Dimensional Field Theory Research Articles

Interactions as intertwiners in 4D QFT

In a recent paper we showed that the correlators of free scalar field theory in four dimensions can be constructed from a two dimensional topological field theory based on so(4,2) equivariant maps (intertwiners). The free field result, along with results of Frenkel and Libine on equivariance properties of Feynman integrals, are developed further in this paper. We show that the coefficient of the log term in the 1-loop 4-point conformal integral is a projector in the tensor product of so(4,2) representations. We also show that the 1-loop 4-point integral can be written as a sum of four terms, each associated with the quantum equation of motion for one of the four external legs. The quantum equation of motion is shown to be related to equivariant maps involving indecomposable representations of so(4,2), a phenomenon which illuminates multiplet recombination. The harmonic expansion method for Feynman integrals is a powerful tool for arriving at these results. The generalization to other interactions and higher loops is discussed.

N = 3 $$ \mathcal{N}=3 $$ four dimensional field theories

We introduce a class of four dimensional field theories constructed by quotienting ordinary $\mathcal{N}=4$ $U(N)$ SYM by particular combinations of R-symmetry and $SL(2,\mathbb{Z})$ automorphisms. These theories appear naturally on the worldvolume of D3 branes probing terminal singularities in F-theory, where they can be thought of as non-perturbative generalizations of the O3 plane. We focus on cases preserving only 12 supercharges, where the quotient gives rise to theories with coupling fixed at a value of order one. These constructions possess an unconventional large $N$ limit described by a non-trivial F-theory fibration with base $AdS_5\times (S^5/\mathbb{Z}_k)$. Upon reduction on a circle the $\mathcal{N}=3$ theories flow to well-known $\mathcal{N}=6$ ABJM theories.

Vortex operator and BKT transition in Abelian duality

We give a new simple derivation for the sine-Gordon description of Berezinskii–Kosterlitz–Thouless (BKT) phase transition. Our derivation is simpler than traditional derivations. Besides, our derivation is a continuous field theoretic derivation by using path integration, different from the traditional derivations which are based on lattice theory or based on Coulomb gas model. Our new derivation relies on Abelian duality of two dimensional quantum field theory. By utilizing this duality in path integration, we find that the vortex configurations are naturally mapped to exponential operators in dual description. Since these operators are the vortex operators that can create vortices, the sine-Gordon description then naturally follows. Our method may be useful for the investigation to the BKT physics of superconductors.

Quantum entanglement of fermionic local operators

In this paper we study the time evolution of (Renyi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Renyi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.

Entanglement and mutual information in two-dimensional nonrelativistic field theories

We carry out a systematic study of entanglement entropy in nonrelativistic conformal field theories via holographic techniques. After a discussion of recent results concerning Galilean conformal field theories, we deduce a novel expression for the entanglement entropy of (1+1)-dimensional Lifshitz field theories --- this is done both at zero and finite temperature. Based on these results, we pose a conjecture for the anomaly coefficient of a Lifshitz field theory dual to new massive gravity. It is found that the Lifshitz entanglement entropy at finite temperature displays a striking similarity with that corresponding to a flat space cosmology in three dimensions. We claim that this structure is an inherent feature of the entanglement entropy for nonrelativistic conformal field theories. We finish by exploring the behavior of the mutual information for such theories.

Gravity as a four dimensional algebraic quantum field theory

Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth $4$-manifold, a manifestly covariant $4$ dimensional and non-perturbative algebraic quantum field theory formulation of gravity is exhibited. More precisely among the bounded linear operators acting on these representation spaces we identify algebraic curvature tensors hence a net of local quantum observables can be constructed from $C^*$-algebras generated by local curvature tensors and vector fields. This algebraic quantum field theory is extracted from structures provided by an oriented smooth $4$-manifold only hence possesses a diffeomorphism symmetry. In this way classical general relativity exactly in $4$ dimensions naturally embeds into a quantum framework. Several Hilbert space representations of the theory are found. First a "tautological representation" of the limiting global $C^*$-algebra is constructed allowing to associate to any oriented smooth $4$-manifold a von Neumann algebra in a canonical fashion. Secondly, influenced by the Dougan--Mason approach to gravitational quasilocal energy-momentum, we construct certain representations what we call "positive mass representations" with unbroken diffeomorphism symmetry. Thirdly, we also obtain "classical representaions" with spontaneously broken diffeomorphism symmetry corresponding to the classical limit of the theory which turns out to be general relativity. Finally we observe that the whole family of "positive mass representations" comprise a $2$ dimensional conformal field theory in the sense of G. Segal.

