key: cord-0104607-tgdokjic
authors: Liu, Chien-Hao; Yau, Shing-Tung
title: Soft noncommutative flag schemes
date: 2022-04-27
journal: nan
DOI: nan
sha: dd5a22c4e0112fbcb4d50a98304e6c6f5c6ae73f
doc_id: 104607
cord_uid: tgdokjic

The construction of soft noncommutative schemes via toric geometry in arXiv:2108.05328 [math.AG] (D(15.1), NCS(1)) can be generalized and applied to a commutative scheme with a distinguished atlas of reasonably good affine local coordinate charts. In the current notes we carry out this exercise for flag varieties.

Beyond toric varieties, another class of varieties that admit versatile use in both mathematics and physics -including Mirror Symmetry -are flag varieties. Indeed, except that the labelling of distinguished charts cannot be realized by a fan, their construction from gluing a finite collection of affine varieties, each isomorphic to an affine space A d for some common d, shares similar features of toric varieties. From this perspective, it is very natural to expect that the construction of soft noncommutative schemes via toric geometry in [L-Y2] (arXiv:2108.05328 [math.AG] (D(15.1), NCS(1))) can be extended to flag varieties as well. The details are carried out in this work. Similar to soft noncommutative toric schemes, this gives a version of 'noncommutative flag schemes' that may serve as target-spaces for dynamical D-branes in String Theory, cf. [L-Y2: Sec. 4].

As a side remark, it should be noted that noncommutatization or quantization of flag manifolds or varieties has been a topic of interest for long, e.g. [M-S] , though there is no obvious connection between soft noncommutative flag schemes constructed in the current work and any previous version of 'noncommutative flag manifolds or varieties' that we know of.

A contravariant construction of a flag variety in terms of a distinguished gluing system of rings is presented in this section. All the material here is standard but the presentation gives an immediate comparison of flag varieties to toric varieties.

A distinguished atlas on a flag variety Let C n be the n-dimensional vector space over C. For integers 0 < d 1 < d 2 < · · · < d r < n, let Fl (d 1 , · · · , d r ; n) denote the flag variety whose C-points are flags of vector subspaces L 1 ⊂ L 2 ⊂ · · · ⊂ L r ⊂ C n with dim L i = d i . Fix an isomorphism C n C ⊕n and the standard basis e 1 , · · · , e n under the isomorphism, with e i := (0, · · · , 0, 1, 0, · · · , 0) where 1 is in the i-th entry. For a nonempty subset I ⊂ {1, · · · , n}, let H I be the vector subspace Span C {e i |i ∈ I} of C n . The canonical decomposition C n = H I ⊕ H I c , where I c := {1, · · · , n} − I, defines a projection map π I : C n → H I . For an inclusion sequence I of subsets

where I j are as above, let

Then, though no longer an affine space in general, U I remains affine and the collection satisfies

Definition 1.1. [admissible sequence and admissible chain] An inclusion sequence I of subsets I 1 ⊂ I 2 ⊂ · · · ⊂ I r ⊂ {1, · · · , n} with cardinality |I i | = d i is called an admissible sequence of subsets of {1, · · · , n}. A collection I = {I 1 , · · · , I l } of distinct admissible sequences of subsets of {1, · · · , n} is a called an admissible chain from {1, · · · , n}.

Note that {U I | I: admissible sequence} is an affine open cover of Fl (d 1 , · · · , d r ; n) and that, for I : I 1 ⊂ · · · ⊂ I r ⊂ {1, · · · , n} an admissible sequence, the flag H I 1 ⊂ · · · ⊂ H Ir ⊂ C n from nested coordinate planes lies only in U I . Thus, this is a minimal cover in the sense that the collection in this cover cannot be reduced to maintain a cover.

