Field Redefinitions, T-duality and Solutions in Closed String Field Theories

Yoji Michishita

REVIEW

Field Redefinitions, T-duality and Solutions in Closed String Field Theories

Yoji Michishita

High Energy Physics - Theory · Submitted 23 Feb 2006 · 28 references

Pith's verdict: A genuinely new T-duality argument fixes part of the field redefinition dictionary in closed string field theory, but the claimed reproduction of the effective solutions depends on an unproven derivative-cancellation assumption.

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arXiv IDhep-th/0602251 (version 3)

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CitationYoji Michishita. "Field Redefinitions, T-duality and Solutions in Closed String Field Theories." arXiv:hep-th/0602251 (2006). https://pith.science/paper/J7W23A7G

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Claim paper

Why Pith says this

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Editor's note

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Josh —

The thing to know: this is a real, defensible paper, and the interesting part is not the solutions but the T-duality argument in Section 4. Michishita shows that requiring the field redefinitions between closed string field theory and the low-energy effective theory to commute with T-duality fixes the nonderivative part up to a small set of terms, and he writes down the result: E_hat = e + (undetermined terms with both 1-1 and 2-2 contractions), phi_hat = c + phi - (1/4) E^mu_mu - (1/2) ln det(1 - E/2) + similar. The induction proof is careful, and the universality claim—independent of interaction details and the integrating-out procedure—looks right. That's new and worth taking seriously.

The paper also constructs explicit closed string field theory solutions, in both bosonic and heterotic theories, order by order from linearized solutions, and shows the massless components don't get corrected. Those constructions are plausible and follow the author's earlier open string work. The comparison to the chiral null model and pp-wave with B-field is where it gets shaky. To identify their SFT fields with the effective fields after the field redefinition, the author has to assume that all higher-derivative terms in the field redefinitions vanish when evaluated on these solutions. That is said explicitly in §5 and §6—'assuming this' and 'we assume that terms with derivatives... cancel each other'—but it is not proved. The stress-test note is right: the fact that F, K, A_i and their derivatives are independent functions means each derivative monomial needs its own vanishing coefficient; that is what needs checking, and it isn't checked. If some survive, the reproduced metric and dilaton would pick up extra derivative terms absent from the effective solutions. So the correspondence is conditional, exactly as the reader's verdict says. The abstract overstates this a little, though the body is honest about it.

Secondary issues: the heterotic dilaton theorem is imported from an unpublished reference [22], and the pp-wave identification is only strictly proven on the effective side for linear profiles. Neither is fatal; both are clearly flagged.

Net: the T-duality result stands on its own as a useful step in the field redefinition program. The solution correspondence is a well-motivated conjecture at this point, not a proof. I'd send it to a serious referee—the method deserves scrutiny and the gap might be closable. But I wouldn't cite it for the correspondence without checking the derivative terms first.

Referee report

Significance

If the T-duality derivation stands, this is a genuinely useful result: it extracts interaction-independent, analytic information about the closed-string-field-theory/effective-theory dictionary, and it provides rare concrete closed string field theory configurations that can be matched to known α'-exact effective backgrounds. The explicit proof that e_μν satisfies the full nonlinear Buscher rules, and the induction that fixes the matrix-product/trace part of the nonderivative dictionary, are checkable and constitute the paper's solid core. The paper is also commendably explicit about the terms its method cannot fix and about the assumptions needed for the solution correspondence. However, the claimed reproduction of the effective solutions is conditional on an unproved cancellation of higher-derivative field-redefinition terms, and the heterotic statements rely on an unpublished result; these points must be resolved or clearly downgraded before the central claim is accepted at face value.

