Chiral Algebras of (0, 2) ModelsBeyond Perturbation Theory

Junya Yagi

March 8, 2010 at Nagoya University

Based on

I M.-C. Tan & JY, arXiv:0801.4782, 0805.1410 [hep-th]

I M.-C. Tan & JY, Lett. Math. Phys. 84 (2008) 257

I JY, arXiv:1001.0118 [hep-th]

Why Consider (0, 2) Supersymmetry?

I Chiral ring of a (2, 2) model (or topological sigma model)I target space X : Kahler manifoldI related to GW invariants, mirror symmetry, etc.I quantum deformations of finite-dim cohomology H∗(X ).

I Elliptic genusI The partition function on a torusI mathematically very interesting, too. (elliptic cohomology,

topological modular forms, etc.)I ∞-dimI uses right-moving supersymmetry only

I Let’s consider theories with (0, 2) supersymmetry.

Chiral Algebras of (0, 2) Models

I ∞-dim analogs of the chiral rings

I ∼ chiral algebras of CFTs (algebra of holomorphic fields)

I related to the geometric Langlands program, pure spinorformulation of the superstring, etc.

I perturbatively described by sheaf theory[Witten, Adv. Theor. Math. Phys. 11 (2007) 1 [hep-th/0504078]]

I Nonperturbatively, not much is understood.

(2, 2) Model

I Bosonic field φ : Σ→ X

ϕ

Σ X

Σ: Riemann surface, X : Kahler manifold

I Fermionic fields ψ−, ψ−, ψ+, ψ+

I Supercharges Q+, Q+, Q−, Q−I Action

S =

∫Σd2z(gi ∂zφ

i∂zφ+igi ψ

i+Dz ψ

++ · · ·︸︷︷︸

terms with ψ−

)+

∫Σφ∗ω

Our (0, 2) Model

I Kill the left-moving fermions:

S =

∫Σd2z(gi ∂zφ

i∂zφ + igi ψ

i+Dz ψ

+) +

∫Σφ∗ω

I Supercharges Q+, Q+:

Q2+ = Q2

+ = 0, {Q+,Q+} = H − P

I U(1) R-symmetry (possibly broken to Zn nonperturbatively):

ψ+ → e−iαψ+, ψ+ → e iαψ+,

Q+ → e−iαQ+, Q+ → e+iαQ+

Q-cohomology of Local Operators

I Let Q = Q+; Q+ = Q†.

I Consider the action of Q on local operators O:

{Q,O} = QO ∓OQ (O bosonic or fermionic).

I The Q-action squares to zero:

{Q, {Q,O}} = {Q2,O} = 0.

I The Q-action increases the R-charge of O by 1.

I Define the Q-cohomology of local operators

I graded by R-charge (possibly broken to a Zn-grading)

Chiral Algebra

I In our model, {Q,Q†} = H − P ∝ ∂z .

I Q-cohomology classes are holomorphic: if O is Q-closed, then

∂zO ∝ {{Q,Q†},O} = {Q, {Q†,O}},

thus ∂zO is Q-exact.

I OPE: if we define [O1] · [O2] = [O1O2], then

[O1(z)] · [O2(z ′)] =∑k

cijk(z − z ′)︸ ︷︷ ︸

can have poles

[Ok(z ′)].

I Holomorphic Q-cohomology + OPE = chiral algebra

A =⊕q

Aq.

Vanishing “Theorem” [JY]

I Suppose X is a compact Kahler manifold of complex dim d ,

I spin, p1(X )/2 = 0, Ric > 0,

I and contains a rational curve C ∼= CP1 such that

I the normal bundle NC/X is trivial (∼= O⊕d−1), where

0 −→ TC −→ TX |C −→ NC/X −→ 0.

I Then A = 0 nonperturbatively.

I Example: X = CP1 (predicted by Witten)

I More generally, any flag manifold X = G/T(also found by Frenkel–Losev–Nekrasov, in preparation?)

