new horizons in inflationary cosmology, stanford march 3, 2017 · 2017. 3. 27. · • many...
TRANSCRIPT
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New Horizons in Inflationary Cosmology, Stanford March 3, 2017
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A. Braun, C. Long, L.M., M. Stillman, B. Sung,
“The Hodge Numbers of Divisors in Calabi-Yau Hypersurfaces”, to appear.
C. Long, L.M., J. Stout,
“Systematics of Axion Inflation in Calabi-Yau Hypersurfaces,” 1603.01259.
T. Bachlechner, C. Long, L.M.,
“Planckian Axions and the Weak Gravity Conjecture,” 1503.07853.
T. Bachlechner, C. Long, L.M.,
“Planckian Axions in String Theory,” 1412.1093.
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• The inflationary models that produce a detectably strong
primordial gravitational wave signal are profoundly
sensitive to quantum gravity.
• How can we maximize the scientific impact of a limit on
(or detection of) primordial B-modes?
– In particular, how can we extract the maximal lessons about
quantum gravity from such a measurement?
• Goal: identify traces of microphysics in effective theories
for large-field inflation derived from string theory.
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• Many scenarios for large-field inflation in effective field
theory, but these rest on implicit assumptions about
symmetries in quantum gravity.
• Promising approaches to deriving such theories in string
theory, but no gold-plated model in an explicit vacuum.
• Leading approaches use axion shift symmetries:
alignment and monodromy.
• Lore: continuous shift symmetries are not exact in
quantum gravity, because of black hole thermodynamics.
• Precise recent formulation: ‘Weak Gravity Conjecture’
(WGC).
Arkani-Hamed, Motl, Nicolis, Vafa; Cheung, Remmen; Heidenreich, Reece, Rudelius
Kim, Nilles, Peloso; Silverstein, Westphal;
L.M., Silverstein, Westphal; Kaloper, Sorbo
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Landscape
(consistent theories)
Swampland
(inconsistent theories)
Possible in QG
Impossible in QG.
Failures of causality, unitarity, etc.
Banks, Dine, Fox, Gorbatov
Vafa
Ooguri, Vafa
Adams et al
Arkani-Hamed, Motl, Nicolis, Vafa
Cheung, Remmen
Saraswat, Sundrum
Rudelius
Heidenreich et al
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• A WGC is a conjecture about the spectrum of charged
black holes.
• Dual description: statements about large-field axion
inflation.
• A WGC can give a conjectural no-go about B-modes
from certain models of axion inflation.
• But:
– there is a proliferation of WGCs, from different premises
– the resulting no-go theorems on the inflationary side exclude
many simple scenarios, but not all.
– a priori reasoning about effective theories of gauge fields and
gravity seems insufficient to exclude B-modes from axion
inflation.
Arkani-Hamed et al; Rudelius; Brown et al.
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• Idea: make progress by enumerating examples of large-
field axion inflation in string compactifications.
– could falsify specific WGCs
– could reveal which mechanisms for large-field inflation are
robust against QG constraints
• Ideally, generate ensemble of models in a class of
compactification manifolds, and study statistics.
• To proceed, need a large-field inflation scenario that is
– theoretically well-grounded
– computationally tractable (no PDEs)
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• Key tool: discrete shift symmetries.
• Multiple sub-Planckian axion periods can be combined
via monodromy or alignment to give a super-Planckian
displacement.
• But gluing together constituents to form a large object
can leave artifacts. Example: resonant contributions to
2-pt and 3-pt function in axion monodromy inflation.
Kim, Nilles, Peloso; Silverstein, Westphal; L.M., Silverstein, Westphal;
Kaloper, Sorbo; Kaloper, Lawrence, Sorbo; L.M., Silverstein, Westphal, Wrase
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Flauger, L.M., Pajer, Westphal, Xu; Flauger and Pajer; Behbahani, Dymarsky, Mirbabayi, Senatore;
Flauger, L.M., Silverstein, Westphal
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• Leading scenarios for large-field inflation in string theory are
limited by quantum gravity constraints on axions.
