On quantization of singular varieties and applications to D-branes

17 Pages·0.18 MB·English

by  D. Melnikov

ITEP-TH-80/01
ITP-2002-1E
hep-th/0201153
On quantization of singular varieties
and applications to D-branes
3
0
0 Dmitry Melnikova,b and Alexander Solovyovc,d
2

n
a
J aITEP, Moscow, Russia;

0
e-mail: [email protected]
1

bMoscow State University, Department of Physics
2
v
3 cBogolyubov ITP, Kiev, Ukraine;
5
e-mail: [email protected]
1
1
0 dKiev National University, Department of Physics
2
0
/
h
t
-
p Abstract
e
h
We calculatethe ringofdifferentialoperatorsonsomesingularaffine varieties(intersect-
:
v
ing stacks, a point on a singular curve or an orbifold). Our results support the proposed
i
X connectionofthe ringofdifferentialoperatorswithgeometryofD-branesin(bosonic)string
r theory. In particular, the answer does know about the resolution of singularities in accor-
a
dance with the string theory predictions.
1 Introduction
Merkulov [7]proposedaconstructionfordeformation quantization of affinevarieties. Inparticu-
lar, he considered quantization of the n-tuple point xn = 0 and has proved the quantum algebra
of functions on the associated phase space to be the matrix algebra Mat(n). Quantization of n
coincident hyperplanes in RN+1 gives the tensor product of the matrix algebra Mat(n) and N
copies of theHeisenbergalgebrarelated todirections along thehyperplanes. Thisresultreminds
the appearance of the non-Abelian degrees of freedom on the stack of n coincident D-branes [8],
[14]. Recently the relevance of this procedure to D-brane physics (in particular, to boundary
string field theory [12], [13], [10], [11]) was established quantitatively in [4].
The present note aims to apply this technique to some singular varieties and to compare the
result to that known from D-brane physics.
The paper is organized as follows. In the second section we describe the procedure of quan-
tization of affine varieties, including singular ones. The methods of algebraic geometry allow
us to perform the quantization. This is given in terms of the ring of differential operators on a
subvariety. We propose the explicit description of this ring in terms of an arbitrary resolvent of
theringof functionsonthesubvariety. According totheproposedconnection withthegeometry
of D-branes, the properties of the ring of differential operators should capture, in particular, the
unbroken gauge symmetry in the open string sector. In the third section we study several ex-
amples and compare the results with the predictions of string theory. In the case of intersecting
stacks we find a good agreement with physics: two subalgebrae responsible for the non-Abelian
degrees of freedom living on the worldvolume of each D-brane and non-local operators (massive
modes of the strings stretching between branes). The structure of the resulting algebra of dif-
ferential operators appears especially transparent in terms of the Mayer-Vietoris sequence. For
a point on a singular curve (“cusp”) the quantum algebra behaves just as if the singularity was
blown up(resolved). In the case of the point on the C2/Z orbifold we finda perfect agreement
m
with the string theory picture: when the point approaches the singularity, the dimension of the
algebra increases m2 times. The same result is obtained using the blow-up.
Thus we found the complete agreement with bosonic string theory picture. It would be
interesting to generalize these considerations to the case of superstrings. This will be done
elsewhere.
2 Reduction of the ring of differential operators onto a subvariety
Algebraic geometry studies geometric concepts using the technique of (commutative) algebra.
An important motivation for such a study is application to reducible or singular varieties
(e.g. [9]). The ring of polynomial functions on an (affine algebraic) variety M embedded into
the affine space AN is given as the factor of the ring of polynomials in x1,...,xN w.r.t. the ideal
generated by the set of equations defining M:
k[M] = k[x1,...,xN]/(φ ,...,φ ), (1)
1 n
2
φ (x) being some polynomials and k being the base field.
α
φ (x) = 0,
1
 (2)
 φ (x··)·= 0
n
are the equations of M. Roughly speaking, k[M] is the restriction of the ring of polynomials