What’s Happening?

I Perturbatively, A is ∞-dim:

bosonic: , , , . . . , fermionic: , , , . . . .

I Instantons pair up the Q-cohomology classes

...

...

I and induce{Q, } =

I is now Q-exact, is no longer Q-closed.

“Corollary” 1: Supersymmetry Breaking

I SUSY states: {Q,Q†} = H − P = 0

H − P = 0(SUSY)

H − P > 0(non-SUSY)

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I Perturbatively, there are ∞ many (P = 0, 1, 2, . . . ).

I Nonperturbatively, there are none if A = 0.

I SUSY is spontaneously broken.

Lifting

I Instantons pair up the perturbative SUSY states

...

...

I and induceQ = 〉∣ 〉∣Q =

I thereby “lifting” all of the perturbative SUSY states at once:

H − P = 0

H − P > 0

...

... ...

...

“Corollary” 2: Geometry of Loop Spaces

I Canonical quantization =⇒ SUSY QM on the loop space LX

X

t

Σ

ϕ

I States are spinors on LX :

{ψi+(σ), ψ+(σ′)} = g i δ(σ − σ′) (cf. {Γi , Γ} = g i )

I SUSY states are harmonic spinors (Q + Q†: Dirac operator)

I If A = 0, then LX has no harmonic spinors (cf. Lichnerowicz)

“Corollary” 3: Hohn–Stolz Conjecture

I If M is compact, spin, has p1(M)/2 = 0 and Ric > 0, thenthe elliptic genus

E(q) = Tr((−1)FRqP

)vanishes. (E(q) is the partition function on the torusC/2π(Z + τZ), with q = e2πiτ ) [Stolz, Math. Ann. 304 (1996) 785]

I E(q) counts

#bosonic SUSY states−#fermionic SUSY states

at each energy level.

I If A = 0, then no SUSY states and E(q) = 0.

I A physics proof of a special case of the HS conjecture

Quasi-topological Twisting

I Essentially only one way to twist:

ψiz ∈ Γ(KΣ ⊗ φ∗TX ), ψı ∈ Γ(φ∗TX ).

I Q is now a scalar on Σ.

I Tzz and Tz z are now Q-exact (but Tzz is not).

I Antiholomorphic reparametrizations act trivially.

I Let’s say O has dimension (n,m) if it transforms as

O(0)→ λ−nλ−mO(0)

under z → λz , z → λz .

I If [O] 6= 0, then m = 0.

Local Operators of R-charge q and Dimension (n, 0)

I Under ψı ↔ dφı, they are identified with (0, q)-forms on X

I with values in a holomorphic vector bundle VX ,n

I n = 0: we have

Oı1···ıq(φ, φ)ψı1 · · · ψıq ;

thus VX ,0 = 1, the trivial bundle.

I n = 1: we have

Oj ı1···ıq∂zφj ψı1 · · · ψıq or Oj

ı1···ıqgj k∂zφk ψı1 · · · ψıq ;

thus VX ,1 = TX ⊕ T∨X .

I VX ,n enter the definition of the elliptic genus.

Classical Chiral Algebra

I {Q, φı} = ψı, {Q, ψi} = −∂zφi , {Q, others} = 0.

I An n = 1 example:

{Q,Oj ı1···ıq∂zφj ψı1 · · · ψıq} = ψı∂ıOj ı1···ıq∂zφ

j ψı1 · · · ψıq .

I The other type:

{Q,Ojı1···ıqgj k∂zφ

k ψı1 · · · ψıq}

= ψı∂ıOjı1···ıq · · ·+Oj

ı1···ıqgj k Dz ψk︸ ︷︷ ︸

0 by EOM

· · · .

I Classically, Q = ψı∂ı = ∂ and

Aq ∼=∞⊕n=0

Hq

∂(X ,VX ,n).

