Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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functions (), g(, x) are holomorphic in , eqn [4]
can be rewritten in terms of (, x):
@½ð; xÞðÞ
@
¼
ZZ
2C
d ^ d
ð; xÞgð; xÞ
Rð; Þg
1
ð; xÞ½6
The right-hand side of [6] is regular; therefore, this
relation is valid for all complex values.
Equation [6] with the boundary condition [5] is
equivalent to the following integral equation:
ð; xÞ¼ðÞþ
1
2i
ZZ
2C
d ^d
ZZ
2C
d ^ d
ð; xÞgð;xÞ
Rð;Þg
1
ð; xÞ½7
This equation can be derived using the generalized
Cauchy formula. Let f (z) be a smooth (not necessa-
rily holomorphic) function in a bounded domain D
in the complex plane. Then
f ðzÞ¼
1
2i
I
@D
d
z
f ðÞ
þ
1
2i
ZZ
D
d ^ d
z
@f ðÞ
@
½8
If the kernel g(, x) R(, )g
1
(, x)is
sufficiently good (e.g., it is sufficient to assume, that
(1 þjj )
1þ
g(, x)R(, )g
1
(, x)(1 þ
)
2
, >0, is
a continuous function at both finite and infinite
points), then we have a Fredholm equation (the
operator on the right-hand side of [7] is compact).
If it has no unit eigenvalues, eqn [7] is uniquely
solvable. But, for some values of x, one of the
eigenvalues may become equal to 1, and it results
in singularities of solutions.
If the norm of the integral operator is smaller than
1, eqn [7] is uniquely solvable. To generate solutions
that are regular for all values of physical variables, it
is natural to restrict the class of admissible spectral
data by assuming the kernel g(, x)R(, )g
1
(, x)
to be bounded in x for all , . In the scalar case
n = 1, this restriction imp lies:
Rð; Þ¼0
for all ; such that gð; xÞg
1
ð; xÞ
is unbounded in x ½9
For specific examples like the Kadomtsev–Petviashvili-II
(KP-II), direct scattering transform automatically
generates spectral data satisfying [9]. In KP-II, [9]
implies
Rð; Þ¼AðÞð ÞþTðÞð
Þ½10
The coefficient A() can be eliminated by multi-
plying the wave function to an appropriate function
of ; therefore, in standard texts, A() 0.
If for every the kernel R(, ) is equal to 0
everywhere except at finite number of points
1
(), ...,
k
(), one has the so-called local
@-problem. Such kernels are rather typical.
Examples of Soliton Systems Integrable
by the
@-Problem Method
Let us discuss some important examples.
The KP-II Hierarchy
The first nontrivial equation from the KP hierarchy
has the following form:
ðu
t
þ 6uu
x
u
xxx
Þ
x
¼ 3
2
u
yy
½11
From a physical point of view, the case of real
2
is
the most interesting one. Equation [11] is called
KP-I if
2
= 1 and KP-II if
2
= þ1. The Lax pair
for KP-II reads:
½L; A ¼0
where
L ¼ @
y
@
2
x
þ uðx; y; tÞ
A ¼@
t
4@
3
x
þ 6uðx; y; tÞ@
x
þ 3u
x
ðx; y; tÞþ3wðx; y; tÞ
½12
The Cauchy prob lem for initial data u(x, y,0)
decaying at infinity is solved by using the nonlocal
Riemann problem for KP-I an d local
@-problem for
KP-II. The wave function is assumed to be scalar
valued (n = 1), and
@-equation [4] takes the follow-
ing form:
@ ð; x; y; tÞ
@
¼ TðÞð
; x; y; tÞ½13
The wave function (, x, y, t) is assumed to be
regular for finite ’s and to have the following
essential singularity as = 1:
ð; x; y; tÞ¼expðx þ
2
y þ 4
3
tÞð1 þ oð1ÞÞ ½ 14
Equivalently, () 1 and the function g(, x, t) has
one essential singularity at = 1,
gð; x; tÞ¼expðx þ
2
y þ 4
3
tÞ½15
36
@-Approach to Integrable Systems
Higher times t
k
from the KP hierarchy are
incorporated into this scheme by assuming that
gð; tÞ¼exp
X
1
k¼1
k
t
k
!
½16
Here x = t
1
, y = t
2
, t = 4t
3
.
Equation [13] was originally derived (
bi
Ablowitz
et al. 1983) from the direct spectral transform. If the
potential u(x, y) is sufficiently small and
u(x, y) = O(1=(x
2
þ y
2
)
1þ
) for x
2
þ y
2
!1, then
the wave function (, x, y) for the L-operator [12]
Lð; x; yÞ¼0
ð; x; yÞ¼expðx þ
2
yÞ½1 þ oð1Þ
for x
2
þ y
2
!1
½17
can be constructed by solving the following integral
equation for the pre-exponent (, x, y) =(, x, y )
exp(x
2
y):
ð; x; yÞ¼1
ZZ
Gð; x x
0
; y y
0
Þuðx; yÞ
ð; x; yÞdx
0
dy
0
½18
where the Green function G(, x, y) is given by
Gð;x;yÞ¼
1
4
2
ZZ
e
iðp
x
xþp
y
yÞ
p
2
x
þip
y
2ip
x
dp
x
dp
y
½19
It is not holomorphic in , but
@G ð; x; yÞ
@
¼
i
2
sgnð<Þe
2i<x4i<=y
½20
The nonholomorphicity of G(, x, y ) results in
the special nonhol omorphicity of (, x, y) of the
form [13].