k[x1,...,xN] onto M having a nontrivial kernel — ideal (φ ,...,φ ). For instance, let X be the
1 n
n-tuple point given by the equation xn = 0 on the affine line A1; then for n = 1
k[M] k, (3)
≃
and for n = 2
k[M] a+bǫ a,b k, ǫ2 =0 . (4)
≃ | ∈
n o
So algebraically a point can be easily distinguished from a double point which would never be
possible in topological approach to the set of solutions of (2).
We would like to define the ring of differential operators on a subvariety. It is natural to use
the construction of the ring of functions on the subvariety in terms of some resolvent. Namely,
let (K·,d ) be a resolution of the structure sheaf k[M] of the subvariety M in the affine space
K
AN with Kp k[x] V , V knp, grd = 1, and H0(K·) k[M]. We define differential
≃ ⊗ np np ≃ K − ≃
0,0
operators on the algebraic variety M as the cohomologies H H of the double sequence
DM δ d
K˜··:
K˜p,q Hom V ,V V V∗ , (5)
≃ D⊗ nq np ≃ np ⊗D⊗ nq
(cid:0) (cid:1) AN
where stands for the ring of differential operators on . The horizontal differential acts
D
according to
dmp,q = d mp,q K˜p−1,q, (6)
K
◦ ∈
and the vertical one
δmp,q = mp,q d K˜p,q+1 (7)
K
◦ ∈
for mp,q Kp,q. Obviously, d and δ commute. The differential d of the resolvent K· is the
K
∈
multiplication by a matrix whose entries are some functions, and we regard those functions as
the zeroth order differential operators. Thereby one needs to consider some matrix differential
operators as elements of K˜·· in order to define differential operators on a subvariety (see (5):
Hom V ,V Mat(n ,n )).
nq np ≃ p q
Note that thus defined differential operators on a subvariety naturally act on the ring of
(cid:0) (cid:1)
functions on this subvariety. It is obvious from the following general consideration. The double
complex just considered is a particular case of the general construction (e.g. [5]). Let (K,d ) be
K
a differential module. These data determine the bicomplex (K˜ = End(K),d,δ) with the two
differentials
d: End(K) m d m, δ :m m d (8)
K K
∋ → ◦ → ◦
3
(note that d End(K)). In the ultimately general case there exists the natural action of
K
∈
H H (K˜) on H(K) described as follows. Take a cohomology class [m] H H (K˜); its repre-
δ d δ d
∈
sentative m satisfies
dm = 0 d m = 0,
K
⇐⇒ ◦ (9)
δm = 0 m d = d m˜ for some m˜
K K
⇐⇒ ◦ ◦
and is determined up to
m m+d m +m d , whered m = 0. (10)
K 1 2 K K 2
→ ◦ ◦ ◦
Given [f +d g] H(K), the natural action
K
∈
[m][f] =[mf] (11)
is defined unambiguously (that is why does act correctly on k[M]). This fact is established
M
D
by the direct calculation. The key feature of the resolvent is its acyclicity everywhere but in
one degree of grading, so the same should apply to the complex just considered: (End(K),d,δ)
also proves to be a resolvent with the nontrivial cohomologies H End(K) (the injectivity
≃
of the just described canonical homomorphism ϕ : H(End(K)) End(H(K)) is proved in
→
Appendix). Note that all the homomorphisms (endomorphisms) are understood as those over
the base field k, Hom( , ) = Hom ( , ).
k
· · · ·
In the above considerations we replace End(k[x]) actually extracting a subclass of
→ D D
all endomorphisms End(k[x]) and call H H (K˜··) differential operators on the subvariety. It is
δ d
completely in the spiritof algebraic geometry — recall the definition (1) of polynomial functions
on an affine variety. Though the reduction procedure has the most natural description in terms
of all endomorphisms End(k[x]), i.e. all integral operators, local differential operators are of
especial physical importance: these are local differential operators that describe massless modes
(see further examples).
Nowassumethatconstrainsφα areregular,i.e.φαdoesnotdividezeroink[x]/(φ1,...,φα−1).
Consider the Koszul resolvent (K·,d ) relevant to this case [6]:
K
ΛnV k[x] dKoszul Λn−1V k[x] dKoszul dKoszul k[x] 0 ; (12)
⊗ −→ ⊗ −→ ··· −→ −→
where V = span e ,...,e is the n-dimensional vector space, v = (φ ,...,φ ); and the differ-
1 n 1 n
{ }
ential d = i is given by the interior differentiation, i e = φ . The reduction procedure is
Koszul v v α α
4
described by the double complex K˜··:
ΛnV ΛnV∗ d Λn−1V ΛnV∗ d d ΛnV∗ 0
⊗D⊗ −→ ⊗D⊗ −→ ··· −→ D⊗ −→
δ δ δ
ΛnV  Λn−1V∗ d Λn−1V  Λn−1V∗ d d Λn−1V∗ 0
  
