1 Azumaya/matrix manifolds with a fundamental module anddifferentiable maps therefrom

Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes

February 2017

arXiv:yymm.nnnnn [hep-th]

D(13.3): standard

Dynamics of D-branes II. The standard action

— an analogue of the Polyakov action for (fundamental, stacked) D-branes


Chien-Hao Liu   and   Shing-Tung Yau

Abstract



We introduce a new action for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This ‘standard action’ is abstractly a non-Abelian gauged sigma model — based on maps from an Azumaya/matrix manifold with a fundamental module with a connection to — enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples to Ramond-Ramond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. With the analysis developed in D(13.1) (arXiv:1606.08529 [hep-th]) for such maps and an improved understanding of the hierarchy of various admissible conditions on the pairs beyond D(13.2.1) (arXiv:1611.09439 [hep-th]) and how they resolve the built-in obstruction to pull-push of covariant tensors under a map from a noncommutative manifold to a commutative manifold, we develop further in this note some covariant differential calculus needed and apply them to work out the first variation — and hence the corresponding equations of motion for D-branes — of the standard action and the second variation of the kinetic term for maps and the dilaton term in this action. Compared with the non-Abelian Dirac-Born-Infeld action constructed in D(13.1) along the same line, the current note brings the Nambu-Goto-string-to-Polyakov-string analogue to D-branes. The current bosonic setting is the first step toward the dynamics of fermionic D-branes (cf. D(11.2): arXiv:1412.0771 [hep-th]) and their quantization as fundamental dynamical objects, in parallel to what happened to the theory of fundamental strings during years 1976–1981.

Key words: D-brane; admissible condition; standard action, enhanced non-Abelian gauged sigma model; Azumaya manifold, -scheme, harmonic map; first and second variation, equations of motion


MSC number 2010: 81T30, 35J20; 16S50, 14A22, 35R01


Acknowledgements. We thank Andrew Strominger, Cumrun Vafa for influence to our understanding of strings, branes, and gravity. C.-H.L. thanks in addition Pei-Ming Ho for a discussion on Ramond-Ramond fields and literature guide and Chenglong Yu for a discussion on admissible conditions; Artan Sheshmani, Brooke Ullery, Ashvin Vishwanath for special/topic/basic courses, spring 2017; Ling-Miao Chou for comments that improve the illustrations and moral support. The project is supported by NSF grants DMS-9803347 and DMS-0074329.

Chien-Hao Liu dedicates this work to

Noel Brady, Hung-Wen Chang, Chongsun Chu, William Grosso, Pei-Ming Ho, Inkang Kim,

who enriched his years at U.C. Berkeley tremendously.

(From C.H.L.)      It was an amazing time when I landed at Berkeley in the 1990s, following Thurston’s transition to Mathematical Sciences Research Institute (M.S.R.I.). On the mathematical side, representing figures on geometry and topology — - and -dimensional geometry and topology and related dynamical system (Andrew Casson, Curtis McMullen, William Thurston), -dimensional geometry and topology (Robion Kirby), -and-above dimensional topology (Morris Hirsch, Stephen Smale), algebraic geometry (Robin Hartshorne), differential and complex geometry (Wu-Yi Hsiang, Shoshichi Kobayashi, Hong-Shi Wu), symplectic geometry and geometric quantization (Alan Weinstein), combinatorial and geometric group theory (relevant to -manifold study via the fundamental groups, John Stallings) — seem to converge at Berkeley. On the physics side, several first-or-second generation string-theorists (Orlando Alvarez, Korkut Bardakci, Martin Halpern) and one of the creators of supersymmetry (Bruno Zumino) were there. It was also a time when enumerative geometry and topology motivated by quantum field and string theory started to emerge and for that there were quantum invariants of -manifolds (Nicolai Reshetikhin) and mirror symmetry (Alexandre Givental) at Berkeley. Through the topic courses almost all of them gave during these years on their related subject and the timed homeworks one at least attempted, one may acquire a very broad foundation toward a cross field between mathematics and physics if one is ambitious and diligent enough.

