(work in progress) Strings 2002member.ipmu.jp/yuji.tachikawa/stringsmirrors/2002/morrison.pdf · 5...
Transcript of (work in progress) Strings 2002member.ipmu.jp/yuji.tachikawa/stringsmirrors/2002/morrison.pdf · 5...
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Half K3 Surfacesor
K3, G2, E8, M, and all that(work in progress)
Strings 2002David R. Morrison, Duke University
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Half K3 Surfacesor
K3, G2, E8, M, and all that(work in progress)
Strings 2002David R. Morrison, Duke University
Thanks to: P. Aspinwall, M. Atiyah, E. Witten.
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Duality in dimension eight
• F-theory in dimension 8: type IIB string compactified
on S2 with 24 D7-branes at points of S2
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• data: elliptically fibered K3 surface with 24 singular
fibers (generically)
y2 = x3 + f8(z)x + g12(z)
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• data: elliptically fibered K3 surface with 24 singular
fibers (generically)
y2 = x3 + f8(z)x + g12(z)
• singular fibers at ∆(z) = 4f(z)3 + 27g(z)2
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• Can take a limit1 in which the S2 stretches into a long
tube with 12 of the D7-branes at one end and 12 at the
other end.
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• Can take a limit1 in which the S2 stretches into a long
tube with 12 of the D7-branes at one end and 12 at the
other end.
• j-invariant of the F-theory data is nearly constant in the
middle of the tube
1Morrison–Vafa, arXiv:hep-th/9603161
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• that constant value determines the T 2 metric for a
heterotic dual description
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• that constant value determines the T 2 metric for a
heterotic dual description
• each set of 12 points determines an E8 gauge
field configuration, e.g., via perturbations of the E8
singularity
y2 = x3 + z5
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• More precise version:2 by rescaling (e.g. x �→ x/z2,
y �→ y/z3) get a limit in which the original S2 splits into
two S2’s meeting at a point
• Over each S2 is a rational elliptic surface with section.(These are NOT the half K3 surfaces of the title.)2Friedman–Morgan–Witten, arXiv:hep-th/9701162
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• Can blow down the section to give a del Pezzo surface
of degree 1
• Looijenga showed that such surfaces are in one-to-one
correspondance with E8 gauge theory on a (fixed) 2-
torus
• The degeneration corresponds to weak heterotic
coupling
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Heterotic string on elliptic fibrations
• If X has an elliptic fibration, the heterotic string on X
should have an F-theory dual, obtained fibrewise
• Base of F-theory has a family of S2’s:
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• weak heterotic coupling limit is dual to a degeneration
• precise correspondance between data; can be used to
investigate many interesting phenomena on both sides
• X re-emerges as the intersection of the two elliptic
fibrations in the limit
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Duality in dimension seven
• We wish to find a similar picture in dimension 7, starting
from M-theory on a K3 surface
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• There are limits in the space of metrics on K3, and
presumably even in the space of Ricci-flat metrics, with
the following behavior:
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• A very long throat of the form T 3 × [0, R] has opened
up in the middle of the K3 surface.
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• A very long throat of the form T 3 × [0, R] has opened
up in the middle of the K3 surface.
• An observer in the middle sees the two complicated ends
recede; they are the half K3 surfaces
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• we will see presently that each half K3 surface accounts
for the data of an E8 gauge field on T 3
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• Explicit realization of M/heterotic duality in dimension
7. Similar to Horava–Witten but with a more geometric
interpretation for the gauge fields at the ends.
• Again corresponds to weak heterotic coupling
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Application: 4-dimensional duality
• As in the previous case, we can put this into families.
To illustrate this, we simplify our stretched K3 to a
cartoon:
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• Now we can put a family of these together:
• The family of T 3’s emerges as the common boundary
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along which the families of half K3’s meet.
• Can be applied to heterotic string on a Calabi–Yau
3-fold, with its supersymmetric T 3-fibration3.
� Fibrewise duality gives M-theory on a family of K3surfaces over S3
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along which the families of half K3’s meet.
• Can be applied to heterotic string on a Calabi–Yau
3-fold, with its supersymmetric T 3-fibration3.
� Fibrewise duality gives M-theory on a family of K3surfaces over S3
� i.e., a fibered G2 manifold
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along which the families of half K3’s meet.
• Can be applied to heterotic string on a Calabi–Yau
3-fold, with its supersymmetric T 3-fibration3.
� Fibrewise duality gives M-theory on a family of K3surfaces over S3
� i.e., a fibered G2 manifold
� weak heterotic coupling leads to a stretching limit
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along which the families of half K3’s meet.
• Can be applied to heterotic string on a Calabi–Yau
3-fold, with its supersymmetric T 3-fibration3.
� Fibrewise duality gives M-theory on a family of K3surfaces over S3
� i.e., a fibered G2 manifold
� weak heterotic coupling leads to a stretching limit
� the half-G2’s are 7-manifolds with boundary, whose
common boundary is the Calabi–Yau 3-fold4
3Strominger–Yau–Zaslow4cf. Donaldson–Thomas
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The half K3 limit
• An observer who stays well within one half of the K3
surface during scaling sees something different:
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• From this perspective, an infinite throat has opened up,
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and the “other” half K3 surface has receded to inifinite
distance.
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and the “other” half K3 surface has receded to inifinite
distance.
• The physics on the two halves does not decouple: this
is heterotic weak coupling
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and the “other” half K3 surface has receded to inifinite
distance.