⊗Dy⊗ −→ ⊗Dy ⊗ −→ ··· −→ D⊗ y −→
δ δ δ
.. .. ..
. . .
y y y
δ δ δ
ΛnV d Λn−1V d d  0
  
y⊗D −→ y ⊗D −→ ··· −→ Dy −→
0 0 0
  
y y y
(13)
The horizontal and vertical differentials d and δ are Koszul differentials, d acting on V and δ
on V∗ (to make notations for δ more convenient, we have replaced ΛkV∗ with its Hodge dual
Λn−kV∗). is a free right or left module over k[x] (but it is not a free bimodule!); so H··(K˜··)
D d
is concentrated in the last column, and H··(K˜··) is concentrated in the last row. These are the
δ
sufficient conditions that provide [2]
H H (K˜··) H (A·) H H (K˜··). (14)
d δ D δ d
≃ ≃
That is why does not depend on whether we work with the right or left ideal (cf. [7]). The
M
D
total complex is obtained via the contraction of the grading:
totK˜·· A· Ak, Ak = K˜p,q. (15)
≃ ≃
k p+q=k
M M
Its differential
D = d+( )pδ. (16)
−
Let us stress that this definition of is equivalent to that of [7]:1
M
D
0,n
H D , (17)
d ≃ φ
α
α D
P
and the ring of differential operators on the subvariety is constructed as
0,n
H H / , (18)
δ d ≃ DM ≃ N I
where the right ideal
= φ . (19)
α
I ( α D)
X
1In [7] Merkulov used deformation quantization and the Moyal ⋆-product. It leads to integral operators in
general,butifwerestrictourselvestopolynomialsinp,theMoyalalgebraisnothingbutthealgebraofdifferential
operators D (see furthercomments).
5
The associated normalizer
= m mφ α (20)
α
N { ∈ D| ⊂ I ∀ }
is actually the maximal subalgebra in which is a two-sided ideal. Introduction of normalizer
I
makes the induced multiplication in correctly defined. Such a prescription is precisely the
M
D
quantum analog of the Hamiltonian reduction [4].
Consider the codimension one example, i.e. a principal ideal (e.g. = (xn)). Contraction
I I
of the grading yields the total complex A· (physicists usually call it BRST):
0 d1 (k k) d2 0,
−→ D −→ D⊗ ⊕ −→ D −→
d f = (xnf,fxn), d (g ,g )= g xn xng , (21)
1 2 1 2 1 2
−
d d = 0.
2 1
◦
What we need is kerd /imd ;
2 1
kerd = (g ,g ) g xn = xng , (22)
2 1 2 1 2
{ | }
so a representative of kerd is uniquely determined by g (of course, the subscript means
2 1 r r
∈N
“right,” whereas means “left”).
l
imd = (xnf,fxn) f , (23)
1
{ | ∈ D}
and, finally,
kerd /imd . (24)
2 1 M
≃D
One can express a representative of any (linear) subspace in modulo as
D I
h(x,∂) = h0(∂)+xh1(∂)+ +xn−1hn−1(∂) (25)
···
for some h = h (∂). For h this sum must also satisfy the equation (“ ” means “equal
i i
∈ N ≡
modulo ,” and standsfor the idealin theringof functions or differential operators depending
I I
on context)
hxn 0 (26)
≡
or, equivalently (we denote ∂ = π for convenience),
l n dn−l+ih (π)
i
= 0, l = 0,...,n 1. (27)
( l i ! dπn−l+i −
Xi=0 −
We have used the commutation relation
n n dkf
f(π)xn = xn−k . (28)
k ! dπk
k=0
X
The general solution is
n−1 i−1 1 i−k−1 i k 1
hi(∂) = ai,k∂k + ( 1)j+1 − − ak,n−i+k+j∂n+j (29)
(i k)!  − j ! 
kX=0 kX=0 − jX=0
 