Remark We see that one function of two real
variables T() is sufficient to solve the Cauchy
problem in the plane. But it is also possible to
construct solutions of KP-II starting from generic
nonlocal kernels R(, ) ( to guarantee at le ast local
existence of solutions, it is enough to assume that
R(, ) is small and has finite support). It looks
like a paradox, but the situation is exactly the
same in the linear case. In the s tandard Fourier
method, only exponents with real momenta are
used, but local solutions can be constructed as
combinations of exponents with arbitrary complex
momenta.
Novikov–Veselov Hierarchy and Two-Dimensional
Schro¨ dinger Operator
Equations from this hierarchy admit representation
in terms of Manakov L–A–B triples,
@L
@t
n
¼½A
n
; LþB
n
L ½21
where
L ¼4@
z
@
z
þ uðz; tÞ
A
n
¼ 2
2nþ1
@
2nþ1
z
þ @
2nþ1
z
þ
½22
The order of B
n
is smaller than 2n þ 1. In particular,
for n = 1,
A
1
¼ 8 @
3
z
þ @
3
z
þ 2ðw @
z
þ
w@
z
Þ
B
1
¼ w
z
þ
w
z
½23
u
t
¼ 8@
3
z
u þ 8@
3
z
u þ 2@
z
ðuwÞþ2@
z
ðu
wÞ½24
where
uðz; tÞ¼
uðz; tÞ;@
z
wðz; tÞ¼3@
z
uðz; tÞ½25
This hierarchy is integrated using the scattering
transform at zero energy for the two-dimensional
Schro¨ dinger ope rator L. If Cauchy data with
asymptotic
uðzÞ!E
0
; wðzÞ!0; for jzj!1 ½26
is considered, the scattering transform for the
operator
~
L = L þ E
0
with the potential
~
u(z) = u(z) þ
E
0
at fixed energy E
0
and decaying at infinity is used.
In fact, the fixed-energy scattering problem is one of
the basic problems of mathematical physics, and the
Novikov–Veselov hierarchy can be treated as an
infinite-dimensional Abel symmetry algebra for this
problem. The scattering transform essentially depends
on the sign of E
0
. The case E
0
= 0, studied by Boiti,
Leon, Manna, and Pempinelli is the most complicated
from the analytic point of view, and we do not
discuss it now.
If E
0
< 0, the wave function satisfies a pure local
@-relation:
@ð; zÞ
@
¼ TðÞ
1
; z
½27
with ( ) 1, and
gð; z Þ¼e
ð=2Þð
zþz=Þ
;
2
¼E
0
½28
Starting from generic spectral data T(), one obtains
a fixed-energy eigenfunction for a second-order
operator,
~
Lð; zÞ¼E
0
ð; zÞ
L ¼4@
z
@
z
þ VðzÞ@
z
þ
~
uðzÞ
½29
To generate pure potential operators (V(z) 0), it is
necessary to impose additional symmetry constraints
of the spectral data (see the sect ion ‘‘Reduct ions on
the
@ da ta’’).
@-Approach to Integrable Systems 37
If E
0
> 0, there are two types of generalized
spectral data –
@-data and nonlocal Riemann
problem data. The wave function satisfies the
@-relation:
@ð; zÞ
@
¼ TðÞ
1
; z
; jj 6¼ 1 ½30
and has a jump at the unit circle jj= 1. The
boundary values
(, z ) =((1 0), z) are
connected by the following relation:
þ
ð; zÞ¼
ð; zÞþ
I
jj¼1
Rð; Þ
ð; zÞjdj½31
gð; z Þ¼e
ið=2Þð
zþz=Þ
;
2
¼ E
0
½32
Constraints on the spectral data associated with
pure potential operators were found by Novikov
and Grinevich for R(, ), and by Manakov and
Grinevich for T(). Existence of two different types
of generalized scattering da ta has a very transparent
physical meaning: there is a one-to-one correspon-
dence between the classical scattering amplitude at
energy E
0
and the nonlocal Riemann problem data
R(, ). The
@-data T() can be treated as a
complete set of additional parameters enumerating
all potentials with a given scattering amplitude at
one energy.
Davey–Stewartson-II and Ishimori-I Equations
The Davey–Stewartson-II (DS-II) equation
i@
t
q þ 2 @
2
z
þ @
2
z
q þðg þ
gÞq ¼ 0 ½33
@
z
g ¼@
z
jqj
2
½34
q ¼ qðz; tÞ; g ¼ gðz; tÞ; z ¼ x þ iy ½35
can be treated as an integrable (2 þ 1)-dimensional
extension of nonlinear Schro¨ dinger equation. The
Ishimori-I equation
@
t
S þS @
2
x
S þ@
2
y
S
þ@
x
w@
y
S þ@
y
w@
x
S ¼0 ½36
@
2
x
w @
2
y
w þ 2Sð@
x
S @
y
SÞ¼0 ½37
S ¼Sðx;y;tÞ; S ¼ðS
1
;S
2
;S
3
Þ; S
2
¼1 ½38
is an integrable (2 þ 1)-dimensional extension of the
Heisenberg magnetic equation. Both systems are
integrated by using the following zero-curvature
representation:
@
z
0
0 @
z
!
¼
1
2
0 qðz; t Þ
qðz; tÞ 0
!