Definition 1.2. [distinguished atlas on Fl (d 1 , · · · , d r ; n)] By construction the finite system U 0 := {U I } I of affine open subsets U I of Fl (d 1 , · · · , d r ; n), where I runs over all admissible chains from {1, · · · , n}, covers Fl (d 1 , · · · , d r ; n) and is closed under taking intersections; it is called the distinguished atlas on Fl (d 1 , · · · , d r ; n) associated to the C-vector-space isomorphism C n C ⊕n . For U I , U J ∈ U 0 , we say that U I is subordinate to U J or interchangeably that U I is a subordinate of U J if U I ⊂ U J . A maximal chart in U 0 with respect to this partial order ⊂ is exactly U I for some admissible sequence I. Remark 1.3. [weight polytope] When Fl (d 1 , · · · , d r ; n) is realized as the orbit of a highest weight vector in the projectivization PV of a complex representation V of the special unitary group SU(n) of rank n − 1, the set II := {I | I is an admissible sequence} that labels maximal charts in U 0 can be identified with the set of vertices of the weight polytope in the dual Carton subalgebra of SU(n) associated to the representation. We shall now describe this distinguished atlas contravariantly in terms of local coordinate rings and their localizations.

The reference chart U I 0 ∈ U 0 and its subordinates Let I 0 be the admissible sequence {1, · · · , d 1 } ⊂ {1, · · · , d 2 } ⊂ · · · ⊂ {1, · · · , d r } ⊂ {1, · · · , n} .

Then the standard coordinate-ring associated to the chart U I 0 is given by the polynomial ring over C R I 0 := C z ij j ∈ {d k + 1, · · · , n} for i ∈ {d k−1 + 1, · · · , d k }, k = 1, · · · , r with d 0 = 0 by convention from the blocked matrix presentation of a flag L : L 1 ⊂ L 2 ⊂ · · · ⊂ L r ⊂ C n that corresponds to a C-point on U I 0

where · the diagonal blocks (in ) from upper-left to lower-right are respectively the identity matrices Id d 1 ×d 1 , Id (d 2 −d 1 )×(d 2 −d 1 ) , · · · , Id (n−dr)×(n−dr) ; · all the blocks (in ) below the diagonal blocks are zero; · for i = 1, · · · , r, L i is the span of the upper d i -many row-vectors of M I 0 (L).

When L varies in U I 0 , the (i, j)-entry of M I 0 (L) that sits in a block (in ) to the right of a diagonal block gives rise to the coordinate function z ij in the coordinate-ring R I 0 of U I 0 . As a bookkeeping device for later discussions, denote M I 0 (z) := the above d r × n matrix with every (i, j)-entry in the -region (i.e. blocks to the right of diagonal blocks) replaced by its associate z ij while keeping all the digonal blocks (cf. the -region) and the zero-blocks (cf. the -region).

Thus, R I 0 is the polynomial ring over C generated by entries of M I 0 (z). In notation,

For a general admissible sequence I : I 1 ⊂ I 2 ⊂ · · · ⊂ I r ⊂ {1, · · · , n} with I 1 = {i 1,1 , · · · , i 1,d 1 }, i 1,1 < · · · < i 1,d 1 ;

. . .

and an r × n matrix M , let us introduce two sets of convenient bookkeeping notations as follows.

(1) The distinguished square submatrices

of M by removing all the j -th rows and the j -th columns of M with j > d j and j / ∈ I j ,

Note that, by construction, M I j −I j−1 is a square submatrix of M I j as well and that the collection

with i r+1,1 < · · · < i r+1,n−dr by convention. Associated to χ I is the characteristic n × n matrix C χ I such that M C χ I = the r × n matrix obtained from M by exchanging the k-th column with χ I (k)-th column, k = 1, · · · , n. Note that C 2 χ I = Id n×n . Let I now be an admissible sequence of subsets of {1, · · · , n} that is distinct from I 0 and consider the affine open subset U {I 0 ,I} := U I 0 ∩ U I of Fl (d 1 , · · · , d r ; n) associated to the chain {I 0 , I}. A flag L ∈ U I 0 lies also in U I if and only if the restriction of the projection map π I j : L j → H I j is an isomorphism for I j ∈ I, j = 1, · · · , r. The latter is true if and only if all the square submatrices M I 0 (L) I 1 , M I 0 (L) I 2 , · · · , M I 0 (L) Ir of M I 0 (L) are invertible, which is equivalent to that all the square submatrices

I} is given by the localization of R I 0 at the multiplicatively closed subset generated by

where det • is the determinant of the square matrix • .