6
with n2 arbitrary integration constants a , 0 i,k n 1; thus dim = n2. The natural
i,k M
≤ ≤ − D
action of on k[M] provides the celebrated isomorphism Mat(n). We see that in the
M M
D D ≃
case of the n-point local differential operators prove to be a sufficient subclass of End(k[x]) to
provide the complete matrix algebra, i.e. all endomorphisms End(k[M]).
D
To complete the picture it remains to determine the cokerd . The obvious relation
2 ≃ Il⊕Ir
n−1 n−1 l n dn−l+ih
xih xn xl i (30)
i! ≡ l i ! dπn−l+i
Xi=0 Xl=0 Xi=0 −
implies
D = 0. (31)
l r
I ⊕I
3 Examples
In this section we study some examples in order to compare the result to that expected from
physics. We find a perfect agreement. In particular, it is clear how the non-Abelian degrees of
freedom appear on each of the intersecting stacks. The quantization procedure also manages to
find the zero modes of the D-instanton. After that we examine the present technique versus the
resolution of singularities, and find that the quantization knows about the resolution. This fact
nicely illustrates the connection of differential operators with string theory.
3.1 Intersecting stacks
Consider the case of A2 and the unique constraint (generator of the ideal) φ = xmyn. The
calculation (see appendix B) reveals that of importance are elements like (in this subsection h’s
are defined by (25),(29))
span h(y,∂ )xmf(x,∂ ) . (32)
y x
N ⊃ { }
They form a closed subalgebra :
y M
A ⊂ D
(h(y,∂ )xmf(x,∂ ))(h˜(y,∂ )xmf˜(x,∂ )) (hh˜)xm(fxmf˜). (33)
y x y x
≡
This subalgebra is localized on the stack along the x axis: all the representatives vanish else-
where. There also exists a similar subalgebra
= span h(x,∂ )ynf(y,∂ ) . (34)
x x y
A { }
The localization of these subalgebrae can be illustrated with the help of the Mayer-Vietoris
sequence for constructed as follows. The two ideals corresponding to the two stacks are
M
D
= (xm), = (yn). (35)
1 2
I I
The exact Mayer-Vietoris sequence for k[M]
embedding difference
0 k[M M ] k[M ] k[M ] k[M M ] 0
1 2 1 2 1 2
−→ ∪ −→ ⊕ −→ ∩ −→
(36)
k k k
k[x] k[x] k[x] k[x]
I1I2 I1 ⊕ I2 I1⊕I2
7
induces the acyclic double complex for End(k[M])
d d
Hom(k[M1∩M2],k[M1∪M2]) −→ Hom(k[M1∩M2],k[M1]⊕k[M2]) −→ End(k[M1∩M2]) −→ 0
δ δ δ
Hom(k[M1]⊕k[M2],k[M1∪M2]) −d→ End(k[M1]⊕k[M2]) −d→ Hom(k[M1]⊕k[M2],k[M1∩M2]) −→ 0
y y y
δ δ δ
End(k[M1∪M2]) −d→ Hom(k[M1∪M2],k[M1]⊕k[M2]) −d→ Hom(k[M1∪M2],k[M1∩M2]) −→ 0
y y y
0 0 0
   (37)
y y y
with d and δ defined by (8).
Remark We must fulfil the condition
= (38)
1 2 1 2
I ∩I I I
for (36) to be exact. That is why such a sequence is of little use say for a double point with
= (x) = .
1 2
I I
The short sequence (36) provides the resolvent
k[M ] k[M ] k[M M ] 0 (39)
1 2 1 2
⊕ −→ ∩ −→
with H1 k[M M ], H2 0. The resolvent for End(k[M M ]) can be built using the
1 2 1 2
≃ ∪ ≃ ∪
general recipe (8):
d
Hom(k[M M ],k[M ] k[M ]) End(k[M M ]) 0
1 2 1 2 1 2
∩ ⊕ −→ ∩ −→
δ δ
End(k[M] k[M ]) d Hom(k[M ] k[M ],k[M M ]) 0 (40)
1 2 1  2 1 2
y ⊕ −→ ⊕ y ∩ −→
0 0
 
