Open strings andnoncommutative
algebraic geometry
David R. Morrison

Open strings and noncommutative
algebraic geometry

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFTDavid R. Morrison
University of California, Santa Barbara

Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence

Colloquium, Institute for the Physics
and Mathematics of the Universe
University of Tokyo
17 November 2008

McKay
correspondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings andnoncommutative
algebraic geometry

I would like to thank my colleagues and collaborators P.
Aspinwall, C. Beil, and D. Berenstein. This talk will discuss
some work in progress, as well as reviewing a number ofolder papers, including:David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories• D. Berenstein, Reverse geometric engineering of
singularities, JHEP 04 (2002) 052, hep-th/0201093.
• P. S. Aspinwall, A point’s point of view of stringy
geometry, JHEP 01 (2003) 002, hep-th/0203111.

AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings andnoncommutative
algebraic geometry

I would like to thank my colleagues and collaborators P.
Aspinwall, C. Beil, and D. Berenstein. This talk will discuss
some work in progress, as well as reviewing a number ofolder papers, including:David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories• D. Berenstein, Reverse geometric engineering of
singularities, JHEP 04 (2002) 052, hep-th/0201093.
• P. S. Aspinwall, A point’s point of view of stringy
geometry, JHEP 01 (2003) 002, hep-th/0203111.
Also relevant are:• M. Van den Bergh, Three-dimensional flops andnoncommutative rings, Duke Math. J. 122 (2004)
423–455, math.AG/0207170.
• V. Ginzburg, Calabi–Yau algebras, math.AG/0612139.

AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

For the past ten years, one of the most exciting branches ofstring theory has been the study of a correspondence
between conformal field theories in 4 dimensions and certain
string compactifications.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

For the past ten years, one of the most exciting branches ofstring theory has been the study of a correspondence
between conformal field theories in 4 dimensions and certain
string compactifications. A quantum field theory is
conformal if, roughly speaking, it behaves the same way at
all distance scales.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

For the past ten years, one of the most exciting branches ofstring theory has been the study of a correspondence
between conformal field theories in 4 dimensions and certain
string compactifications. A quantum field theory is
conformal if, roughly speaking, it behaves the same way at
all distance scales. Interesting examples of conformal
behavior are found in many places in science andmathematics.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

For the past ten years, one of the most exciting branches ofstring theory has been the study of a correspondence
between conformal field theories in 4 dimensions and certain
string compactifications. A quantum field theory is
conformal if, roughly speaking, it behaves the same way at
all distance scales. Interesting examples of conformal
behavior are found in many places in science andmathematics. For example, the theory of fractals exhibits
this kind of behavior: a fractal is self-similar over a broad
range of distance scales.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

For the past ten years, one of the most exciting branches ofstring theory has been the study of a correspondence
between conformal field theories in 4 dimensions and certain
string compactifications. A quantum field theory is
conformal if, roughly speaking, it behaves the same way at
all distance scales. Interesting examples of conformal
behavior are found in many places in science andmathematics. For example, the theory of fractals exhibits
this kind of behavior: a fractal is self-similar over a broad
range of distance scales. Many other examples are available
from condensed matter physics, where phase transitions
between different states of matter are often characterized by
self-similar behavior.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings and D-branes

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTThe string theory model for conformal field theories isrelated to open strings in the so-called type IIB string
theory.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings and D-branes

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTThe string theory model for conformal field theories isrelated to open strings in the so-called type IIB string
theory. This is one of the five types of string theory in ten
dimensions, which was originally believed to involve closed
strings only.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings and D-branes

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTThe string theory model for conformal field theories isrelated to open strings in the so-called type IIB string
theory. This is one of the five types of string theory in ten
dimensions, which was originally believed to involve closed
strings only. However, in the mid-1990’s it was recognizedthat open strings also play a rôle in this theory, with theendpoints of the open strings confined to a subspace known
as a D-brane.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings and D-branes

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTThe string theory model for conformal field theories isrelated to open strings in the so-called type IIB string
theory. This is one of the five types of string theory in ten
dimensions, which was originally believed to involve closed
strings only. However, in the mid-1990’s it was recognizedthat open strings also play a rôle in this theory, with theendpoints of the open strings confined to a subspace known
as a D-brane. The case of relevance for today’s story is a
D-brane with 3 spatial dimensions (and 1 temporaldimension)—a so-called D3-brane.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Gauge theories
The open strings must carry gauge fields at their endpoints,
and these are provided by gauge fields on the D3-branes.

