Dynamics of Dbranes II. The standard action  an analogue of the Polyakov action for (fundamental, stacked) Dbranes
February 2017
arXiv:yymm.nnnnn [hepth]
D(13.3): standard
Dynamics of Dbranes II. The standard action
— an analogue of the Polyakov action for (fundamental, stacked) Dbranes
ChienHao Liu and ShingTung Yau
Abstract
We introduce a new action for Dbranes that is to Dbranes as the Polyakov action is to fundamental strings. This ‘standard action’ is abstractly a nonAbelian 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 gaugetheory term, and the ChernSimons/WessZumino term that couples to RamondRamond 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 [hepth]) 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 [hepth]) and how they resolve the builtin obstruction to pullpush 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 Dbranes — of the standard action and the second variation of the kinetic term for maps and the dilaton term in this action. Compared with the nonAbelian DiracBornInfeld action constructed in D(13.1) along the same line, the current note brings the NambuGotostringtoPolyakovstring analogue to Dbranes. The current bosonic setting is the first step toward the dynamics of fermionic Dbranes (cf. D(11.2): arXiv:1412.0771 [hepth]) and their quantization as fundamental dynamical objects, in parallel to what happened to the theory of fundamental strings during years 1976–1981.
Key words: Dbrane; admissible condition; standard action, enhanced nonAbelian 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 PeiMing Ho for a discussion on RamondRamond 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; LingMiao Chou for comments that improve the illustrations and moral support. The project is supported by NSF grants DMS9803347 and DMS0074329.
ChienHao Liu dedicates this work to
Noel Brady, HungWen Chang, Chongsun Chu, William Grosso, PeiMing 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), andabove dimensional topology (Morris Hirsch, Stephen Smale), algebraic geometry (Robin Hartshorne), differential and complex geometry (WuYi Hsiang, Shoshichi Kobayashi, HongShi 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 firstorsecond generation stringtheorists (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 HungWen (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 threemanifolds, 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 thoughtprovoking French book

M. Gromov, Structures métrique pour les variétés riemanniennes, rédigé par J. Lafontaine et P. Pansu, Text Math. 1, Cedic/FernandNathan, Paris, 1980.
for a summer. The numerous other afterclass 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 PeiMing (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 threetofour 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 takehome 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 Dbranes, 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 [hepth]) and D(13.2.1) (arXiv:1611.09439 [hepth]) and along the line of our understanding of the basic structures on Dbranes 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 Dbranes that is to Dbranes as the (BrinkDi VecchiaHowe/DeserZumino/)Polyakov action is to fundamental strings. This action depends both on the (dilaton field , metric ) on the underlying topology of the Dbrane worldvolume and on the background (dilaton field , metric , field , RamondRamond field ) on the target spacetime ; and is naturally a nonAbelian 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 gaugetheory term that couples to , and the ChernSimons/WessZumino term that couples to in our standard action .
Before one can do so, one needs to resolve the builtin obstruction of pullpush of covariant tensors under a map from a noncommutative manifold to a commutative manifold. Such issue already appeared in the construction of the nonAbelian DiracBornInfeld 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 openstring 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 Dbranes (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 Dbranes — 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 nonAbelian DiracBornInfeld 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 YangMills theory or other kinds of applicable gauge theory) with a nilpotent type fuzzy extension. The current bosonic setting is the first step toward fermionic Dbranes (cf. D(11.2): arXiv:1412.0771 [hepth]) and their quantization as fundamental dynamical objects, in parallel to what happened for fundamental superstrings during 1976–1981; (the roadmap at the end: ’Where we are’).
Convention.
References for standard notations, terminology, operations and facts are
(1) Azumaya/matrix algebra: [Ar], [Az], [ANT];
(2) sheaves and bundles: [HL]; with connection: [Bl], [BB], [DK], [Ko];
(3) algebraic geometry: [Ha]; algebraic geometry: [Jo];
(4) differential geometry: [Eis], [GHL], [Hi], [HE], [KN];
(5) noncommutative differential geometry: [GBVF];
(6) string theory and Dbranes: [GSW], [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 uplow 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 Dbranes

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

sheaves , vs. curvature tensor , gaugesymmetry group

dilaton field vs. representation of a gaugesymmetry group

The ’support’ of a quasicoherent sheaf on a scheme in algebraic geometry or on a scheme in algebraic geometry means the schemetheoretical 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 .

coordinatefunction 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

[LY8] Dynamics of Dbranes I. The nonAbelian DiracBornInfeld action, its first variation, and the equations of motion for Dbranes — with remarks on the nonAbelian ChernSimons/WessZumino term, arXiv:1606.08529 [hepth]. (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