½39
@
t
¼
2i@
2
z
þ ig iq
z
iq@
z
i
q
z
þ iq@
z
2i@
2
z
ig
0
@
1
A
½40
The wave function satisfies the following ‘‘scatter-
ing’’ equation:
@
k
0
0 @
k
0
@
1
A
T
¼
0
bðkÞ
bðkÞ 0
0
@
1
A
T
½41
Here
T
denotes the transposed matrix. Let us point
out the amazing symm etry between the direct and
inverse transforms.
Discrete Systems
In the examples discussed above, con tinuous vari-
ables are ‘‘encoded’’ in essential singularities of
g(, x). Discrete variables correspond to orders of
zeros and poles. For example, assuming that the
function g(, t) in the KP integration scheme
depends on extra continuous variables t
1
, t
2
, ...,
t
n
, ... and discrete variable t
0
= n,
gð; tÞ¼
t
0
exp
X
1
k6¼0
k
t
k
0
@
1
A
½42
one obtains solutions of the so-called two-dimensional
Toda–KP hierarchy.
Assume that we have a nonlocal
@-problem for a
scalar function with 1 and
gð; n
1
; ...; n
k
Þ¼
Y
k
j¼1
P
k
Q
k
n
k
½43
The wave function defines a map Z
k
! C
N
,
ðn
1
; ...; n
k
Þ!ðð
1
; n
1
; ...; n
k
Þ; ...;
ð
N
; n
1
; ...; n
k
ÞÞ ½44
where
1
, ...,
N
are some points in C. This
construction generates the so-called quadrilateral
lattices (each two-dimensional face is planar).
Multidimensional Problems
The
@-approach can also be applied to multidimen-
sional inverse-scattering problems, but typically the
scattering data are overdetermined and satisfy
additional nonlinear compatibility conditions. For
38
@-Approach to Integrable Systems
example, the Faddeev wave functions for the
n-dimensional stationary Schr
¨
odinger operator
@
2
x
1
@
2
x
n
þ uðxÞ
ðk;xÞ¼ðk kÞðk;xÞ½45
ðk; xÞ¼e
ikx
ð1 þ oð1ÞÞ ½46
in the nonphysical domain k
I
6¼ 0(k
R
and k
I
denote
the real and imaginary parts of k, respectively)
satisfy the following
@-equation:
@ðk; xÞ
@
k
j
¼2
Z
x2R
n
j
hðk; k
R
þ xÞðk þ x; xÞ
ðx x þ 2k x Þd
1
d
n
½47
The characterization of admissible spectral data
h(k, l), k 2 C
n
, l 2 R
n
is based on the following
compatibility equation:
@hðk; lÞ
@
k
j
þ
1
2
@hðk; lÞ
@l
j
¼2
Z
x2R
n
j
hðk; k
R
þ xÞhðk þ x; lÞ
ðx x þ 2k x Þd
1
d
n
½48
More details can be found in
bi
Novikov and Henkin
(1987).
Reductions on the
@-Data
The most generic
@-data usually result in solutions
from wrong functional class (they may, e.g., be
complex or singular), or extra constraints on the
auxiliary linear operators are necessary to obtain
solutions of the zero-curvature representation. For
example, to obtain real KP-II solutions using the
local
@-problem [13], the following reduction on the
@-data should be implied:
Tð
Þ¼
TðÞ½49
It can be easily derived from the direct transform.
But it is not always the case. For example, the
selection of pure potential two-di mensional
Schro¨ dinger operators originally was not so evident.
To formulate the answer, it is convenient to
introduce a new function b(), T() = b()
sgn(
1)=
.
For E
0
< 0, the following constraints select real
potential operators:
b
1
¼ bðÞ; b
1
¼
bðÞ½50
In some situations, the problem of finding appro-
priate reductions is the most difficult part of the
integration procedure. It is true not only for the
@-approach, but also for other techniques including
the finite-gap method. For example, the inverse-
spectral transform for the two-dimensional
Schro¨ dinger operator was first developed for
finite-gap (quasiperiodic) potentials and only later
for the decaying ones. For operators with finite gap
at one energy the first-order terms were constructed
by Dubrovin, Krichever, and Novikov in 1976, but
only in 1984 the potentiality reduction was found by
Novikov and Veselov.
Nonsingular Solutions
As mentioned above, one can construct regular
solutions by choosing sufficiently small (in an
appropriate norm) scattering data. But for some
special systems the regularity follows automatically
from reality reductions. For example, for arbitrary
large
@-data, real KP-II solutions constructed by the
local
@-problem [13] with reduction [49] are regular.
The proof is based on the theory of generalized
analytic functions (in the Vekua sense). Another
example is the two-dimensional Schr
¨
odinger opera-
tor at a fixed negative energy E
0
< 0. The potenti-
ality and reality constraints imply that the potential
is nonsingular for arbitrary large T(). But, unfortu-
nately, the
@-problem with regular data covers only
a part of the space of potentials. In fact, each such
operator possesses a strictly positive real eigenfunc-
tion at the level E
0
, exponentially growing in all
directions (it also follows from the generalized
analytic functions theory). Existence of such func-
tion implies that the whole discrete spectrum is
located above the energy E
0
, and it gives a
restriction on the potential. (For more details, see
the review by
bi
Grinevich (2000).)