Major objections (3)

[§5 (after Eq. (192)) and §6 (after Eq. (218))] The correspondence claimed in Eqs. (193)-(194) and (221)-(222) depends on the assumption that all higher-derivative terms in the field redefinitions cancel when evaluated on the string field theory solutions. The text states this directly ('Assuming this...' in §5 and 'we assume that terms with derivatives in the field redefinitions cancel each other' in §6), but no proof is given. Since F, K, A_i and B_i are arbitrary harmonic functions, a surviving term such as ∂_i∂^i F, ∂_-∂^i A_i, or (∂_i A_j)^2 would introduce derivatives into Ê_μν or φ̂ that are absent from the target expressions (173) and (210). The independence argument shows that derivative monomials are independent functions, but it does not show that their coefficients in the field redefinitions vanish. This is the decisive unchecked link between the T-duality-determined dictionary and the solution correspondence, and it should be proved, or the correspondence should be explicitly presented as conditional on this assumption.
[§3 (paragraph beginning 'We can expect that the dilaton theorem...')] The heterotic version of the field-redefinition dictionary assumes the heterotic dilaton theorem of reference [22], which is cited only as 'to appear'. Because equations (60)-(61) and their T-duality application to the heterotic theory depend on this theorem, the heterotic claims are contingent on an unpublished external result. Please supply a proof or an available preprint, or explicitly state the heterotic results as conditional on this theorem.
[§6 (last paragraph)] The paper itself notes that α'-exactness of the pp-wave solution has been proven only when Â_{i1} and B̂_{i1} are linear in x^{i2}, whereas the string field theory construction in §6 allows arbitrary solutions of the Laplace equations (207)-(209). The claimed reproduction of (210) via (221) therefore requires the derivative-cancellation assumption at least for the linear-profile subclass; otherwise the correspondence should be restricted explicitly to that subclass, and the statement that the general harmonic-profile string field theory solution is reproduced should be removed or qualified.

Minor objections (5)

[Eq. (124)] The string '+and' in the parenthetical 'terms which contain at least one 1-1 contraction +and at least one 2-2 contraction' appears to be a typographical artifact; the 'and' should be removed.
[Eqs. (105)-(108)] The matrix notation in the Buscher rules is compressed, especially in Eq. (108), where the contraction index on (δ_ij + Ê_ij)^{-1}_{ij} is written with repeated ij; using explicit matrix indices such as ((δ + Ê)^{-1})^{jk} would remove ambiguity.
[§5 (paragraph beginning 'Since F, K and Ai are restricted...')] The claim that F, K, A_i and their derivatives are independent functions would benefit from a precise functional-independence statement for generic solutions of (184)-(187); as written, the conclusion that derivative terms 'should cancel' does not follow directly from the preceding sentence alone.
[§6 (beginning of the section)] The split coordinates x^{i1} and x^{i2} are used in the ansatz before their index ranges are explicitly defined; the ranges should be stated at the first occurrence in Eq. (196).
[Abstract and Introduction] The abstract and introduction state that the effective solutions 'are reproduced' from the string field theory solutions without mentioning that the reproduction assumes the cancellation of higher-derivative field-redefinition terms; this assumption should appear in the summary of the paper's claims.

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The paper

The paper claims

T-duality constrains the field-redefinition dictionary in closed string field theory and reproduces known exact solutions.

The reading

This paper tries to establish a concrete dictionary between the massless fields of closed string field theory and those of the low-energy effective string theory. The nonderivative part of the field redefinitions is fixed by requiring that T-duality act consistently on the two sides, without needing the details of the interaction terms or the integrating-out procedure. Applying that dictionary to explicitly constructed solutions reproduces the chiral null model and the pp-wave solution with a B-field, both previously known as $\alpha'$-exact solutions of the effective theory. If the dictionary is right, geometric low-energy solutions can be lifted into closed string field theory, and the remaining ambiguity in the dictionary is confined to terms that vanish on the profiles studied here.