However, despite such an amazing time and an intellectually enriching environment, it would be extremely difficult, if not impossible, to learn at least some good part of them without a friend — well, of course, unless one is a genius, which I am painfully not. For that, on the mathematics side, I thank Hung-Wen (Weinstein and Givental’s student) who said hello to me in our first encounter at the elevator at the 9th floor in Evans Hall and influenced my understanding of many topics — particularly symplectic geometry, quantum mechanics and gauge theory — due to his unconventional background from physics, electrical engineering, to mathematics; Bill (Stallings’ student) and Inkang (Casson’s student) for suggesting me a weekly group meeting on Thurston’s lecture notes at Princeton on -manifolds

  • W.P. Thurston, The geometry and topology of three-manifolds, typed manuscript, Department of Mathematics, Princeton University, 1979.

which lasted for one and a half year and tremendously influenced the depth of my understanding of Thurston’s work; Noel (Stallings’ student) for a reading seminar on the very thought-provoking French book

  • M. Gromov, Structures métrique pour les variétés riemanniennes, rédigé par J. Lafontaine et P. Pansu, Text Math. 1, Cedic/Fernand-Nathan, Paris, 1980.

for a summer. The numerous other after-class discussions for classes some of them and I happened to sit in shaped a large part of my knowledge pool, even for today. Incidentally, thanks to Prof. Kirby, who served as the Chair for the Graduate Students in the Department at that time. I remember his remark to his staff when I arrived at Berkeley and reported to him after meeting Prof. Thurston: “We have to treat them [referring to all Thurston’s then students] as nice as our own”

On the physics side, thanks to Chongsun and Pei-Ming (both Zumino’s students) for helping me understand the very challenging topic: quantum field theory, first when we all attended Prof. Bardakci’s course Phys 230A, Quantum Field Theory, based closely on the book

  • L.H. Ryder, Quantum field theory, Cambridge Univ. Press, 1985.

and a second time a year later when we repeated it through the same course given by Prof. Alvarez, followed by a reading group meeting on Quantum Field Theory guided by Prof. Alvarez — another person who I forever have to thank and another event which changed the course of my life permanently. The former with Prof. Bardakci was a semester I had to spend at least three-to-four days a week just on this course: attending lectures, understanding the notes, reading the corresponding chapters or sections of the book, doing the homeworks, occasionally looking into literatures to figure out some of the homeworks, and correcting the mistakes I had made on the returned homework. I even turned in the take-home final. Amusingly, due to the free style of the Department of Mathematics at Princeton University and the visiting student status at U.C. Berkeley, that is the first of the only three courses and only two semesters throughout my graduate student years for which I ever honestly did like a student: do the homework, turn in to get graded, do the final, and in the end get a semester grade back. Special thanks to Prof. Bardakci and the TA for this course, Bogdan Morariu, for grading whatever I turned in, though I was not an officially registered student in that course.

That was a time when I could study something purely for the beauty, mystery, and/or joy of it. That was a time when the future before me seemed unbounded. That was a time when I did not think too much about the less pleasant side of doing research: competitions, publications, credits, . That was a time I was surrounded by friends, though only limitedly many, in all the best senses the word ‘friend’ can carry. Alas, that wonderful time, with such a luxurious leisure, is gone forever!

Dynamics of D-branes, II: The Standard Action

0. Introduction and outline

In this sequel to D(11.1) (arXiv:1406.0929 [math.DG]), D(11.3.1) (arXiv:1508.02347 [math.DG]), D(13.1) (arXiv:1606.08529 [hep-th]) and D(13.2.1) (arXiv:1611.09439 [hep-th]) and along the line of our understanding of the basic structures on D-branes in Polchinski’s TASI 1996 Lecture Notes from the aspect of Grothendieck’s modern Algebraic Geometry initiated in D(1) (arXiv:0709.1515 [math.AG]), we introduce a new action for D-branes that is to D-branes as the (Brink-Di Vecchia-Howe/Deser-Zumino/)Polyakov action is to fundamental strings. This action depends both on the (dilaton field , metric ) on the underlying topology of the D-brane world-volume and on the background (dilaton field , metric , -field , Ramond-Ramond field ) on the target space-time ; and is naturally a non-Abelian gauged sigma model — based on maps from an Azumaya/matrix manifold with a fundamental module with a connection to — enhanced by the dilaton term that couples to , the -coupled gauge-theory term that couples to , and the Chern-Simons/Wess-Zumino term that couples to in our standard action .

