled to non-Abelian gauge theories can undergo an Anderson-type localization transition. This transition affects eigenmodes in the lowest part of the Dirac spectrum, the ones most relevant to the low-energy physics of these models. Here we review several aspects of this phenomenon, mostly using the tools of lattice gauge theory. In particular, we discuss how the transition is related to the finite-temperature transitions leading to the deconfinement of fermions, as well as to the restoration of chiral symmetry that is spontaneously broken at low temperature. Other topics we touch upon are the universality of the transition, and its connection to topological excitations (instantons) of the gauge field and the associated fermionic zero modes. While the main focus is on Quantum Chromodynamics, we also discuss how the localization transition appears in other related models with different fermionic contents (including the quenched approximation), gauge groups, and in different space-time dimensions. Finally, we offer some speculations about the physical relevance of the localization transition in these models.

Electroweak axion string and superconductivity

We study the axion strings with the electroweak gauge flux in the DFSZ axion model and show that these strings, called the electroweak axion strings, can exhibit superconductivity without fermionic zeromodes. We construct three types of electroweak axion string solutions. Among them, the string with W-flux can be lightest in some parameter space, which leads to a stable superconducting cosmic string. We also show that a large electric current can flow along the string due to the Peccei-Quinn scale much higher than the electroweak scale. This large current induces a net attractive force between the axion strings with the same topological charge, which opens a novel possibility that the axion strings form Y-junctions in the early universe.

Robust zero modes in disordered two-dimensional honeycomb lattice with Kekulé bond ordering

Robustness of zero-modes of two-dimensional Dirac fermions is examined numerically for the honeycomb lattice in the presence of Kekulé bond ordering. The split n=0 Landau levels in a magnetic field as well as the zero-modes generated by topological defects in the Kekulé ordering are shown to exhibit anomalous robustness against disorder when the chiral symmetry is respected.

Horizon states and the sign of their index in mathcal{N} = 4 dyons

Classical single centered solutions of 1/4 BPS dyons in mathcal{N} = 4 theories are usually constructed in duality frames which contain non-trivial hair degrees of freedom localized outside the horizon. These modes are in addition to the fermionic zero modes associated with broken supersymmetry. Identifying and removing the hair from the 1/4 BPS index allows us to isolate the degrees of freedom associated with the horizon. The spherical symmetry of the horizon then ensures that index of the horizon states has to be positive. We verify that this is indeed the case for the canonical example of dyons in type IIB theory on K3 × T2 and prove this property holds for a class of states. We generalise this observation to all CHL orbifolds, this involves identifying the hair and isolating the horizon degrees of freedom. We then identify the horizon states for 1/4 BPS dyons in mathcal{N} = 4 models obtained by freely acting ℤ2 and ℤ3 orbifolds of type IIB theory compactified on T6 and observe that the index is again positive for single centred black holes. This observation coupled with the fact the 1/4 BPS index of single centred solutions without removal of the hair violates positivity indicates that there exists no duality frame in these models without non-trivial hair.

Fermion number 1/2 of sphalerons and spectral mirror symmetry

We present a rederivation of the baryon and lepton numbers frac{1}{2} of the SU(2)_L S sphaleron of the standard electroweak theory based on spectral mirror symmetry. We explore the properties of a fermionic Hamiltonian under discrete transformations along a noncontractible loop of field configurations that passes through the sphaleron and whose endpoints are the vacuum. As is well known, CP transformation is not a symmetry of the system anywhere on the loop, except at the endpoints. By augmenting CP with a chirality transformation, we observe that the Dirac Hamiltonian is odd under the new transformation precisely at the sphaleron, and this ensures the mirror symmetry of the spectrum, including the continua. As a consistency check, we show that the fermionic zero mode presented by Ringwald in the sphaleron background is invariant under the new transformation. The spectral mirror symmetry which we establish here, together with the presence of the zero mode, are the two necessary conditions whence the fermion number frac{1}{2} of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency frac{1}{2} of the Dirac sea. The relevance of this analysis to other solutions is also discussed.