BCOV on the Large Hilbert Space

BCOV theory is the closed string field theory of the topological B-model. Since the B-model localizes to constant maps, it is reasonable to expect to describe (at least some significant portion of) the same physics by studying the topological worldline model obtained by reducing the B-model to one dimension. The paper under review studies such a worldline approach and suggests a connection between issues related to picture number and picture-changing in traditional formulations of string field theory (on the one hand) and issues related to gravitational descendants and the degeneracy of the shifted Poisson structure on BCOV theory (on the other).

Here is a rough overview of the structure of the paper: Section 2 reviews the earlier (constrained) formulations of BCOV theory due to Bershadsky-Cecotti-Ooguri-Vafa and Barannikov-Kontsevich, as well as Costello and Li's formulation. Section 3 is devoted to constructing an N=(2,2) supersymmetric worldline model whose observables are identified with holomorphic polyvector fields on the target space. Section 4 discusses the "large Hilbert space" of this model, obtained by inverting a particular ghost field. The very brief Section 5 gives a remark on the role of the holomorphic volume form, and Section 6 offers some outlook.

The idea of using a worldine (rather than worldsheet) model is interesting, as is the idea of relating pictures in string field theory to the degeneracy of the shifted Poisson structure on BCOV theory. These ideas give the paper scientific merit. Each of them raises some immediate questions. The first, for example, leads one to wonder about the relationship between the E1-algebra structure on a topological worldline model and the E2-algebra structure in topological string theory. Both of these structures exist on general grounds, and presumably are related in some fashion in the example the authors consider. The second suggests an intimate connection between the gravitational descendant parameter in BCOV theory and the "picture" field gamma, about which I will say more later in the report. Unfortunately, neither of these ideas is explored further in the paper.

The current version of the paper suffers from a lack of clarity in several places: sometimes scientifically, sometimes just as regards the exposition. As such, I find it difficult to recommend publication in SciPost in its present form. But, as mentioned above, the paper does raise some interesting scientific ideas, and I would be happy to revise my opinion after substantial revisions that address the points detailed in the remainder of this report.

The first point is a stylistic quibble, but nevertheless feels important to mention: The text is prone to side remarks and rarely adheres to a straightforward linear pattern argument, which frequently makes the discussion hard to parse. To take a minor example at random, I quote from the first paragraph of section 2: "Kodaira–Spencer or BCOV is the field theory on the target space of closed topological strings (for the open string case in the A-model, the target space field theory is Chern–Simons theory instead). The worldsheet theory is known as B-model." Do the authors expect the reader to know what the A-model and B-model are? If so, we don't need the last sentence; if not, the parenthetical remark is unhelpful and can't be parsed. And open string field theory is (as far as this referee can tell) never really the topic of discussion anywhere in the paper, so it's not clear why it's being mentioned here at all.

Another random example: In the third paragraph of the introduction, the reader is told: "Regarding the off-shell formulation, the tricky business is usually in the identification of a bilinear form that builds up the kinetic term in the action functional in Lagrangian formalism. In BCOV, an extra complication comes from the fact that there is no degree-preserving pairing for holomorphic vector fields. Instead the minimal model to the cohomology (quasi-isomorphic L-infinity algebra) was recently worked out in [12]." How does the third sentence relate to the rest of what is being said? And why would a "degree-preserving pairing on holomorphic vector fields" be relevant? One expects, at best, a shifted symplectic structure on a complex resolving (some variant of) holomorphic vector fields, which would appear in an action of Maurer-Cartan type for a nondegenerate BV theory. But BCOV theory in the Costello-Li formulation is a well-defined, local, degenerate BV theory; as such, it doesn't suffer from any "complication" related to a nonexistent pairing, and the fact that the minimal model has been worked out explicitly isn't pertinent at all.

This leads me to an important physics point, which is not addressed clearly anywhere in the text. Nondegenerate BV theories (with honest (-1)-shifted symplectic structures) correspond one-to-one, at least locally, to variational formal moduli problems, and thus to "Lagrangian theories" in the fullest sense of the word. The degeneracy of a field theory is a model-independent feature, and is measured by the "universal bulk" of the theory, which is trivial precisely for nondegenerate theories (work of Butson and Yoo). Degenerate theories can be either Poisson (typical for "theories of field strengths") or presymplectic (typical for "theories of gauge potentials.") BCOV theory is a Poisson degenerate BV theory, and its degrees of freedom are, morally speaking, gauge-invariant field strengths. Various non-quasi-isomorphic reformulations or "theories of potentials" exist, and have been studied in the twisted supergravity literature. Depending on the dimension, a nondegenerate theory of potentials may or may not exist; in complex dimension five, it does not (just as type IIB supergravity contains a self-dual four-form, and can thus be formulated either as a presymplectic or as a Poisson BV theory, but not as a nondegenerate one). As such, I don't know how to interpret the authors' claim on page 2 that their construction "results in BV-theory for complex structure deformation with auxiliary fields and a local kinetic term where, however, only the auxiliary fields enjoy a non-linear gauge redundancy." A new version of the paper should take these well-known facts into account and clarify how its constructions stand in relation to them.