Before one can do so, one needs to resolve the built-in obstruction of pull-push of covariant tensors under a map from a noncommutative manifold to a commutative manifold. Such issue already appeared in the construction of the non-Abelian Dirac-Born-Infeld action (D(13.1) ). In this note, we give a hierarchy of various admissible conditions on the pairs that are enough to resolve the issue while being open-string compatible (Sec.2). This improves our understanding of admissible conditions beyond D(13.2.1). With the noncommutative analysis developed in D(13.1), we develop further in this note some covariant differential calculus for such maps (Sec.3) and use it to define the standard action for D-branes (Sec.4). After promoting the setting to a family version (Sec.5), we work out the first variation — and hence the corresponding equations of motion for D-branes — of the standard action (Sec.6) and the second variation of the kinetic term for maps and the dilaton term in this action (Sec.7).

Compared with the non-Abelian Dirac-Born-Infeld action constructed in D(13.1) along the same line, the current standard action is clearly much more manageable. Classically and mathematically and in the special case where the background on is set to vanish, this new theory is a merging of the theory of harmonic maps and a gauge theory (free to choose either a Yang-Mills theory or other kinds of applicable gauge theory) with a nilpotent type fuzzy extension. The current bosonic setting is the first step toward fermionic D-branes (cf. D(11.2): arXiv:1412.0771 [hep-th]) and their quantization as fundamental dynamical objects, in parallel to what happened for fundamental superstrings during 1976–1981; (the road-map at the end: ’Where we are’).


Convention. References for standard notations, terminology, operations and facts are
(1) Azumaya/matrix algebra: [Ar], [Az], [A-N-T];    (2) sheaves and bundles: [H-L]; with connection: [Bl], [B-B], [D-K], [Ko];    (3) algebraic geometry: [Ha]; algebraic geometry: [Jo];    (4) differential geometry: [Eis], [G-H-L], [Hi], [H-E], [K-N];    (5) noncommutative differential geometry: [GB-V-F];    (6) string theory and D-branes: [G-S-W], [Po2], [Po3].

  • For clarity, the real line as a real -dimensional manifold is denoted by , while the field of real numbers is denoted by . Similarly, the complex line as a complex -dimensional manifold is denoted by , while the field of complex numbers is denoted by .

  • The inclusion ’’ is referred to the field extension of to by adding , unless otherwise noted.

  • All manifolds are paracompact, Hausdorff, and admitting a (locally finite) partition of unity. We adopt the index convention for tensors from differential geometry. In particular, the tuple coordinate functions on an -manifold is denoted by, for example, . However, no up-low index summation convention is used.

  • For this note, ’differentiable’, ’smooth’, and are taken as synonyms.

  • matrix vs. manifold of dimension

  • the Regge slope vs. dummy labelling index vs. covariant tensor

  • section of a sheaf or vector bundle vs. dummy labelling index

  • algebra vs. connection -form

  • ring vs. -th remainder vs. Riemann curvature tensor

  • boundary of an open set vs. partial differentiations ,

  • () of a commutative Noetherian ring in algebraic geometry
    vs.  of a -ring ()

  • morphism between schemes in algebraic geometry vs. -map between -manifolds or -schemes in differential topology and geometry or -algebraic geometry

  • group action vs. action functional for D-branes

  • metric tensor vs. element in a group vs. gauge coupling constant

  • sheaves , vs. curvature tensor , gauge-symmetry group

  • dilaton field vs. representation of a gauge-symmetry group

  • The ’support of a quasi-coherent sheaf on a scheme in algebraic geometry or on a -scheme in -algebraic geometry means the scheme-theoretical support of unless otherwise noted; denotes the ideal sheaf of a (resp. -)subscheme of of a (resp. -)scheme ; denotes the length of a coherent sheaf of dimension .

  • For a sheaf on a topological space , the notation ’’ means a local section for some open set .

  • For an -module , the fiber of at is denoted while the stalk of at is denoted .

  • coordinate-function index, e.g.  for a real manifold vs. the exponent of a power, e.g. .