Some Explicit Solutions
The generic
@-problem results in potentials that
could not be expressed in terms of elementary or
standard special functions. But for degenerate
kernels, a solution of the inverse-spectral problem
can be written explicitly. For example, if
Rð; Þ¼
X
n
k¼1
r
k
ðÞs
k
ðÞ½51
the wave function and solutions can be expressed in
quadratures.
In particular, if all r
k
() and s
k
() are -functions,
r
k
() = R
k
(
k
) and s
k
() = S
k
(
k
), the
wave function is rational in and can be expressed
as a rational combination of exponents of x
k
. If for
some k and l,
k
=
l
, this procedure needs some
@-Approach to Integrable Systems 39
regularization. For exampl e, it is possible to assume,
that (
0
)=(
0
) = 0.
If for all k,
k
=
k
, the
@-problem genera tes
rational in x solutions (lumps). It is possible to show
that, the Novikov–Veselov real rational solitons for
E
0
> 0 are always nonsingular, decay at 1 as
1=(x
2
þ y
2
), and the potential u(z) has zero scatter-
ing in all directions for the waves with energy E
0
.
The
@-Problem on Riemann Surfaces
In all examples discussed above, the spectral vari-
able is defined in a Riemann sphere. It is natural to
generalize it by considering wave functions depend-
ing on a spectral parameter defined on a Riemann
surface of higher genus. Spectral transforms of such
type arise in the theory of localized pertur bation of
periodic solutions. Assume that the KP-II potential
u(x, y) has the form
uðx; yÞ¼u
0
ðx; yÞþu
1
ðx; yÞ½52
where u
0
(x, y) is a real nonsingular finite-gap
potential and u
1
(x, y) decays sufficiently fast at
infinity. Denote by
0
(, x, y) the wave function of
the operator L
0
= @
y
@
2
x
þ u
0
(x, y), where 2 ,
the spectral curve is a compact Riemann surface of
genus g with a distinguished point 1. In addition to
essential singularity at the point 1, the wave
function
0
(, x, y) has g simple poles at points
1
, ...,
g
and is holomorphic in outside these
singular points. For a real nonsingular potent ial, is
an M-curve, that is, there exists an antiholomorphic
involution : ! , 1 = 1, the set of fixed
points form g þ 1 ovals a
0
, ..., a
g
, 12a
0
,
k
2 a
k
.
The wave function (, x, y) of the perturbed
operator L = @
y
@
2
x
þ u(x, y) is defined at the
same spectral curve , but it is not holomorphic
any more. It has the following properties:
1. At the point 1, the wave function (, x, y) has
an essential singularity: (, x, y) =
0
(, x, y)
(1 þ o(1)).
2. In the neighborhoods of the points
k
, (, x, y)
can be written as a product of a continuous
function by a simple pole at
k
.
3. The wave function (, x, y) satisfies the
@
equation
@ð; x; y; t Þ
@
d
¼ TðÞð;x; y; tÞ½53
where the (0, 1)-form T() = t()d
is regular
outside the divisor points
k
and in the neighbor-
hood of
k
it possible to define local coo rdinate
such that t() = sgn(=)t
1
()(
k
)=(
k
), t
1
()
is regular.
A solution of the inverse problem can be obtai ned
by using appropriate analogs of Cauchy kernels on
Riemann surfaces.
Quasiclassical Limit
The systems integrable by the
@-method usually
describe integrable systems with high-order deriva-
tives. It is well known that by applying some
limiting procedures to integrable systems one can
construct new completely integrable equations, but
integration methods for these equations are based on
completely different analytic tools. One of most
important examples is the theory of dispersionless
hierarchies. The limiting procedure for the
@-
problem (quasiclassical @-problem) was developed
by Konopelchenko and collaborators. In the KP
case, the quasiclassical limit of the wave function
(, t) is assumed to have the following form:
;
t
¼
^
;t;ðÞexp
Sð; tÞ
½54
It is possible to show that the function S(, t)
satisfies a Beltrami-type equ ation:
Sð; tÞ
@
¼ W ;
Sð; tÞ
@
½55
which is treated as a dispersionless limit of [13].
Higher-order corrections were also discussed in the
literature (see
bi
Konopelchenko and Moro (2003)).
See also: Boundary-Value Problems for Integrable
Equations; Integrable Systems and Algebraic Geometry;
Integrable Systems and the Inverse Scattering Method;
Integrable Systems: Overview; Integrable Systems and
Discrete Geometry; Riemann–Hilbert Methods in
Integrable Systems.
Further Reading
Ablowitz MJ, Bar Jaakov D, and Fokas AS (1983) On the inverse
scattering of the time-dependent Schro¨ dinger equation and the
associated Kadomtsev–Petviashvili equation. Studies in
Applied Mathematics 69(2): 135–143.
Beals R and Coifman RR (1981–82) Scattering, transformations
spectrales et equations d’evolution nonlineare. I, II: Seminaire
Goulaouic – Meyer–Schwartz – Exp. 22; Exp. 21. Palaiseau:
Ecole Polytechnique.
Beals R and Coifman RR (1985) Multidimensional inverse
scattering and nonlinear partial differential equations.
Proceedings of Symposia in Pure Mathematics 43: 45–70.
Bogdanov LV and Konopelchenko BG (1995) Lattice and
q-difference Darboux–Zakharov–Manakov systems via
@-dressing method. Journal of Physics A 28(5): L173–L178.
