One-loop corrections to heterotic flux compactifications,

One-loop corrections to heterotic flux compactifications Dan Israël Imperial College, May 14th 2016 New supersymmetric index of heterotic compactifications with torsion, D.I. and Matthieu Sarkis, JHEP 1512, 069 (2015) One-loop corrections to heterotic flux compactifications, C. Angelantonj, D.I., M. Sarkis, to appear

Introduction

Heterotic compactifications with torsion Supersymmetric heterotic compactifications Killing spinor eqn. : (ω 1 2 H) ɛ = 0 + O(α 2 ) Gauge, tangent bundles and 3-form H tied together by the Bianchi identity: dh = α 4 (tr R( T ) R( T ) Tr F F) + O(α 2 ) Non-Kähler geometry { Complex manifold M General SUSY compactification: + Holomorphic vector bundle V 3-form flux H Non-Kähler manifold H = i( )J 0 Supergravity limit of flux compactifications? In Bianchi: connection with torsion, e.g. T = + = (ω + 1 2 H) If H 0 at leading order no large-volume limit 2d worldsheet description more appropriate. Fortunately, no RR fluxes! general framework: (0, 2) superconformal theories in 2d Dan Israël One-loop corrections Imperial College, May 14th 2016 1 / 25

Important questions Are compactifications with torsion consistent beyond O(α )? What are their quantum symmetries? What are their moduli spaces? How to compute the four-dimensional effective action? Dan Israël One-loop corrections Imperial College, May 14th 2016 2 / 25

Gauged linear sigma-model (GLSM) approach Basic idea: simple 2d QFT in the same universality class as NLSM (Witten, 96) UV-free 2d Abelian gauge theory low energy = NLSM on (non)kähler manifold Toy example with (2,2) supersymmetry U(1) theory with N chiral fields Φ i of charge Q = 1 = Classical vacua Supersymmetric non-linear sigma-model on CP N 1 (φ i : homogeneous coords) non-conformal! Strategy: find supersymmetry protected quantities invariant under RG flow In this talk How to extend this formalism to non-kähler geometries? RG-invariant partition function for these theories SUSY index One-loop corrections to 4d effective action Dan Israël One-loop corrections Imperial College, May 14th 2016 3 / 25

Outline 1 Gauged linear sigma models basics 2 GLSMs with torsion 3 Supersymmetric index 4 Supersymmetric localization 5 Mathematical formulation 6 Threshold corrections Dan Israël One-loop corrections Imperial College, May 14th 2016 4 / 25

Gauged linear sigma models basics

(0,2) superspace and superfields in two dimensions Superspace coordinates (x ±, θ +, θ + ): D + = θ + i θ + +, D + = θ + + iθ+ + Superfields Chiral superfield: D+ Φ = 0 = Φ = φ + θ + ψ + + Fermi superfield: D+ Γ = 0 = Γ = γ + θ + G + auxiliary fields Vector superfields Chiral field-strength: Υ = µ iθ + (D if 01 ) + Lagrangian L = L kin 1 8e 2 d 2 θ + ῩΥ dθ + Γ a J a(φ I ) + t 4 }{{} Superpotential dθ + Υ }{{} Fayet-Iliopoulos term +h.c. = L 1 ( ) kin+ 2i µ +µ 2e 2 + D 2 + F01 2 + G a J a(φ i ) γ ψ a + i i J a + r D θ 2π F01+h.c. with t = ir + θ 2π Dan Israël One-loop corrections Imperial College, May 14th 2016 5 / 25

Calabi-Yau NLSM from GLSM: K3 surface example Fermat quartic L 0 = Γ 0 }{{} q 0= 4 (Φ1 4 + }{{} Q i =1 +Φ 4 4 ) F-term: φ 4 1 + + φ 4 4 = 0 in CP 3 Gauge bundle: L G = Φ }{{} 0 J m (Φ I ) }{{} q m Q 0<0 Γ m Fermat quartic D-term: i φ i 2 = r + 4 φ 0 2 In Calabi-Yau regime r 1 : φ 0 = 0 CP 3 of size r Yukawa couplings J m(φ I )γ m ψ 0 + Massless left-moving fermions: sections of the holomorphic vector bundle V 0 V O(q m) Jm O( Q 0 ) 0 Dan Israël One-loop corrections Imperial College, May 14th 2016 6 / 25