Witten’s Approach to Perturbative Chiral Algebra

I Perturbatively, Q is deformed: Q = ∂ + α′Q1 + (α′)2Q2 + · · ·I Perturbative A is reconstructed by gluing free βγ CFTs,

β(z)γ(z ′) ∼ − 1

z − z ′,

over X :

Xβγ

βγβγ

...

I Aq ∼= Hq(X ,DchX ), where Dch

X is a sheaf of CDO on X .[Witten, Adv. Theor. Math. Phys. 11 (2007) 1 [hep-th/0504078]].

CP1 Model

I X = CP1:

I R-charge q = 0 or 1; A = A0 ⊕A1

I The simplest and most important example

I Basis for our vanishing “theorem”

CP1 Model: Perturbative Chiral Algebra (Charge 0)

I Q-closed charge 0, dimension n local operator:

Oi1i2...j1j2...(φ)︸ ︷︷ ︸

holomorphic

∂m1z φi1∂m2

z φi2 · · · gj1k1∂n1z φ

k1gj2k1∂n2z φ

k2 · · ·︸ ︷︷ ︸n ∂z s

I Dimension 0: holomorphic functions, thus [1]

I Dimension 1: [J−], [J3], [J+], where

J− = gφφ∂z φ, J3 = φgφφ∂z φ, J− = φ2gφφ∂z φ,

correspond to 3 holomorphic vector fields

I Js generate the SL2 current algebra at the critical level −2.

I The list continues.

CP1 Model: Perturbative Chiral Algebra (Charge 1)

I Q-closed charge 1, dimension n local operator:

Oi1i2...j1j2...

l(φ)∂m1z φi1 · · · gj1k1

∂n1z φ

k1 · · · ψ l

I Dimension 0: none

I Dimension 1: [θ], where

θ ∝ Ri ∂zφi ψ

I Dimension 2: [J−θ], [J3θ], [J+θ]

I Is [∂zθ] nonzero? Classically, yes. Perturbatively, no:

[Q,Tzz ] = ∂zθ.

I No energy-momentum tensor if c1(X ) 6= 0!

CP1 Model: Perturbative Chiral Algebra (Summary)

I Compare A0 and A1:

dim A0 classes dim A1 classes...

......

...1 [J−], [J3], [J+] 2 [J−θ], [J3θ], [J+θ]0 [1] 1 [θ]

0 none

I Looks like the spectrum of a free string

I [1] and [θ] are “ground states”

I on which elements of A0 acts as “creation operators.”

I A0 ∼= A1 via [O] 7→ [Oθ][Malikov et al., Comm. Math. Phys. 204 (1999) 439 [math/9803041]]

Instanton Tunneling?

I Instantons: holomorphic maps from Σ to X (like A-model)

I Instantons “tunnel” between the “ground states” [1] and [θ]?

I If yes, they will induce

{Q, θ} ∼ 1.

I R-charge: 2 (LHS) vs. 0 (RHS)

I For X = CP1, instantons break the R-symmetry to Z2.

Computation of {Q, θ}

I Compute the matrix elements of {Q, θ} on Σ = S1 × R.

I Compactify and map to correlation functions on CP1:

〈j |{Q, θ(0)}|i〉 =⟨Oj(∞)

∮dz G (z)θ(1)Oi (0)

⟩.

I Compute in the 1-instanton background (Mobius CP1 → CP1)

I Express the result as a 0-instanton correlation function.

I Found⟨Oj(∞){Q, θ(1)}Oi (0)

⟩1-instanton

∼⟨Oj(∞)Oi (0)

⟩0-instanton

I Therefore, {Q, θ} ∼ 1.

The Chiral Algebra Vanishes

I More precisely,

{Q, θ} = e−t(1 + Q-exact local operator).

I t: area of X = CP1 (cf. [H]2 = e−t [1] of the A-ring)

I θ recovers in the chiral algebra at t =∞.

I 1 = 0 in the Q-cohomology!

