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New Horizons in Inflationary Cosmology, Stanford March 3, 2017
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.
• 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.
• 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
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
• 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.
• 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)
• 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
Flauger, L.M., Pajer, Westphal, Xu; Flauger and Pajer; Behbahani, Dymarsky, Mirbabayi, Senatore;
Flauger, L.M., Silverstein, Westphal
• 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?
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?
• 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
How large can K,N be?
How ‘aligned’ can the charge matrix Q be?
Bachlechner, Dias, Frazer, L.M.; Bachlechner, Long, L.M.
• 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.
• 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.
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.
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
• 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
• 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.
Triangulation
=
Images: Florian Frick; Peter Lindstrom; Simons Center
Triangulation
• 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.
• By triangulating polytopes, our work becomes
combinatorics, rather than 6d real analysis!
Enumeration comparatively straightforward.
analysis
algebraic
geometry
combinatorics
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.
h1,1 h2,1
Triangulation.
Finding all (‘star, fine, regular’) triangulations of a polytope is costly at h1,1>10. Sage fails.
For 10
h1,1 h2,1
hardest
easiest
Prior capability:
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
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 .
• 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
Braun, Long, L.M., Stillman, Sung
h1,1 h2,1
hardest
easiest
Prior capability:Our capability:
• So far, have studied 4,390 examples, which is everything
at h1,1 ≤ 4.
• Remaining 473,796,386: work in progress.
Field ranges in Planck units
Alignment
No alignment in 2,180 cases
Anti-alignment in 1,716 cases
Alignment in 494 cases Max alignment = 2.55
Example with Alignment
Alignment by factor extends range from
• 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.
• 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
?
• 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.
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