Symplectic Fermions Research Articles

Double wreath quasi-Hopf algebras

For a quasi-bialgebra H, we show that the category C of H-bimodules is duoidal and that the so called u-coaugmented bimonoids in C are exactly the quasi-bialgebras with a coalgebra projection. When H is a quasi-Hopf algebra with bijective antipode, we prove that the u-coaugmented bimonoids in C can also be described as what we will call double wreath quasi-Hopf algebras, objects determined by H and pre-bialgebras R within the category of Yetter-Drinfeld modules over H. A particular class of double wreath quasi-Hopf algebras is obtained by deforming with a 2-cocycle the multiplication of a Radford biproduct quasi-Hopf algebra. Other classes of this type are given by the symplectic fermion quasi-Hopf algebras.

Discrete Symplectic Fermions on Double Dimers and Their Virasoro Representation

AbstractA discrete version of the conformal field theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge $$-2$$ - 2 .

Spectrum of mesons in quenched Sp(2N) gauge theories

We report the findings of our extensive study of the spectra of flavored mesons in lattice gauge theories with symplectic gauge group and fermion matter content treated in the quenched approximation. For the Sp(4), Sp(6), and Sp(8) gauge groups, the (Dirac) fermions transform in either the fundamental, or the 2-index, antisymmetric or symmetric, representations. This study sets the stage for future precision calculations with dynamical fermions in the low-mass region of lattice parameter space. Our results have potential phenomenological applications ranging from composite Higgs models, to top (partial) compositeness, to dark matter models with composite, strong-coupling dynamical origin. Having adopted the Wilson flow as a scale-setting procedure, we apply Wilson chiral perturbation theory to extract the continuum and massless limits for the observables of interest. The resulting measurements are used to perform a simplified extrapolation to the large-N limit, hence drawing a preliminary connection with gauge theories with unitary groups. We conclude with a brief discussion of the Weinberg sum rules. Published by the American Physical Society 2024

Generalized unitary evolution for symplectic scalar fermions

The theory of symplectic scalar fermion of LeClair and Neubert is studied. The theory evades the conventional spin-statistics theorem because its Hamiltonian is pseudo Hermitian. The definition of pseudo Hermiticity is examined in the interacting and the Heisenberg picture. For states that evolve under pseudo Hermitian Hamiltonians, we define the appropriate inner-product and matrix element of operators that preserve time translation symmetry. The resulting S-matrix is shown to satisfy the generalized unitarity relation. We clarify the derivation of the symplectic currents and charges. By demanding the currents and charges to be pseudo Hermitian, the global symmetry of the free Lagrangian density reduces from Sp(2, ℂ) to SU(2). By explicit calculations, we show that the LeClair-Neubert model of N quartic self-interacting scalar fermions admits generalized unitary evolution.

Boundaries and interfaces with localized cubic interactions in the O(N) model

We explore a new approach to boundaries and interfaces in the O(N) model where we add certain localized cubic interactions. These operators are nearly marginal when the bulk dimension is 4 − ϵ, and they explicitly break the O(N) symmetry of the bulk theory down to O(N − 1). We show that the one-loop beta functions of the cubic couplings are affected by the quartic bulk interactions. For the interfaces, we find real fixed points up to the critical value Ncrit ≈ 7, while for N > 4 there are IR stable fixed points with purely imaginary values of the cubic couplings. For the boundaries, there are real fixed points for all N, but we don’t find any purely imaginary fixed points. We also consider the theories of M pairs of symplectic fermions and one real scalar, which have quartic OSp(1|2M) invariant interactions in the bulk. We then add the Sp(2M) invariant localized cubic interactions. The beta functions for these theories are related to those in the O(N) model via the replacement of N by 1 − 2M. In the special case M = 1, there are boundary or interface fixed points that preserve the OSp(1|2) symmetry, as well as other fixed points that break it.

A duality between vertex superalgebras [formula omitted] and [formula omitted] and generalizations to logarithmic vertex algebras

We introduce a certain subalgebra F‾ of the Clifford vertex superalgebra (bc system), which is completely reducible (as a LVir(−2,0)–module) and C2–cofinite, but not conformal (and not isomorphic to the symplectic fermion algebra SF(1)). We show that there is an interesting duality between SF(1) and F‾, given by the fact that F‾ can be equipped with the structure of a SF(1)–module and vice versa.Using the decomposition of F‾ and a free-field realization from [3], we decompose Lk(osp(1|2)) at the critical level k=−3/2 as a module for Lk(sl(2)). The decomposition of Lk(osp(1|2)) is exactly the same as that of the N=4 superconformal vertex algebra with central charge c=−9, denoted by V(2). Using the duality between F‾ and SF(1), we prove that Lk(osp(1|2)) and V(2) are in a duality of the same type. As an application, we construct and classify all irreducible Lk(osp(1|2))–modules in the category O and the category R, which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra N−3/2(osp(1|2)) as a N−3/2(sl(2))–module.Extending this example, we introduce for each p≥2 a non-conformal vertex algebra Anew(p) and show that Anew(p) is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra Vnew(p), which is isomorphic to the logarithmic vertex algebra V(p) as a module for slˆ(2).We also conjecture the existence of the conformal vertex algebra V(sp(2n)) which is in a duality with the affine vertex algebra L−n−1/2(osp(1|2n)).