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Gauge theories
The open strings must carry gauge fields at their endpoints,
and these are provided by gauge fields on the D3-branes.
The gauge group in question is a unitary group U(N), and N
has an intrinsic definition as a “D-brane charge”.

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Gauge theories
The open strings must carry gauge fields at their endpoints,
and these are provided by gauge fields on the D3-branes.
The gauge group in question is a unitary group U(N), and N
has an intrinsic definition as a “D-brane charge”. (For thisreason, one often pictures a D3-brane with gauge group
U(N) as being a “stack” of N D3-branes located in the same
position.)Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Gauge theories
The open strings must carry gauge fields at their endpoints,
and these are provided by gauge fields on the D3-branes.
The gauge group in question is a unitary group U(N), and N
has an intrinsic definition as a “D-brane charge”. (For thisreason, one often pictures a D3-brane with gauge group
U(N) as being a “stack” of N D3-branes located in the same
position.) Considering the “D-brane charge” in more detail
gives the first hint about the AdS/CFT correspondence.

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Gauge theories
The open strings must carry gauge fields at their endpoints,
and these are provided by gauge fields on the D3-branes.
The gauge group in question is a unitary group U(N), and N
has an intrinsic definition as a “D-brane charge”. (For thisreason, one often pictures a D3-brane with gauge group
U(N) as being a “stack” of N D3-branes located in the same
position.) Considering the “D-brane charge” in more detail
gives the first hint about the AdS/CFT correspondence. Just
as the electric charge at a point can be determined byintegrating the electric field over a sphere surrounding thepoint, the D-brane charge can be determined by integrating
an appropriate field over a sphere in the transverse space.

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Gauge theories
The open strings must carry gauge fields at their endpoints,
and these are provided by gauge fields on the D3-branes.
The gauge group in question is a unitary group U(N), and N
has an intrinsic definition as a “D-brane charge”. (For thisreason, one often pictures a D3-brane with gauge group
U(N) as being a “stack” of N D3-branes located in the same
position.) Considering the “D-brane charge” in more detail
gives the first hint about the AdS/CFT correspondence. Just
as the electric charge at a point can be determined byintegrating the electric field over a sphere surrounding thepoint, the D-brane charge can be determined by integrating
an appropriate field over a sphere in the transverse space.
The D3-brane is a 4-dimensional spacetime embedded within
a 10-dimensional spacetime, so the transverse space is
6-dimensional.

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Gauge theories
The open strings must carry gauge fields at their endpoints,
and these are provided by gauge fields on the D3-branes.
The gauge group in question is a unitary group U(N), and N
has an intrinsic definition as a “D-brane charge”. (For thisreason, one often pictures a D3-brane with gauge group
U(N) as being a “stack” of N D3-branes located in the same
position.) Considering the “D-brane charge” in more detail
gives the first hint about the AdS/CFT correspondence. Just
as the electric charge at a point can be determined byintegrating the electric field over a sphere surrounding thepoint, the D-brane charge can be determined by integrating
an appropriate field over a sphere in the transverse space.
The D3-brane is a 4-dimensional spacetime embedded within
a 10-dimensional spacetime, so the transverse space is
6-dimensional. Thus, we integrate an appropriate field over a
5-dimensional sphere to determine the D-brane charge.

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The AdS/CFT correspondence is a duality between string
theory formulated on the product of a 5-sphere and a5-dimesional anti-de Sitter spacetime (with N units of
D3-brane charge), and the conformal field theory with gauge
group U(N) and maximal supersymmetry.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The AdS/CFT correspondence is a duality between string
theory formulated on the product of a 5-sphere and a5-dimesional anti-de Sitter spacetime (with N units of
D3-brane charge), and the conformal field theory with gauge
group U(N) and maximal supersymmetry. In the “weakform” of the correspondence, the duality is expected to holdin the large N limit.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The AdS/CFT correspondence is a duality between string
theory formulated on the product of a 5-sphere and a5-dimesional anti-de Sitter spacetime (with N units of
D3-brane charge), and the conformal field theory with gauge
group U(N) and maximal supersymmetry. In the “weakform” of the correspondence, the duality is expected to holdin the large N limit. In the strong form, the string theory
and the gauge theory should be equiavlent for any value ofN.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The AdS/CFT correspondence is a duality between string
theory formulated on the product of a 5-sphere and a5-dimesional anti-de Sitter spacetime (with N units of
D3-brane charge), and the conformal field theory with gauge
group U(N) and maximal supersymmetry. In the “weakform” of the correspondence, the duality is expected to holdin the large N limit. In the strong form, the string theory
and the gauge theory should be equiavlent for any value of
N. In either form, one must take a limit which focusses onan asymptotic neighborhood of the D3-brane and ignoresfeatures far away.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryAdS/CFT

