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Surface Operators and W-Algebras Surface operators and W-algebras Yuji Tachikawa (IPMU&IAS) based on various papers with the help of various people Most recently with H. Kanno, [1105.0357] May 2011 May 2011 1 / 33 4d instantons , W-algebras. May 2011 2 / 33 What is the phenomenon? (no string theory necessary) Why does it happen? (string theory provides the explanation ) How should we generalize? (W-algebras come back with vengeance.) May 2011 3 / 33 Contents 1. What is the phenomenon? 2. Why does it happen? 3. How should we generalize? May 2011 4 / 33 Virasoro coherent state • Consider the Virasoro algebra c 3 [Lm;Ln] = (m − n)Lm+n + δm;−n(m − m) 12 • and the Verma module j i j i 2 j i j i 3 j i ∆ ;L−1 ∆ ;L−1 ∆ ;L−2 ∆ ;L−1 ∆ ;::: j i • ∆ annihilated by all annihilation operators: L0j∆i = ∆j∆i;Lnj∆i = 0; n > 0 May 2011 5 / 33 • We’d like to consider a nice coherent state of the Virasoro algebra: L1jq; ∆i = qjq; ∆i;Ln≥2jq; ∆i = 0; normalized to have jq; ∆i = j∆i + descendants: qL−1 j i • It’s not e ∆ , but can be determined order by order: q jq; ∆i = j∆i + L−1j∆i + ··· 2∆ with the norm q2 q4(c + 8∆) hq; ∆jq; ∆i = 1 + + + ··· 2∆ 4∆((1 + 2∆)c − 10∆ + 16∆2) May 2011 6 / 33 Instantons • Consider an instanton F = dA + A ^ A; F = − ?F where A is an (2) gauge field on R4 with SU Z 1 k = F ^ F : positive integer 8π2 tr • The space of parameters : M2;k, dimR = 8k Position Gauge Dir. Position Gauge Dir. Size Size May 2011 7 / 33 • Consider a toy statistical mechanical model of instantons: X Z Z = Λ4k dvol k M2;k R4 • Of course it diverges due to the volume of . Need to put the system into a “box”. May 2011 8 / 33 • Consider a very stupid integral Z 1 Z 1 R2 Area of = dx1 dx2: x2 −∞ −∞ • Regularize it via Z Z 1 1 x −πϵ(x2+x2) 1 1 dx1 dx2 e 1 2 = : −∞ −∞ ϵ f g • Let x1; x2 P.B. = 1. 2 2 • H = x1 + x2 is the rotation generator • ϵ is the chemical potential. May 2011 9 / 33 R4 f g f g • On , let x1; x2 P.B. = x3; x4 P.B. = 1. • M2;k inherits the Poisson bracket. • Introduce chemical potentials / Hamiltonians ϵ1 H1 for the rotation of x1-x2 plane, ϵ2 H2 for the rotation of x3-x4 plane, 3 a J for the σ of the global SU(2) rotation. • Our toy model is then X Z 4k −π(ϵ1H1+ϵ2H2+aJ) Z = Λ Zk where Zk = e dvol: k M2;k • It’s just an ensemble of instantons with chemical potentials for all the conserved quantities. • Exactly calculable. [Moore-Nekrasov-Shatashvili] May 2011 10 / 33 Statistical mechanical toy model of 4d instantons We consider X Z 4k −π(ϵ1H1+ϵ2H2+aJ) Z(Λ; ϵ1; ϵ2; a) = Λ e dvol k M2;k Explicitly, we have 2 Z = 1 + Λ4 + 2 2 ϵ1ϵ2((ϵ1 + ϵ2) − 4a ) 2 2 2 2 ((ϵ1 + ϵ2) + ϵ1ϵ2/8 − a )/(8ϵ ϵ ) + Λ8 1 2 + ··· ϵ1+ϵ2 2 − 2 ϵ2 2 − 2 ϵ1 2 − 2 (( 2 ) a )((ϵ1 + 2 ) a )(( 2 + ϵ2) a ) May 2011 11 / 33 Norm of Virasoro coherent state We take L1jq; ∆i = qjq; ∆i;Ln≥2jq; ∆i = 0; normalized to have jq; ∆i = j∆i + descendants: The norm was q2 q4(c + 8∆) chq; ∆jq; ∆ic = 1 + + + ··· 2∆ 4∆((1 + 2∆)c − 10∆ + 16∆2) May 2011 12 / 33 There’s an equality [Alday-Gaiotto-YT], [Gaiotto] chq; ∆jq; ∆ic = Z(Λ; ϵ1; ϵ2; a) under the identification 2 (ϵ1 + ϵ2) c = 1 + 6 ; ϵ1ϵ2 2 2 (ϵ1 + ϵ2) a ∆ = − ; 4ϵ1ϵ2 ϵ1ϵ2 Λ2 q = : ϵ1ϵ2 May 2011 13 / 33 h j i • Note that it’s not that strange that Z = : X Z − Z = Λ4k e π(ϵ1H1+ϵ2H2+aJ)dvol k M2;k 4 8 = 1 + Λ h 1j 1i + Λ h 2j 2i + ··· where π − (ϵ1H1(X1;:::;X8k)+··· ) k(X1;:::;X8k) = exp 2 is a wavefunction in H(M2;k) : the Hilbert sp.of QM on M2;k. • Therefore, 2 Z = h j i where j i = j 0i ⊕ Λ j 1i ⊕ · · · May 2011 14 / 33 • So, to have hq; ∆jq; ∆i = Z(Λ; ϵ1; ϵ2; a) = h j i; we just need jq; ∆i = j i: $ • It’s just like Heisenberg’s matrix mechanics Schrödinger’s wave mechanics. • We need Dirac’s transformation theory. May 2011 15 / 33 Contents 1. What is the phenomenon? 2. Why does it happen? 3. How should we generalize? May 2011 16 / 33 IIA realization and M-theory lift • Witten’s IIA construction. N • 4d = 2 SU(2) theory. 2 D4s 4 • At constant ‘time’, R × the BPS config. is instantons. ⊕ ∗ A BPS state is in kHG(M2;k). NS5 NS5 • NS5-branes at the start/end of ‘time’ prepare the same state j i. Consider this direction as the time! h j i • So Z = . May 2011 17 / 33 • Lift the system to M-theory. 