Crete 22/03/11 A Non-Compact Elliptic Genus Sujay Ashok Jan Troost 1004.3649 and CNRS Ecole Normale Superieure 1101.1059 with Sujay Ashok
The Plan Sketch of various Contexts Introduction of a Simple Model The calculation of a non-compact elliptic genus Lessons and open problems
Broader Context Holography is an important and general property of quantum gravity. We exploit well the holographic map between quantum gravity in anti-deSitter space and conformal field theories. Other non-compact space-times ? Applications are even broader: QCD, SQCD, CM, ..
Further Context A quantum theory of gravity : string theory Perturbative string theory : can be described in terms of two-dimensional conformal field theories on the worldsheet of the string Non-compact curved target spaces will lead to interacting two-dimensional conformal field theories with continuous spectrum
The string worldsheet is a Riemann surface in the euclidean. At one loop it is a torus. Large diffeomorphisms on the torus act as the modular group SL(2,Z) on the lattice defining the torus in the complex plane. Understanding the modular properties of the one-loop amplitude are a necessary ingredient to understanding string theory.
Spectrum of Conformal Dimensions Continuum Bound States Ground States
In a theory with continuous spectrum, a lot of interesting physics is related to the appearance of bound states. Example: D-branes encountering NS5-branes.
A Point In this talk, I wish to exhibit a first example in which the presence of bound states in the spectrum of an interacting conformal field theory with continuous spectrum is taken into account consistently with modularity of a one-loop vacuum amplitude. As a bonus, we will learn to interpret aspects of mock modular forms in terms of elementary physics.
Mock Modular Forms Ramanujan’s last letter to Hardy (1920) contained 17 “Mock Theta functions” with relations and properties. Dyson estimated that the realm of Mock Modular forms is big : Theta Functions Modular Forms Mock Theta Functions Mock Modular Forms
Andrews, Selberg, Watson, ... 2002 : Definition of Mock Modular Forms by Zwegers in his PhD thesis (under the guidance of Don Zagier). lead to a modular and non-holomorphic completion of Mock Modular Forms. Opened gate to number theoretic exploitation.
Even more context: Invariants of three-manifolds Characters of Super Lie Algebras D-brane bound state counting functions on non-compact Calabi-Yau manifolds
The Model We concentrate on a conformal field theory in two dimensions with two left-moving and two right-moving supersymmetries. It can either be thought off as a free field supplemented with an exponential superpotential or as a supersymmetric non-linear sigma-model on a cigar geometry.
The simple model: a non-compact supercoset a non-linear sigma-model with target group manifold SL(2,R), with Wess-Zumino term add fermions supersymmetrically and gauge a U(1) adjoint action SL(2,R)/U(1) Kazama-Suzuki model with two left- and two right-moving supersymmetries.
c = 3 + 6 It has central charge k where k is the level of the Wess-Zumino-Witten model.
A Cigar Target
There is a continuous spectrum of modes which travel in from infinity, reflect off the potential or off the tip of the cigar, with calculable reflection coefficient. They are in long multiplets.
There are bound states, localized inside the strong- coupling region. They are non-local winding states on the cigar. They are in short multiplets.
The elliptic genus A twisted toroidal partition sum: F q L 0 − c Trperiodic( − 1) F + ˜ ˜ 24 z J R L 0 − c 0 y Q 24 ¯ χ ( q, z, y ) = q left-moving fermion number F left-moving scaling dimension L 0 J R left-moving R-charge 0 Q angular momentum
Under the assumption of having a discrete spectrum, we can prove that the elliptic genus is holomorphic in the modular parameter: The right-moving supercharge commutes with the Hamiltonian. At non-zero right-moving conformal dimension, each state necessarily has a superpartner with opposite right-moving fermion number. The contribution of the pair of states is zero. There are then only contributions from right-moving ground states, which implies holomorphy.
Example: the elliptic genus of N=2 minimal models with central charge 3 - 6/k. Witten A path integral calculation gives: ∞ (1 − zq n )(1 − z − 1 q n ) 1 − z k − 2 � z − χ min( q, z, 1) = 2( k − 1) 1 1 1 k − 1 q n )(1 − z − k − 1 q n ) 1 − z (1 − z k − 1 n =1 and this agrees with the counting of left-moving characters that correspond to right-moving ground states. Identity between elliptic modular forms. Evidence for IR f.p. of LG model.
