Elliptic Genera of non-compact CFTs

Transcription

1 Elliptic Genera of non-compact CFTs Sujay K. Ashok IMSc, Chennai (work done with Jan Troost) Indian Strings Meeting, December 2012 arxiv: [hep-th] arxiv: [hep-th] arxiv:1212.xxxx [hep-th]

2 Introduction and motivation Asymptotically linear dilaton (ALD) spaces with non-trivial metric and dilaton. [Kiritsis, Kounnas, Lüst] ds 2 = g N(Y ) dy N 2 g N (Y ) (dψ + NA FS) 2 + 2Yds 2 CP N 1 Φ = NY k. Kähler manifolds: N = 2 CFTs with ĉ = N(1 + 2N k ). Tensored with minimal models, these are T (ψ+β) -dual to NS5 branes wrapped on CP N 1 [Hori-Kapustin] Aim: Study elliptic genera of worldsheet CFT, Result: N = 1 case (cigar metric): EG is a Jacobi form that is a modular completion of a mock-modular form. Work in progress: Elliptic genera for N > 1 via SQM.

3 Plan of the talk N = 1: A calculation of the elliptic genus using free fields Liouville theory: holomorphic but not modular [Troost 10] A path integral derivation: modular but non-holomorphic Spectral asymmetry as the cause for non-holomorphicity Applications Elliptic genera of products and orbifolds: mirror pairs of noncompact CY n. Higher dimensional generalizations.

4 Elliptic genus Definiton: χ(q, z) = Tr H ( 1) F z J 0 q L 0 c 24 q L0 c 24 (1) The trace projects onto right moving ground states. Consider Landau-Ginzburg models described by W (φ) with λw (φ) = W (λ ω φ). (2) The elliptic genus is also an index of the right-moving supercharge. It is unchanged if W ɛw, with ɛ 0. So use free fields. All one needs is the R-charge. [Witten 93] Alternately, as a sum over Ramond sector characters. These two calculations match [di-francesco and Yankielowicz]. Result: Modular form that was holomorphic.

5 Noncompact LG models: Liouville Theory Simplest example: super-liouville theory with Supersymmetry variations: k W = e 2 Φ. (3) δφ = ɛ ψ + + ɛ + ψ δψ + = ( )φ ɛ + ɛ + W δψ = ( 0 1 )φ ɛ + + ɛ W (4) The R-charges are inferred from the susy variations ψ + ψ + ψ e iα k ψ e 2 φ e iα k e 2 φ (5) φ = ρ + iθ. Charge is carried by the angular component ψ: e i θ 2k has charge 1 k. (6)

6 Liouville: free field calculation of EG Contribution from zero winding sector: χ L,0 (q, z) = iθ [ ] 11(q, z) 1 η 3 (q) 1 z 1 k Bosonic zero mode with charge 1 k. Fermion zero mode with unit charge. Bosonic and fermionic non-zero modes. Use spectral flow to obtain contributions from non-zero winding sectors. One uses just the asymptotic N = 2 SCFA [Murthy, 03]. T T + w J + c 6 w 2 (7) J J + c 6 w. (8) χ L,hol = iθ 11(q, z) η 3 (q) m Z z 2m q km2 1 z 1 k q m. (9)

7 Free field calculation continued... So far the discussion is parallel to that of the compact case where we would get χ MM = θ 11(q, z 1 1 k ) θ 11 (q, z 1 k ) (10) χ hol has a simple expression in terms of the twisted Ramond sector (extended) characters of an N = 2 superconformal theory with c = k. χ L,hol (q, z) = But, χ L,hol is NOT modular. k 1 2j 1=0 Ch R d (j, 1 ; q, z). (11) 2 χ L,hol ( 1 τ, α τ ) = e πiĉα2 τ χ L,hol (τ, α) + anomaly (12)

8 Path integral evaluation of elliptic genus Path integrals on genus 1 Riemann surface (x 1, x 2 ) such that x 1 x 1 + m and x 2 x 2 + n, with m, n integers. So we need to choose boundary conditions on the fields involved. χ(q, z) = Tr H ( 1) F z J 0 q L 0 c 24 q L0 c 24 (13) Here q = e 2πiτ and z = e 2πiα. H is an untwisted Hilbert space periodic boundary conditions along x 1. ( 1) F alone would have meant periodic boundary conditions for bosons and fermions along x 2. But extra insertion e 2πiαJ 0 means twisted boundary conditions for the fields. First construct EG of the Z k orbifold of (super-) Liouville. This is a coset CFT that describes a cigar type background: ( ) SL(2, R) U(1) k

9 Path integral for Gauged WZW model Write the coset as an axially gauged WZW model. The action is schematically given by S A = S g,wzw + S g,a + S f (ψ ±, A) (14) The elliptic genus is given by the path integral χ cos (q, z, y) = [Dg] d 2 u [DX DY ] Σ [Dψ ± D ψ ± ] e κsa(g,a) e S f (ψ ±, ψ ±,A), (15) TPC The gauge field integral is broken up into two scalar field integrals and a holonomy integral where u = s 1 τ + s 2. A z = X i Y ū 2τ 2, (16)

10 Path integral continued... It is possible to show that the path integral factorizes via change of variables, Polyakov-Wiegmann identities etc. [Eguchi-Sugawara, S.A-Troost 11] χ cos (τ, α) = d 2 u Z g (u, τ) Z Y (u, τ) Z f (u, τ) Z gh (τ). (17) Σ What makes the calculation possible is the exact answer for the gauged WZW model when the group is SL(2, R). [Gawedzki] The final answer is χ cos (τ, α) = k 1 ds 1,2 0 m,w Z Clearly not holomorphic. But modular. θ 11 (s 1 τ + s 2 α k+1 k, τ) θ 11 (s 1 τ + s 2 α k, τ) e 2πiαw e kπ τ 2 (m+s 2 )+(w+s 1 )τ 2. (18)

11 Properties of elliptic genus Modular and elliptic properties: χ cos (τ + 1, α) = χ cos (τ, α) χ cos ( 1 τ, α τ ) = eπi c α 2 3 τ χcos (τ, α) χ cos (τ, α + k) = ( 1) c 3 k χ cos (τ, α) χ cos (τ, α + kτ, β) = ( 1) c 3 k e πi c 3 (k2 τ+2kα) χ cos (τ, α). Jacobi form of index k2 c 6. How is this related to the holomorphic free field result we obtained for Liouville theory?