  • The current Note D(13.3) continues the study in

    • [L-Y8]    Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes — with remarks on the non-Abelian Chern-Simons/Wess-Zumino term, arXiv:1606.08529 [hep-th]. (D(13.1))

    Notations and conventions follow ibidem when applicable.

Outline

  • Introduction.

  • Azumaya/matrix manifolds with a fundamental module and differentiable maps therefrom

    • Azumaya/matrix manifolds with a fundamental module

    • When is equipped with a connection

    • Differentiable maps from

    • Compatibility between the map and the connection

  • Pull-push of tensors and admissible conditions on

    • Admissible conditions on and the resolution of the pull-push issue

    • Admissible conditions from the aspect of open strings

  • The differential of and its decomposition, the three basic -modules, induced structures, and some covariant calculus

    • The differential of and its decomposition induced by

    • The three basic -modules relevant to , with induced structures

      • The -valued cotangent sheaf of , and beyond

      • The pull-back tangent sheaf

      • The -module , where lives

  • The standard action for D-branes

    • The gauge-symmetry group

    • The standard action for D-branes

    • The standard action as an enhanced non-Abelian gauged sigma model

  • Admissible family of admissible pairs

    • Basic setup and the notion of admissible families of admissible pairs

    • Three basic -modules with induced structures

    • Curvature tensors with and other order-switching formulae

    • Two-parameter admissible families of admissible pairs

  • The first variation of the enhanced kinetic term for maps and ……

    • The first variation of the kinetic term for maps

    • The first variation of the dilaton term

    • The first variation of the gauge/Yang-Mills term and the Chern-Simons/Wess-Zumino term

      • The first variation of the gauge/Yang-Mills term

      • The first variation of the Chern-Simons/Wess-Zumino term for lower-dimensional
        D-branes

        • D-brane world-point

        • D-particle world-line

        • D-string world-sheet

        • D-membrane world-volume

  • The second variation of the enhanced kinetic term for maps

    • The second variation of the kinetic term for maps

    • The second variation of the dilaton term

  • Where we are

1 Azumaya/matrix manifolds with a fundamental module and differentiable maps therefrom

Basics of maps from an Azumaya/matrix manifold with a fundamental module needed for the current note are collected in this section to fix terminology, notations, and conventions. Readers are referred to [L-Y1] (D(1)), [L-L-S-Y] (D(2)), [L-Y5] (D(11.1)) and [L-Y7] (D(11.3.1)) for details; in particular, why this is a most natural description of D-branes when Polchinski’s TASI 1996 Lecture Note is read from the aspect of Grothendieck’s modern Algebraic Geometry. See also [H-W] and [Wi2].


Azumaya/matrix manifolds with a fundamental module

From the viewpoint of Algebraic Geometry, a D-brane world-volume is a manifold equipped with a noncommutative structure sheaf of a special type dictated by (oriented) open strings.


Definition 1.1.

[Azumaya/matrix manifold with fundamental module] Let be a (real, smooth) manifold and be a (smooth) complex vector bundle over . Let

  • be the structure sheaf of (smooth functions on) ,

  • be its complexification,

  • be the sheaf of (smooth) sections of , (it’s an -module), and

  • be the endomorphism sheaf of as an -module
    (i.e. the sheaf of sections of the endomorphism bundle of ).

Then, the (noncommutative-)ringed topological space

is called an Azumaya manifold1(or synonymously, a matrix manifold to be more concrete to string-theorists.) It is important to note that non-isomorphic complex vector bundles may give rise to isomorphic endomorphism bundles and from the string-theory origin of the setting, in which plays the role of a Chan-Paton bundle on a D-brane world-volume, we always want to record as a part of the data in defining . Thus, we call the pair (or in bundle notation) an Azumaya/matrix manifold with a fundamental module.


While it may be hard to visualize geometrically, there in general is an abundant family of commutative -subalgebras

that define an abundant family of -schemes

each finite and germwise algebraic over . They may help visualize geometrically.


Definition 1.2.

[(commutative) surrogate of ] Such is called a (commutative) surrogate of (the noncommutative manifold) . Cf.Figure 1-1.