2 M5s 4 M-theory × circle R1,2 Consider this direction as the time! • We get some 2d theory. This has Virasoro symmetry. ⊕ ∗ • That’s why kHG(M2;k) becomes Virasoro Verma module. ∼ • L0 KK momenta along the M-theory circle. • which are D0s in the IIA description • which are instantons on D4s! May 2011 18 / 33 Consider this direction as the time! Consider this direction as the time! 4 originally, × 2 D4s R M-theory circle NS5 NS5 At "constant time", we have instantons At "constant time", we have the Virasoro Verma module May 2011 19 / 33 • Why do we have coherent states at the tips? • M5-branes are wrapped on the Seiberg-Witten curve, given by 2 2 2 2 Λ dz λ − ϕ2(z) = 0; ϕ2(z) = (Λ z + u + ) : z z2 h i 2 ! • The general rule is that ϵ1ϵ2 T (z) dz ϕ2(z): • Recall L0 L1 T (z) ∼ · · · + + + ··· : z2 z3 Then we see u Λ2 L0j i ∼ j i;L1j i ∼ j i: ϵ1ϵ2 ϵ1ϵ2 May 2011 20 / 33 Contents 1. What is the phenomenon? 2. Why does it happen? 3. How should we generalize? May 2011 21 / 33 SU(2) Consider this direction as the time! Consider this direction as the time! 4 originally, × 2 D4s R M-theory circle NS5 NS5 At "constant time", we have instantons At "constant time", we have the Virasoro Verma module May 2011 22 / 33 SU(N) Consider this direction as the time! Consider this direction as the time! 4 originally, × N D4s R M-theory circle NS5 NS5 At "constant time", we have instantons At "constant time", we have the W-algebra Verma module May 2011 23 / 33 • For SU(N), the statistical-mechanical model is Z X P − Z = Λ2Nk e π(ϵ1H1+ϵ2H2+ aiJi)dvol k MN;k where MN;k : the moduli space of k instantons of SU(N) (ai): chemical potentials for SU(N) (Ji): Hamiltonians for SU(N) May 2011 24 / 33 ^ • Virasoro algebra = W2 = W (SU(2)) had T (z) corresponding to the Seiberg-Witten curve 2 λ + ϕ2(z) = 0 ^ • WN = W (SU(N)) algebra has W2(z);W3(z);:::;WN (z) corresponding to the Seiberg-Witten curve N N−2 λ + ϕ2(z)λ + ··· + ϕN (z) = 0 May 2011 25 / 33 • The instanton partition function Z X P − Z = Λ2Nk e π(ϵ1H1+ϵ2H2+ aiJi)dvol k MN;k = h j i is the norm of the coherent state j i : ΛN W j i ∼ j i N;1 N/2 (ϵ1ϵ2) • For N = 3, checked by [Taki]. For N ≥ 4, not done yet. I don’t know why... • Rumored that [Maulik-Okounkov] have a geometric proof. May 2011 26 / 33 More general W-algebras There are more general W -algebras [de Boer-Tjin ’93] ^ ··· W (SU(N); [n1; : : : ; nM ]) where n1 + + nM = N with the special cases ^ WN = W (SU(N); [N]); ^ ^ SU(N) = W (SU(N); [1;:::; 1]) May 2011 27 / 33 z }|m { z }|n { ^ • E.g. W (SU(N); [2;:::; 2; 1;:::; 1]) has a i spin one Jb (z); jj(z); a ~i spin 3/2 Gi (z); Ga(z) a spin 2 Sb (z) where a; b = 1; : : : ; m and i; j = 1; : : : ; n. • W ([2; 1]) is called Bershadsky-Polyakov algebra; W ([2; 1;:::; 1]) is Romans’ quasisuperconformal algebra. (both 1991) • Do they appear in 4d gauge theory? May 2011 28 / 33 • The answer is Yes! [Braverman-Feigin-Finkelberg-Rybnikov], [Wyllard], [Kanno-YT] • Consider N D4-branes on Rt × Cz × (Cw/ZM ) where ZM acts on the Chan-Paton indices as 2 2 M M (|!; :{z : : ; !}; |! ;:::;!{z };:::; |! ;:::;!{z }) n1 n2 nM • Bulk D0s + fractional D0s bound at the surface operator w = 0 May 2011 29 / 33 • As before, the BPS part of the dynamics reduces to susy QM on the (fractional) instantons, with the Hilbert space ∗ H = H (moduli space of instantons on Cz × (Cw/ZM )) • This equals the Verma module of the W-algebra ^ W (SU(N); [n1; : : : ; nM ]) • Moreover, the instanton partition function = the norm of the coherent state • (only explicitly checked for W ([2;:::; 2; 1;:::; 1]), though) May 2011 30 / 33 Summary Consider this direction as the time! Consider this direction as the time! ZM orbifold N D4s M-theory circle At "constant time", we have instantons At "constant time", we have the general W-algebra Verma module May 2011 31 / 33 • WN -algebra reappeared also in the context of higher-spin AdS3/CFT2 last year.
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