Bogdanov LV and Manakov SV (1988) The nonlocal
@ problem
and (2 þ 1)-dimensional soliton equations. Journal of Physics
A 21(10): L537–L544.
40
@-Approach to Integrable Systems
Grinevich PG (2000) The scattering transform for the
two-dimensional Schrodinger operator with a potential that
decreases at infinity at fixed nonzero energy. Russian
Mathematical Surveys 55(6): 1015–1083.
Konopelchenko B and Moro A (2003) Quasi-classical
@-dressing
approach to the weakly dispersive KP hierarchy. Journal of
Physics A 36(47): 11837–11851.
Nachman AJ and Ablowitz MJ (1984) A multidimensional
inverse-scattering method. Studies in Applied Mathematics
71(3): 243–250.
Novikov RG and Henkin GM (1987) The @-equation in the
multidimensional inverse scattering problem. Russian Mathe-
matical Surveys 42(4): 109–180.
Derived Categories
E R Sharpe, University of Utah, Salt Lake City,
UT, USA
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
In this article we shall briefly outline derived
categories and their relevance for physics. Derived
categories (and their enhancements) classify off-shell
states in a two-dimensional topological field theory
on Riemann surfaces with boundary known as the
open-string B model. We briefly review pertinent
aspects of that topological field theory and its
relation to derived categories, the Bondal–Kapranov
enhancement and its relation to the open-string B
model, as well as B model twists of two-dimensional
theories known as Landau–Ginzburg models, and
how information concerning stability of D-branes is
encoded in this language. We concentrate on more
physical aspects of derived categories; for a very
readable short review concentrating on the mathe-
matics, see, for example, Thomas (2000).
Sheaves and Derived Categories
in the Open-String B Model
Derived categories are mathematical constructions
which are believed to be related to D-branes in the
open-string B model. We shall begin by briefly
reviewing the B model, as well as D-branes.
The A and B models are two-dimensional topolo-
gical field theories, closely related to nonlinear
sigma models, which are supersymmetrizations of
theories summing over maps from a Riemann
surface (the world sheet of the string) into some
‘‘target space’’ X. In both the A and B model s, one
considers only certain special correlation functions,
involving correlators closed under the action of a
nilpotent scalar operator known as the ‘‘BRST
operator,’’ Q, which is part of the original super-
symmetry transformations. In considering the perti-
nent correlation functions, only certain types of
maps contribute. The A model has the properties of
being invariant under complex structure deforma-
tions of the target space X, and its pertinent
correlation functions are computed by summing
over holomorphic maps into the target X. The A
model will not be relevant for us here. The B model
has the properties of being invariant under Ka¨hler
moduli of the target X, and its pertinent correlation
functions are computed by summing over constant
maps into the target X. In the closed-string B model,
the states of the theory are counted by the
cohomology groups H
(X,
TX), where X is con-
strained to be Calabi–Yau. The BRST operator in
the B model Q can be identified with
@ for many
purposes. The open-string B model is the same
topological field theory, but now defined on a
Riemann surface with boundaries. As with all
open-string theories, we specify boundary conditions
on the fields, which force the ends of the string to
live on some submanifold of the target, and we
associate to the boundaries degrees of freedom
(known as the Chan–Paton factors) which describe
a (possibly twisted) vector bundle over the submani-
fold. In the case of the B model, the submanifold is a
complex submanifold, and the vector bundle is
forced to be a holomorphic vector bundle over that
submanifold.
To lowest order, that combi nation of a submani-
fold S of X together with a (possibly twisted)
holomorphic vector submanifold, is a ‘‘D-brane’’ in
the open-string B model. We shall denote the twisted
bundle by E
ffiffiffiffiffiffi
K
S
p
, where K
S
is the canonical
bundle of S, and the
ffiffiffiffiffiffi
K
S
p
factor is an explicit
incorporation of something known as the Freed–
Witten anomaly. Now, if i : S ,!X is the inclusion
map, then to this D-brane we can associate a
sheaf i
E.
Technically, a sheaf is defined by associ ating sets,
or modules, or rings, to each open set on the
underlying space, together with restriction maps
saying how data associated to larger open sets
restricts to smaller open sets, obeying the obvious
consistency conditions, together with some gluing
conditions that say how local sections can be
patched back together. A vector bundle defines a
Derived Categories 41
sheaf by associating to any open set sections of the
bundle over that open set. Sheaves of the form ‘‘i
E’’
look like, intuitively, vector bundles over submani-
folds, with vanishing fibers off the submanifold.
A more detailed discussion of sheaves is beyond the
scope of this article; see instead, for example, Sharpe
(2003).
To ‘‘associate a sheaf’’ means finding a sheaf such
that physical properties of the D-brane system are
well modeled by mathematics of the sheaf. (In
particular, the physical definition of D-brane has,
on the face of it, nothing at all to do with the
mathematical definition of a sheaf, so one cannot
directly argue that they are the same, but one can
still use one to give a mathematical model of the
other.) For example, the spectrum of open-string
states in the B model stretched between two
D-branes, associated to sheaves i
E and j
F, turn
out to be calculated by a cohomology group known
as Ext
X
(i
E, j
F).
There are many more sheaves not of the form i
E,
that is, that do not look like vector bundles over
submanifolds. It is not known in genera l whether
they also correspond to (on-shell) D-branes , but in
some special cases the an swer has been worked out.
For example, structure sheaves of nonreduced
schemes turn out to correspond to D-branes with
nonzero nilpotent Higgs vevs.