• These scenarios involve special parameter values (e.g.,
axion charges).
• The parameters are fundamentally discrete, and
correspond to topological data of a compactification.
• Aim: enumerate models of large-field inflation in explicit
compactifications. Survey the quantized parameters.
– Do QG constraints exclude any EFT scenarios?
– Can we establish definitively that string theory admits solutions that
can be ruled out by upper limits on B-modes?
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In EFT: p,q are real-valued parameters.
In string theory: p,q are integers determined by topological
data. (e.g., by which divisors are rigid)
How large can p,q be?
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• Inflaton is a linear combination of N≥2 axions.
• Individual axions have decay constants
• Potential generated by instantons (without monodromy).
• In favorable cases, alignment occurs: the periodicity
along the longest direction satisfies
• Unbounded in principle from EFT perspective,
but would yield an exact global symmetry.
• Quantum gravity must limit , but where exactly is the
bound?
Kim, Nilles, Peloso 2004
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How large can K,N be?
How ‘aligned’ can the charge matrix Q be?
Bachlechner, Dias, Frazer, L.M.; Bachlechner, Long, L.M.
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• Consider type IIB string theory compactified on an O3/O7
orientifold of a Calabi-Yau threefold (CY3), X.
• The RR 4-form gives rise to axions,
which are the imaginary parts of the Kähler moduli:
• We will take the as candidates for aligned natural
inflation.
basis for
GKP, KKLT, BBCQ, CQS, BBKR, et seq.
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• Shift symmetry is unbroken perturbatively,
and broken to by Euclidean D3-branes.
• The effective theory takes the form
with
where are the divisors that support ED3.
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A Euclidean D3-brane wrapping the cycle
is an instanton with charge under the shift of the axion .
The instanton charge matrix :
• Dictates degree of alignment, and field range:
• Is determined by which integer linear combinations of divisors
support ED3-branes.
The instanton charges are topological data: they are integers
determined by the properties of the divisors of X.
We can compute the maximal degree of alignment by
determining which divisors support ED3-branes.
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Consider a Euclidean M5-brane wrapping a divisor D in a CY4, X.
The worldvolume Dirac operator has two universal zero modes from the
supersymmetries broken by the M5-brane.
These saturate the Grassmann integral
The M5-brane will give a nonvanishing instanton contribution to W
provided that
1. There are no additional fermionic zero modes (which would
necessarily give W=0);
2. Integration over bosonic moduli, if any, does not give a zero;
3. There is no anomaly forbidding W.
Conditions (1),(2),(3) can be checked from topological data of D: in
particular, the Hodge numbers
Sufficient condition: D is rigid, Witten
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• Complications can arise from
– Failure of X or D to be smooth
– Worldvolume fluxes and bulk fluxes
which can change the Dirac operator’s zero modes.
• These can be accounted for in terms of additional
topological data.
• In any case, the key step toward computing the
Euclidean brane superpotential is to compute the Hodge
numbers of divisors D in X.
Kallosh, Sorokin; Kallosh, Kashani-Poor, Tomasiello;
Lüst et al; Bianchi, Collinucci, Martucci
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• Task: study the statistics of aligned inflation,
by computing the Hodge numbers of divisors
in an ensemble of geometries.
• Idea: study Calabi-Yau hypersurfaces in toric varieties V.
• Toric varieties are very nice spaces that admit a
combinatorial description in terms of triangulations of
polytopes.
• A polytope is the n-dimensional generalization of a
polygon.
• A triangulation of a polytope Δ is a division of Δ into
simplices.
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Triangulation
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=
Images: Florian Frick; Peter Lindstrom; Simons Center
Triangulation
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• Given a suitable (“reflexive”) polytope Δ one can construct an
associated toric variety V by triangulating Δ.
• We will study CY3 that are hypersurfaces in varieties V4determined by 4d reflexive polytopes.
• 4d reflexive polytopes have been classified.
Kreuzer and Skarke
There are 473,800,776 of them.