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The AdS/CFT correspondence is a duality between string
theory formulated on the product of a 5-sphere and a5-dimesional anti-de Sitter spacetime (with N units of
D3-brane charge), and the conformal field theory with gauge
group U(N) and maximal supersymmetry. In the “weakform” of the correspondence, the duality is expected to holdin the large N limit. In the strong form, the string theory
and the gauge theory should be equiavlent for any value of
N. In either form, one must take a limit which focusses onan asymptotic neighborhood of the D3-brane and ignoresfeatures far away. As a consequence, the correspondence
makes sense for arbitrary transverse 6-dimensional spaces; inparticular, for Calabi–Yau manifolds.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

String theory gets its connection to algebraic geometry
through compactification from 10 to 4 spacetime dimensions.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

String theory gets its connection to algebraic geometry
through compactification from 10 to 4 spacetime dimensions.
The 6 dimensions used for compactification have restrictive
geometric properties, in order to preserve certain features of
the 10 dimensional theory (such as supersymmetry) in 4
dimensions.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

String theory gets its connection to algebraic geometry
through compactification from 10 to 4 spacetime dimensions.
The 6 dimensions used for compactification have restrictive
geometric properties, in order to preserve certain features of
the 10 dimensional theory (such as supersymmetry) in 4
dimensions. The Riemannian metrics satisfying those
restrictive properties are hard to describe explicitly, but
thanks to Yau’s solution to the Calabi conjecture, a large
class of these metrics can be described indirectly using
algebraic geometry.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

String theory gets its connection to algebraic geometry
through compactification from 10 to 4 spacetime dimensions.
The 6 dimensions used for compactification have restrictive
geometric properties, in order to preserve certain features of
the 10 dimensional theory (such as supersymmetry) in 4
dimensions. The Riemannian metrics satisfying those
restrictive properties are hard to describe explicitly, but
thanks to Yau’s solution to the Calabi conjecture, a large
class of these metrics can be described indirectly using
algebraic geometry. The study of these Calabi–Yau
manifolds and their properties has provided one of the major
themes in algebraic geometry over the past 20 years.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryNon-spherical horizons
Early in the study of the AdS/CFT correspondence, it was
realized that the same “AdS/CFT” idea could be applied if
the branes were located at a singular point of spacetime, inwhich the transverse space is a (real) cone over a “horizon
manifold ” H which need not be a sphere. In fact, it had
long been known that some spacetime singularities (such as
orbifolds) are just as well-behaved in string theory as are
nonsingular points, and the AdS/CFT idea gave a new
approach to understanding why that might be true. In thisapproach, on the “AdS” side we need to be concerned withthe differential geometry of the horizon manifold H.

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryNon-spherical horizons
Early in the study of the AdS/CFT correspondence, it was
realized that the same “AdS/CFT” idea could be applied if
the branes were located at a singular point of spacetime, inwhich the transverse space is a (real) cone over a “horizon
manifold ” H which need not be a sphere. In fact, it had
long been known that some spacetime singularities (such as
orbifolds) are just as well-behaved in string theory as are
nonsingular points, and the AdS/CFT idea gave a new
approach to understanding why that might be true. In thisapproach, on the “AdS” side we need to be concerned withthe differential geometry of the horizon manifold H.
Today, I will discuss a different point of view on thiscorrespondence, in which a careful analysis of the “CFT”
side produces some novel (noncommutative) algebraic
structures which—at least in many examples—permit one torecover the singular Calabi–Yau manifold used on the “AdS”
side (and hence the horizon H).

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryD-Branes and categories

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The original AdS/CFT correspondence is not simply astatement about N D3-branes for a fixed N, but is rather astatement about all possible stacks of branes located at thegiven point.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryD-Branes and categories

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The original AdS/CFT correspondence is not simply astatement about N D3-branes for a fixed N, but is rather astatement about all possible stacks of branes located at thegiven point.
In the generalizations in which the point is singular, there isa phenomenon of brane fractionation discovered by Douglas,
Diaconescu, and Gomis: an “ordinary” stack of branes at P
can break up into a collection of different types of branes (alllocated at P).

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometryD-Branes and categories

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The original AdS/CFT correspondence is not simply astatement about N D3-branes for a fixed N, but is rather astatement about all possible stacks of branes located at thegiven point.
In the generalizations in which the point is singular, there isa phenomenon of brane fractionation discovered by Douglas,
Diaconescu, and Gomis: an “ordinary” stack of branes at P
can break up into a collection of different types of branes (alllocated at P).
To achieve a full understanding of these brane states, weneed to introduce a category of D-branes. The objects inthe category will be all possible (stacks of) D-branes locatedat P, and the morphisms in the category describe the open
string states, where the open strings are stretched between
one kind of brane and another.

AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings andnoncommutative
algebraic geometry

For stacks of branes located at a single point P, thiscategorical description is rather elementary: we simply get
the category of finite-dimensional complex vector spaces. A
vector space of dimension N corresponds to a stack of N
D3-branes, and the strings from N branes to M branes aredescribed by linear transformations T : CN → CM .

David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings andnoncommutative
algebraic geometry

For stacks of branes located at a single point P, thiscategorical description is rather elementary: we simply get
the category of finite-dimensional complex vector spaces. A
vector space of dimension N corresponds to a stack of N
D3-branes, and the strings from N branes to M branes aredescribed by linear transformations T : CN → CM . (Thisdescription oversimplies things a bit, since it ignores thepossibility that branes may move away from the given point.
Our refined categorical description later on will take this into
account.)

David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Open strings andnoncommutative
algebraic geometry

For stacks of branes located at a single point P, thiscategorical description is rather elementary: we simply get
the category of finite-dimensional complex vector spaces. A
vector space of dimension N corresponds to a stack of N
D3-branes, and the strings from N branes to M branes aredescribed by linear transformations T : CN → CM . (Thisdescription oversimplies things a bit, since it ignores thepossibility that branes may move away from the given point.
Our refined categorical description later on will take this into
account.)
In the more general case with a singularity at P, for many
types of singularities we will be able to describe thecorresponding D-brane category as the category of (left)
A-modules for a particular algebra A associated to the pointP.

David R. Morrison
AdS/CFTOpen strings andD-branes
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Noncommutative
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Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

There is an existing description of many of these D-brane
categories in terms of algebraic geometry.

Open strings andD-branes
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Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

There is an existing description of many of these D-brane
categories in terms of algebraic geometry. If X denotes the
singular Calabi–Yau manifold, and π : Y → X is a crepant
resolution of X , then the category of coherent sheaves
Coh(Y ) captures many of the features of the D-brane
category.

Open strings andD-branes
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Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

There is an existing description of many of these D-brane
categories in terms of algebraic geometry. If X denotes the
singular Calabi–Yau manifold, and π : Y → X is a crepant
resolution of X , then the category of coherent sheaves
Coh(Y ) captures many of the features of the D-brane
category. More precisely, one should consider the boundedderived category D b (Coh(Y )), but that is too technical fortoday’s talk!

Open strings andD-branes
Gauge theories
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ExamplesAlgebraic
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Morita equivalenceOpen strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOne fundamental notion in the theory of noncommutative
rings and algebras is the notion of Morita equivalence. Tworings A and B are said to be Morita equivalent if there is anequivalence of categories between the category of left
A-modules and the category of left B-modules.

Open strings andD-branes
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Morita equivalenceOpen strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOne fundamental notion in the theory of noncommutative
rings and algebras is the notion of Morita equivalence. Tworings A and B are said to be Morita equivalent if there is anequivalence of categories between the category of left
A-modules and the category of left B-modules. For example,
a ring R is Morita equivalent to the ring Mn (R) of n × nmatrices over R.

Open strings andD-branes
Gauge theories
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Morita equivalenceOpen strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOne fundamental notion in the theory of noncommutative
rings and algebras is the notion of Morita equivalence. Tworings A and B are said to be Morita equivalent if there is anequivalence of categories between the category of left
A-modules and the category of left B-modules. For example,
a ring R is Morita equivalent to the ring Mn (R) of n × nmatrices over R.
A key fact is that if A and B are Morita equivalent, then thecenters of the rings Z(A) and Z(B) are isomorphic.

Open strings andD-branes
Gauge theories
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Morita equivalenceOpen strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOne fundamental notion in the theory of noncommutative
rings and algebras is the notion of Morita equivalence. Tworings A and B are said to be Morita equivalent if there is anequivalence of categories between the category of left
A-modules and the category of left B-modules. For example,
a ring R is Morita equivalent to the ring Mn (R) of n × nmatrices over R.
A key fact is that if A and B are Morita equivalent, then thecenters of the rings Z(A) and Z(B) are isomorphic. (This is
obvious in the case of R and Mn (R).)