For a set of ordinary D-branes, the description
above suffices. However, more generally one would
like to describe collections of D-branes and anti-
D-branes, and tachyon s. An anti-D-brane has all
the same physical properties as an ordinary D-
brane, modulo the fact that they try to annihilate
each other. The open-string spectrum between
coincident D-branes and anti-D-branes contains
tachyons. One can give an (off-shell) vacuum
expectation value to such tachyons, and then the
unstable brane–antibrane–tachyon system will
evolve to some other, usually simpler, configura-
tion. For example, given a single D-brane wrapped
on a curve, with trivial line bundle, and an anti-D-
brane wrapped on the same curve, with line bundle
O(1), and a nonzero tachyon O(1) !O, then
one expects that the system will dynamically evolve
to a smaller D-brane sitting at a point on the curve.
Now, one would like to find some mathematics
that describes such systems, and gives information
about the e ndpoints of their evolution. Techni-
cally, one would like to classify universality classes
of world-sheet boundary renormalization group
flow.
It has been conjectured that derived categories of
sheaves provide such a classification. To properly
explain derived categories is well beyond the scope
of this article (see instead the ‘‘Furt her reading’’
section at the end), but we shall give a short outline
below.
Mathematically, derived categories of sheaves
concern complexes of sheaves, that is, sets of
sheaves E
i
together with maps d
i
: E
i
!E
iþ1
!E
i
!
d
i
E
iþ1
!
d
iþ1
E
iþ2
!
d
iþ2
such that d
iþ1
d
i
= 0. A cate gory is defined by a
collection of ‘‘objects’’ together with maps between
the objects, known as morphisms. In a derived
category of coherent sheaves, the objects are com-
plexes of sheaves, and the maps are equivalence
classes of maps between complexes.
Physically, if the complex consists of locally free
sheaves (equivalently, vector bundles), then we can
associate a brane/antibrane/tachyon system, by iden-
tifying the E
i
for i even, say, with D-branes, and the
E
i
for i odd with anti-D-branes. If the E
i
are all
locally free sheaves, then there are tachyons between
the branes and antibranes, and we can identify the
d
i
’s with those tachyons. In the open-string world-
sheet theory, giving a tachyon a vacuum expectation
value modifies the BRST operator Q, and a necessary
condition for the new theory to still be a topological
field theory is that Q
2
= 0, a condition which turns
out to imply that d
iþ1
d
i
= 0.
To re-create the structure of a derived category,
we need to impose some equivalence relations. To
see what sorts of equivalence relations one would
like to impose, note the following. Physically , we
would like to identify, for example, a configurat ion
consisting of a brane, an antibrane, and a tachyon,
which we can describe as a complex
OðDÞ!O
with a one-element complex
O
D
corresponding to the D-brane which we believe is
the endpoint of the evolution of the brane/antibrane
configuration.
One natural mathematical way to create identifi-
cations of this form is to identify complexes that
differ by ‘‘quasi-isomorphisms,’’ meaning, a set of
maps (f
n
: C
n
!D
n
) compatible with d’s, and
inducing an isomorphism
~
f
n
: H
n
(C) ffi H
n
(D)on
the cohomologies of the complexes. In particular,
in the example above, there is a natural set of maps
(–D)
0
D
42 Derived Categories
that define a quasi-isomorphism. More generally, in
homological algebra, one typically does computations
by replacing ordinary objects with projective or
injective resolutions, that is, complexes with special
properties, in which the desired computation
becomes trivial, and defining the result for the
original object to be the same as the result for the
resolution. To formalize this procedure, one would
like a mathematical setup in which objects and their
projective and injective resolutions are isomorphic.
However, to define an equivalence relation, one
usually needs an isomorphism, and the quasi-
isomorphisms above are not, in general, isomorphisms.
Creating an equivalence from nonisomorphisms,
to resolve this problem, can be done through a
process known as ‘‘localization’’ (generalizing the
notion of localization in commutative algebra).
The resulting equivalence relations on maps
between complexes define the derived category.
The derived category is a category whose objects
are complexes, and whose morphisms C
!D
are
equivalence classes of pairs (s, t) where s : G
!C
is
a quasi-isomorphism between C
and another com-
plex G
, and t : G
!D
is a map of complexes. We
take two such pairs (s, t), (s
0
, t
0
) to be equivalent if
there exists another pair (r, h) between the auxiliary
complexes G
, G
0
, making the obvious diagram
commute. This is, in a nutshell, what is meant by
localization, and by working with such equivalence
classes, this allows us to formally invert maps that
are otherwise noninvertible. (We encourage the
reader to consul t the ‘‘Furt her readi ng’’ sect ion for
more details.)
Mathematically, this technology gives a very
elegant way to rethink, for example, homological
algebra. There is a notion of a derived functor, a
special kind of functor between derived categori es,
and notions from homological algebra such as Ext
and Tor can be re-expressed as cohomologies of the
image complexes under the action of a derived
functor, thus replacing cohomologies with
complexes.
Physically, looking back at the physical realization
of complexes, we see a basic problem: different
representatives of (isomorphic) objects in the de rived
category are described by very different physical
theories. For example, the sheaf O
D
corresponds to a
single D-brane, defined by a two-dimensional
boundary conformal field theory (CFT), whereas
the brane/antibrane/tachyon collection O(D) !O
is defined by a massive nonconformal two-
dimensional theory. These are very different physical
theories. If we want ‘‘localization on quasi-
isomorphisms’’ to happen in physics, we have to
explain which properties of the physical theories we
are interested in, because clearly the entire physical
theories are not and cannot be isomorphi c.