Some confusion in the non-compact realm : Proposal for a holomorphic (part of the) elliptic genus, with only right-moving ground state contributions, with anomalous modular properties : a mock modular form. Eguchi and Sugawara Mock modular forms are known to have non- holomorphic modular completions (real Jacobi forms). Zwegers / Zagier
The Difficulty / The Resolution The bulk partition function suffers from an infrared divergence due to the non-compact nature of the target space of the sigma-model. The symmetry group of the model is smaller than the volume of the target space manifold. No modular invariant regulator known for the (untwisted) bulk partition function.
Idea Firstly, think of the elliptic genus as an infrared regulator. It suppresses the contribution of the infinite volume by (tentatively) projecting onto short multiplets. Secondly, evaluate the elliptic genus through a path integral calculation, thus rendering its modular properties manifest
Evaluation of the path integral on the torus Main technical steps Identify U(1) R-symmetry / global symmetries Gauge fix and introduce ghosts Decouple bosons and fermions Use good coordinate choice on SL(2,R) Gawedzki Evaluate standard twisted partition functions for decoupled factors
q = e 2 π i τ Shape of the torus z = e 2 π i α U(1) R chemical potential y = e 2 π i β Global U(1) chemical potential � u = s 1 τ + s 2 Holonomies of the gauge field Current algebra levels κ = k − 2
2 π (Im u )2 √ k κ e τ 2 Z group = √ τ 2 τ , u − α τ , u − α � � � � k + β ¯ k + β θ 11 θ 11 � � √ k τ 2 | ( w + s 1 ) τ +( m + s 2 ) | 2 e − π k � Z co-exact = √ τ 2 | η ( τ ) | 2 m,n ∈ Z θ 11 ( τ , u − α ( k +1) τ , u − α + β ) θ 11 (¯ k + β ) Z ferm = 1 κ e − i 2 π s 1 α e − 2 π (Im u )2 k τ 2 | η ( τ ) | 2 Volume divergence Z ghost = τ 2 | η ( τ ) | 4 cancels against right-moving Fermion zeromode
The path integral is equal to: � − S susy coset χ = e
� 1 � χ ( τ , α , β ) = k ds 1 , 2 0 m,w ∈ Z θ 11 ( s 1 τ + s 2 − α k +1 + β , τ ) e 2 π i α w k θ 11 ( s 1 τ + s 2 − α k + β , τ ) τ 2 | ( m + s 2 )+( w + s 1 ) τ | 2 e − k π We stilll need to perform the finite dimensional integral over the holonomies.
The integral over one holonomy is trivial. The integration over the second holonomy is trivialized by the introduction of an integral over the radial momentum on the cigar.
Radial Momentum Complex Plane Im Path Integral Contour Bound States Re Continuous Spectrum
Pole / right-moving ground state contributions : q kw 2 q − w γ z 2 w − γ i θ 11 ( q, z ) k � � y γ − kw = χ hol η 3 1 − zq kw − γ w ∈ Z γ ∈ { 0 ,...,k − 1 } Integral over continuous real momentum : � + ∞− i ǫ ( − 1) m ds − 1 � = χ compl πη 3 2 is + n + kw −∞− i ǫ m,n,w ∈ Z ( m − 1 / 2)2 k + ( n − kw )2 k + ( n + kw )2 s 2 s 2 kw − n z m − 1 2 y n q ¯ q z q 2 4 k k 4 k
The path integral result is a mock modular form (which is a generalized Appell-Lerch sum), plus a modular completion. After integration over the radial momentum, this agrees with the formulas of Zwegers. Therefore, it is a rigorous fact that the twisted elliptic genus is a Jacobi form in three variables. Alternatively, we can read this off directly from the modular properties of the path integral expression.
Modular Properties of the elliptic genus Jacobi form χ ( τ + 1 , α , β ) = χ ( τ , α , β ) χ ( − 1 τ , α τ , β 3 α 2 / τ − 2 π i αβ / τ χ ( τ , α , β ) e π i c τ ) = c 3 k χ ( τ , α , β ) χ ( τ , α + k, β ) = ( − 1) 3 ( k 2 τ +2 k α ) e 2 π i β k χ ( τ , α , β ) c 3 k e − π i c χ ( τ , α + k τ , β ) = ( − 1) χ ( τ , α , β + 1) = χ ( τ , α , β ) e 2 π i α χ ( τ , α , β ) χ ( τ , α , β + τ ) =
Due to the cancellation of a volume divergence with a fermion zero mode, we can get non- holomorphic contributions to the elliptic genus. We want to obtain an even more elementary understanding of the non-holomorphic contributions.
When we concentrate on the right-movers we compute the Witten index of a supersymmetric quantum mechanics with a continuous part to its spectrum.
The Spectrum of Right-Movers Difference in Spectral Densities Unpaired Ground States Mock
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