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• By triangulating polytopes, our work becomes
combinatorics, rather than 6d real analysis!
Enumeration comparatively straightforward.
analysis
algebraic
geometry
combinatorics
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Process:
1. Select a reflexive polytope from Kreuzer-Skarke list.
2. Triangulate to reach a toric variety V with at most pointlike
singularities. Anticanonical hypersurface in V is a CY3, X.
3. Compute Kähler cone + intersection numbers of X.
4. Search cone of effective divisors for rigid divisors.
5. Compute field range and amount of alignment.
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h1,1 h2,1
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Triangulation.
Finding all (‘star, fine, regular’) triangulations of a polytope is costly at h1,1>10. Sage fails.
For 10
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h1,1 h2,1
hardest
easiest
Prior capability:
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For larger h1,1, a new approach is required.
Idea: we show that divisors correspond to graphs
on the 2-dimensional faces of the (dual) polytope .
We can then write down a simple formula for the Hodge
numbers in terms of the data of the graph.
This is a complete and extremely efficient solution in terms
of combinatorial data.
Braun, Long, L.M., Stillman, Sung
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A square-free effective divisor D
corresponds to a choice of vertices
in the 2d faces of .
D is connected only if all chosen
vertices are connected by edges of
the triangulation.
Divisors connected graphs on 2d faces of .
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• Corresponding to each connected square-free divisor D is a
connected lattice graph GD on the 2d faces of .
• Study V directly from combinatorial data.
• Compute topology of prime toric divisors (lattice points of graph) via
stratification.
• Use Koszul sequence to descend from V to X to D.
• Use Mayer-Vietoris sequence to compute cohomology of sums of
prime toric divisors.
• Use structure of graph to assemble the summands.
• The ordinary Hodge numbers of GD then appear in the answer!
Braun, Long, L.M., Stillman, Sung
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Braun, Long, L.M., Stillman, Sung
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h1,1 h2,1
hardest
easiest
Prior capability:Our capability:
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• So far, have studied 4,390 examples, which is everything
at h1,1 ≤ 4.
• Remaining 473,796,386: work in progress.
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Field ranges in Planck units
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Alignment
No alignment in 2,180 cases
Anti-alignment in 1,716 cases
Alignment in 494 cases Max alignment = 2.55
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Example with Alignment
Alignment by factor extends range from
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• We have ignored corrections to the fermion zero-mode counting from:
– self-intersections of the divisors (normal crossings)
– D7-branes and orientifold planes
– worldvolume fluxes and bulk fluxes
• In toric analysis:
– Have not yet carefully kept track of redundancies in set of threefolds
– Considered only favorable hypersurfaces.
– Worked with the Mori cone of V, not X; possibly too restrictive.
– Only estimated perturbative corrections to K.
– Only studied square-free divisors.
• We computed superpotential terms. Kähler potential terms remain to be obtained.
• These caveats can in principle be dealt with by generalizing our computation, except
for perturbative (quantum) corrections to K.
• As the axions are displaced, the saxions shift, and the metric changes. Instabilities
can arise.
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• We have obtained a formula for the Hodge numbers of
divisors D in CY3 hypersurfaces in toric varieties.
• The coefficients of D in an integral basis are the axion
charges of a Euclidean D3-brane wrapping D.
• When these charges are special, the axion field space
enjoys alignment, and the diameter is enlarged.
• We have begun to determine the statistics of aligned
axion inflation in Calabi-Yau hypersurfaces. Very
modest alignment so far.
• Are there examples with large alignment, and large r, at
?
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• Obtaining the classical geometric data for compactification
on any CY3 hypersurface in the Kreuzer-Skarke list is
straightforward, thanks to improved triangulation methods.
• At this level the Kähler moduli of IIB are unfixed.
• Program: determine leading quantum effects, and
enumerate stabilized vacua.
• So far: ED3-brane superpotential, with some limitations.
• Systematic enumeration of stabilized vacua may be
achievable.
• In time we may build an ensemble of inflationary solutions
of string theory.