Open strings andD-branes
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The McKay correspondence
An important example of the types of algebras we willencounter is the “twisted group algebra”. Let G be a finite
subgroup of SU(2) so that G acts on the polynomial algebraC[x, y ]. The twisted group algebra C[x, y ] ? G consists ofpairs (f (x), g ) and a multiplication

Open strings andnoncommutative
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David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT(f (x), g ) · (φ(x), γ) = (f (x) · φg (x), g ◦ γ),

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The McKay correspondence
An important example of the types of algebras we willencounter is the “twisted group algebra”. Let G be a finite
subgroup of SU(2) so that G acts on the polynomial algebraC[x, y ]. The twisted group algebra C[x, y ] ? G consists ofpairs (f (x), g ) and a multiplication

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT(f (x), g ) · (φ(x), γ) = (f (x) · φg (x), g ◦ γ),
where φg (x) is the function obtained from acting on φ(x) by
g . Kapranov and Vasserot interpreted the McKay
correspondence as a statement about the structure of thisalgebra: we can writeMC[x, y ] ? G =Mρ ⊗ ρ
ρ∈Irrep(G )and describe the algebra in terms of the modules Mρ . Its
structure is determined by the graph of representations
which McKay used:

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ρi−2

ρi+3
V

V

xi−2



yi−2

ρi−1 g
xi−1

yi+2
yi+1

yi−1

(ρ u

yi

i

xi



3 ρi+1



xi+2

ρi+2
C

xi+1

In the An case, for example, the graph forms a cycle and one
has relations xi yi = yi−1 xi−1 .

Open strings andnoncommutative
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David R. Morrison
AdS/CFTOpen strings andD-branes
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ρi−2

ρi+3
V

V

xi−2



yi−2

ρi−1 g
xi−1

yi+2
yi+1

yi−1

(ρ u

yi

i

xi



3 ρi+1



xi+2

ρi+2
C

xi+1

In the An case, for example, the graph forms a cycle and one
has relations xi yi = yi−1 xi−1 .
Its not too hard to determine the center of the algebra fromthis description: it is generated byOpen strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
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X = x0 · · · xn−1 + cyclic permutations

D-brane algebras

Y = y0 · · · yn−1 + cyclic permutations

ExamplesZ = xi yi + cyclic permutations

Algebraic
AdS/CFT
correspondence

subject to the relation Z n = XY . Of course, the center canalso be identified with C[x, y ]G , so this was not unexpected.

More examples
Noncommutative
algebraic geometryD-brane algebras

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

How do we associate an algebra to the category of D-branes
at a singular point P? In every known example, thecollection of possible D-branes at P can be described as acollection of quantum field theory with the same Lagrangian
for each of the theories.

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algebraic geometryD-brane algebras

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

How do we associate an algebra to the category of D-branes
at a singular point P? In every known example, thecollection of possible D-branes at P can be described as acollection of quantum field theory with the same Lagrangian
for each of the theories.
More precisely, one does not specify in advance whichunitary groups Gi = U(Ni ) take part in the theory, but one
does specify the matter representation (as a collection ofadjoint and bifundamental fields for the gauge groups Gi )and one specifies a superpotential W which is the trace of apolynomial in the matter fields.

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To such data, we can first assign a quiver, that is, a directed
graph whose vertices label the groups Gi and whose directededges specify the bifundamental and adjoint fields in thematter representation.

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David R. Morrison
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Noncommutative
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To such data, we can first assign a quiver, that is, a directed
graph whose vertices label the groups Gi and whose directededges specify the bifundamental and adjoint fields in thematter representation.
From the quiver, we directly get the path algebra, which isthe algebra of all paths on the quiver (i.e., all orderedmonomials in matter fields). However, a universal feature ofthis family of theories is the relations in the path algebra
determined by what are called “F-term constraints” inphysics: these are the algebra relations dictated by ∂W /∂Xj .

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
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Noncommutative
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To such data, we can first assign a quiver, that is, a directed
graph whose vertices label the groups Gi and whose directededges specify the bifundamental and adjoint fields in thematter representation.
From the quiver, we directly get the path algebra, which isthe algebra of all paths on the quiver (i.e., all orderedmonomials in matter fields). However, a universal feature ofthis family of theories is the relations in the path algebra
determined by what are called “F-term constraints” inphysics: these are the algebra relations dictated by ∂W /∂Xj .
So, given a field theory description of the family of D-branes
in the form above, the D-brane algebra isOpen strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
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D-brane algebras

A = path algebra of quiver/(∂W /∂Xj ).

