Homotopy Transfer Research Articles

Algebraic structures in closed superstring field theory, homotopy transfer, and effective actions

A consistent action for heterotic and type II superstring field theory was recently proposed by Sen. We give an algebraic formulation of this action in terms of certain twisted L∞ algebra. We further show that Sen’s Wilsonian effective superstring field action can be obtained using homotopy transfer and the effective theory also possesses the algebraic structure of a twisted L∞ algebra. Published by the American Physical Society 2024

Holography as homotopy

We give an interpretation of holography in the form of the AdS/CFT correspondence in terms of homotopy algebras. A field theory such as a bulk gravity theory can be viewed as a homotopy Lie or L∞ algebra. We extend this dictionary to theories defined on manifolds with a boundary, including the conformal boundary of AdS, taking into account the cyclic structure needed to define an action with the correct boundary terms. Projecting fields to their boundary values then defines a homotopy retract, which in turn implies that the cyclic L∞ algebra of the bulk theory is equivalent, up to homotopy, to a cyclic L∞ algebra on the boundary. The resulting action is the ‘on-shell action’ conventionally computed via Witten diagrams that, according to AdS/CFT, yields the generating functional for the correlation functions of the dual CFT. These results are established with the help of new techniques regarding the homotopy transfer of cyclic L∞ algebras.

Open string stub as an auxiliary string field

Witten’s open string field theory with a generalized version of stubs is reformulated as a cubic string field theory using an auxiliary string field. The gauge symmetries and equations of motion as well as the associative algebra of the resulting theory are investigated. Integrating out either the original or auxiliary field is shown to recover the conventional cubic theory. Our analysis demonstrates that deformations due to the stubs can be described as a homotopy transfer purely in the context of strong deformation retract. We also discuss to what extent the vertex regions resulting from stubs provide a model for the elementary interactions of closed string field theory.

Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes

Asymptotic observables in quantum field theory beyond the familiar S-matrix have recently attracted much interest, for instance in the context of gravity waveforms. Such observables can be understood in terms of Schwinger-Keldysh-type ‘amplitudes’ computed by a set of modified Feynman rules involving cut internal legs and external legs labelled by time-folds.In parallel, a homotopy-algebraic understanding of perturbative quantum field theory has emerged in recent years. In particular, passing through homotopy transfer, the S-matrix of a perturbative quantum field theory can be understood as the minimal model of an associated (quantum) L∞-algebra.Here we bring these two developments together. In particular, we show that Schwinger-Keldysh amplitudes are naturally encoded in an L∞-algebra, similar to ordinary scattering amplitudes. As before, they are computed via homotopy transfer, but using deformation-retract data that are not canonical (in contrast to the conventional S-matrix). We further show that the L∞-algebras encoding Schwinger-Keldysh amplitudes and ordinary amplitudes are quasi-isomorphic (meaning, in a suitable sense, equivalent). This entails a set of recursion relations that enable one to compute Schwinger-Keldysh amplitudes in terms of ordinary amplitudes or vice versa.

Field theory equivalences as spans of L∞ -algebras

Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying L∞ -algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of L∞ -algebras in which the quasi-isomorphic L∞ -algebras are obtained from a correspondence L∞ -algebra by a homotopy transfer. Our construction is very useful: homotopy transfer is computationally tractable, and physically, it amounts to integrating out fields in a Feynman diagram expansion. Spans of L∞ -algebras allow for a clean definition of quasi-isomorphisms of cyclic L∞ -algebras. Furthermore, they appear naturally in many contexts within physics. As examples, we first consider scalar field theory with interaction vertices blown up in different ways. We then show that (non-Abelian) T-duality can be seen as a span of L∞ -algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of L∞ -algebras for the Penrose–Ward transform in the context of self-dual Yang–Mills theory and Bogomolny monopoles.

