Topological Conformal Field Theories Research Articles

Measurement as a Shortcut to Long-Range Entangled Quantum Matter

The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback (``adaptive circuits'') can evade such restrictions. We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs). The three classes are inspired by distinct physical insights, including tensor-network constructions, multiscale entanglement renormalization ansatz (MERA), and parton constructions. A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and one-dimensional CFT states and non-abelian topological orders with both solvable and non-solvable groups can be prepared in depth scaling logarithmically with system size. We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary CSS (Calderbank-Shor-Steane) codes. Our work illustrates the practical and conceptual versatility of measurement for state preparation.

The Trace Field Theory of a Finite Tensor Category

Given a finite tensor category $\mathcal {C}$ , we prove that a modified trace on the tensor ideal of projective objects can be obtained from a suitable trivialization of the Nakayama functor as right $\mathcal {C}$ -module functor. Using a result of Costello, this allows us to associate to any finite tensor category equipped with such a trivialization of the Nakayama functor a chain complex valued topological conformal field theory, the trace field theory. The trace field theory topologically describes the modified trace, the Hattori-Stallings trace, and also the structures induced by them on the Hochschild complex of $\mathcal {C}$ . In this article, we focus on implications in the linear (as opposed to differential graded) setting: We use the trace field theory to define a non-unital homotopy commutative product on the Hochschild chains in degree zero. This product is block diagonal and can be described through the handle elements of the trace field theory. Taking the modified trace of the handle elements recovers the Cartan matrix of $\mathcal {C}$

Nonliquid cellular states: Gluing gauge-higher-symmetry-breaking versus gauge-higher-symmetry-extension interfacial defects

The existence of quantum nonliquid states and fracton orders, both gapped and gapless states, challenges our understanding of phases of entangled matter. We generalize the cellular topological states to liquid or nonliquid cellular states. We propose a mechanism to construct more general non-Abelian states by gluing gauge-symmetry-breaking versus gauge-symmetry-extension interfaces as extended defects in a cellular network, including defects of higher symmetries, in any dimension. Our approach also naturally incorporates the anyonic particle/string condensations and composite string (related to particle-string or p-string)/membrane condensations. This approach shows gluing the familiar extended topological quantum field theory or conformal field theory data via topology, geometry, and renormalization consistency criteria (via certain modified group cohomology or cobordism theory data) in a tensor network can still guide us to analyze the nonliquid states. (Part of the Abelian construction can be understood from the $K$-matrix Chern-Simons theory approach and the coupled-layer-by-junction constructions.) This approach may also lead us toward a unifying framework for quantum systems of both higher symmetries and subsystem/subdimensional symmetries.

On the structure of open equivariant topological conformal field theories

A classification of open equivariant topological conformal field theories in terms of Calabi-Yau $A_\infty $-categories endowed with a group action is presented.

The topological symmetric orbifold

We analyze topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the cohomology of the Hilbert scheme on the boundary.

Two-Dimensional Non-abelian BF Theory in Lorenz Gauge as a Solvable Logarithmic TCFT

We study two-dimensional non-abelian BF theory in Lorenz gauge and prove that it is a topological conformal field theory. This opens the possibility to compute topological string amplitudes (Gromov-Witten invariants). We found that the theory is exactly solvable in the sense that all correlators are given by finite-dimensional convergent integrals. Surprisingly, this theory turns out to be logarithmic in the sense that there are correlators given by polylogarithms and powers of logarithms. Furthermore, we found fields with "logarithmic conformal dimension" (elements of a Jordan cell for $L_0$). We also found certain vertex operators with anomalous dimensions that depend on the non-abelian coupling constant. The shift of dimension of composite fields may be understood as arising from the dependence of subtracted singular terms on local coordinates. This generalizes the well-known explanation of anomalous dimensions of vertex operators in the free scalar field theory.

