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: Kim, Nilles, Peloso; Silverstein, Westphal; alignment and monodromy. L.M., Silverstein, Westphal; Kaloper, Sorbo • 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 Possible in QG

Landscape (consistent theories) Banks, Dine, Fox, Gorbatov Vafa Ooguri, Vafa Adams et al Arkani-Hamed, Motl, Nicolis, Vafa Cheung, Remmen Saraswat, Sundrum Rudelius Swampland Heidenreich et al (inconsistent theories)

Impossible in QG. Failures of causality, unitarity, etc. • A WGC is a conjecture about the spectrum of charged black holes. • Dual description: statements about large-field axion inflation. Arkani-Hamed et al; Rudelius; Brown et al. • 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. • 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 Kim, Nilles, Peloso; Silverstein, Westphal; L.M., Silverstein, Westphal; displacement. Kaloper, Sorbo; Kaloper, Lawrence, Sorbo; L.M., Silverstein, Westphal, Wrase • 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. 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? Kim, Nilles, Peloso 2004 • 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?

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,

basis for

which are the imaginary parts of the Kähler moduli:

• We will take the as candidates for aligned natural inflation.

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 Kallosh, Sorokin; Kallosh, Kashani-Poor, Tomasiello; – Worldvolume fluxes and bulk fluxes Lüst et al; Bianchi, Collinucci, Martucci 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. • 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 Triangulation

=

Images: Florian Frick; Peter Lindstrom; Simons Center • 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 V4 determined by 4d reflexive polytopes. • 4d reflexive polytopes have been classified. There are 473,800,776 of them.

Kreuzer and Skarke • 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

Divisor topology. For h1,1<5, Cohomcalg (Blumenhagen et al ) can compute Hodge numbers, but: 1,1 • Most CY3 have much larger h • Questions about super-Planckian displacements in aligned axion inflation, and about the WGC, require data at . Bachlechner, Long, L.M. cf. Douglas Denef Floria Grassi Kachru hardest

h1,1 h2,1

Prior capability:

easiest 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 hardest

h1,1 h2,1

Our capability: Prior capability:

easiest • 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.