A second major physics point, expressed a bit roughly for the sake of brevity: A topological string theory is defined by a choice of Calabi-Yau category; for the B-model, this is coherent sheaves on the target. The target-space fields are given by the Hochschild cohomology of this category, which governs its deformations as a category; there is a natural Gerstenhaber algebra structure on Hochschild cochains (Deligne's conjecture, proved by Tamarkin). In the example of the B-model, the Hochschild-Kostant-Rosenberg theorem identifies Hochschild cohomology with holomorphic polyvector fields on the target.

Coupling to topological gravity (to pass to a topological string theory), the corresponding string field theory is described by the cyclic cohomology, which governs deformations as a Calabi-Yau category, and which is described by the derived invariants of the circle action on Hochschild cochains. This is where Costello and Li's space of fields (PV[[u]], u div) arises. Cyclic homology is naturally a Lagrangian in periodic cyclic homology, which has a similar description but with the "u" parameter inverted. Any Lagrangian in a (-n)-shifted symplectic space canonically carries an (n-1)-shifted Poisson structure (Melani-Safronov). Conversely, any (n-1)-shifted Poisson structure arises in universal fashion from such a Lagrangian map (roughly the construction of Butson and Yoo mentioned above). The Calabi-Yau structure equips periodic cyclic homology with an even symplectic structure, and therefore equips BCOV theory with an odd-shifted Poisson structure.

Some version of this construction seems to be what is happening in the section on the large Hilbert space: the ghost field denoted "gamma" is being identified, in some sense, with the gravitational descendant parameter u in periodic cyclic homology. Of course I understand that making this identification is not the primary focus of the paper, but it would still be good to at least comment on connections of the authors' constructions to the well-developed body of prior work on BCOV theory, especially in regard to such key points such as the degeneracy of the shifted Poisson structure.

Here are the major points that I would like to see addressed:

— Every time a formulation of a theory is introduced or modified, the authors should give a clear and complete list of all of the fields and their names, the action functional or L-infinity structure being considered, and any supplemental maneuvers such as additional constraints (which, strictly speaking, are not allowed in a BV theory, whether degenerate or not). Each distinct formulation should ideally be given an unambiguous name that allows the reader to track what is being talked about. It should be made abundantly clear when the object of discussion is replaced or modified, and this should not happen in flowing text in the middle of a paragraph (as occurs a few times in the present manuscript).
— For each theory considered, the authors should state clearly whether it is odd-shifted symplectic, odd-shifted Poisson, odd-shifted presymplectic, or something else.
— The relationship of the worldline model the authors consider to the dimensional reduction of the topological B-model should be made clear, both with and without ghosts.
— A coherent set of grading conventions should be unambiguously stated at the beginning of the article and adhered to uniformly throughout. The absence of this makes various things hard to parse. For example, I do not understand what an "even differential" is. Usually, in this setting, a differential is a homological vector field. Does an "even differential" mean a self-commuting vector field (so any even vector field)? Or does it mean an vector field whose *square* is zero as a differential operator?
— The same applies to notation, which should be streamlined. As an example, there are at least four different notation in use for the divergence operator on holomorphic polyvector fields: q, div, \del, \del_\Upsilon. I can understand the desire to notationally distinguish the supersymmetry q from the geometric operator, corresponding to the notational distinction being made in the isomorphism in (3.1). But the remainder should be dispensed with.
— The discussion should be revised throughout to use consistent, unambiguous, and meaningful language and to proceed in as linear or clearly structured a fashion as is possible. Remarks that cannot be made to adhere to this standard should be omitted. (As an example, I cite the last sentence of section 3: "Note that although S_BV is a functional, we are effectively taking derivatives of the arguments of the BV action and shoving them in." What does this mean?)
— A quick review of the notion of "picture," including definitions of "picture," "picture number," and "picture changing" that are appropriate to the applications here, should be given.

Finally, here are a couple of minor remarks:

— Top of page 4: "An observation due to Costello and Li fixes this gap, at least on-shell." How is "on-shell" being used here? One major point of Costello and Li's construction is to write a genuine degenerate BV theory (without constrained fields). By comparison, it would be reasonable to call the formulation of Bershadsky et al. "on-shell," if the "shell" is interpreted as meaning the kernel of the divergence operator.
— Below (2.8): "Costello and Li suggested a cubic interaction term." The BCOV action is not cubic, but contains terms of all orders, as is clear from the unnumbered equation defining S_int just below.
— On page 11 (and some other places): An object "H_delbar(PV)" is introduced without definition, and claimed to admit "a quasi-iso to (ker div, dbar)." How is this object defined? It is not the cohomology of the Dolbeault resolution of holomorphic polyvector fields with respect to the delbar operator, since no Calabi-Yau structure (and hence no divergence operator) appear there. It is not the cohomology of the totalization of the dbar and div operators on the same complex, since that is just equivalent to de Rham cohomology. The authors presumably want something like PV[[u]], u*div; this is, in a certain sense, a model for divergence-free holomorphic polyvector fields, but it isn't quasi-isomorphic (there are infinitely many extra copies of the de Rham complex around in higher and higher degrees, coming from the unbounded powers of u).

I thank the authors in advance for their attention to the points raised in this report. I know that extensive and time-consuming revisions are a painful exercise, but I genuinely believe that the paper could be greatly improved and attract a much wider audience if given some attention of this sort. As it stands, the novel ideas it contains risk going unnoticed, which would be unfortunate indeed.

Ask for major revision

I recommend for publication in “SciPost Physics” after the authors reviewed the attached report.

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Publish (easily meets expectations and criteria for this Journal; among top 50%)