Progress in supersymmetric lattice theories Simon Catterall (Syracuse) YITP , Kyoto, 21 July arXiv:1410.6971, arXiv:1411.0166, arXiv:1505.03135, arXiv:1505.00467 & ... with Poul Damgaard, Tom DeGrand, Joel Giedt, David Schaich and Aarti Veernala Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 1 / 28
Outline: Brief review: constructing lattice actions with exact supersymmetry N = 4 Yang-Mills on the lattice Flat directions and how to lift them ... improved action Real space RG Recent results: Konishi anomalous dimension and static potential Generalizations: lattice quivers and 2d super QCD. Dynamical susy breaking. Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 2 / 28
Motivations and difficulties of lattice supersymmetry Much interesting physics in 4D supersymmetric gauge theories: dualities, holography, conformality, BSM, . . . Lattice promises non-perturbative insights from first principles Problem: Discrete spacetime breaks supersymmetry algebra � � J = 2 δ IJ σ µ Q I α P µ where I , J = 1 , · · · , N α , Q α ˙ α ˙ = ⇒ Impractical fine-tuning generally required to restore susy, especially for scalar fields (from matter multiplets or N > 1) Solution: Preserve (some subset of) the susy algebra on the lattice Possible for N = 4 supersymmetric Yang–Mills (SYM) Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 3 / 28
Brief review of N = 4 SYM N = 4 SYM is a particularly interesting theory —AdS/CFT correspondence —Testing ground for reformulations of scattering amplitudes —Arguably simplest non-trivial field theory in four dimensions Basic features: SU( N ) gauge theory with four fermions Ψ I and six scalars Φ IJ , all massless and in adjoint rep. Action consists of kinetic, Yukawa and four-scalar terms with coefficients related by symmetries I Supersymmetric: 16 supercharges Q I α with I = 1 , · · · , 4 α and Q ˙ Fields and Q ’s transform under global SU(4) ≃ SO(6) R symmetry Conformal: β function is zero for any ’t Hooft coupling λ Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 4 / 28
Topological twisting − → exact susy on the lattice What is special about N = 4 SYM I The 16 spinor supercharges Q I α and Q α fill a Kähler–Dirac multiplet: ˙ Q 1 Q 2 Q 3 Q 4 = Q + Q µ γ µ + Q µν γ µ γ ν + Q µ γ µ γ 5 + Q γ 5 α α α α − → Q + γ a Q a + γ a γ b Q ab 1 2 3 4 Q Q Q Q α ˙ α ˙ α ˙ α ˙ with a , b = 1 , · · · , 5 Q ’s transform with integer spin under “twisted rotation group” � � SO(4) tw ≡ diag SO(4) euc ⊗ SO(4) R SO(4) R ⊂ SO(6) R This change of variables gives a susy subalgebra {Q , Q} = 2 Q 2 = 0 This subalgebra can be exactly preserved on the lattice Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 5 / 28
Twisted N = 4 SYM fields and Q Everything transforms with integer spin under SO(4) tw — no spinors I Q I α and Q α − → Q , Q a and Q ab ˙ I − Ψ I and Ψ → η, ψ a and χ ab A µ and Φ IJ − → A a = ( A µ , φ ) + i ( B µ , φ ) and A a The twisted-scalar supersymmetry Q acts as Q A a = ψ a Q ψ a = 0 Q χ ab = −F ab Q A a = 0 Q η = d Q d = 0 տ bosonic auxiliary field with e.o.m. d = D a A a Scalars → vectors under twisted group. Combine with gauge fields 1 The susy subalgebra Q 2 · = 0 is manifest 2 Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 6 / 28
Twisted N = 4 action Obtain by dimensional reduction of N = 2 Yang-Mills in five dimensions: � � � N χ ab F ab + η [ D a , D a ] − 1 S = 2 λ Q 2 η d M 4 × S 1 Tr � − N . M 4 × S 1 ǫ abcde Tr χ ab D c χ de 8 λ Q 2 = 0 and Bianchi guarantee supersymmetry independent of metric of M 4 Marcus/GL twist of N = 4. Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 7 / 28
Lattice N = 4 SYM fields and Q The lattice theory is very nearly a direct transcription Covariant derivatives − → finite difference operators eg. D a ψ b → U a ( x ) ψ b ( x + a ) − ψ b ( x ) U a ( x + b ) Gauge fields A a − → gauge links U a Q A a − →Q U a = ψ a Q ψ a = 0 Q χ ab = −F ab Q A a − →Q U a = 0 Q η = d Q d = 0 Geometrical formulation facilitates discretization η live on lattice sites ψ a live on links χ ab face links Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 8 / 28