y y
Using (37), one easily computes the necessary cohomologies
End(k[M M ]) H2,1H dHom(k[M1]⊕k[M2],k[M1∪M2]) Hom(k[M1]⊕k[M2],k[M1∪M2]). (41)
1∪ 2 ≃ δ d ≃ δdHom(k[M1∩M2],k[M1∪M2]) ≃ δHom(k[M1∩M2],k[M1∪M2])
In the case of the finite order operators we have
span (h h + ,h˜ h˜ + )
x y x y x y
End(k[M M ]) { A A } ; (42)
1 2
∪ ≃ span (h h ,h h )
x y x y
{ }
and that is why we associate , with the two stacks and the finite-dimensional part h h
x y x y
A A { }
with the intersection.
A very important property is
= 0 = . (43)
x y y x
A A A A
8
It allows to identify , with the non-Abelian degrees of freedom living on the worldvolume
x y
A A
ofthecorrespondingstack of D-branes. Inthecaseof intersecting stacks thefollowing excitation
modes appear [14], [1], [3]:
Mat(m) Ψ
. (44)
Ψ Mat(n) !
Orthogonal subalgebrae Mat(m), Mat(n) are identified with , . Ψ corresponds to massive
x y
A A
strings stretching between the two stacks and may be interpreted in terms of the non-local
operators which appear if we omit the locality condition in our construction.2
If the two stacks are intersecting at an arbitrary non-zero angle, the result does not depend
on its value. It also agrees well with the answer known from physics: the structure of massless
modes does not depend on the non-zero angle between branes.
For an arbitrary number of stacks intersecting at a point the generator is given by φ =
xmyn ri=3φi with φ1 = xm, φ2 = yn, φi = (αix+y)βi. The straightforward generalization is to
consider
Q
r
= span yn φ h(x,∂ )g(y,∂ ) . (45)
x i x y
A ( ! )
i=3
Y
It is a closed subalgebra localized on the first stack. This fact can be proved, for example, using
(91). Just as in the case of the two stacks, the orthogonality relation (43) is satisfied for any
pair of stacks. Denote
= φ ...φ ...φ ...φ H (x,y,∂) , (46)
Aij 1 ˆi ˆj r ij
n o
where H satisfy
ij
H φ φ = φ φ H˜ (47)
ij i j i j ij
for some H˜ (x,y,∂) . Obviously, ( was defined in (45)). Some orthogonality
ij i ij i
∈ D A ⊂ A A
relations hold for , e.g.
ij
A
= span (φ φ φ ...φ H )(φ φ φ ...φ H ) = 0. (48)
12 34 3 4 5 r 12 1 2 5 r 34
A A { }
So one can associate with the pair of stacks (i,j).
ij
A
3.2 A line with a double point
The constraints are
xy = 0,
(49)
( y2 = 0,
determining the ring of functions
k[M] f(x)+Aǫ ǫ2 = 0 . (50)
≃ |
n o
The subring f(x) is the ring of functions on the affine line, and the nilpotent is responsible
{ }
for the D-instanton located at the origin.
2Of course, ourresults apply tothe non-supersymmetric(purely bosonic) theory.
9
Let us determine . Any operator modulo can be brought to the form
M
D I
h= xnf (π ,π )+yg(π ,π ). (51)
n x y x y
n
X
For convenience we define π = ∂ , π = ∂ . Belonging of h to requires
x x y y
N
∂2fn = 0, n = 0,1,...,
∂π2
y
 ∂2g +2∂f0 = 0,
 ∂∂∂πf∂nπy22−yf01 +=∂∂ππ∂0yx2,f∂nπy = 0, n = 1,2,..., (52)
∂πx∂πy
The general solution in the clas∂sπ∂ox2∂fgπfiyn+ite∂∂πof0xrd=er0o.perators is
h= f(x,∂ )(1 y∂ )+yg(∂ )+Cy∂ . (53)
x y x y
−
Obviously, f(x,∂ )(1 y∂ ) is the subring of differential operators on the line, which can
x y
{ − }
be established through its action on k[M]:
f(x,∂ )(1 y∂ )φ(x) = f(x,∂ )φ(x),
x y x
− (54)
f(x,∂ )(1 y∂ )y = 0.
x y
−
It describes the original D-brane’s degrees of freedom. Analogously, y∂ is related to the D-
y
instanton,
y∂ φ(x) = 0,
y
y∂ y = y, (55)
y
(y∂ ) (y∂ ) = y∂ .
y y y
◦
At last, zero modes of the D-instanton are the physical assignment of yg(∂ ) .
x
{ }
3.3 A point on the cusp
The cusp is defined by
y2 x3 = 0. (56)
−
Additional equation x = a = 0 sets two different points on this curve. The related algebra is
6
the matrix algebra Mat(2), off-diagonal elements being represented by non-local operators (shift
operators). When a 0 these two points glue together to form a double point with the ring of
→
functions (4) and the related algebra of differential operators Mat(2).
M
D ≃
Let us resolve the singularity. This goal is reached blowing up the origin, i.e. saying that the
good coordinate is s = y instead of y; then the equation of the curve (56) takes the form
x
x2(x s2) =0. (57)
−
10