Anomalies Gauge anomaly: δ Ξ L eff = A 8 dθ + Ξ Υ + c.c., with A = Q i Q i q nq n Consistent models need also 2 non-anomalous global symmetries: Right-moving U(1) R Left-moving U(1) L IR N = 2 superconformal algebra GSO projection Towards GLSM with torsion Interpretation of gauge anomaly: A ch 2 (T M ) ch 2 (V) Geometry of Fayet-Iliopoulos terms L fi = a t a }{{} Υ a [h a ] H (1,1) (M) Complexified Kähler class [J] = a t a[h a ] (Re J B-field) Idea: t a t a (Φ i ) gives dj 0 non-zero torsion H Dan Israël One-loop corrections Imperial College, May 14th 2016 7 / 25

GLSMs with torsion

Introducing the torsion multiplet Strategy: cancel gauge anomaly of K3 GLSM with axion fields (Adams et al. 06) Torsion chiral multiplet Θ = (ϑ, χ) ϑ : coordinate on T 2 with moduli (T, U) metric ds 2 = U2 T 2 dx 1 + T dx 2 2 Gauge transformation: Θ Axial coupling Ξ Θ M Ξ with M Z + T Z Linear field-dependent Fayet Iliopoulos term (gauge-variant) L = M U2 2T 2 dθ + ΘΥ + h.c. M U2 2T 2 ϑ F 01 Cancels the gauge anomaly if A + 2U2 T 2 M 2 = 0 Important consistency condition (single-valuedness of action/duality covariance): Moduli of T 2 such that the free boson ϑ defines a rational conformal field theory (DI, 13) Dan Israël One-loop corrections Imperial College, May 14th 2016 8 / 25

Non-Kähler Geometry: Fu-Yau manifolds At low energies: flows to a NLSM on a class of non-kähler manifolds Principal T 2 bundles over a K3 surface S: (Dasgupta et al., 99, Fu and Yau 2006) T 2 M 6 π S Metric: ds 2 = e 2A(y) ds 2 (S) + U 2 T 2 dx 1 + T dx 2 + π α 2 dα = ω anti-self dual (1, 1) form, ω = ω 1 + T ω 2 with 1 2π ω i H 2 (S, Z) Gauge bundle V : Hermitian-Yang-Mills connection over S: F 0,2 = F 2,0 = 0, J F = 0 Bianchi identity: topological condition S ( ) U2 ω ω ch 2 (V) + ch 2 (T M) = 0 anomaly condition in GLSM T 2 Dan Israël One-loop corrections Imperial College, May 14th 2016 9 / 25

Lessons Class of GLSMs with torsion low energy = Conformal NLSM on Fu-Yau manifolds Most generic compactifications of heterotic strings with N = 2 SUSY in 4d A subset of those are the well-studied K3 T 2 compactifications Many far-reaching insights made for K3 T 2 compactifications including: Duality: Heterotic on K3 T 2 IIA on elliptically fibered Calabi-Yau 3-fold Mathieu moonshine: hidden symmetry (Mathieu group M 24) of NLSMs on K3 due to N = 2 SUSY constraints, many exact statements possible. General understanding should include Fu-Yau manifolds (generic case) Dan Israël One-loop corrections Imperial College, May 14th 2016 10 / 25

Supersymmetric index

Supersymmetric partition function Partition function of (2, 2) superconformal NLSM on Calabi-Yau: elliptic genus Z e (q, y) = Tr p (q q ) ( 1) F e 2iπyJR 0, with J R 0 zero-mode of the left-moving R-current in superconformal algebra vanishes for K3 T 2 because of T 2 fermionic zero-modes (0, 2) NLSM on Fu-Yau zero-modes (χ 0, χ 0 ) from torsion multiplet Assuming there exists a left current J L (and right-moving J R ), define (DI & Sarkis, 15) ( Z fy (q, q, y) = Tr p q q ) ( 1) F e 2iπyJL 0 JR 0, We have J R = χχ + }{{} terms with no χ zero-modes As we will argue, can be computed exactly in the GLSM thanks to SUSY! Dan Israël One-loop corrections Imperial College, May 14th 2016 11 / 25