Hidden Kac-Moody structures in the fermionic sector of five-dimensional supergravity

We study the supersymmetric quantum dynamics of the cosmological models obtained by reducing $D=5$ supergravity to one timelike dimension. This consistent truncation has fourteen bosonic degrees of freedom, while the quantization of the homogeneous gravitino field leads to a $2^{16}$--dimensional fermionic Hilbert space. We construct a consistent quantization of the model in which the wave function of the Universe is a $2^{16}$--component spinor %\textcolor{red}{of Spin(24,8)} depending on fourteen continuous coordinates, which satisfies eight Dirac-like wave equations (supersymmetry constraints) and one Klein-Gordon-like equation (Hamiltonian constraint). The fermionic part of the quantum Hamiltonian is built from operators that generate a $2^{16}$-dimensional representation of the (infinite-dimensional) maximally compact sub-algebra $K(G_2^{++})$ of the rank-4 hyperbolic Kac--Moody algebra $G_2^{++}$. The (quartic-in-fermions) squared-mass term $\widehat \mu^2$ entering the Klein-Gordon-like equation has several remarkable properties: (i) it commutes with the generators of $K(G_2^{++})$; and (ii) it is a quadratic polynomial in the fermion number $N_F \sim \overline\Psi \Psi$, and a symplectic fermion bilinear $C_F \sim \Psi C\Psi$. Some aspects of the structure of the solutions of our model are discussed, and notably the Kac-Moody meaning of the operators describing the reflection of the wave function on the fermion-dependent potential walls ("quantum fermionic Kac-Moody billiard").

The symplectic fermion ribbon quasi-Hopf algebra and the [formula omitted]-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras Q(N) for N a positive integer: as an algebra, Q(N) is the semidirect product of CZ2 with the direct sum of a Graßmann and a Clifford algebra in 2N generators. We show that RepQ(N) is ribbon equivalent to the symplectic fermion category SF(N) that was computed in [54] from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of Vev, the even part of the symplectic fermion vertex operator super algebra.Using the formalism developed in [20] we compute the projective SL(2,Z)-action on the centre of Q(N) as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the SL(2,Z)-action computed from Q(N) agrees projectively with that on pseudo trace functions of Vev.

Geometric free field realization for the genus-two class S theory of type a1

We present a free field realisation for the vertex operator algebra associated to the genus-two, class $\mathcal{S}$ superconformal field theory of type $\mathfrak{a}_1$. The free field realisation is in the style of recent work by the authors, and is formulated in terms of a one-dimensional isotropic lattice vertex algebra along with two pairs of symplectic fermions. Our realisation makes manifest an enhanced ${\rm USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an ${\rm SU(2)}$ outer automorphism that has been observed in recent work of Kiyoshige and Nishinaka and significantly simplifies the structure of the algebra. Along the way, we also produce a realisation of the generic subregular Drinfel'd-Sokolov $\mathcal{W}$ algebra of type $\mathcal{c}_2$ in terms of the generic principle $\mathcal{W}$ algebra of type $\mathfrak{c}_2$ and a one-dimensional isotropic lattice vertex algebra.

Gentle introduction to rigorous Renormalization Group: a worked fermionic example

Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non- perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal (“long-range”) kinetic term depending on a parameter ε and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with ϕ4 or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in ε, a somewhat surprising fact relying on the fermionic nature of the problem.

On Zhu's algebra and C2–algebra for symplectic fermion vertex algebra SF(d)+

In this paper, we study the family of vertex operator algebras SF(d)+, known as symplectic fermions. This family is of a particular interest because these VOAs are irrational and C2-cofinite. We determine Zhu's algebra A(SF(d)+) and show that the equality of dimensions of A(SF(d)+) and the C2–algebra P(SF(d)+) holds for d≥2 (the case of d=1 was treated by T. Abe in [1]). We use these results to prove a conjecture by Y. Arike and K. Nagatomo ([8]) on the dimension of the space of one-point functions on SF(d)+.

Projective objects and the modified trace in factorisable finite tensor categories

For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1)${\mathcal{C}}$ always contains a simple projective object;(2)if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;(3)the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;(4)the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

Modified traces for quasi-Hopf algebras

Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified traces are non-degenerate, and we show that non-degenerate left/right modified traces can only exist for unimodular H. This generalises results of Beliakova, Blanchet, and Gainutdinov [1] from Hopf algebras to quasi-Hopf algebras. As an example we compute cointegrals and modified traces for the family of symplectic fermion quasi-Hopf algebras.

Free field realizations from the Higgs branch

We present free field realizations for the associated vertex operator algebras of a number of four-dimensional mathcal{N} = 2 superconformal field theories. Our constructions utilize an exceptionally small set of chiral bosons whose number matches the complex dimensionality of the Higgs branch of the superconformal field theory. In the case of theories whose Higgs branches support additional degrees of freedom (free vector multiplets or decoupled interacting SCFTs), the corresponding “free field realizations” include additional ingredients: symplectic fermions in the case of vector multiplets and a C2 co-finite VOA in the case of a residual interacting SCFT. The resulting picture is that the associated VOA can be constructed from the Higgs branch effective theory via free field realization. Our constructions also provide a natural realization of the R-filtration of the associated VOA.