Loops in exceptional field theory

We study certain four-graviton amplitudes in exceptional field theory in dimensions $D\geq 4$ up to two loops. As the formulation is manifestly invariant under the U-duality group $E_{11-D}(\mathbb{Z})$, our resulting expressions can be expressed in terms of automorphic forms. In the low energy expansion, we find terms in the M-theory effective action of type $R^4$, $\nabla^4R^4$ and $\nabla^6 R^4$ with automorphic coefficient functions in agreement with independent derivations from string theory. This provides in particular an explicit integral formula for the exact string theory $\nabla^6 R^4$ threshold function. We exhibit moreover that the usual supergravity logarithmic divergences cancel out in the full exceptional field theory amplitude, within an appropriately defined dimensional regularisation scheme. We also comment on terms of higher derivative order and the role of the section constraint for possible counterterms.

Defects in cohomological gauge theory and Donaldson–Thomas invariants

Donaldson-Thomas theory on a Calabi-Yau can be described in terms of a certain sixdimensional cohomological gauge theory. We introduce a certain class of defects in this gauge theory which generalize surface defects in four dimensions. These defects are associated with divisors and are defined by prescribing certain boundary conditions for the gauge fields. We discuss generalized instanton moduli spaces when the theory is defined with a defect and propose a generalization of Donaldson-Thomas invariants. These invariants arise by studying torsion free coherent sheaves on Calabi-Yau varieties with a certain parabolic structure along a divisor, determined by the defect. We discuss the case of the affine space as a concrete example. In this case the moduli space of parabolic sheaves admits an alternative description in terms of the representation theory of a certain quiver. The latter can be used to compute the invariants explicitly via equivariant localization. We also briefly discuss extensions of our work to other higher dimensional field theories.

Double Ramification Cycles and Quantum Integrable Systems

In this paper we define a quantization of the Double Ramification Hierarchies of [Bur15b] and [BR14], using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with the first descendant of the unit of the cohomological field theory only. We study various examples which provide, in very explicit form, new $(1+1)$-dimensional integrable quantum field theories whose classical limits are well-known integrable hierarchies such as KdV, Intermediate Long Wave, Extended Toda, etc. Finally we prove polynomiality in the ramification multiplicities of the integral of any tautological class over the double ramification cycle.

Flowing to higher dimensions: a new strongly-coupled phase on M2 branes

We describe a one-parameter family of new holographic RG flows that start from $AdS_4 \times S^7$ and go to $\widehat{AdS_5} \times {\cal B}_6$, where ${\cal B}_6$ is conformal to a K\"ahler manifold and $\widehat{AdS_5}$ is Poincar\'e $AdS_5$ with one spatial direction compactified and fibered over ${\cal B}_6$. The new solutions "flow up dimensions," going from the $(2+1)$-dimensional conformal field theory on M2 branes in the UV to a $(3+1)$-dimensional field theory on intersecting M5 branes in the infra-red. The M2 branes completely polarize into M5 branes along the flow and the Poincar\'e sections of the $\widehat{AdS_5}$ are the $(3+1)$-dimensional common intersection of the M5 branes. The emergence of the extra dimension in the infra-red suggests a new strongly-coupled phase of the M2 brane and ABJM theories in which charged solitons are becoming massless. The flow solution is first analyzed by finding a four-dimensional $N \! = \! 2$ supersymmetric flow in $N \! = \! 8$ gauged supergravity. This is then generalized to a one parameter family of non-supersymmetric flows. The infra-red limit of the solutions appears to be quite singular in four dimensions but the uplift to eleven-dimensional supergravity is remarkable and regular (up to orbifolding). Our construction is a non-trivial application of the recently derived uplift formulae for fluxes, going well beyond the earlier constructions of stationary points solutions. The eleven-dimensional supersymmetry is also analyzed and shows how, for the supersymmetric flow, the M2-brane supersymmetry in the UV is polarized entirely into M5-brane supersymmetry in the infra-red.

A Lorentz covariant holoraumy-induced “gadget” from minimal off-shell 4D, N = 1 $$ \mathcal{N}=1 $$ supermultiplets

Starting from three minimal off-shell 4D, $\cal N$ = 1 supermultiplets, using constructions solely defined within the confines of the four dimensional field theory we show the existence of a "gadget" - a member of a class of metrics on the representation space of the supermultiplets - whose values directly and completely correspond to the values of a metric defined on the 1d, $N$ = 4 adinkra networks adjacency matrices corresponding to the projections of the four dimensional supermultiplets.

Lorentz violation correction to the Aharonov-Bohm scattering

In this paper, using a (2+1)-dimensional field theory approach, we study the Aharonov-Bohm (AB) scattering with Lorentz symmetry breaking. We obtain the modified scattering amplitude to the AB effect due to the small Lorentz violation correction in the breaking parameter and prove that up to one loop the model is free from ultraviolet divergences.

Geometries from field theories

We propose a method to define a $d+1$ dimensional geometry from a $d$ dimensional quantum field theory in the $1/N$ expansion. We first construct a $d+1$ dimensional field theory from the $d$ dimensional one via the gradient flow equation, whose flow time $t$ represents the energy scale of the system such that $t\rightarrow 0$ corresponds to the ultra-violet (UV) while $t\rightarrow\infty$ to the infra-red (IR). We then define the induced metric from $d+1$ dimensional field operators. We show that the metric defined in this way becomes classical in the large $N$ limit, in a sense that quantum fluctuations of the metric are suppressed as $1/N$ due to the large $N$ factorization property. As a concrete example, we apply our method to the O(N) non-linear $\sigma$ model in two dimensions. We calculate the three dimensional induced metric, which is shown to describe an AdS space in the massless limit. We finally discuss several open issues in future studies.

