An introduction

1
An introduction to topological field theory and
topological strings
Marcel Vonk
19th Nordic Network Meeting, Uppsala, November
2004
2
Notes, slides references

• Notes and slides at
• http//www.teorfys.uu.se/people/marcel
• References
• Witten, 1988-1991
• Book Mirror Symmetry
• Recent notes by Marino and Neitzke Vafa
• Bershadsky, Cecotti, Ooguri Vafa, 1993

3
Motivation

• Topological theories are
• easy to calculate with
• nice toy models
• describing aspects of the real world
• very interesting mathematically

4
Overview

• Topology
• Topological field theories
• Cohomological field theories
• Topological strings
• Applications

• Chern-Simons theory

• The A-model
• The B-model

5
Topology
Topology studies the invariant properties
of object under continuous deformations
e
d
8 e gt 0, 9 d gt 0 y-x lt d ) f(y)-f(x) lt e
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Topology
Topology studies the invariant properties
of object under continuous deformations
S
f-1(S)
8 S open ) f-1(S) open
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Topology
To define properties which are invariant under
continuous deformations, one does not need a
metric!
Mathematically, a topological space consists of a
set and its collection of open subsets.
We will always use manifolds, which locally look
like Rd or Cd
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Topology
The aim of topology is to look for topological
invariants.
?

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Topological invariants
d0 Number of points d1 Number of lines,
circles, etc. d2 For each component number of
holes, boundaries and crosscaps dgt2 ???
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Topological invariants
General strategy construct a vector bundle over
the manifold (or over an associated one) and
study its global properties.

• Tangent bundle
• Moduli spaces

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Topological invariants
Often, the global data are encoded in a
connection (gauge field), which tells us how to
parallel transport vectors.
Field strength ) curvature ) Chern classes
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Topological invariants
Other key example bundle of differential forms
The exterior derivative staisfies d20, so
one can construct the cohomology groups
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Topological invariants
These examples are related The connection is
not a differential form, but the trace of its
field strength is. Chern classes
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Topological invariants
Third example homology
Homology is Poincaré dual to cohomology
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Topological field theories
How can we calculate topological invariants using
physics? (Lecture 1 and 2) Topological field
theories What can we learn about real
physics in this way? (Lecture 3)
independent of g
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Topological field theories
Trivial example Chern-Simons theory
invariant!
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Topological field theories
Observables in Chern-Simons theory
Holonomy Wilson loop
In this way, one finds knot invariants! (Witten,
1989)
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Cohomological field theories
Cohomological field theories satisfy four
properties
(1) There is an operator Q satisfying
Q2 0

• BRST-symmetry
• Supersymmetry

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Cohomological field theories
Cohomological field theories satisfy four
properties
(2) The physical operators satisfy

• Q is a symmetry operator
• The physical operators are invariant
• Gauge symmetries (BRST)

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Cohomological field theories
Cohomological field theories satisfy four
properties
(3) The vacuum is Q-symmetric

• Commute Q towards the vacuum
• (1), (2) and (3) ) cohomology

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Cohomological field theories
Cohomological field theories satisfy four
properties
(4) The energy-momentum is Q-exact

• Hamiltonian and momentum are Q-closed
• The theory is now g-independent!

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Cohomological field theories
A standard way to achieve (4) is to choose
Quantum measure
This also implies
In general, topological theories are not very
sensitive to parameters in the initial action.
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Cohomological field theories
Descent equations
)
From this, one derives
with
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Cohomological field theories
Descent equations
So
is physical (but nonlocal)
Continuing like this, construct p-forms
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Cohomological field theories
Descent equations
So
is physical (but nonlocal)
Continuing like this, construct p-forms
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Cohomological field theories
d2 recursion relations