Formula omitted]-action for ribbon quasi-Hopf algebras

We study the universal Hopf algebra L of Majid and Lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi-Hopf algebra A. We show that L=A⁎ with coadjoint action and compute the Hopf algebra structure morphisms of L in terms of the defining data of A. We give explicitly the condition on A which makes Rep A factorisable and compute Lyubashenko's projective SL(2,Z)-action on the centre of A in this case.The point of this exercise is to provide the groundwork for the applications to ribbon categories arising in logarithmic conformal field theories – in particular symplectic fermions and Wp-models – and to test a conjectural non-semisimple Verlinde formula.

The non-semisimple Verlinde formula and pseudo-trace functions

Using results of Shimizu on internal characters we prove a useful non-semisimple variant of the categorical Verlinde formula for factorisable finite tensor categories. Conjecturally, examples of such categories are given by the representations RepV of a vertex operator algebra V subject to certain finiteness conditions. Combining this with results on pseudo-trace functions by Miyamoto and Arike–Nagatomo, one can make a precise conjecture for a non-semisimple modular Verlinde formula which relates modular properties of pseudo-trace functions for V and the product in the Grothendieck ring of RepV. We test this conjecture in the example of the vertex operator algebra of N pairs of symplectic fermions by explicitly computing the modular S-transformation of the pseudo-trace functions.

Logarithmic conformal field theories of type B n, ℓ = 4 and symplectic fermions

There are important conjectures about logarithmic conformal field theories (LCFT’s), which are constructed as a kernel of screening operators acting on the vertex algebra of the rescaled root lattice of a finite-dimensional semisimple complex Lie algebra. In particular, their representation theory should be equivalent to the representation theory of an associated small quantum group. This article solves the case of the rescaled root lattice Bn/2 as a first working example beyond A1/p. We discuss the kernel of short screening operators, its representations, and graded characters. Our main result is that this vertex algebra is isomorphic to a well-known example: The even part of n pairs of symplectic fermions. In the screening operator approach, this vertex algebra appears as an extension of the vertex algebra associated with A1n/2, which are n copies of the even part of one pair of symplectic fermions. The new long screenings give the new global Cn-symmetry. The extension is due to a degeneracy in this particular case: Rescaled long roots still have an even integer norm. For the associated quantum group of divided powers, the first author has previously encountered matching degeneracies: It contains the small quantum group of type A1n and the Lie algebra Cn. Recent results by Farsad, Gainutdinov, and Runkel on symplectic fermions suggest finally the conjectured category equivalence to this quantum group. We also study the other degenerate cases of a quantum group, giving extensions of LCFT’s of type Dn, D4, A2 with larger global symmetry Bn, F4, G2.

Yang–Baxter solution of dimers as a free-fermion six-vertex model

It is shown that Dimers is Yang–Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter . A one-to-many mapping of vertices onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we establish and solve inversion identities for Dimers with periodic boundary conditions on the cylinder. In the particle representation, the local face tile operators give a representation of the fermion algebra, and the fermion particle trajectories play the role of nonlocal (logarithmic) degrees of freedom. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . At the isotropic point, the exact solution allows for the explicit counting of rotated dimer configurations on a periodic rectangular lattice. We show that the modular invariant partition function on the torus is the same as that of symplectic fermions and critical dense polymers. We also show that nontrivial Jordan cells appear for the dimer Hamiltonian on the strip with vacuum boundary conditions. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a logarithmic conformal field theory with central charge , minimal conformal weight and effective central charge .

Logarithmic conformal field theory, log-modular tensor categories and modular forms

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters, and the interpretation of its category of modules as a modular tensor category. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and C2-cofinite vertex operator algebras. We suggest logarithmic variants of those pillars and of Verlinde’s formula. We illustrate our ideas with the -triplet algebras and the symplectic fermions.

The second bosonization of the CKP hierarchy

In this paper we discuss the second bosonization of the Hirota bilinear equation for the CKP hierarchy introduced in the work of Date et al. [J. Phys. Soc. Jpn. 50(11), 3813–3818 (1981)]. We show that there is a second, untwisted, Heisenberg action on the Fock space, in addition to the twisted Heisenberg action suggested by Date et al. [J. Phys. Soc. Jpn. 50(11), 3813–3818 (1981)] and studied in the work of van de Leur et al. [SIGMA 8, 28 (2012)]. We derive the decomposition of the Fock space into irreducible Heisenberg modules under this action. We show that the vector space spanned by the highest weight vectors of the irreducible Heisenberg modules has a structure of a super vertex algebra, specifically the symplectic fermion vertex algebra. We complete the second bosonization of the CKP Hirota equation by expressing the generating field via exponentiated boson vertex operators acting on a polynomial algebra with two infinite sets of variables.