Higher derivative terms in three dimensional supersymmetric theories

In this work, we systematically analyze higher derivative terms in the supersymmetric effective actions for three dimensional scalar field theories using $\mathcal{N} =1$ superspace formalism. In these effective actions, we show that auxiliary fields do not propagate and their effective actions can be expressed in terms of the physical fields. So, the theory does not change its field content upon addition of higher derivative terms. We use derivative expansion to generate four, five and six dimensional terms for an interacting scalar field theory with $\mathcal{N} =1$ supersymmetry. We show that along with pure fermionic and bosonic terms, there are various five and six dimensional topological terms that mix bosonic and fermionic fields. Finally, we use these results to obtain higher derivative topological terms in the effective action for two M2-branes.

Anomalies, Chern-Simons terms and black hole entropy

Recent derivations of Cardy-like formulae in higher dimensional field theories have opened up a way of computing, via AdS/CFT, universal contributions to black hole entropy from gravitational Chern-Simons terms. Based on the manifestly covariant formulation of the differential Noether charge for Chern-Simons terms proposed in arXiv:1407.6364, we compute the entropy and asymptotic charges for the rotating charged AdS black holes in higher dimensions at leading order of the fluid/gravity derivative expansion in the Einstein-Maxwell-Chern-Simons system. This gives a result that exactly matches the field theory predictions from Cardy-like formulae.

Entanglement entropy of non-unitary integrable quantum field theory

In this paper we study the simplest massive 1+1 dimensional integrable quantum field theory which can be described as a perturbation of a non-unitary minimal conformal field theory: the Lee–Yang model. We are particularly interested in the features of the bi-partite entanglement entropy for this model and on building blocks thereof, namely twist field form factors. Non-unitarity selects out a new type of twist field as the operator whose two-point function (appropriately normalized) yields the entanglement entropy. We compute this two-point function both from a form factor expansion and by means of perturbed conformal field theory. We find good agreement with CFT predictions put forward in a recent work involving the present authors. In particular, our results are consistent with a scaling of the entanglement entropy given by ceff3log⁡ℓ where ceff is the effective central charge of the theory (a positive number related to the central charge) and ℓ is the size of the region. Furthermore the form factor expansion of twist fields allows us to explore the large region limit of the entanglement entropy and find the next-to-leading order correction to saturation. We find that this correction is very different from its counterpart in unitary models. Whereas in the latter case, it had a form depending only on few parameters of the model (the particle spectrum), it appears to be much more model-dependent for non-unitary models.

Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level $m$, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for $n$-characters on $\infty$-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for $n+1$-dimensional TQFTs produce $n$-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.

P-wave holographic superconductors from Born-Infeld black holes

We obtain (2+1) dimensional p-wave holographic superconductors from charged Born-Infeld black holes in the presence of massive charged vector fields in a bulk AdS4 Einstein-Born-Infeld theory through the AdS4-CF T3 correspondence. Below a certain critical transition temperature the charged black hole develops vector hair that corresponds to charged vector condensate in the strongly coupled (2+1) dimensional boundary field theory that breaks both the U(1) symmetry as well as the rotational invariance. The holographic free energy is computed for the boundary field theory which shows that the vector order parameter exhibits a rich phase structure involving zeroth order, first order, second order and retrograde phase transitions for different values of the backreaction and the Born-Infeld parameters. We numerically compute the ac conductivity for the p-wave superconducting phase of the strongly coupled (2+1) dimensional boundary field theory which also depends on the relative values of the parameters in the theory.

Computations and generation of elements on the Hopf algebra of Feynman graphs

Two programs, feyngen and feyncop, were developed. feyngen is designed to generate high loop order Feynman graphs for Yang-Mills, QED and ϕk theories. feyncop can compute the coproduct of these graphs on the underlying Hopf algebra of Feynman graphs. The programs can be validated by exploiting zero dimensional field theory combinatorics and identities on the Hopf algebra which follow from the renormalizability of the theories. A benchmark for both programs was made.

Entanglement entropy in Galilean conformal field theories and flat holography.

We present the analytical calculation of entanglement entropy for a class of two-dimensional field theories governed by the symmetries of the Galilean conformal algebra, thus providing a rare example of such an exact computation. These field theories are the putative holographic duals to theories of gravity in three-dimensional asymptotically flat spacetimes. We provide a check of our field theory answers by an analysis of geodesics. We also exploit the Chern-Simons formulation of three-dimensional gravity and adapt recent proposals of calculating entanglement entropy by Wilson lines in this context to find an independent confirmation of our results from holography.