With slight abuse of notation, we shall denote this localized ring as

Hence

Here, the numerator is the polynomial ring generated by the variables z ij 's and w k 's as specified, the denominator is the ideal of the numerator generated by the elements as indicated, and I 0 is the empty set by convention. The same discussion applies to the affine open subset U I of U I 0 ⊂ Fl (d 1 , · · · , d r ; n) associated to a higher chain I = {I 0 , I 1 , · · · , I l }; which gives: · The function-ring R I of U I is given by the localization of R I 0 at the multiplicatively closed subset generated by

where I i,• ∈ I i and I• ,0 are the empty set by convention. I.e. (with slight abuse of notation)

Here, notice that there could be repeating elements in the set S I 0 ;{I 1 ,··· ,I l } , which are redundant as long as localizations of R I 0 are concerned, and S I 0 ;{I 1 ,··· ,I l },• is the pruned S I 0 ;{I 1 ,··· ,I l } that removes all such repeating elements. In particular, 

as a set of admissible sequences and is the the maximal admissible chain. We shall call

the master ring for the flag variety Fl (d 1 , · · · , d r ; n). This is the function-ring of the smallest affine open subset I∈II U I in the distinguished atlas U 0 on Fl (d 1 , · · · , d r ; n). 

, i = 1, · · · , r, d 0 = 0 by convention. For convenience, we shall call any of the latter square submatrices a distinguished submatrix of M I 0 (z).

A general chart U I ∈ U 0 and its subordinates For a general admissible sequence I, recall the characteristic map χ I and the characteristic matrix C χ I . Then, similar to the function-ring R I 0 of U 0 , the function-ring R I of U I is given by the polynomial ring over C generated by the entries of where C I;J is the unique r × r matrix with entries in R I [S −1 I;J ] such that C I;J M I (x)C χ J resumes the blocked matrix form M I 0 (L) for a flag L ∈ U 0 . Completely analogously for a deeper chain {I 1 , · · · , I l+1 }, all the isomorphisms

are explicitly constructible this way for any decompositions {I 1 , · · · , I l+1 } = I 1 ∪ I 2 = I 1 ∪ I 2 of {I 1 , · · · , I l+1 } into disjoint unions.

Realization of all R • in the master ring R II for Fl (d 1 , · · · , d r ; n)

The built-in C-algebra-homomorphisms

for all admissible sequence I, allow one to identify all the function-rings R I , and hence all R J , where J is an admissible chain, from localizations of some R I as well, as C-subalgebras of the master C-algebra R II for Fl (d 1 , · · · , d r ; n). Indeed, from the discussions in the previous two themes and with the notations therein, we shall make the following explicit identifications:

· For I an admissible sequence, R I = the C-subalgebra of R I 0 [S −1 I 0 ;I ] ⊂ R II generated by the entries of the r × n matrix C I 0 ;I M I 0 (z). · For I = {I 1 , · · · , I l } an admissible chain, R I = the C-subalgebra of R II generated by R I 1 ∪ · · · ∪ R I l . and take this as the starting point to construct soft noncommutative flag schemes.

Notation 1.6. [contravariant description of Fl (d 1 , · · · , d r ; n) as 2 II -system of subrings] One thus has a contravariant description of the flag variety Fl (d 1 , · · · , d r ; n) as a 2 II -system of C-subalgebras in R II . We shall denote this system by

where O U 0 is the structure sheaf of Fl (d 1 , · · · , d r ; n) adapted to U 0 , which assigns the subring R I ⊂ R II to the chart U I ∈ U 0 , for I ⊂ II.

Remark 1.7. [comparison to toric variety ] The master ring R II for a flag variety will play the role of C[z 1 , · · · , z n , z −1 1 , · · · , z −1 n ] for some n (i.e. the function ring of the n-torus T n C := (C × ) n over C or equivalently the group algebra C[M ] associated to a lattice M of rank n for some n) in the case of toric varieties. In particular, the distinguished gluing system of C-algebras associated to a flag variety is simply a special system of C-subalgebras of the master C-algebra R II labelled by admissible chains just like the distinguished gluing system of function-rings associated of a toric variety X ∆ /C being a system of C-subalgebras in C[z 1 , · · · , z n , z −1 1 , · · · , z −1 n ] labelled by cones in the fan ∆.