Weakly constrained double field theory as the double copy of Yang-Mills theory

The weakly constrained double field theory, in the sense of Hull and Zwiebach, captures the subsector of string theory on toroidal backgrounds that includes gravity, B-field, and dilaton together with all of their massive Kaluza-Klein and winding modes, which are encoded in doubled coordinates subject to the “weak constraint.” Due to the complications of the weak constraint, this theory was only known to cubic order. Here we construct the quartic interactions for the case that all dimensions are toroidal and doubled. Starting from the kinematic C∞ algebra K of pure Yang-Mills theory and its hidden Lie-type algebra, we construct the L∞ algebra of weakly constrained double field theory on a subspace of the “double copied” tensor product space K⊗K¯, by doing homotopy transfer to the weakly constrained subspace and performing a nonlocal shift that is well-defined on the torus. We test the resulting three-brackets and establish their uniqueness up to cohomologically trivial terms, by verifying the Jacobi identities up to homotopy for the gauge sector. Published by the American Physical Society 2024

Differential Graded Manifolds of Finite Positive Amplitude

Abstract We prove that dg manifolds of finite positive amplitude, that is, bundles of positively graded curved $L_{\infty }[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved $L_{\infty }[1]$-algebras. As an application, we study the derived intersections of manifolds.

Homological quantum mechanics

We provide a formulation of quantum mechanics based on the cohomology of the Batalin-Vilkovisky (BV) algebra. Focusing on quantum-mechanical systems without gauge symmetry we introduce a homotopy retract from the chain complex of the harmonic oscillator to finite-dimensional phase space. This induces a homotopy transfer from the BV algebra to the algebra of functions on phase space. Quantum expectation values for a given operator or functional are computed by the function whose pullback gives a functional in the same cohomology class. This statement is proved in perturbation theory by relating the perturbation lemma to Wick’s theorem. We test this method by computing two-point functions for the harmonic oscillator for position eigenstates and coherent states. Finally, we derive the Unruh effect, illustrating that these methods are applicable to quantum field theory.

Homotopy transfer for QFT on non-compact manifolds with boundary: A case study

In this work we report a homological perturbation calculation to construct effective theories of topological quantum mechanics on R⩾0. Such calculation can be regarded as a generalization of Feynman graph computations. The resulting effective theories fit into a derived BV algebra structure, which generalizes BV quantization. Besides, our construction may serve as the simplest example of a process called “boundary transfer”, which may help study bulk-boundary correspondence.

Open-closed string field theory in the large N limit

We use the new nilpotent formulation of open-closed string field theory to explore the limit where the number N of identical D-branes of the starting background is large. By reformulating the theory in terms of the ’t Hooft coupling λ := κN, where κ is the string coupling constant, we explicitly see that at large N only genus zero vertices with arbitrary number of boundaries survive. After discussing the homotopy structure of the obtained large N open-closed theory we discuss the possibility of integrating out the open string sector with a quantum but planar homotopy transfer. As a result we end up with a classical closed string field theory, described by a weak L∞-algebra containing a tree-level tadpole which, to first order in λ, is given by the initial boundary state. We discuss the possibility of removing the tadpole with a closed string vacuum shift solution, to end up with a new classical closed string background, where the initial D-branes have been turned into pure closed-string backreaction.

The teleparallel complex

We formalise the teleparallel version of extended geometry (including gravity) by the introduction of a complex, the differential of which provides the linearised dynamics. The main point is the natural replacement of the two-derivative equations of motion by a differential which only contains terms of order 0 and 1 in derivatives. Second derivatives arise from homotopy transfer (elimination of fields with algebraic equations of motion). The formalism has the advantage of providing a clear consistency relation for the algebraic part of the differential, the “dualisation”, which then defines the dynamics of physical fields. It remains unmodified in the interacting BV theory, and the full non-linear models arise from covariantisation. A consequence of the use of the complex is that symmetry under local rotations becomes as good as manifest, instead of arising for a specific combination of tensorial terms, for less obvious reasons. We illustrate with a derivation of teleparallel Ehlers geometry, where the extended coordinate module is the adjoint module of a finite-dimensional simple Lie group.