Although the entire physical theories are not
isomorphic, we can hope that under renormalization
group flow, the theories will become isomorphic.
That is certainly the physical content of the statement
that the brane/antibrane system O(D) !O should
describe the D-brane corresponding to O
D
–after
world-sheet boundary renormalization group flow,
the nonconformal two-dimensional theory describing
the brane/antibrane system becomes a CFT describing
a single D-brane.
More globally, this is the general prescription for
finding physical meanings of m any categories: we
can associate physical theoriestoparticulartypes
of representatives of isomorphism classes of
objects, and then although distinct r epresentatives
of the same object may give rise to very different
physical theories, those physical theories at least lie
in the same universality class of world-sheet
renormalization group flow. In oth er words,
(equivalence classes of) objects are in one-to-one
correspondence with universality classes of physical
theories.
Showing such a statement directly is usually not
possible – it is usually technically impractical to
follow renormalization group flow explicitly. There
is no symmetry reason or other basic physics reason
why renormalization group flow must respect quasi-
isomorphism. The strongest constraint that is clearly
applied by physics is that renormalization group
flow must preserve D-brane charges (Chern char-
acters, or more properly, K-theory), but objects in a
derived category contain much more information
than that.
However, indirect tests can be performed, and
because many indirect tests are satisfied, the result is
generally believed.
The reader might ask why it is not more efficient
to just work with the cohomology complexes
H
(C) themselves, rather than the original com-
plexes. One reason is that the original complexes
contain more information than the cohomology –
passing to cohomology loses information. For
example, there exist examples of complexes that
have the same cohomology, ye t are not quasi-
isomorphic, and so are not identified within the
derived category, and so physically are believed
to lie in different universality classes of boundary
renormalization group flow.
Another motivation for relating physics to derived
categories is Kontsevich’s approach to mirror sym-
metry. Mirror symmetry relates pairs of Calabi–Yau
manifolds, of the same dimension, in a fashion such
that easy classical computations in one Calabi–Yau
Derived Categories 43
are mapped to difficult ‘‘quantum’’ computations
involving sums over holom orphic curves in the other
Calabi–Yau. Because of this property, mirror sym-
metry has proved a fertile ground for algebraic
geometers to study. Kontsevich proposed that mirror
symmetry should be understood as a relation
between derived categories of coherent sheaves on
one Calabi–Yau and derived Fukaya categories on
the other Calabi–Yau. At the time he made this
proposal, no one had any idea how either could be
realized in physics, but since that time, physicists
have come to believe that Kontsevich was secretly
talking about D-branes in the A and B models.
Bondal–Kapranov Enhancements
Mathematically derived categories are not quite as
ideal as one would like. For example, the cone
construction used in triangulated categories does not
behave functorially – the cone depends upon the
representative of the equivalence clas s defining an
object in a derived category, and not just the object
itself.
Physically, our discussion of brane/antibrane
systems was not the most general possible. One
can give vacuum expectation values to more general
vertex operators, not just the tachyons.
Curiously, these two issues solve each other. By
incorporating a more general class of boundary vertex
operators, one realizes a more general mathematical
structure, due to Bondal and Kapranov, which repairs
many of the technical deficiencies of ordinary derived
categories. Ordinary complexes are replaced by gen-
eralized complexes in which arrows can map between
non-neighboring elements of the complex. Schemati-
cally, the BRST operator is deformed by boundary
vacuum expectation values to the form
Q ¼
@ þ
X
a
a
and demandi ng that the BRST operator square to
zero implies that
X
a
@
a
þ
X
a;b
b
a
¼ 0
which is the same as the condition for a generalized
complex. Note that for ordinary complexes, the
condition above factors into
@
n
¼ 0
nþ1
n
¼ 0
which yields an ordinary complex of sheaves
(Figure 1).
Landau–Ginzburg Models
So far we have described how derived categories are
relevant to geome tric compactifications, that is,
sigma models on Calabi–Yau manifolds. However,
there are also ‘‘nongeometric’’ theories – CFTs that
do not come from sigma models on manifolds, of
which Landau–Ginzbur g models and their orbifolds
are prominent examples. Landau–Ginzburg models
can also be twisted into topological field theories,
and the B-type topological twist of (an orbifold of) a
Landau–Ginzburg model is believed to be iso-
morphic, as a topological field theory, to the B
model obtained from a nonlinear sigma model, of
the form we outlined earlier. Landau–Ginzburg
models have a very differe nt form than nonlinear
sigma models, and so sometimes there can be
practical computational advantages to working
with one rather than the other.
A Landau–Ginzburg model is an ungauged sigma
model with a nonzero superpotential (a holo-
morphic function over the target space that defines
a bosonic potential and Yukawa couplings). (In
‘‘typical’’ cases, the target space is a vector space.)
Because of the superpotential, a Landau–Ginzburg
model is a massive theory – not itself a CFT, but
many Landau–Ginzburg models are believed to flow
to CFTs under the renormalization group.