With the preparation in Sec. 1, we can now proceed in the same way as [L-Y2] (D(15.1), NCS(1)), where the toric case is studied, to construct 'soft noncommutative flag schemes' and their 'soft subschemes'. The details and an example are given in this section.

be the (noncommutative) associative C-algebra freely generated by the variable entries z ij 's of M I 0 (z), with the built-in C-algebra-epimorphism π 0 from commutatization that sendsz ij ∈Ȓ I 0 to z ij ∈ R I 0 ; ·SI 0 ;II ,• be a lifting of S I 0 ;II ,• inȒ I 0 under π 0 , (i.e.S I 0 ;II ,• ⊂ π −1 0 (S I 0 ;II ,• ) such that the restriction π 0 :S I 0 ;II ,• → S I 0 ;II ,• is a set-isomorphism); and

Here, · recall that S I 0 ;II ,• ⊂ C[M I 0 (z)] is the set of the determinant of all distinguished submatrices of M I 0 (z), now written as {m k (z) | k = 1, · · · , n d 1 n d 2 −d 1 · · · n dr−d r−1 }; ·mk (z) ∈S I 0 ;II ,• is the specified lifting of m k (z); · C z ij ,w k | ij, k = C z ij ,w k ij,k is the associative C-algebra freely generated by the set {z ij ,w k } ij,k ; and

By construction, π 0 :Ȓ I 0 → R I 0 extends to a C-algebra epimorphism from commutatization π 0 :Ȓ II −→ R II .

Definition 2.1. [noncommutative master ring for Fl (d 1 , · · · , d r ; n)]Ȓ II is called the noncommutative master ring for the flag variety Fl (d 1 , · · · , d r ; n).

R II plays the same role in the notion and construction of soft noncommutative flag schemes associated to Fl (d 1 , · · · , d r ; n) as the group-algebra over C of a finitely-generated non-Abelian free group in the toric case, cf. [L-Y2: Example 1.7] (D(15.1), NCS(1)).

Definition 2.2. [soft noncommutative flag scheme] A 2 II -system of C-subalgebras ofȒ II is a collection {Ȓ I } I⊂II of C-subalgebras ofȒ II specified for each admissible chain I ⊂ II such that if I ⊂ J, thenȒ I ⊂Ȓ J inȒ II . A such system is called a soft noncommutative flag scheme if, in addition, the built-in C-algebra epimorphism via commutatization π 0 :Ȓ II → R II restricts a C-algebra epimorphism π 0 :Ȓ I → R I for all I ⊂ II. We shall denote a soft noncommutative flag scheme associated to Fl (d 1 , · · · , d r ; n) by

whereȎ U 0 is its structure sheaf adapted to the distinguished atlas, defined by the assignment U I →Ȓ I , I ⊂ II. In this case, for I ⊂ J, we shall denote the inclusion mapȒ I →Ȓ J of C-algebras by ι JI , where ι JI : U J → U I is the inclusion of charts in U 0 .

Definition 2.3. [softening of noncommutative flag scheme] LetX II = (U 0 ,Ȏ U 0 ) and X II = (U 0 ,Ȏ U 0 ) be two soft noncommutative flag schemes associated to Fl (d 1 , · · · , d r ; n). We say thatX II is a softening ofX II or interchangeably thatX

for all I ⊂ II. This defines a morphismX II →X II of soft noncommutative flag schemes, named a softening morphism.

Lemma 2.4. [common softening] LetX II andX II be two soft noncommutative flag schemes associated to Fl (d 1 , · · · , d r ; n). Then there exists another soft noncommutative flag schemeX II associated to Fl (d 1 , · · · , d r ; n) that softens bothX II andX II .

Proof. For example, take O U 0 (U I ) to be the C-subalgebra ofȒ II generated by O U 0 (U I ) ∪ O U 0 (U I ) for I ⊂ II.

In the toric case, there are monoidal structures that one would like to keep track, making the construction of a soft noncommutative toric scheme slightly involved. Here, there is no similar built-in structure from the flag variety Fl (d 1 , · · · , d r ; n). Thus the task of the construction becomes indeed light.