Path integral formulation On a two-torus of complex structure τ (q = exp 2iπτ) we have: Z fy (τ, τ, y) = Da z Da z DµD µdd e 1 e 2 Svector[a,µ,D] t Sfi(a,D) Dφ i D φ i Dψ i D ψ i e 1 g 2 S chiral[φ i,ψ i,a,d,a l] Dγ a D γ a DG a DḠ a e 1 f 2 S fermi[γ a,g a,a,a l] S j[γ a,g a,φ i,ψ i ] with background gauge field for the U(1) l global symmetry DϑD ϑdχd χ e S torsion[ϑ,χ,a,a l] d 2 z χχ, 2τ 2 a l = πy 2iτ 2 (dz d z) Interacting theory with coupling constants 1/e 2, t, 1/f 2, 1/g 2 Dan Israël One-loop corrections Imperial College, May 14th 2016 12 / 25

Supersymmetric localization

Supersymmetric localization I: the idea Choose a supercharge Q of a supersymmetric QFT Q S[Φ] = 0 and Q-invariant measure Aim : compute O with QO = 0 Find functional P[Φ] such that Q P bos positive definite Deformed path integral O l = DΦ O(Φ) e S[φ] lqp As l O l = Q ( DΦ O(Φ) e S[φ] lqp P ) = 0 compute in the l limit Exact result: one-loop fluctuations around the saddle points QP(Φ 0 ) = 0 Z = dφ 0 e S[Φ0] O class [Φ 0 ] Z one loop Φ 0,QP(Φ 0)=0 Dan Israël One-loop corrections Imperial College, May 14th 2016 13 / 25

Supersymmetric localization II: why we can In the (0, 2) GLSM natural choice of supercharge Q = Q + + Q + Good news Whole vector, chiral and fermi multiplets actions are Q-exact 1/e 2 limit for the gauge fields keep only gauge holonomies (a z z = 0) 1/f 2, 1/g 2 limit superpotential vanishes Bad news? 1 O = d 2 z 2τ 2 χχ not Q-invariant 2 Torsion multiplet action not gauge-invariant not Q-invariant 3 Gauge anomaly measure D Φ i DΓ a not Q-invariant 1 QO doesn t contain enough zero-modes (χ 0, χ 0 ) to contribute } 2 In anomaly-free models Q ( ) DΦ i D Γ a e S torsion = 0 3 Dan Israël One-loop corrections Imperial College, May 14th 2016 14 / 25

Supersymmetric localization III: one-loop determinants Localisation locus e 0 limit flat connections on worldsheet torus : a = πū 2iτ 2 dz πu 2iτ 2 d z f, g 0 limit free chiral and Fermi multiplets Zero modes (generically) : aux. field D 0, gaugino zero-modes λ 0, λ 0 One-loop determinants ambiguities in chiral determinant choice consistent with torsion multiplet Det (u) = e π τ (u 2 uū) 2 ϑ 1 (τ u) holomorphic section of holomorphic line bundle over space of gauge connections (Witten, 96) Chiral : Z Φ = e π τ 2 (υ 2 υῡ) i η(τ) ϑ 1(τ υ), υ = Q iu + q l i y Fermi : Z Γ = e π τ 2 (υ 2 υῡ) ϑ 1(τ υ) i η(τ) U(1) vector : Z A (τ, y) = 2iπη(τ) 2 du Dan Israël One-loop corrections Imperial College, May 14th 2016 15 / 25

Supersymmetric localization IV: the short story Torsion multiplet action not Q-exact Z fy depends on torus moduli T, U Torsion multiplet maps to a rational two-torus defined by two isomorphic rank two even lattices Γ l = Γ 2,2 (T, U) R 2,0, Γ r = Γ 2,2 (T, U) R 0,2 Integral over gauge holonomies u l C/(Z + τz) poles in Z one loop due to accidental bosonic zero modes Turn d 2 u l into contour integral around the poles Jeffrey-Kirwan residues (Benini et al. 2013) Z fy (τ, τ, y) has an intrinsic mathematical definition, as an non-holomorphic elliptic genus that I will use to present the result Dan Israël One-loop corrections Imperial College, May 14th 2016 16 / 25