What carries the argument

The load-bearing object is the T-duality transformation used as a constraint. On the string-field side T-duality acts on zero-momentum massless fields as $E'_{\mu a}=E_{\mu a}$, $E'_{\mu i}=-E_{\mu i}$, $\phi'=\phi-\tfrac12 E_{ii}$, while on the effective side it acts through the standard effective-side T-duality rules. Requiring the field-redefinition dictionary to conjugate these two actions fixes the nonderivative terms order by order in $E_{\mu\nu}$. The technical engine is the identity that $e_{\mu\nu}=((\delta-\tfrac12 E)^{-1}E)_{\mu\nu}$ satisfies the effective-side T-duality rules exactly, together with an induction showing that only terms containing at least one 1-1 contraction and at least one 2-2 contraction can escape the constraint; here 1-1 means contracting first indices of two $E$'s and 2-2 means contracting their second indices.

Extended reading notes

Core claim

The central claim is that the nonderivative part of the field redefinition relating the string-field massless field $E_{\mu\nu}$ to the effective-theory field $\hat E_{\mu\nu}$ is forced by T-duality to take the form $\hat E_{\mu\nu}=e_{\mu\nu}+$ (terms with both 1-1 and 2-2 contractions), where $e_{\mu\nu}=E_{\mu\nu}+\tfrac12 E^{\lambda}_{\ \mu}E_{\lambda\nu}+\cdots$ is the matrix-product series $((\delta-\tfrac12 E)^{-1}E)_{\mu\nu}$. The dilaton is similarly fixed as $\hat\phi=c+\phi-\tfrac14 E^{\mu}_{\ \mu}-\tfrac12\ln\det(\delta-\tfrac12 E)+$ (terms with both 1-1 and 2-2 contractions), with $c$ a constant. The undetermined terms are precisely those that cannot be written purely as matrix products and traces of $E_{\mu\nu}$. On the chiral null and pp-wave profiles those undetermined contractions vanish, and the remaining formulas reproduce the known effective solutions: $\hat F=-1/(1+F/4)$, $\hat K=K$, $\hat A_i=A_i$ for the chiral null model, and $\hat E_{\mu\nu}=E_{\mu\nu}+\tfrac12 E_{\mu\lambda}E^{\lambda}_{\ \nu}$, $\hat\phi=c+\phi$ for the pp-wave with B-field.

Load-bearing premise

The argument assumes, without deriving it, that all higher-derivative terms in the field redefinitions cancel when evaluated on the constructed solutions, so that only the nonderivative T-duality-determined terms survive.

Editorial extensions

If this is right

The chiral null model and the pp-wave solution with B-field, both known as $\alpha'$-exact effective solutions, are realized inside closed bosonic and heterotic string field theory as order-by-order solutions.
The nonderivative part of the massless field redefinition is universal: it depends only on T-duality and Lorentz covariance, not on the choice of interaction vertices or the integrating-out scheme.
On these string-field solutions the massless components receive no higher-order corrections, so the comparison with effective-theory solutions is controlled by the nonderivative dictionary alone.
The undetermined 1-1 and 2-2 contraction terms carry the scheme dependence; fixing them requires a direct integrating-out computation, which the paper leaves to future work.
Once a consistent type II closed string field theory exists, the same T-duality argument should give the same nonderivative dictionary for the NS-NS sector.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same T-duality-consistency method could be applied to the tachyon and to the heterotic gauge field, extending the dictionary beyond the NS-NS massless sector that the paper fixes.
Because the undetermined terms are field redefinitions within the effective theory, comparing different closed string field theory schemes would turn those terms into a diagnostic of scheme dependence.
The derivative-cancellation assumption is testable: if one constructs a solution whose profile functions vary nontrivially in the light-cone coordinate so derivatives cannot be tuned away, the higher-derivative T-duality corrections would have to be added and the present dictionary would acquire extra terms.