Pullpush of tensors and admissible conditions on

Admissible conditions on and the resolution of the pullpush 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 pullback tangent sheaf

The module , where lives



The standard action for Dbranes

The gaugesymmetry group

The standard action for Dbranes

The standard action as an enhanced nonAbelian 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 orderswitching formulae

Twoparameter 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/YangMills term and the ChernSimons/WessZumino term

The first variation of the gauge/YangMills term

The first variation of the ChernSimons/WessZumino term for lowerdimensional
Dbranes
Dbrane worldpoint

Dparticle worldline

Dstring worldsheet

Dmembrane worldvolume




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 [LY1] (D(1)), [LLSY] (D(2)), [LY5] (D(11.1)) and [LY7] (D(11.3.1)) for details; in particular, why this is a most natural description of Dbranes when Polchinski’s TASI 1996 Lecture Note is read from the aspect of Grothendieck’s modern Algebraic Geometry. See also [HW] and [Wi2].
Azumaya/matrix manifolds with a fundamental module
From the viewpoint of Algebraic Geometry, a Dbrane worldvolume 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 manifold
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 11.
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 11). 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 ChanPaton bundle on the Dbrane worldvolume, 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 ] ([DVM].) 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 spacetime, a Dbrane moving in a spacetime is realized by a map from a Dbrane worldvolume to . Back to our language, we need thus a notion of a ’map from to ’ that is compatible with the behavior of Dbranes 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 ringhomomorphism
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 ringhomomorphisms 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 ringhomomorphisms (over or , whichever is applicable) between equivalence classes of gluing systems of rings, with
a commutative diagram of equivalence classes of ringhomomorphisms between equivalence classes of gluing systems of rings. Here, and are the projection maps, follows from the inclusion of the center of , and
(Cf. [LY7: 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 pushforward of under is called the graph of . It is an module. Its schemetheoretical 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 12 and Figure 13.
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 differentialtopological level. However, when one moves on to address the issue of constructing an action functional for as in [LY8] (D(13.1)), one immediately realizes that,

Due to a builtin 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 Dbrane worldvolume as ‘seen’ by open strings in through should be massless.
We address such compatibility condition on systematically in the next section.
2 Pullpush of tensors and admissible conditions on
When one attempts to construct an action functional for a theory that involves maps from a worldvolume to a target spacetime, 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 spacetime to the worldvolume’. In the case where only maps from a commutative worldvolume to a (commutative) spacetime are involved, this is a wellestablished standard notion from differential topology. However, in a case, like ours, where maps from a noncommutative worldvolume to a (commutative) spacetime is involved,
with the accompanying contravariant ringhomomorphism
there is a builtin 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 functorialinthecategoryrings) bimodule of Kähler differentials is naturally defined to be
while for the commutative ring its (standard and functorialinthecategoryofcommutativerings) (left) module of Kähler differentials is naturally defined to be
with the convention that to turn it to a bimodule as well. Treating as a ring (that happens to be commutative), it has also the (standard and functorialinthecategoryrings) bimodule of Kähler differentials
exactly like for . There is a builtin tautological quotient homomorphism as bimodules
whose kernel is generated by . Given the map , one has the following builtin 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., [LY5: Example 4.1.20] (D(11.1)) for an explicit counterexample. When is a ring, e.g. the functionring 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. [LY5: Sec.4.1] (D(11.1))). The issue becomes even more involved. In particular, as the counterexample ibidem shows

[builtin mathematical obstruction of pullback] For a map , there is no way to define functorially a pullback 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 [LY8] (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 [LY9] (D(13.2.1)), we propose (Sec.2.1) in this section a stillnaturalbutmuchweaker admissible condition on that bypasses even the surrogate but is still robust enough to construct naturally a pullpush 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 pullpush 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 algebraisomorphisms, 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 [LY8: Definition 2.2.1] (D(13.1)) to define the DiracBornInfeld 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 pullpush issue under Admissible Condition
The current theme is devoted to the proof of the following theorem:
Theorem 2.1.5.
[pullpush under admissible ] Let be admissible. Then the assignment
is welldefined.
The study in [LY8: Sec. 4] (D(13.1)) allows one to express locally explicit enough so that one can check that is welldefined when is admissible. Note that, with Lemma 2.1.2, this implies that if is either  or admissible, then is also welldefined. 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 germwiseover polynomial in with coefficients in from [LY8: 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 ringhomomorphism
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 [LY8: 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.[LY8:Sec.4.2.3] (D(13.1)).) Then