Mathematical formulation

Geometrical data (0, 2) Non-linear sigma-model on Fu-Yau manifold characterized by Holomorphic tangent bundle T S over the base, with c 1 (T S ) = 0 splitting principle c(t S ) = 2 i=1 (1 + ν i) A rank r holomorphic vector bundle V over S, with c 1 (V) = 0 splitting principle c(v ) = r a=1 (1 + ξ a) A rational Narain lattice in R 2,2 defined by a pair of isomorphic rank two lattices Γ l, Γ r and a gluing map ϕ : Γ l/γ l Γ r/γ r A pair of anti-self-dual (1,1)-forms ω 1 and ω 2 in H 2 (S, Z) characterizing the torus bundle Constraint U2 T 2 ω ω ch 2 (V) + ch 2 (T M ) = 0 in cohomology Dan Israël One-loop corrections Imperial College, May 14th 2016 17 / 25

Non-holomorphic elliptic genus Z fy (M, V, ω τ, τ, y) = 1 η(τ) 2 S Notations η(τ): Dedekind eta-function θ 1 (τ y) : Jacobi theta-function r iθ 1 (τ ξa 2iπ y ) a=1 η(τ) Θ Γl µ µ Γ l /Γl 2 i=1 ( τ η(τ)ν i iθ 1 (τ ν i p ω 2iπ 2iπ ) ) Θ Γr ϕ(µ) ( τ Θ Γ µ : theta-function on lattice Γ Θ Γ µ(τ λ) = γ Γ+µ q 1 2 γ,γ Γ e 2iπ γ,λ Γ p ω : two-dimensional vector of two-forms with p ω, p ω Γl = 2U2 T 2 ω ω belongs to formal extension of Γ L over H 2 (S, Z) H 2 (S, Z) Coincides with path integral computation (checked in simple examples) ) 0 Dan Israël One-loop corrections Imperial College, May 14th 2016 18 / 25

Properties In physics Exact path-integral computation valid for the superconformal non-linear sigma-model emerging in the IR Seed for computing the one-loop corrections to the 4d effective action Allows for quantitative tests of non-perturbative Heterotic/type II duality In mathematics: generalized elliptic genus when ch 2 (V ) ch 2 (T S ) η 2 Z fy (τ, τ, y) transforms as weak Jacobi form of weight 0 and index r 2 η( a τ+b c τ+d ) 2 Z fy ( aτ+b cτ+d, a τ+b c τ+d, y cτ+d ) = e iπ r cy 2 cτ+d η( τ) 2 Z fy (τ, τ, y) ( a b c d) SL(2, Z) Terms with p L = 0 generating function for indices of Dirac operators in representations given by the formal power series V = 1 e 2iπy V + q(t S + T S e2iπy V e 2iπy V ) + In general: Dirac operator with non-zero KK momentum along T 2 fiber? Dan Israël One-loop corrections Imperial College, May 14th 2016 19 / 25

Adding line bundles General SUSY gauge bundles in FY compactifications Pullback of an HYM gauge bundle over the base S : F (2,0) = F (0,2) = 0 and J F = 0 Abelian bundles over the total space (would be Wilson lines for K3 T 2 ) A = T a Re ( Va ι ) with ι = dx 1 + T dx 2 + π α, globally defined on M 6 Abelian bundles in the GLSM Neutral Fermi multiplets (γ,a, G a ) bosonize into chiral multiplets B a = (b a, ψ,a, ψ +,a ) with Im (b a ) at fermionic radius Consider a larger torus T 2+n with V a off-diagonal terms in the metric No single-valuedness constr. vector moduli space (S, V a) in N = 2 SUGRA Contribution to the index Kill superfluous fermionic zero-modes (J 0 ) 16 insertion in the path integral SUSY localization argument still holds Factor out non-compact & right-moving compact bosons in B a Dan Israël One-loop corrections Imperial College, May 14th 2016 20 / 25