• The physics on the two halves does not decouple: this
is heterotic weak coupling
• The physics of the other half K3 surface must be
encoded in the boundary of the manifold-with-boundary
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Variant: K3 surfaces with frozensingularities
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• There are M-theory compactifications on certain K3
surfaces with ADE singularities with 3-form flux at the
singular points5
• Allowed values of 3-form flux: k/N (mod Z) where N
is one of the multiplicities in the longest root of the
corresponding root system
• At most 4 singularities, and the total 3-form flux
vanishes
• These can similarly be cut in half along T 3, giving a half
K3 with frozen singularities5Witten, de Boer et al. (Triples, fluxes, and strings), Atiyah–Witten
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• We can extend T 3 to a long throat as before
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• and also replace the singular point by an asymptotic
tube whose boundary is S3/Γ
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M-theory on manifolds with boundary
• Half K3 surfaces provide an interesting laboratory for
studying M-theory
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M-theory on manifolds with boundary
• Half K3 surfaces provide an interesting laboratory for
studying M-theory
• Horava–Witten: M-theory on a manifold with boundary
must have an E8 gauge field on the boundary, in addition
to bulk fields
• We use coordinates with x ≥ 0 near the boundary; the
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metric is
ds2 =(
dx
x
)2
+ ds2
and the 3-form can be written
C =dx
x∧ α + β
• Horava–Witten anomaly analysis for the boundary
theory says:
� The limit of C on the boundary (caputured by β)
equals CS(A3) − 12CS(R3) for the gauge connection
A3 and curvature R3 on the boundary
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� α restricted to the boundary provides the B-field there
� the familiar∫
C ∧ G ∧ G term, together with a new∫C ∧ X8 term, gives rise to the Green–Schwarz
mechanism on the boundary
• To describe our M-theory vacuum on the half K3 surface,
we need a metric and 3-form field there, as well as an
E8 gauge field on the boundary T 3
• The E8 gauge field on T 3 is the remnant of physics on
the “other half” of the original K3
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Metrics on half K3
• Write X = X+ ∪ X− with X+ ∩ X− = T 3 × [0, 1]
•→ H1(T 3) → H2(X) → H2(X+)⊕H2(X−) → H2(T 3) →
• H2(X+) has rank 11, and a degenerate negative semi-
definite intersection form with kernel of rank 3
• Moduli: Γ\Gr(16, R0,8,3), with Γ = ΓE8; precisely the
moduli of an E8 gauge field on T 3
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The 3-form field on half K3
• Morgan–Mrowka–Ruberman: L2 gauge fields on a 4-
manifold with boundary, whose boundary is T 3
• any gauge group G
• anti-self-dual connections correspond to flows in the
space of G-connections; gradient flow for the Chern–
Simons functional
• the limit is a flat connection on T 3
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The 3-form field on half K3
• Morgan–Mrowka–Ruberman: L2 gauge fields on a 4-
manifold with boundary, whose boundary is T 3
• any gauge group G
• anti-self-dual connections correspond to flows in the
space of G-connections; gradient flow for the Chern–
Simons functional
• the limit is a flat connection on T 3
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• thus, a 3-form with the properties we need can be
obtained from an anti-self-dual connection A on the
half K3, via
C = CS(A) − 12CS(R)
where A and R are now connections on the 4-manifold.
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A bold proposal
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A bold proposal
• The three-form field in M-theory can be written in terms
of a (non-propagating) E8-connection A, with
C = CS(A) − 12CS(R)
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• Not new:
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• Not new:
� Witten, Flux Quantization in M-Theory,
arXiv:hep-th/9609122
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• Not new:
� Witten, Flux Quantization in M-Theory,
arXiv:hep-th/9609122� Diaconescu–Moore–Witten, arXiv:hep-th/0005090
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• Not new:
� Witten, Flux Quantization in M-Theory,
arXiv:hep-th/9609122� Diaconescu–Moore–Witten, arXiv:hep-th/0005090
� Adams–Evslin, arXiv:hep-th/0203218
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• Not new:
� Witten, Flux Quantization in M-Theory,
arXiv:hep-th/9609122� Diaconescu–Moore–Witten, arXiv:hep-th/0005090
� Adams–Evslin, arXiv:hep-th/0203218
• The kinetic term for C becomes∫‖ 130
tr(F 2) − 12tr(R2)‖
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quartic in the curvature F
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quartic in the curvature F
• Need a new kind of gauge invariance: two A’s are gauge
equivalent if they lead to the same C (in bulk)
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quartic in the curvature F
• Need a new kind of gauge invariance: two A’s are gauge
equivalent if they lead to the same C (in bulk)
• The heterotic CS(A) matches across the two half K3’s
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quartic in the curvature F
• Need a new kind of gauge invariance: two A’s are gauge
equivalent if they lead to the same C (in bulk)
• The heterotic CS(A) matches across the two half K3’s
• In the frozen singularity models, the 3-form flux should
be interpreted as CS(A) for an E8 connection A
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quartic in the curvature F
• Need a new kind of gauge invariance: two A’s are gauge
equivalent if they lead to the same C (in bulk)
• The heterotic CS(A) matches across the two half K3’s
• In the frozen singularity models, the 3-form flux should
be interpreted as CS(A) for an E8 connection A
• This matches the observation in “Triples, fluxes, and
strings” that 3-form flux at one half matches CS(A)for corresponding data on T 3 (up to sign)
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• Can we characterize frozen singularities in terms of data
of this sort? (Atiyah)
Preliminary evidence says yes
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• Can we characterize frozen singularities in terms of data
of this sort? (Atiyah)
Preliminary evidence says yes
• Can every 3-form be written in terms of some A? Or do
we perhaps need to go to an infinite-dimensional group
containing E8? (Witten)
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• Can we characterize frozen singularities in terms of data
of this sort? (Atiyah)
Preliminary evidence says yes
• Can every 3-form be written in terms of some A? Or do
we perhaps need to go to an infinite-dimensional group
containing E8? (Witten)
• Perhaps this only makes sense as a quantum theory?
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M