HIERARCY OF WDVV ASSOCIATIVITY EQUATIONS FOR n = 3 AND N = 2 CASE WHEN = 0 V0 WITH NEW SYSTEM t t t a ,b ,c

We investigate solutions of Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) equations. The articlediscusses nonlinear equations of the third order for a function f = f(x,t)) of two independent variables x,t. Theequations of associativity reduce to the nonlinear equations of the third order for a function f = f(x,t)) whenprepotential F dependet of the metric η. In this work we consider the WDVV equation for n = 3 case with anantidiagonal metric η. The solution of some cases of hierarchy equations of associativity illustrated. Lax pairs for thesystem of three equations, that contains the equation of associativity are written to find the hierarchy of associativityequation. Using the compatibility condition are found the relations between the matrices U, V2, V1. The elements ofmatrix V2 are found with the expression of zij and independent and dependent variables for the matrix V2. Alsosolving elements of matrix V1 expressed through yij and independent and dependent variables for the matrix V1. Weaccepted that elements of matrix V0 are zero. In the physical setting the solutions of WDVV describe moduli spaceof topological conformal field theories [1, 2]. Let us introduce new variables a, b, c. In the above variables thenonlinear equations of the third order for a function f = f(x,t)) we rewritten as a new system of three equations.Expressed are variables at ,bt ,ct of three equations are written with the help of matrix elements zij ,yij.

Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum

Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking ‘Kitaev chain’ superconductor (symmetry class D), the Haldane phase, and the classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.

Quantum Hall physics: Hierarchies and conformal field theory techniques

The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past thirty years, has generated many groundbreaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the thirty years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states, and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of nonabelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.

On the structure of unoriented topological conformal field theories

We give a classification of open Klein topological conformal field theories in terms of Calabi–Yau $$A_\infty $$ -categories endowed with an involution. Given an open Klein topological conformal field theory, there is a universal open-closed extension whose closed part is the involutive variant of the Hochschild chains of the open part.

Flat coordinates for Saito Frobenius manifolds and string theory

It was shown in \cite{DVV} for $2d$ topological Conformal field theory (TCFT) \cite{EY,W} and more recently in \cite{BSZ}-\cite{BB2} for the non-critical String theory \cite{P}-\cite{BAlZ} that a number of models of these two types can be exactly solved using their connection with the Frobenius manifold (FM) structure introduced by Dubrovin\cite{Dub}. More precisely these models are connected with a special case of FMs, so called Saito Frobenius manifolds (SFM)\cite{Saito} (originally called Flat structure together with the Flat coordinate system), which arise on the space of the versal deformations of the isolated Singularities after choosing of a suitabe so-called Primitive form, and which also arises on the quotient spaces by reflection groups. In this paper we explore the connection of the models of TCFT and non-critical String theory with SFM. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of SFMs as functions of the parameters of the deformed singularity. We suggest a direct way to find the flat coordinates, using the integral representation for the solutions of Gauss-Manin system connected with the corresponding SFM for a simple singularity. Also, we address the possible generalization of our approach for the models investigated in \cite{Gep} which are ${SU(N)_k}/({SU(N-1)_{k+1} \times U(1)})$ Kazama-Suzuki theories \cite{KS}.

Flat structures on the deformations of Gepner chiral rings

We propose a simple method for the computation of the flat coordinates and Saito primitive forms on Frobenius manifolds of the deformations of Jacobi rings associated with isolated singularities. The method is based on using a conjecture about integral representations for the flat coordinates and on the Saito cohomology theory. This reduces the computation to a simple linear problem. We consider the case of the deformed Gepner chiral rings. The knowledge of the flat structures of Frobenius manifolds can be used for exact solution of the models of the topological conformal field theories corresponding to these chiral rings.

Relation d’équivalence sur les algèbres quasi-Batalin–Vilkovisky différentielles

Differential quasi-Batalin–Vilkovisky algebras are generalisations of differential Batalin–Vilkovisky algebras, introduced by Getzler for applications to topological conformal field theory. As for Lie quasi-bialgebra structures on a finite-dimensional vector space \(\mathcal{F}\), we define an equivalence relation on unimodular differential quasi-Batalin–Vilkovisky algebra structures on \(\Lambda \mathcal{F}^*\), the exterior algebra of the dual of \(\mathcal{F}\). Therefore, using the relation between Lie quasi-bialgebras and quasi-Batalin–Vilkovisky algebras, we show that unimodular differential quasi-Batalin–Vilkovisky algebra structures on \(\Lambda \mathcal{F}^*\) are equivalent if and only if the corresponding Lie quasi-bialgebra structures on \(\mathcal{F}\) are equivalent.