A ∗ 4 lattice with five links in four dimensions Maximize global symmetries of lattice theory if treat all five U a symmetrically ( S 5 symmetry) —Start with hypercubic lattice in 5d momentum space — Symmetric constraint � a ∂ a = 0 projects to 4d momentum space —Result is A 4 lattice → dual A ∗ − 4 lattice in real space Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 9 / 28
Novel features of lattice construction Fermions live on links not sites To keep Q -susy (complex) gauge links U a must also live in algebra like the fermions. Employ flat not Haar measure D U D U . Still gauge invariant! Correct naive continuum limit forces use of complexified U ( N ) theory. Allows for expansion around U a = I + A a + . . . Exact lattice symmetries strongly constrain renormalization of lattice theory. Can show only single marginal coupling remains to be tuned ! Not quite suitable for numerical calculations Exact 0 modes/flat directions must be regulated especially the U(1) In the past instabilities in scalar U ( 1 ) mode regulated with a soft scalar mass term. We add such a term with coeff µ 2 but this is not enough .... Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 10 / 28
Lattice monopole instabilities Flat directions in U(1) gauge field sector can induce transition to confined phase at strong coupling This lattice artifact is not present in continuum N = 4 SYM Around λ lat ≈ 2. . . Left: Polyakov loop falls towards zero Center: Plaquette determinant falls towards zero Right: Density of U(1) monopole world lines becomes non-zero Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 11 / 28 2
Supersymmetric lifting of the U(1) flat directions arXiv:1505.03135 Better: modify e.o.m for auxiliary field d to add new moduli space condition det P ab = 1 � � N χ ab F ab + ↓ − 1 N ǫ abcde χ ab D c χ de + µ 2 V S = Q 2 η d − 2 λ lat 8 λ lat � � � D a U a + G [ det P − 1 ] I N η P Scalar potential softly breaks Q , much less than old non-susy det P ( ∼ 500 × smaller lattice artifacts for L = 16) Effective O ( a ) improvement since Q forbids all dim-5 operators Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 12 / 28
Update on observables with improved action Static potential. Anomalous dimensions. Latter rely in part on a recently formulated real space RG which respects the lattice Q -symmetry (arXiv:1408.7067) Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 13 / 28
Static potential Previously reported Coulombic static potential V ( r ) at all λ Currently confirming and extending with improved action Left: Agreement with perturbation theory for N = 2, λ � 2 √ Right: Tantalizing λ -like behavior for N = 3, λ � 1, √ possibly approaching large- N AdS/CFT prediction C ( λ ) ∝ λ Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 14 / 28
Konishi operator scaling dimension N = 4 SYM is conformal at all λ − → power-law decay for all correlation functions The Konishi operator is the simplest conformal primary operator � � Φ I Φ I � C K ( r ) ≡ O K ( x + r ) O K ( x ) ∝ r − 2 ∆ K O K = Tr I There are many predictions for the scaling dim. ∆ K ( λ ) = 2 + γ K ( λ ) From weak-coupling perturbation theory (2-4 loops) From holography for N → ∞ and λ → ∞ but λ ≪ N Upper bounds from the conformal bootstrap program S duality: 4 π N λ ← → 4 π N λ Only lattice gauge theory can access nonperturbative λ at moderate N Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 15 / 28
Real space RG for susy lattices Exact lattice symmetries ( Q , S 5 , ghost number, gauge invariance) + power counting lead to remarkable result: only a single marginal coupling needs to be tuned for lattice theory to flow to continuum N = 4 theory as L → ∞ , g = fixed . (arXiv:1408.7067) However This analysis implicitly assumes existence of RG that preserves Q One simple blocking exists: a ( x ′ ) a ′ = 2 a U ′ = ξ U a ( x ) U a ( x + a ) ψ ′ a = ξ ( ψ a ( x ) U a ( x + a ) + U a ( x ) ψ a ( x + a )) ..... ξ is free parameter obtained by matching vevs of observables computed on initial and blocked lattices. RG also yields a tool for extracting beta functions and anomalous dimensions from Monte Carlo data Simon Catterall (Syracuse) Progress in supersymmetric lattice theories YITP , Kyoto, 21 July 16 / 28
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