ExamplesAlgebraic
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correspondence
More examples
Noncommutative
algebraic geometry

To such data, we can first assign a quiver, that is, a directed
graph whose vertices label the groups Gi and whose directededges specify the bifundamental and adjoint fields in thematter representation.
From the quiver, we directly get the path algebra, which isthe algebra of all paths on the quiver (i.e., all orderedmonomials in matter fields). However, a universal feature ofthis family of theories is the relations in the path algebra
determined by what are called “F-term constraints” inphysics: these are the algebra relations dictated by ∂W /∂Xj .
So, given a field theory description of the family of D-branes
in the form above, the D-brane algebra isOpen strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
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McKaycorrespondence
D-brane algebras

A = path algebra of quiver/(∂W /∂Xj ).
This is sometimes called a superpotential algebra, or aCalabi–Yau algebra.

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Example 1

Open strings andnoncommutative
algebraic geometry

As a first example, we consider the case in which P is asmooth point. In physics language, the conformal field
theory is the N = 4 super-Yang–Mills theory, here written in
N = 1 language. The N = 4 gauge multiplet decomposes
as an N = 1 gauge multiplet plus three complex scalar fields
X , Y , Z , each transforming in the adjoint representation of
the group. The superpotential is
W = tr (X (YZ − ZY )) .

David R. Morrison
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Example 1

Open strings andnoncommutative
algebraic geometry

As a first example, we consider the case in which P is asmooth point. In physics language, the conformal field
theory is the N = 4 super-Yang–Mills theory, here written in
N = 1 language. The N = 4 gauge multiplet decomposes
as an N = 1 gauge multiplet plus three complex scalar fields
X , Y , Z , each transforming in the adjoint representation of
the group. The superpotential is
W = tr (X (YZ − ZY )) .

David R. Morrison
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The F-term constraint in this case tells us YZ = YZ ,XZ = ZX and XY = YX . Thus, we find

McKaycorrespondence
D-brane algebras

A = C[X , Y , Z ],the (commutative) polynomial algebra in three variables.

ExamplesAlgebraic
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Example 1

Open strings andnoncommutative
algebraic geometry

As a first example, we consider the case in which P is asmooth point. In physics language, the conformal field
theory is the N = 4 super-Yang–Mills theory, here written in
N = 1 language. The N = 4 gauge multiplet decomposes
as an N = 1 gauge multiplet plus three complex scalar fields
X , Y , Z , each transforming in the adjoint representation of
the group. The superpotential is
W = tr (X (YZ − ZY )) .

David R. Morrison
AdS/CFTOpen strings andD-branes
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The F-term constraint in this case tells us YZ = YZ ,XZ = ZX and XY = YX . Thus, we find

McKaycorrespondence
D-brane algebras

A = C[X , Y , Z ],the (commutative) polynomial algebra in three variables. (Sowe get the category of C[X , Y , Z ]-modules; if the branes are
fixed at P, this is just the category of C-vector spaces asremarked earlier.)

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Example 2Open strings andnoncommutative
algebraic geometry
David R. Morrison

cn in this context, we need to
To realize the McKay quiver A
add loops at each of the vertices, represented by fields φi ;then the superpotential
W = tr (φi (xi yi − yi−1 xi−1 ))gives the same relations as before. (This is known to be theappropriate field theory description when P lies on a curve ofAn−1 singularities.)

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Example 2Open strings andnoncommutative
algebraic geometry
David R. Morrison

cn in this context, we need to
To realize the McKay quiver A
add loops at each of the vertices, represented by fields φi ;then the superpotential
W = tr (φi (xi yi − yi−1 xi−1 ))gives the same relations as before. (This is known to be theappropriate field theory description when P lies on a curve ofAn−1 singularities.) The algebra is the twisted group algebra,
tensored with the algebra of the φi ’s, which have no
relations among them. The center is:G

Z(A) = C[x, y ] ⊗ C[Φ],where Φ =P

φi .

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Algebraic AdS/CFT correspondenceOpen strings andnoncommutative
algebraic geometry
David R. Morrison

The algebraic version of the AdS/CFT correspondence which
we would like to formulate relates a point P of a Calabi–Yauspace X on the AdS side to a field theoretic description of
the D-branes at P on the CFT side. To recover the standard
AdS picture, one needs to know about metrics near P andtake a scaling limit; instead, we will focus on the formal
completion of the power series ring, we carries the local
algebraic information about the Calabi–Yau space.

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Algebraic AdS/CFT correspondenceOpen strings andnoncommutative
algebraic geometry
David R. Morrison

The algebraic version of the AdS/CFT correspondence which
we would like to formulate relates a point P of a Calabi–Yauspace X on the AdS side to a field theoretic description of
the D-branes at P on the CFT side. To recover the standard
AdS picture, one needs to know about metrics near P andtake a scaling limit; instead, we will focus on the formal
completion of the power series ring, we carries the local
algebraic information about the Calabi–Yau space.