Recall the 2 II -system X II = (U 0 , O U 0 ) of rings that represents Fl (d 1 , · · · , d r ; n) contravariantly. A soft noncommutative flag schemeX II = (U 0 ,Ȏ U 0 ) that is obtained by soft-gluings of |II|-many copies of noncommutative affine spaces over C, cf. [L-Y2: Definition 2.1.1] (D(15.1), NCS(1)), can be constructed as follows:

· For I ∈ II an admissible sequence, let G I ⊂ R I 0 [S −1 I 0 ;I ] be the set of the entries of the r × n matrix C I 0 ;I M I 0 (z) that are not 0 or 1. Recall that R I ⊂ R II is the polynomial ring over C generated by G I . LetG I be a lifting of G I inȒ II via π 0 ; i.e.,G I ⊂ π −1 0 (G I ) such that the restriction π 0 :G I → G I is a set-isomorphism. Then, set O U 0 (U I ) =Ȓ I = the C-subalgebra ofȒ II generated byG I . · For I = {I 1 , · · · , I l } ⊂ II an admissible chain, set O U 0 (U I ) =Ȓ I = the C-subalgebra ofȒ II generated byȒ I 1 ∪ · · · ∪Ȓ I l .

By construction, for I ⊂ J, one hasȒ I ⊂Ȓ J . Thus, the system defines a soft noncommutative flag scheme over C. Furthermore, for I ⊂ II, π 0 :Ȓ I → R I is the commutatization ofȒ I and, for I ⊂ J, the following diagrams commutȇ

Thus, Fl (d 1 , · · · , d r ; n) →X II as a maximal commutative subscheme.

For a closed subscheme Z ⊂ X, one can realize Z as a 2 II -system

LetȊ I ⊂Ȓ I be a two-sided ideal, for I ⊂ II, such that π 0 (Ȋ I ) = I I for all I and that ι JI (Ȋ I ) ⊂Ȋ J for I ⊂ J. Then, the C-algebra-homomorphism ι JI :Ȓ I →Ȓ J descend to quotients (with slight abuse of notation) ι JI :Ȓ I /Ȋ I →Ȓ J /Ȋ J for I ⊂ J.

Definition 2.5. [soft noncommutative closed subscheme ofX II associated to Z ⊂ X] The 2 II -system of quotient C-algebrasZ

as above is called a soft noncommutative closed subscheme ofX II associated to Z ⊂ X. Here, O U Z 0 is the structure sheaf ofZ II , defined by the specificationȎ U Z 0 (U I ) =Ȓ I /Ȋ I for I ⊂ II.

Lemma 2.6.

[existence] Let Z be a closed subscheme of Fl (d 1 , · · · , d r ; n) andX II be a soft noncommutative scheme associated to Fl (d 1 , · · · , d r ; n). Then there exists a soft noncommutative closed subschemeZ II ofX II associated to Z.

Proof. Continuing the notation in the current theme. Note that the ideal I I associated to Z ∩ U I ⊂ U I is finitely generated, for I ∈ II. LetȊ I be the two-sided ideal ofȒ I generated by a lifting of a finite generating set of I I toȒ I . For general I = {I 1 , · · · , I l } ⊂ II, letȊ I be the two-sided ideal inȒ I generated by l j=1Ȋ I j ⊂Ȓ I .

and 1)), [Liu] . This allows stacked D-branes to get mapped to noncommutative target-spaces. On the other hand, a theory of localizations of general noncommutative rings is problematic, which leads to the notion of noncommutative schemes that directly generalizes Grothendieck's work for (Commutative) Algebraic Geometry impossible. Yet, for the purpose of serving as target spaces of dynamical D-branes, one only needs a gluing system of charts. This motivates the notion of 'soft noncommutative schemes'. Beginning with a (commutative) scheme X, we may cover it by an atlas U 0 of simple enough, reasonably good affine charts. One then extends this inclusion system of charts to a gluing system systemX := (U 0 ,Ȏ U0 ) of noncommutative charts, in which inclusions of commutative charts extend but relax to morphisms −→ of noncommutative charts that satisfy cocycle conditions for gluing. By construction, X naturally embeds inX. X as constructed can be "very fluffy around X". Other inputs from String Theory, particularly D-brane probes, should be taken into account to make a good notion of 'soft noncommutative Calabi-Yau schemes' when X is a Calabi-Yau space.