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Cohomological field theories
d2 recursion relations
All information at fixed t is contained in the
two- and three-point functions on the sphere
t-derivatives
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N(2,2) sigma-models
Superspace
Superfields
Supertranslations and -rotations
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N(2,2) sigma-models
Chiral superfields
This leaves four degrees of freedom
Since D,Q0, actions with chiral
superfields have N(2,2) supersymmetry.
D-terms F-terms
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N(2,2) sigma-models
D-terms
Integrate over q
Where
We find a supersymmetric sigma-model with a
Kähler manifold as a target space
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R-symmetries
R-symmetries are rotations of the fermionic
coordinates
Is the path integral measure invariant?
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R-symmetries
In the manifestly invariant measure, one has to
absorb zero-modes
One can show that
The measure is only FA-invariant if c1(M)0
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Calabi-Yau manifolds
A Kähler manifold with c1(M)0 is called a
Calabi-Yau manifold
Theorem (Yau) Given a complex structure, there
is a unique Ricci-flat Kähler metric in the same
Kähler class
Consequence the moduli space of Calabi-Yau
manifolds consists of pairs of (compatible)
Kähler classes and complex structures.
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Calabi-Yau moduli
In a given complex structure, the Kähler form is
a (1,1)-form.
The complex structure is equivalent to specifying
a (d,0)-form. Locally
Infinitesimal changes change W by a (d-1,1)-form.
The moduli space is a subspace of H2 Hd
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Calabi-Yau moduli
Local coordinates for the complex structure
moduli
Redundant coordinates
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Twisting N(2,2) theories
The energy and momentum can be written as
Q-commutators for either
or
Does this mean we have constructed our first
topological field theories?
No! The generators are fermionic, and
the symmetry is broken on a curved background.
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Twisting N(2,2) theories
Solution twist the theories
Now define
Then
A global QA-symmetry can now be defined on any
curved Kähler manifold
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Twisting N(2,2) theories
Solution twist the theories
Now define
Then
A global QB-symmetry can now be defined on any
curved Calabi-Yau manifold
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The A-model
Twisted fields
Lagrangean
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The A-model
Lagrangean
This is nearly a QA-commutator
The difference in the actions is
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The A-model

• We see that
• The A-model is topological
• Up to a simple t-dependent factor, the
• model is semi-classical
• The results will depend on the Kähler class,
• but not on other parameters
• In particular, there is no dependence on the
• complex structure

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The A-model
Local observables
Since
we find that
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The A-model
The observables correspond to cohomology classes.
A zero-mode analysis shows that we need to
include kd(1-g) operators c ) (k,k)-forms

• ggt1 no nonzero correlators
• g1 nonzero partition function
• g0 nonzero correlation functions

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The A-model
Semiclassical (exact!) treatment the path
integral localizes to
Holomorphic instantons. There is a
finite-dimensional space of these.
In fact, the dimension equals d(1-g) as well!
The path integral is an integral over this moduli
space. What does it compute?
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The A-model
Poincaré-duality
One finds
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The A-model
Special case t ! 1
The general case can be viewed as quantum
intersection numbers
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The B-model
Twisted fields
Lagrangean
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The B-model
Lagrangean
This is nearly a QB-commutator
where W is linear in q.
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The B-model

• We see that
• The B-model is completely semi-classical
• Moreover, one can show that
• The B-model is topological
• The results will depend on the complex
• structure, but not on the Kähler class

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The B-model
Local observables
Since
we find that
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The B-model
The observables correspond to Dolbeault
cohomology classes with values in
A zero-mode analysis shows that we need to
include kd(1-g) operators q, h ) k-vector valued
k-forms

• ggt1 no nonzero correlators
• g1 nonzero partition function
• g0 nonzero correlation functions

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The B-model
Semiclassical (exact!) treatment the path
integral localizes to
Constant maps. The moduli space is simply M
itself!
The path integral is an integral over
space-time. What does it compute?
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The B-model
How to integrate a k-vector valued k-form over M?
This gives a (k,k)-form, which can be integrated.
Note the dependence on the complex structure!
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Mirror symmetry
A-model ) easy interpretation (integers) B-model
) easy calculation
For every Calabi-Yau X, one can find a mirror
Calabi-Yau Y, with H1,1 and Hd-1,1 interchanged.
Moreover, the A-model on X is equivalent to the
B-model on Y, and vice versa.
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Topological strings

• To couple a theory to gravity
• Covariantize the action
• Introduce kinetic terms
• Integrate over metrics

Here, we will only focus on the last step.
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Topological strings
Why do we integrate over metrics at all?
???