Original abstract

We investigate classical solutions in closed bosonic string field theory and heterotic string field theory that are obtained order by order starting from solutions of the linearized equations of motion, and we discuss the ``field redefinitions'' which relate massless fields in the string field theory side and the low energy effective theory side. Massless components of the string field theory solutions are not corrected and from them we can infer corresponding solutions in the effective theory: the chiral null model and the pp-wave solution with B-field, which have been known to be alpha'-exact. These two sets of solutions in the two sides look slightly different because of the field redefinitions. It turns out that T-duality is a useful tool to determine them: We show that some part of the field redefinitions can be determined by using the correspondence between T-duality rules in the two sides, irrespective of the detail of the interaction terms and the integrating-out procedure. Applying the field redefinitions, we see that the solutions in the effective theory side are reproduced from the string field theory solutions.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.69 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We investigate classical solutions in closed bosonic string field theory and heterotic string field theory that are obtained order by order starting from solutions of the linearized equations of motion, and we discuss the ``field redefinitions'' which relate massless fields in the string field theory side and the low energy effective theory side. Massless components of the string field theory solutions are not corrected and from them we can infer corresponding solutions in the effective theory: the chiral null model and the pp-wave solution with B-field, which have been known to be alpha'-exact. These two sets of solutions in the two sides look slightly different because of the field redefinitions. It turns out that T-duality is a useful tool to determine them: We show that some part of the field redefinitions can be determined by using the correspondence between T-duality rules in the two sides, irrespective of the detail of the interaction terms and the integrating-out procedure. Applying the field redefinitions, we see that the solutions in the effective theory side are reproduced from the string field theory solutions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate classical solutions in closed bosonic string field theory and heterotic string field theory that are obtained order by order starting from solutions of the linearized equations of motion, and we discuss the ``field redefinitions'' which relate massless fields in the string field theory side and the low energy effective theory side.	contribution 0.99	0.93	0.12	asserted 0.97
s1	Massless components of the string field theory solutions are not corrected and from them we can infer corresponding solutions in the effective theory: the chiral null model and the pp-wave solution with B-field, which have been known to be alpha'-exact.	result 0.95	0.75	0.27	asserted 0.98
s2	These two sets of solutions in the two sides look slightly different because of the field redefinitions.	result 0.66	0.51	0.08	asserted 0.98
s3	It turns out that T-duality is a useful tool to determine them: We show that some part of the field redefinitions can be determined by using the correspondence between T-duality rules in the two sides, irrespective of the detail of the interaction terms and the integrating-out procedure.	method 0.83	0.90	0.36	asserted 1.00
s4	Applying the field redefinitions, we see that the solutions in the effective theory side are reproduced from the string field theory solutions.	result 0.98	0.92	0.41	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.75 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v3

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full text
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levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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A complete, T-duality-invariant, background-independent action for double field theory is constructed and shown to reduce to Einstein gravity coupled to a two-form and dilaton.
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hep-th 2009-04 accept novelty 8.0 of 10

A gauge-invariant field theory of gravity, a Kalb-Ramond field, and a dilaton on a doubled torus, with doubled diffeomorphism symmetry, is constructed to cubic order from closed string field theory.
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hep-th 2014-07 conditional novelty 7.0 of 10

Bosonic string alpha-prime corrections require a distinct, Z2-even deformation of the double field theory gauge algebra, and a cubic four-derivative action invariant under it exists.
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For fields restricted to a null subspace, the gauge algebra of double field theory closes with field-independent structure constants and its bracket is exactly the Courant bracket of generalized geometry.
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How the literature uses it

8 arXiv papers cite this in 16 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 12 sentences without a confident label

Compares with it 1 paper

The result is[A similar relation has been discussed in [ref] and [ref] .

The classical cosmological constant of open-closed string field theory · Comparison with the $g$-function deformation · 0.84

Cites it as background 2 papers

Earlier work in this direction includes that of [ref] , which discussed general coordinate invariance in closed string field theory and [ref] , which studied the constraints that T-duality imposes on the relation between closed string fields and sigma model fields.

Double Field Theory · Introduction and summary · 0.93
The function ([ref]) first arose in the work of Michishita [ref] .[ This function was proposed in [ref] as a simple ansatz for the relation between the conventional variable $$ and the string field variable that satisfies a number of constraints but is not uniquely selected.