12 Holomorphic part of the elliptic genus Expanding the θ functions, introducing integral over radial momentum and integrating over the holonomies one can show that where χ cos,hol = iθ 11(τ, α) η 3 χ cos,rem = 1 πη 3 χ cos = χ cos,hol + χ cos,rem. (19) m,n,w γ {0,...,k 1} + iɛ iɛ Compare with Liouville answer: χ L,hol = iθ 11(q, z) η 3 (q) w z 2w γ k qkw2 q wγ 1 zq kw γ ( 1) m ds 2is + n + kw q (m 1/2) q s2 k + (n kw)2 4k m Z z 2m q km2 1 z 1 k q m 2 2 z m 1 2 z kw n k q s2 k + (n+kw)2 4k.

13 Orbifolds Doing a Z k orbifold of the coset elliptic genus we recover the elliptic genus of Liouville theory: χ cos,orb = χ L (20) What we have is a Jacobi form written down by Zwegers χ L = iθ 11(q, z) η(q) 3 Â 2k (z 1 k, z 2 ; q) (21) The path integral therefore provides the modular completion of the free field calculation. Clearly we missed some states in the free field counting. Where do they come from? What is the source of the non-holomorphic contribution?

14 Spectral asymmetry Consier the remainder function once again: χ cos,rem = 1 πη 3 p,n,w + iɛ iɛ ( 1) p ds 2is + n + kw q (p 1/2) q s2 k + (n kw)2 4k 2 2 z p 1 2 z kw n k q s2 k + (n+kw)2 4k. (22) This arises from the continuum of states in the theory (noncompactness). On right-movers, the elliptic genus reduces to an index. We can obtain a contribution to the index from a continuum of modes due to a mismatch in the spectral density of boson and fermions. [Comtet-Akhoury] The relative spectral density between the two right-moving Ramond sectors can be read off from the ratio of reflection amplitudes in these sectors.

15 Spectral asymmetry continued... The spectral asymmetry in the two right-moving Ramond sectors is given by ρ(s) = 1 d R+ log 2πi ds R = 1 ( 1 2π is m 1 ).(23) is + m This is a spectral measure on the half-line s [0, [. For even functions the measure on the full line becomes 1 1 2π is + m The origin of the remainder contribution is therefore a mismatch in the spectral density of right-moving bosons and right-moving fermions. (24)

16 Summary so far ( ) We calculated the elliptic genus of the coset SL(2,R) U(1) and k via orbifolding, that of super Liouville theory. [S.A, Troost 11] The elliptic genus is modular. However it is not holomorphic. The holomorphic piece is captured by a free field analysis. To find the modular completion we used the path integral. Exact evaluation possible because the SL(2, R) component already done. [Gawedzki, Ray-Singer] The remainder term arises from the continuum of states; the spectral asymmetry between the bosons and fermions in the continuum feeds into the witten index calculation. What s next?

17 Tensor products and orbifolds Tensor these together with minimal models, do a GSO orbifold to calculate elliptic genus of noncompact Calabi-Yau ( X k + Y k) /Z k c = 6 ( ) X k + Y1 2k + Y2 2k /Z 2k c = 9 ( X1 2k + X2 2k + Y k) /Z 2k c = 9 (25) Constructive way to find noncompact mirror pairs using/generalzing Greene-Plesser techniques by orbifolding by subgroups of GSO orbifold group. [S.A-Troost 12]

18 Higher dimensional generalizations Let us return to the higher dimensional KKL backgrounds: ds 2 = g N(Y ) dy N 2 g N (Y ) (dψ + NA FS) 2 + 2Yds 2 CP N 1 Φ = NY k. What is the elliptic genus of these worldsheet theories? Path integral is hard to do; no known exact CFT description. q 0 limit of elliptic genus is the Dirac index on the given space. For Liouville: mock part obtained by spectral flow on zero-winding sector.

19 Supersymmetric quantum mechanics Use isometry ψ and Scherk-Schwarz compactify: super quantum mechanics with supercharge Q γ µ (D µ K µ ) (26) Example: Taub-NUT Taub-NUT with self-dual gauge field. Problem reduces to calculating Dirac zero modes of wound strings in these (regularized) KKL backgrounds. [S.A., Nampuri, Troost] We get an index χ(y 1, y 2 ) = w n<m(w) n=1 w Choose weights, spectral flow and... 1 n= M(w) d(n)y n 1 y w 2 (27)

20 Work in progress and future directions For KKL2(=cigar) we recover χ hol from the quantum mechanics by choosing appropriate weights. For higher N, we hope to obtain a higher-pole mock-modular forms. e.g. double pole mock-modular form obtained by Dabholkar-Murthy-Zagier. The non-holomorphic piece should follow if we can recover the spectral asymmetry. Cigar/Liouville can be realized as moduli space of domain walls in 3d gauge theories. [Tong] Higher dimensional generalizations? If yes Dirac zero modes count domain wall bound states. These spaces have a GLSM realization [Hori-Kapustin]. Can we do a path integral in that formulation?