AdS/CFTOpen strings andD-branes
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Our conjectured correspondence relates the D-brane algebra
A (such that the D-brane category is the category of left
A-modules) to the formal completion of the coordinate ringof X at P by asserting that the center Z(A) and thecoordinate ring of X at P have isomorphic formal
completions.

McKay
correspondence
D-brane algebras
ExamplesAlgebraic
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correspondence
More examples
Noncommutative
algebraic geometryRemarks:
1. The algebra A is not uniquely determined by the
D-brane category (this is related to Seiberg duality andhas been studied extensively); however, Morita
equivalence guarantees that the center does not change.

Open strings andnoncommutative
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David R. Morrison
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Noncommutative
algebraic geometryRemarks:
1. The algebra A is not uniquely determined by the
D-brane category (this is related to Seiberg duality andhas been studied extensively); however, Morita
equivalence guarantees that the center does not change.
2. As stated, this provides a way of passing from a
Lagrangian description of a family of field theories tothe algebro-geometric structure near the AdS dual. It isalso possible to go backwards (work of Aspinwall andcollaborators): given P ∈ X , one studies the (derived)
category of coherent sheaves on X supported at P, anddetermines a so-called tilting module for the category.
That tilting module, and some further computations of
Ext groups of the sheaves, gives a Lagrangian
description for the family of D-branes (including a
superpotential).

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3. To the best of my knowledge, there is no known
algorithm for computing the center of A. There is also
no known algorithm for producing a tilting module forthe derived category of coherent sheaves supported at
X . And of course, there is no proof at present that theformal completion of Z(A) is isomorphic to the formal
completion of the coordinate ring of X at P.

Open strings andnoncommutative
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3. To the best of my knowledge, there is no known
algorithm for computing the center of A. There is also
no known algorithm for producing a tilting module forthe derived category of coherent sheaves supported at
X . And of course, there is no proof at present that theformal completion of Z(A) is isomorphic to the formal
completion of the coordinate ring of X at P.

Open strings andnoncommutative
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David R. Morrison
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4. Van den Bergh has conjectured that a “canonicalsingularity” P ∈ X in dimension 3 admits an algebra A
whose category of modules describe the coherent
sheaves supported at P if and only if there is a
resolution of singularities π : Y → X which is relatively
Calabi–Yau (that is, no zeros are introduced into theholomorphic 3-form), and he has proven this conjecture
in a number of cases. However, neither his conjecturenor his proof address the question of whether A can be
written as a superpotential algebra.

Non-sphericalhorizons
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Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories

5. Ginzburg has introduced a more general notion of
Calabi–Yau algebras, which suggests that these ideas
may be true more globally, but it is fair to say that the
study of such algebras is still in its infancy.

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Algebraic geometryNon-sphericalhorizons
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algebraic geometry

Example 3 (Cachazo–Katz–Vafa;
Aspinwall–Katz)
c1 by adding another term to
Let us modify the theory for A
the superpotential. That is, we have fields xi , yi , φi for
i = 0, 1 with superpotential
W = tr (φ0 (x0 y0 − y1 x1 ) + φ1 (x1 y1 − y0 x0 ) + P(φ0 ) + P(φ1 ))Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
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Algebraic geometry

where P is some fixed polynomial.
The F-term relations are:Non-sphericalhorizons
D-Brane categories

y0 φ0 = φ1 y0

Morita equivalence

φ0 y1 = y1 φ1

McKay
correspondence

φ0 x0 = x0 φ1
x1 φ0 = φ1 x1
x0 y0 = y1 x1 − P 0 (φ0 )x1 y1 = y0 x0 − P 0 (φ1 )D-brane algebras
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y0 φ0 = φ1 y0
φ0 y1 = y1 φ1
φ0 x0 = x0 φ1
x1 φ0 = φ1 x1
x0 y0 = y1 x1 − P 0 (φ0 )x1 y1 = y0 x0 − P 0 (φ1 )One can see that the following elements are central:

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X = x0 x1 + x1 x0

Morita equivalence

Y = y1 y0 + y0 y1

McKay
correspondence

Z = x0 y0 + x1 y1
Φ = φ0 + φ1
and satisfy the relation

D-brane algebras
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Z 2 = XY + ZP 0 (Φ).

Noncommutative
algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes

These singularities are closely related to flops: by an old
result of Reid, the blowdown of a flop with normal bundle
OP1 ⊕ OP1 (−2) always has an equation of the above form,
where P has a zero at Φ = 0 of order at least 3.