Figure 1-1. The noncommutative manifold has an abundant collection of -schemes as its commutative surrogates. See [L-Y8: Figure 2-1-1: caption] (D(13.1)) for more details.



Without loss of generality, one may assume that is connected. However even so, a surrogate of in general is disconnected locally over (and can be disconnected globally as well; cf. Figure 1-1). To keep track of this algebraically, recall the following definition:


Definition 1.3.

[complete set of orthogonal idempotents] (Cf. e.g. [Ei].) Let be an (associative, unital) ring, with the identity element . A set of elements is called a complete set of orthogonal idempotents if the following three conditions are satisfied

  • idempotent

    , .

  • orthogonal

    for .

  • complete

    .

A complete set orthogonal idempotents is called maximal if no in the set can be further decomposed into a summation of two orthogonal idempotents.


Let be a commutative -subalgebra of and the associate surrogate of . Then, for an open set, there is a unique maximal complete set of orthogonal idempotents of the -ring and it corresponds to the set of connected components of . Up to a relabelling, corresponds the function on that is constant on the -th connected component and on all other connected components.


Finally, we recall also the tangent sheaf and the cotangent sheaf of .


Definition 1.4.

[tangent sheaf, cotangent sheaf, inner derivations on ] The sheaf of (left) derivations on is denoted by and is called the tangent sheaf of . The sheaf of Kähler differentials of is dented by and is called the cotangent sheaf of . is naturally a (left) -module while is naturally a (left) -module. For our purpose, we treat both as -modules. There is a natural -module homomorphism

where acts on by . The image of this homomorphism is called the sheaf/-module of inner derivations on and is denoted by or . The kernel of the above map is exactly the center , canonically identified with , of . When the choice of a representative of an element of by an element in is irrelevant to an issue, we’ll represent elements of simply by elements in .


When is equipped with a connection

From the stringy origin of the setting with serving as the Chan-Paton bundle on the D-brane world-volume, is equipped with a gauge field (i.e. a connection) created by massless excitations of open strings. Thus, let be a connection on . Then induces a connection on . With respect to a local trivialization of , , where is an -valued -form on . Then on under the induced local trivialization. As a consequence, leaves the center of invariant and restrict to the usual differential on .

Once having the induced connection on , one has then -module homomorphism


Lemma 1.5.

[-induced decomposition of ] ([DV-M].) One has the short exact sequence

split by the above map.


The following two lemmas address the issue of when an idempotent in can be constant under a derivation .


Lemma 1.6.

[(local) idempotent under ] With the above notations, let be an open set, a vector field on , and be a complete set of orthogonal idempotents of . Assume that, say, commutes with all , . Then .

Proof.

Since , . If in addition and commute, then one has . The multiplication from the left by gives then ; i.e. . If, furthermore, commutes also with all , then , for . The multiplication from the left by gives then , for . It follows that .


Lemma 1.7.

[(local) idempotent under inner derivation] With the above notations, let be an open set, represent an inner derivation of , and be a complete set of orthogonal idempotents of . Assume that, say, commutes with all , . Then .

Proof.

Note that the proof of Lemma 1.6 uses only the Leibnitz rule property of on and the commutativity property of with . Since satisfies also the Leibniz rule property on and by assumption commutes with , the same proof goes through.

The contraction defines a trace map

One has

where is the ordinary differential on .


Differentiable maps from

As a dynamical object in space-time, a D-brane moving in a space-time is realized by a map from a D-brane world-volume to . Back to our language, we need thus a notion of a ’map from to ’ that is compatible with the behavior of D-branes in string theory.


Definition 1.8.

[map from Azumaya/matrix manifold] Let be a (real, smooth) manifold, be a complex vector bundle of rank over , and be the associated Azumaya/matrix manifold with a fundamental module. A map (synonymously, differentiable map, smooth map)

from to a (real, smooth) manifold is defined contravariantly by a ring-homomorphism

Equivalently in terms of sheaf language, let be the structure sheaf of . Regard both and as equivalence classes of gluing system of rings over the topological space and respectively. Then the above specifies an equivalence class of gluing systems of ring-homomorphisms over

which we will still denote by .