L∞-formality check for the Hochschild complex of certain universal enveloping algebras

We study the L∞-formality problem for the Hochschild complex of the universal enveloping algebra of some examples of Lie algebras such as Cartan-3-regular quadratic Lie algebras (for example semisimple Lie algebras and in more detail so(3)), and free Lie algebras. We show that for these examples formality in Kontsevich's sense does NOT hold, but we compute the L∞ structure on the cohomology given by homotopy transfer in certain cases.

New Objects in Scattering Theory with Symmetries

We consider 1D quantum scattering problem for a Hamiltonian with symmetries. We show that the proper treatment of symmetries in the spirit of homological algebra leads to new objects, generalizing the well known T- and K-matrices. Homological treatment implies that old objects and new ones are be combined in a differential. This differential arises from homotopy transfer of induced interaction and symmetries on solutions of free equations of motion. Therefore, old and new objects satisfy remarkable quadratic equations. We construct an explicit example in SUSY QM on $S^1$ demonstrating nontriviality of the above relation.

The inverse function theorem for curved $L$-infinity spaces

In this paper, we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved L_\infty spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for L_\infty algebras. The main ingredients are the obstruction theory for L_\infty homomorphisms (in the curved setting) and the homotopy transfer theorem for curved L_\infty algebras. Both techniques work in the A_\infty case as well.

Perspectives on the pure spinor superfield formalism

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions.

Filtered 𝐴-infinity structures in complex geometry

We prove a filtered version of the Homotopy Transfer Theorem which gives anAA-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory.

QFT with stubs

The BV-Laplacian ∆ in quantum field theory is singular, by construction, but can be regularized by deforming the classical BV-action. Taking inspiration from string theory we describe a non-local deformation of the latter by adding stubs to the interaction vertices while keeping classical BV-invariance manifest. This is achieved using a version of homotopy transfer resulting in a non-polynomial action for which the quantum master equation is now well defined and will be satisfied by adding additional vertices at loop level. The latter can be defined with the help of standard regularization schemes and is independent of the definition of ∆. In particular, the determination of anomalies reduces to the standard text-book calculation. Finally, we describe how the deformed (quantum) action can be obtained as a canonical transformation. As an example, we illustrate this procedure for quantum electrodynamics.

Homotopy transfer and formality

Homotopy transfer and formality

Homotopy Transfer and Effective Field Theory II: Strings and Double Field Theory

AbstractWe continue our study of effective field theory via homotopy transfer of ‐algebras, and apply it to tree‐level non‐Wilsonian effective actions of the kind discussed by Sen in which the modes integrated out are comparable in mass to the modes that are kept. We focus on the construction of effective actions for string states at fixed levels and in particular on the construction of weakly constrained double field theory. With these examples in mind, we discuss closed string theory on toroidal backgrounds and resolve some subtle issues involving vertex operators, including the proper form of cocycle factors and of the reflector state. This resolves outstanding issues concerning the construction of covariant closed string field theory on toroidal backgrounds. The weakly constrained double field theory is formally obtained from closed string field theory on a toroidal background by integrating out all but the ‘doubly massless’ states and homotopy transfer then gives a prescription for determining the theory's vertices and symmetries. We also discuss consistent truncation in the context of homotopy transfer.

Homotopy Transfer and Effective Field Theory I: Tree‐level

AbstractWe use the dictionary between general field theories and strongly homotopy algebras to provide an algebraic formulation of the procedure of integrating out of degrees of freedom in terms of homotopy transfer. This includes more general effective theories in which some massive modes are kept while other modes of a comparable mass scale are integrated out, as first explored by Sen in the context of closed string field theory. We treat ‐algebras both in terms of a nilpotent coderivation and, on the dual space, in terms of a nilpotent derivation (corresponding to the BRST charge of the field theory) and provide explicit formulas for homotopy transfer. These are then shown to govern the integrating out of degrees of freedom at tree level, while the generalization to loop level will be explored in a sequel to this paper.