I Thus[O] = [1] · [O] = 0 · [O] = 0.

I Any Q-closed local operator is Q-exact.

I Therefore, the chiral algebra is nonperturbatively trivial.

Supersymmetry Breaking

I Q2 = 0 =⇒ define the Q-cohomology of states

I It is isomorphic to the space of SUSY states

I and a module over the chiral algebra:

[O] · [|Ψ〉] = [O|Ψ〉].

I If A = 0, then

[|Ψ〉] = [1] · [|Ψ〉] = 0 · [|Ψ〉] = 0.

I Therefore, the Q-cohomology of states is trivial

I and SUSY is spontaneously broken.

Vanishing “Theorem”: Conditions

I A = 0 if the following conditions are satisfied:

1. X is spin ⇐⇒ fermion # is well defined mod 22. p1(X )/2 = 0 ⇐⇒ sigma model anomalies cancel3. Ric > 0 ⇐⇒ the model is asymptotically free4. contains C ∼= CP1 ⇐⇒ there are instanton corrections5. NC/X

∼= O⊕d−1 ⇐⇒ R-symmetry is broken to Z2

by min # of fermion zero modesI 1 & 2 are mandatory

I 3 has a natural interpretation

I 4 follows from 3

I 5 can be removed?

Sketch of a “Proof”

I Say {Q, θ} ∼ e−tO. Then O is a classically Q-closed local op.

I The charge 0, dim 0 part of O is a holomorphic function f .

I X is compact, so f = const.

I Since NC/X is trivial, near C we have a CP1 model:

{Q, θ} = e−t(χC + Q-exact + other instantons).

Here χC = 1 on C and 0 elsewhere.

I We have found f 6= 0:

{Q, θ} = e−t(1 + · · · ).

One can show · · · are all Q-exact.

I Therefore, [1] = 0 and A = 0.

I X is covered by CP1s.

Holomorphic Morse Theory on Loop Space

I Morse function on LX : symplectic action functional afI Deformation by af introduces a potential in H − P:

N

S

I S : approx SUSY states of fermion # 0 (Morse index 0)

I N: approx SUSY states of fermion # 1 (Morse index 1)

Worldline Instantons

I propagate as a point particle:

N

S

I induce Q|S〉 ∼ |N〉I lift all the states that do not enter the classical Q-cohomology.

I Perturbative corrections lift some of the remaining states.

Worldsheet Instantons

I propagate in an essentially stringy way:

N

S

I violate fermion # by 2

I induce Q|N〉 ∼ |S〉I lift all the rest.

Summary

I Vanishing “theorem”I the chiral algebra is perturbatively ∞-dimI but vanishes nonperturbatively under certain conditions

I “Corollary” 1: supersymmetry is spontaneously brokenI ∞ many perturbative supersymmetric statesI Instantons pair up and lift all of them at once.

I “Corollary” 2: LX has no harmonic spinorsI Ric > 0 means the scalar curvature is positive on LX?I If so, this is the loop-space version of the Lichnerowicz

theorem.

I “Corollary” 3: The elliptic genus vanishesI vanishes already at the classical levelI Kahler case of the Hohn–Stolz conjecture

Future Research

I Extension of the vanishing “theorem”I All Kahler target spaces with c1(X ) > 0?I A proof: combination of perturbative analysis and RG flow?

I Sheaf-theoretic formulation of the exact chiral algebraI Perturbatively, defined on the 0-inst moduli space M0

∼= XI Nonperturbatively, defined on the whole inst moduli space M?I Impact on the study of (0, 2) mirror symmetry etc.

I Relation to 4d super Yang–Mills; compactifyI N = 2 SYM → (2, 2) model: Donaldson → chiral ring

[Bershadsky–Johansen–Sadov–Vafa]

I N = 1 SYM → (0, 2) model: ? → chiral algebra

I Applications to the geometric Langlands, pure spinorformalism, heterotic compactification, etc.