Result for the index with most general gauge bundle Z fy(x, V, ω τ, τ, z) = r S a=1 iθ 1 ( τ ξ a 2iπ z ) η(τ) 2 η(τ)ν i iθ 1(τ ν i i=1 (p l,p r) Γ 10,2 exp 2iπ ) ( p ω, p ) l Va=0 q 1 4 pl 2 q η(τ) 18 1 4 pr 2 η( τ) 2 With standard expressions for (10, 2) lattice (however T, U quantized here): 1 1 ( 4 pr 2 = 4 ( n1t T 2U 2 a (V 2 a)2) + n 2 + w 1U + w 2 TU (V a ) 2) + N a av a 2, 1 4 pl 2 = 1 4 pr 2 + n 1w 1 + n 2w 2 + 1 4 NaNa Dan Israël One-loop corrections Imperial College, May 14th 2016 21 / 25

Threshold corrections

Threshold corrections: generalities One-loop corrections to the gauge couplings : G = a G a 4π 2 g 2 a (µ 2 ) 2 one-loop = k aim(s) + ba 4 log M2 s /µ 2 }{{} one-loop β-fct + 1 4 a(m, M) + universal term }{{} moduli dependence N = 2, d = 4 heterotic compactification: b a 4 log M2 s µ + 1 2 4 a(m, M) = i [( ) ] d 2 τ 1 4 F τ 2 η 2 Tr R Qa 2 ka q L 0 c24 q L 0 c24 e iπ J R0 JR0 2πτ 2 modified SUSY index with insertion of Casimir Q 2 a for G a factor Heterotic/type II duality Heterotic threshold corrections one-loop prepotential for vector multiplets F h = γ ij SV i V j + F 1-loop h (V i ) + CY 3 Type IIA dual: K3 fibration with a compatible T 2 fibration perturbative het. limit large P 1 limit in IIA Expansion of CY 3 vector multiplet prepotential qualitative duality check Dan Israël One-loop corrections Imperial College, May 14th 2016 22 / 25

Threshold corrections for FY compactifications Simplify geometric formulation of the index using Gritsenko s formula θ 1 (τ z + ξ) = exp π2 6 E 2(τ)ξ 2 + θ 1 (τ z) θ 1 (τ z) ξ (l 2) (τ, z) ξl l! θ 1(τ z), (l) := l z l l 2 Assume rank two bundle V over K3 in E 8 with S ch 2(V ) = n Correction to hidden E 8 gauge coupling E8 = F d 2 τ Ê 2 E 4 E 6 { q 1 2 p2 1 l q 2 p2 r n τ 2 24 12 E 6 + n 24 } Ê 2 E 6 f (p l ω)e 4 12 p l,p r with f (p l ω) := Each term separately modular invariant S p l, p ω 2 Γ l 1 4πτ 2 p ω, p ω Γl Likewise, one-loop corrections for other gauge factors and gravitational thresholds Computation of the modular integral over F? Dan Israël One-loop corrections Imperial College, May 14th 2016 23 / 25

Modular integral and final result Modular integral Expand (weakly almost holomorphic) modular forms in e8 into Niebur-Poincaré series (Angelantonj, Florakis, Pioline 12) F(s, w, z) = 1 2 γ Γ \Γ M s( Iz)e 2iπRz w γ Unfold the modular integral against the F(s, w, z) (rather than unfolding against the Narain lattice) appropriate here because momentum-dependent insertion f (p l ω) Final result for E 8 threshold e8 = bps { 1 + n 24 6p 2 + 1 4 l ( 1 1 3p 2 l + (p 2 l /2 1) log(1 2/p 2 l ) 1 ) + (p 2 pl 2 l /2 1) log(1 2/pl 2 ) p l, p ω 2 S + 2(n + 12) log 4πe γ T 2U 2 η(u)η(t ) 4} Dan Israël One-loop corrections Imperial College, May 14th 2016 24 / 25

Conclusions

Summary What has been understood Non-Kähler manifolds gauge-variant couplings GLSMs for Fu-Yau manifolds generic N = 2 heterotic vacua Supersymmetric localization exact supersymmetric partition function Defines a modified non-holomorphic elliptic genus, interesting on its own right Computation of threshold corrections for these flux compactifications Ongoing work and directions for future studies Heterotic/type II duality beyond K3 T 2 (Melnikov, Minasian, Theisen 14) Thresholds: instanton corrections from Fourier expansion (T 2 fiber is not a 2-cycle in Fu-Yau geometries!) Generalization of K3 T 2 Mathieu moonshine to FY? More general non-kähler compactifications Dan Israël One-loop corrections Imperial College, May 14th 2016 25 / 25