Together with the proof of Lemma 2.6, this gives a direct association of a subscheme V of P 5 with a soft noncommutative closed subschemeZ II ofX II associated to V ∩ Gr (2; 4). In particular, for Z a subscheme in Gr (2; 4) arising from a generic quartic hypersurface in P 5 described by a homogeneous polynomial f ∈ Γ (P 5 , O P 5 (4)), one has a soft noncommutative schemeZ II that contains the Calabi-Yau 3-fold Z as its master commutative subscheme

NCS(1)).) What is the correct notion/definition of soft noncommutative Calabi-Yau spaces in the current context? From pure mathematical generalization of the commutative case? From world-volume conformal invariance or supersymmetry of D-branes? What is the mirror symmetry phenomenon in this context? References [B-CF-K-vS

Noncommutative algebraic geometry

Differential forms in algebraic topology

Commutative algebra -with a view toward algebraic geometry

The geometry of schemes

Intersection theory

Introduction to toric varieties

Representation theory -A first course

Principles of algebraic geometry

Algebraic geometry

Azumaya noncommutative geometry and D-branes -an origin of the master nature of Dbranes, lecture given at the workshop Noncommutative algebraic geometry and D-branes

Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpoint

Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -(Dynamical, complex algebraic) D-branes on a soft noncommutative space

Quantization of flag manifolds and their supersymmetric extensions

Theory of non-commutative polynomials

Example: Gr (2; 4)The Grassmann variety Gr (2; 4) of 2-planes in C 4 is the simplest flag variety that is not a projective space itself. The Plücker embedding realizes it as a nonsingular quadric hypersurface in CP 5 , whose intersection with a generic quartic hypersurface is a Calabi-Yau 3-fold Z, (e.g. [B-CF-K-vS] ). The distinguished atlas U 0 for Gr (2; 4) contains 4 2 = 6 maximal charts.· The reference chart U I 0 = U {1,2} :R {1,2} is the polynomial ring over C generated by the non-0-nor-1 entries ofThe master ring for Gr (2; 4) is given by the following localization of R {1,2} :The function ring of other maximal charts in U 0 , as a C-subalgebra of R II , are given below.· U {1,3} :R {1,3} is the polynomial ring over C generated by the non-0-nor-1 entries ofis the polynomial ring over C generated by the non-0-nor-1 entries ofis the polynomial ring over C generated by the non-0-nor-1 entries ofis the polynomial ring over C generated by the non-0-nor-1 entries ofis the polynomial ring over C generated by the non-0-nor-1 entries ofWithout further input from mathematics or D-brane probe in string theory, a soft noncommutative flag scheme can be obtained very flexibly or randomly. For example, here for Gr (2; 4), a formal replacement z ij −→z ij while keeping all the expressions above intact with the knowing thatz ij 's are noncommutative gives rise to a soft noncommutative Grassmann schemeX II associated to Gr (2; 4) by the recipe in the previous theme.Soft noncommutative Calabi-Yau schemes inX II associated to Gr (2; 4)?Let · P 5 be the projective 5-space over C with the homogeneous coordinates [y 0 : y 1 : · · · : y 5 ]; · V i A 5 (the affine 5-space over C) be the complement of the hyperplane {y i = 0} in P 5 , for i = 0, · · · , 5, that together produce an atlas on P 5 ; · Ψ : Gr (2; 4) → P 5 be the Plücker embedding of Gr (2; 4) in P 5 given by the correspondencewhere M is a 2 × 4 matrix with coefficients in C and |M I |, I ∈ II, is the determinant of the 2 × 2 submatrix of M specified by I.Then, in terms of X II , Ψ : Gr (2; 4) → P 5 is given by a gluing of embeddings of affine schemesWith slight abuse of notations, these Ψ I , I ∈ II, can be described by C-vector-space-homomorphisms Ψ I : Γ (P 5 , O P 5 (n)) −→ R I , n = 1, 2, · · · . Explicitly, for a homogeneous polynomial of degree n f := f (y 0 , y 1 , y 2 , y 3 , y 4 , y 5 ) ∈ Γ (P 5 , O P 5 (n)), one has = f (−z −1 13 , −z −1 13 z 23 , −z −1 13 z 24 , 1, z −1 13 z 14 , −z 24 + z −1 13 z 14 z 23 ),