Flat coordinates of topological conformal field theory and solutions of the Gauss–Manin system

It was shown many years ago by Dijkgraaf, Velinde, and Verlinde for two-dimensional topological conformal field theory and more recently for the non-critical String theory that some models of these two types can be solved using their connection to the special case of Frobenius manifolds—the so-called Saito Frobenius manifolds connected to a deformed singularity. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of Saito Frobenius manifolds as functions of the parameters of the deformed singularity. We suggest a direct way to find the flat coordinates, using the integral representation for the solutions of the Gauss–Manin system connected to the corresponding Saito Frobenius manifold for the singularity.

Morse homotopy and topological conformal field theory

By studying spaces of flow graphs in a closed oriented manifold, we equip the Morse complex with the operations of an open topological conformal field theory. This complements previous constructions due to R. Cohen et al., K. Costello, K. Fukaya and M. Kontsevich and is also the Morse theoretic counterpart to a conjectural construction of operations on the chain complex of the Lagrangian Floer homology of the zero section of a cotangent bundle, obtained by studying uncompactified moduli spaces of higher genus pseudoholomorphic curves.

Surface defects and symmetries

In quantum field theory, defects of various codimensions are natural ingredients and carry a lot of interesting information. In this contribution we concentrate on topological quantum field theories in three dimensions, with a particular focus on Dijkgraaf-Witten theories with abelian gauge group. Surface defects in Dijkgraaf-Witten theories have applications in solid state physics, topological quantum computing and conformal field theory. We explain that symmetries in these topological field theories are naturally defined in terms of invertible topological surface defects and are thus Brauer-Picard groups.

Universal operations in Hochschild homology

AbstractWe provide a general method for finding all natural operations on the Hochschild complex ofℰ${\mathcal{E}}$-algebras, whereℰ${\mathcal{E}}$is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative orA∞${A_{\infty}}$-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex offormal operations, for which we provide an explicit model that we can calculate in a number of cases. Whenℰ${\mathcal{E}}$encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich–Soibelman via different methods identify with all formal operations. Whenℰ${\mathcal{E}}$encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler–Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.

Involutive $$A_\infty $$ -algebras and dihedral cohomology

We define and study the cohomology theories associated to $$A_\infty $$ -algebras and cyclic $$A_\infty $$ -algebras equipped with an involution, generalising dihedral cohomology to the $$A_\infty $$ context. Such algebras arise, for example, as unoriented versions of topological conformal field theories. It is well known that Hochschild cohomology and cyclic cohomology govern, in a precise sense, the deformation theory of $$A_\infty $$ -algebras and cyclic $$A_\infty $$ -algebras and we give analogous results for the deformation theory in the presence of an involution. We also briefly discuss generalisations of these constructions and results to homotopy algebras over Koszul operads, such as $$L_\infty $$ -algebras or $$C_\infty $$ -algebras equipped with an involution.

Moduli spaces of Klein surfaces and related operads

We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul and we identify the dual dg operad governing A-infinity algebras with involution in terms of Mobius graphs which are a generalisation of ribbon graphs. We then generalise open topological conformal field theories to open Klein topological conformal field theories and give a generators and relations description of the open KTCFT operad. We deduce an analogue of the ribbon graph decomposition of the moduli spaces of Riemann surfaces: a Mobius graph decomposition of the moduli spaces of Klein surfaces (real algebraic curves). The Mobius graph complex then computes the homology of these moduli spaces. We also obtain a different graph complex computing the homology of the moduli spaces of admissible stable symmetric Riemann surfaces which are partial compactifications of the moduli spaces of Klein surfaces.

Homotopy Batalin–Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin–Vilkovisky algebras. Thus it defines the notion of homotopy Batalin–Vilkovisky algebras with the required homotopy properties. To define this resolution, we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin–Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincaré–Birkhoff–Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad \mathcal{BV} . We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.