In formulating open strings based on Landau–
Ginzburg models, naive attempts fail because of
something known as the Warner problem: if the
superpotential is nonzero, then the obvious ways to
try to define the theory on a Riemann surface with
boundary have the undesirable property that the
supersymmetry transformations only close up to a
nonzero boundary term, proportional to derivatives
of the superpotential. In order to find a description
of open strings in which the supersymmetry trans-
formations close, one must take a very nonobvious
formulation of the boundary data. Specifically, to
solve the Warner problem, one is led to work with
pairs of matrices whose product is proportional to
the superpotential.
This method of solving the Warner problem is
known as matrix factorization, and D-branes in this
theory are defined by the factorization chosen, that
is, the choice of pairs of matrices. In simple cases,
we can be more explicit as follows. Choose a set of
3
2
00000
1
Figure 1 1. Example of generalized complex. Each arrow is
labeled by the degree of the corresponding vertex operator.
44 Derived Categories
polynomials F
, G
such that the Landau–Ginzburg
superpotential W is given by
W ¼
X
F
G
þ constant:
The F
and G
are used to define the boundary
action – the F’s appear as part of the boundary
superpotential and the G’s appear as part of the
supersymmetry transformations of boundary fermi
multiplets. The F
and G
, that is, the factorization
of W, determine the D-brane in the Landau–
Ginzburg theory. We can also think of having a
pair of holomorphic vector bundles E
1
, E
2
of the
same rank, and interpret F and G as holomorphic
sections of E
_
1
E
2
and E
_
2
E
1
, respectively, obey-
ing FG / W Id and GF / W Id, up to additive
constants.
Although a Landau–Ginzburg model is not the
same thing as a sigma model on a Calabi–Yau,
orbifolds of Landau–Ginzburg models are often on
the same Ka¨ hler moduli space. Perhaps, the most
famous example of this relates sigma models on
quintic hypersurfaces in P
4
to a Z
5
orbifold of a
Landau–Ginzburg model over C
5
with five chiral
superfields x
1
, x
2
, x
3
, x
4
, x
5
, and a superpotential of
the form
W ¼x
5
1
þ x
5
2
þ x
5
3
þ x
5
4
þ x
5
5
þ x
1
x
2
x
3
x
4
x
5
for a complex number, corresponding to the
equation of the degree-5 hypersurface in P
4
. The
(complexified) Ka¨hler moduli space in this example
is a P
1
, with the sigma model on the quintic a t one
pole, the zero-volume limit of the sigma model along
the equator, and the Landau–Ginzburg orbifo ld at
the opposite pole.
Since the closed-string topological B model is
independent of Ka¨ hler moduli, and the sigma model
on the quintic and the Landau–Ginzburg orbifold
above lie on the same Ka¨ hler moduli space, one
would expect them both to have the same spectrum
of D-branes, and indeed this is believed to be true.
Pi-Stability
So far we have discussed D-branes in the topological
B model, a topological twist of a physical sigma
model. If we untwist back to a physical sigma
model, then the stability of those D-branes becomes
an issue.
To begin to understand what we mean by stability
in this context, consider a set of N D-branes
wrapped on, say, a K3 surface, at large radius (so
that world-sheet instanton corrections are small).
On the world volume of the D-branes, we have a
rank-N vector bundle, and in the physical theory on
that world volume we have a consistency condition
for supersymmetric vacua, that the vector bundle be
‘‘Mumford–Takemoto stable.’’ To understand what
is meant by this condition on a Ka¨hler manifold, let
! denote the Ka¨ hler form, and define the ‘‘slope’’
of a vector bundle E on a manifold X of complex
dimension n to be given by
ðEÞ ¼
R
X
!
n1
^ c
1
ðEÞ
rank E
where ! is the Ka¨ hler form. Then, we say that E is
(semi-)stable if for all subsheaves F satisf ying
certain consistency conditions, (F)() <(E).
Since the slope of a bundle depends upon the
Ka¨ hler form, whether a given bundle is Mumford–
Takemoto stable depends upon the metric. In
general, on a Ka¨ hler manifold, the Ka¨ hler cone
breaks up into subcones, with a different moduli
space of (stable) holomorphic vector bundles in each
subcone.
This is a mathematical notion of stability, but it also
corresponds to physical stability, at least in a regime in
which quantum corrections are small. If a given
bundle is only stable in a proper subset of the Ka¨hler
cone, then when it reaches the boundary of the
subcone in which it is stable, the gauge field config-
uration that satisfies the Donaldson–Uhlenbeck–Yau
partial differential equation splits into a sum of two
separate bundles. In a heterotic string compactifica-
tion, this leads to a low-energy enhanced U(1) gauge
symmetry and D-terms which realize the change in
moduli space. In D-branes, this means the formerly
bound state of D-branes (described by an irreducible
holomorphic vector bundle) becomes only marginally
bound; a decay becomes possible.
Pi-stability is a proposal for genera lizing the
considerations above to D-branes no longer wrap-
ping the entire Calabi–Yau, and including quantum
corrections.
In order to define pi-stability, we must first
introduce a notion of grading ’ of a D-brane.
Specifically, for a D-brane wrapped on the entire
Calabi–Yau X with holomorphic vector bundle E,
the grading is defined as the mirror to the expression
R
X
ch(E) ^ , where encodes the periods. Close to
the large-radius limit, this has the form:
’ðEÞ ¼
1
Im log
Z
X
exp B þ i!ðÞ^ch EðÞ
^
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tdðTXÞ
q
þ
where B is a 2-form, the ‘‘B field.’’ As defined ’ is
clearly S
1
-valued; however, we must choose a
Derived Categories 45