The gauge algebra of double field theory and Courant brackets · Introduction and main results · 0.80

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

E. Witten, “ Noncommutative geometry and string ﬁeld theory ”, Nucl. Phys. B268 (1986) 253

View on Pith 1986

[2] [2]

B. Zwiebach, “ Closed string ﬁeld theory: Quantum action and the B-V master equation”, hep-th/9206084, Nucl. Phys. B390 (1993) 33

View on Pith Pith review arXiv 1993

[3] [3]

M. Saadi and B. Zwiebach, “ Closed String Field Theory from Polyhedra ”, Ann. Phys. (N.Y.) 192 (1989) 213; T. Kugo, H. Kunitomo and K. Suehiro, “ Nonpolynomial Closed String Field Theory ”, Phys. Lett. B226 (1989) 48; T. Kugo and K. Suehiro, “ Nonpolyno- mial Closed String Field Theory: Action and Its Gauge Invari ance”, Nucl. Phys. B337 (1990) 434; M. Ka...

View on Pith 1989

[4] [4]

Y. Okawa and B. Zwiebach, “ Heterotic String Field Theory ”, hep-th/0406212, JHEP 0407 (2004) 042

View on Pith Pith review arXiv 2004

[5] [5]

N. Berkovits, Y. Okawa and B. Zwiebach, “ WZW-like Action for Heterotic String Field Theory”, hep-th/0409018, JHEP 0411 (2004) 038

View on Pith Pith review arXiv 2004

[6] [6]

Y. Michishita, “ Fundamental String Solutions in Open String Field Theories ”, hep-th/0511167, Phys. Rev. D73 (2006) 046002

View on Pith Pith review arXiv 2006

[7] [7]

G. T. Horowitz and A. A. Tseytlin, “ A New Class of Exact Solutions in String Theory ”, hep-th/9409021, Phys. Rev. D51 (1995) 2896

View on Pith Pith review arXiv 1995

[8] [8]

C. G. Callan, Jr., J. M. Maldacena and A. W. Peet, “ Extremal Black Holes As Funda- mental Strings ”, hep-th/9510134, Nucl. Phys. B475 (1996) 645

View on Pith Pith review arXiv 1996

[9] [9]

A. A. Tseytlin, “ String Vacuum Backgrounds with Covariantly Constant Null K illing Vector and 2d Quantum Gravity ”, hep-th/9209023, Nucl. Phys. B390 (1993) 153. 37

View on Pith Pith review arXiv 1993

[10] [10]

C. Duval, Z. Horv´ ath and P. A. Horv´ athy, “ Strings in Plane-Fronted Gravitational Waves”, hep-th/0602128, Mod. Phys. Lett. A8 (1993) 3749

View on Pith Pith review arXiv 1993

[11] [11]

N. Berkovits, “ Super-Poincar´ e Invariant Superstring Field Theory ”, hep-th/9503099, Nucl. Phys. B450 (1995) 90

View on Pith Pith review arXiv 1995

[12] [12]

C. G. Callan Jr. and J. M. Maldacena, “ Brane Dynamics From the Born-Infeld Action ”, hep-th/9708147, Nucl. Phys. B513 (1998) 198

View on Pith Pith review arXiv 1998

[13] [13]

C. G. Callan Jr. and I. R. Klebanov, “ D-brane Boundary State Dynamics ”, hep-th/9511173, Nucl. Phys. B465 (1996) 473

View on Pith Pith review arXiv 1996

[14] [14]

C. Bachas and C. Hull, “ Null Brane Intersections ”, hep-th/0210269, JHEP 0212 (2002) 035

View on Pith Pith review arXiv 2002

[15] [15]

K. Skenderis and M. Taylor, “ Open Strings in the Plane Wave Background I: Quantization and Symmetries ”, hep-th/0211011, Nucl. Phys. B665 (2003) 3

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