Through the Generalized Division Lemma à la Malgrange, one can show that extends to a commutative diagram

of equivalence classes of ring-homomorphisms (over or , whichever is applicable) between equivalence classes of gluing systems of rings, with

a commutative diagram of equivalence classes of ring-homomorphisms between equivalence classes of gluing systems of -rings. Here, and are the projection maps, follows from the inclusion of the center of , and

(Cf. [L-Y7: Theorem 3.1.1] (D(11.3.1)).)

In terms of spaces, one has the following equivalent diagram of maps

where is the -scheme

associated to .


Definition 1.9.

[graph of ] The push-forward of under is called the graph of . It is an -module. Its -scheme-theoretical support is denoted by .


Definition 1.10.

[surrogate of specified by ] The -scheme is called the surrogate of specified by .


is finite and germwise algebraic over and, by construction, it admits a canonical embedding into as a -subscheme. The image is identical to . Cf. Figure 1-2 and Figure 1-3.




Figure 1-2. A map specifies a surrogate of over . is a -scheme that may not be reduced (i.e. it may have some nilpotent fuzzy structure thereon). It on one hand is dominated by and on the other dominates and is finte and germwise algebraic over .


Figure 1-3. The equivalence between a map from an Azumaya manifold with a fundamental module to a manifold and a special kind of Fourier-Mukai transform from to . Here, is the category of -modules.



Compatibility between the map and the connection

Up to this point, the map to and the connection on are quite independent objects. A priori, there doesn’t seem to be any reason why they should constrain or influence each other at the current purely differential-topological level. However, when one moves on to address the issue of constructing an action functional for as in [L-Y8] (D(13.1)), one immediately realizes that,

  • Due to a built-in mathematical obstruction in the problem, one needs some compatibility condition between and before one can even begin the attempt to construct an action functional for .

Furthermore, as a hindsight, that there needs to be a compatibility condition on is also implied by string theory:

  • We need a condition on to encode the stringy fact that the gauge field on the D-brane world-volume as ‘seen’ by open strings in through should be massless.

We address such compatibility condition on systematically in the next section.


2 Pull-push of tensors and admissible conditions on

When one attempts to construct an action functional for a theory that involves maps from a world-volume to a target space-time, one unavoidably has to come across the notion of ’pulling back a (covariant) tensor, for example, the metric tensor or a differential form on the target space-time to the world-volume’. In the case where only maps from a commutative world-volume to a (commutative) space-time are involved, this is a well-established standard notion from differential topology. However, in a case, like ours, where maps from a noncommutative world-volume to a (commutative) space-time is involved,

with the accompanying contravariant ring-homomorphism

there is a built-in mathematical obstruction to such a notion. Here, is an (associative, unital) noncommutative ring, is a (associative, unital) commutative ring, and and are the topological spaces whose function rings are and respectively.

For the noncommutative ring , its (standard and functorial-in-the-category-rings) bi--module of Kähler differentials is naturally defined to be

while for the commutative ring its (standard and functorial-in-the-category-of-commutative-rings) (left) -module of Kähler differentials is naturally defined to be

with the convention that to turn it to a bi--module as well. Treating as a ring (that happens to be commutative), it has also the (standard and functorial-in-the-category-rings) bi--module of Kähler differentials

exactly like for . There is a built-in tautological quotient homomorphism as bi--modules

whose kernel is generated by . Given the map , one has the following built-in diagram

where . The issue is now whether one can extend the above diagram to the following commutative diagram

The answer is No, in general. See, e.g., [L-Y5: Example 4.1.20] (D(11.1)) for an explicit counterexample. When is a -ring, e.g. the function-ring of a smooth manifold , then the -module of differentials of is a further quotient of the above module of Kähler differentials by additional relations generated by applications of the chain rule on the transcendental smooth operations in the -ring structure of ([Jo]; cf. [L-Y5: Sec.4.1] (D(11.1))). The issue becomes even more involved. In particular, as the counterexample ibidem shows

  • [built-in mathematical obstruction of pullback] For a map , there is no way to define functorially a pull-back map that takes a (covariant) -tensor on to a -tensor on . As a consequence, there is no functorial way to pull back a (covariant) tensor on to a tensor on .