• No, since
• There may be large transformations
• There may be anomalies
• The volume is infinite

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Topological strings

• How do we integrate over all metrics? Note that
  the sigma-model is conformal
• First integrate over the conformal group
• Then integrate over the remaining (finite
• dimensional) moduli space.

To integrate over conformally equivalent metrics,
one usually has to worry about conformal
anomalies.
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Topological strings
Superconformal algebra
The twisting amounts to
which implies
) no anomaly!
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Topological strings
However, we still have to integrate over the
space of (complex structure) moduli. What is a
good measure?
So one can define
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Topological strings
Recall that the axial anomaly was 2d(g-1)

• Therefore, the anomaly cancels if d3! This can
  be considered a critical dimension
• g0 Three-point function
• g1 One-point function
• ggt1 Partition function

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Topological strings
What about other n-point functions? We must
insert nonlocal operators.
etc.
One can show that
satisfies
B-model F0 is the prepotential!
so
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The holomorphic anomaly
Using the descent equations, one shows
The complex conjugate equation gives
Therefore
seems to decouple from the A-model
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The holomorphic anomaly
Naively, the models are holomorphic in t.
However, note that
So we find a total derivative
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The holomorphic anomaly
Boundary of moduli space
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The holomorphic anomaly
Since the genus g partition functions are nearly
holomoprphic, they can be determined up to a
finite number of constants. This is most easily
calculated in the B-model. Then, doing a mirror
symmetry and using integrality in the A-model,
one can often completely fix the constants.
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Applications

• N2 F-terms
• NS 5-brane partition function
• 5-d and 4-d supersymmetric black holes
• Geometric transitions
• Matrix models () black holes)
• N1 F-terms
• Cristal melting

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N2 F-terms
Consider a type II string theory on M4
CY6 This is described by a CFT4 CFT6 The CFT6
is precisely the N(2,2) sigma model!
We may expect physical results depending only on
these fields to be calculable in the topological
string theory
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N2 F-terms
Compactification on CY6 leads to a 4-dimensional
N2 supergravity theory. h2,1 complex structure
moduli XI(x) h1,1 Kähler moduli Ya(x) XI(x) )
vector multiplets (in IIB) Ya(x) )
hypermultiplets (in IIB)
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N2 F-terms
Supergravity theories are constructed from a
small number of building blocks. F-term
holomorphic superpotential F0(XI) There is an
Sp(2,R)-symmetry acting on XI and FI IF0.
After quantization Sp(2,Z) This determines F0
) prepotential!
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N2 F-terms
What about Fg for ggt0? Antoniadis, Gava, Narain,
Taylor (1993)
Effective action
Or using superfields
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Black holes in 4 dimensions
Now take a spacetime of the form BH4 CY6 The
black hole has charges (PI, QI) Attractor
equations at the horizon
Entropy
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Black holes in 4 dimensions
Corrections to this formula from the F-terms
Legendre transforming this result leads to
(Ooguri, Strominger, Vafa - 2004)
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Geometric transitions
Simple noncompact Calabi-Yau the conifold
S3
S2
One can deform or resolve the conifold
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Geometric transitions

• Topological D-branes can be constructed
• A-model 3-dimensional Lagrangean cycles
• B-model 2p-dimensional holomorphic
• cycles
• A-model on deformed conifold

Wrap N D-branes on S3
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Geometric transitions
What is the AdS/CFT dual geometry? Natural
guess the resolved conifold with
Using a sigma model which has two phases, this
can indeed be proven. (Gopakumar, Vafa 1998)
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N1 F-terms
Construct N1, d4 gauge theories wrap D5-branes
on 2-cycles. (B-model) Geometric transition
this is equivalent to resolved geometries with
fluxes.
)
(N1)
(N2)
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N1 F-terms
Fluxes through dual (noncompact) cycles as well
In the dual N1 SYM, in the IR
Dijkgraaf, Vafa (2002) W Weff
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N0 ???
Topological M-theory?
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N0 ???
Topological M-theory?
?
?
?
The End