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Example 4: the conifold

Open strings andnoncommutative
algebraic geometry
David R. Morrison

The previous example also describes the conifold, if we takeP(Φ) = 12 Φ2 . The standard physics approach to this is to“integrate out” φi , using the F-term equations to solve
φ0 = y1 x1 − x0 y0 , φi = y0 x0 − x1 y1 . The new superpotentialis
W = tr (y1 x1 − x0 y0 )(x0 y0 − y1 x1 ) +AdS/CFTOpen strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories(y0 x0 − x1 y1 )(x1 y1 − y0 x0 )+1
1(y1 x1 − x0 y0 )2 + (y0 x0 − x1 y1 )2
2
2

Morita equivalence



McKay
correspondence
D-brane algebras
ExamplesThis is the standard superpotential for the conifold, firstfound by Klebanov and Witten.

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AdS/CFT
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Example 5: the suspended pinch point

Open strings andnoncommutative
algebraic geometry
David R. Morrison

As one final example, we will compute the superpotential
algebra and its centered for the “suspended pinch point”(first considered in hep-th/9810201). The method which
Plesser and I used to find the superpotential in this case wastoric geometry. The suspended pinch point singularity canbe described torically as the cone over the following lattice
polyhedron:

w1

AdS/CFTOpen strings andD-branes
Gauge theories
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Algebraic geometryNon-sphericalhorizons
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w0

McKay
correspondence
D-brane algebras
Examples

v0

w2

v1

Algebraic
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Open strings andnoncommutative
algebraic geometry

Plesser and I calculated a field theory dual, which can be
expressed in terms of the quiverDavid R. Morrison
AdS/CFTOpen strings andD-branesvF 1 g

x

φ

 s

7 v3

x̃

y

Gauge theoriesz̃

AdS/CFT

z

'
3 v2

ỹ

Algebraic geometryNon-sphericalhorizons
D-Brane categories

and superpotential
W = tr φ(Ỹ Y − X̃ X ) + λ(Z Z̃ X X̃ − Z̃ Z Ỹ Y ) .

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ExamplesAlgebraic
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The F-term constraints become:Ỹ Y = X̃ X
φX̃ = λZ Z̃ X
X φ = λX Z̃ X

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFTOpen strings andD-branes
Gauge theories

φỸ = λỸ Z̃ Z
Y φ = λZ̃ ZY
λZ̃ X Z̃ = λY Ỹ Z̃
λX X̃ Z = λZY Ỹ

AdS/CFT
Algebraic geometryNon-sphericalhorizons
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Morita equivalence
McKaycorrespondence

There are central elements A = φ + λZ Z̃ + λZ̃ Z ,B = X̃ X + X X̃ + Y Ỹ , C = Ỹ Z̃ X + Z̃ X Ỹ + X Ỹ Z̃ ,D = X̃ ZY + ZY X̃ + Y X̃ Z .
The relation is:
AB 2 = λCD.

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ExamplesAlgebraic
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Noncommutative algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

There have been many efforts in the past few decades toextend the familiar notions of commutative algebraic
geometry to noncommutative settings.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Noncommutative algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

There have been many efforts in the past few decades toextend the familiar notions of commutative algebraic
geometry to noncommutative settings. This work has met
with considerable skepticism from the algebraic geometrycommunity.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Noncommutative algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

There have been many efforts in the past few decades toextend the familiar notions of commutative algebraic
geometry to noncommutative settings. This work has met
with considerable skepticism from the algebraic geometrycommunity. However, the corner of noncommutative
algebraic geometry which I described today seems to be very
close to central issues in (commutative) algebraic geometry.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry

Noncommutative algebraic geometry

Open strings andnoncommutative
algebraic geometry
David R. Morrison
AdS/CFT

There have been many efforts in the past few decades toextend the familiar notions of commutative algebraic
geometry to noncommutative settings. This work has met
with considerable skepticism from the algebraic geometrycommunity. However, the corner of noncommutative
algebraic geometry which I described today seems to be very
close to central issues in (commutative) algebraic geometry.
One can hope that further exploration of these ideas willlead to both progress in the physics, and also to a deeper
understanding of the noncommutative aspects of algebraic
geometry per se.

Open strings andD-branes
Gauge theories
AdS/CFT
Algebraic geometryNon-sphericalhorizons
D-Brane categories
Morita equivalence
McKaycorrespondence
D-brane algebras
ExamplesAlgebraic
AdS/CFT
correspondence
More examples
Noncommutative
algebraic geometry