Before the attempt to construct an action functional that involves such maps , one has to resolve the above obstruction first. In [L-Y8] (D(13.1)), we learned how to use the connection to impose a natural admissible condition on so that the above obstruction is bypassed through the surrogate of specified by . With the lesson learned therefrom and further thought beyond [L-Y9] (D(13.2.1)), we propose (Sec.2.1) in this section a still-natural-but-much-weaker admissible condition on that bypasses even the surrogate but is still robust enough to construct naturally a pull-push map we need on tensors. It turns out that this much weaker admissible condition remains to be compatible with open strings (Sec.2.2).


2.1 Admissible conditions on and the resolution of the pull-push issue

A hierarchy of admissible conditions on is introduced. A theorem on how even the weakest admissible condition in the hierarchy can resolve the above obstruction in our case is proved.


Three hierarchical admissible conditions

Definition 2.1.1.

[admissible connection on ] Let be a map. For a connection on , let be its induced connection on . A connection on is called

  • -admissible to     if ;

  • -admissible to     if ;

  • -admissible to     if

for all . Here, denotes the commutant of in .

When is -admissible to , we will take the following as synonyms:

  • is an -admissible pair,

  • is -admissible to ,

  • is -admissible.

Similarly, for -admissible pair and -admissible pair , … , etc..

Lemma 2.1.2.

[hierarchy of admissible conditions]

Proof.

Admissible Condition says that the -subalgebra is invariant under -parallel transports along paths on . Since -parallel transports on are algebra-isomorphisms, if is -invariant, the -subalgebra of must also be -invariant since it is determined by fiberwise algebraically. In other words, Admissible Condition Admissible Condition .

Since is commutative, . Thus, the inclusion always holds. This implies that Admissible Condition Admissible Condition .


Definition 2.1.3.

[strict admissible connection on ] Continuing Definition 2.1.1. Let be the curvature tensor of . It is an -valued -form on . Then, for , is called strictly -admissible to if

  • is -admissible to and takes values in .

In this case, is said to be a strictly -admissible pair.


Clearly, the same hierarchy holds for strict admissible conditions:

The Strict -Admissible Condition on was introduced in [L-Y8: Definition 2.2.1] (D(13.1)) to define the Dirac-Born-Infeld action for .


Lemma 2.1.4.

[commutativity under admissible condition] Let be a map. If is -admissible, then for all and . If is -admissible, then for all and .

Proof.

Statement (1) is the -Admissible Condition itself.

For Statement (2), let and . Then since . Thus, applying to both sides,

The -Admissible Condition implies that

And, hence, . Statement (2) follows.


Resolution of the pull-push issue under Admissible Condition

The current theme is devoted to the proof of the following theorem:


Theorem 2.1.5.

[pull-push under -admissible ] Let be -admissible. Then the assignment

is well-defined.


The study in [L-Y8: Sec. 4] (D(13.1)) allows one to express locally explicit enough so that one can check that is well-defined when is -admissible. Note that, with Lemma 2.1.2, this implies that if is either - or -admissible, then is also well-defined. We now proceed to prove the theorem.


Lemma 2.1.6.

[local expression of for -admissible , I] Let be -admissible; i.e.  for all . Let be a small enough open set so that is contained in a coordinate chart of , with coordinate . For , recall the germwise-over- polynomial in with coefficients in from [L-Y8: Sec. 4 & Remark/Notation 4.2.3.5] (D.13.1). Then, for a vector field on and , and at the level of germs over ,

Here, for a multiple degree , , and
means ‘replacing by ’.

Proof.

Denote the coordinate chart of in the Statement by . Let , be the projection maps. Recall the induced ring-homomorphism
over and the graph of and its support on . Denote still by when there is no confusion. For clarity, we proceed the proof of the Statement in three steps.


Step How is constructed in [L-Y8: Sec. 4] (D(13.1))       For any , let be a neighborhood of in over which the Generalized Division Lemma à la Malgrange is applied to on a neighborhood of in with respect to the characteristic polynomials , . Passing to a smaller open subset if necessary, one may assume that is a disjoint union with a neighborhood of and the closure are all disjoint from each other. Let be a smooth functions on that takes the value on and the value zero on , , . (Cf.[L-Y8:Sec.4.2.3] (D(13.1)).) Then