Scattering in Large N Matter Chern Simons Theory
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Scattering in large N matter Chern Simons Theory Shiraz Minwalla Department of Theoretical Physics Tata Institute of Fundamental Research, Mumbai. Developments in M-theory, Korea Jan 15 Shiraz Minwalla References S. Jain, M. Manglik, S. M, S. Wadia, S. Yokoyama : ArXiv 1404.6373 Y. Dandekar, M. Manglik, S.M. ArXiv 1407.1322 K. Inbasekar, S. Jain, S. Mazumdar, S.M. V. Umesh, Y. Yokoyama: in progress Y. Dandekar, M. Manglik, S.M. : in progress Shiraz Minwalla Introduction The last 20 years has seen the discovery of many strong weak coupling dualities in supersymmetric quantum field theories in 2, 3 and 4 dimensions. It thus seems clear that field theory-field theory dualities are common in the space of susy quantum field theories. Shiraz Minwalla Introduction Why have all dualities discovered over the last 20 years been supersymmetric? Exp 1: Susy is an essential ingredient in field theory field theory dualities, atleast in dimensions greater that two. Exp 2: Field theory - field theory dualities are common in the space of all quauntum field theories. Dualities have been discovered only in susy qfts as these are the only theories in which we can do exact computations. How to distinguish 1 from 2? One strategy : find a context in which exact computations are possible at strong coupling without susy. If you find a lot of duality, suggests 2 is the correct explanation. One example : this talk Shiraz Minwalla Chern Simons Theory CS theory. Gauge theory in 3 dimensions. κ Z 2 S = Tr AdA + A3 + S 4π 3 matter Note that: Equation of motion κF = J. No local degrees of freedom in the gauge field. Action is gauge invariant even though Lagrangian is not. This requires κ = integer. N Effective coupling constant λ = κ . Dimensionless. Discrete. Cannot be continuously renormalized. Consequence no running of the gauge coupling. Easy to construct superconformal field theories without susy. Effective fixed lines in the large N t’ Hooft limit. Shiraz Minwalla Chern Simons Theory: Notation We pause to define notation. The integer behind the Chern Simons action, κ depends on regulation scheme and needs a clear definition. Throughout this talk we will use the symbol k for the level of the WZW theory obtained in studying Chern Simons theory on a manifold with a boundary. κ = k + sgn(k)N Under level rank duality duality of pure Chern Simons theory, k 0 = −sgn(k)N; N` = jkj This implies κ0 = −κ, N0 = jκj − N. In other words λ0 = λ − sgn(λ) Shiraz Minwalla Chern Simons theory coupled to fermionic matter Simple example of conformal theories. Z ¯ µ Smatter = (Dµγ + mF ) where is a fermion in any representation of the gauge group. Atleast at small enough λF this theory is conformal at mF = 0 in dim reg. Proof. Gauge coupling cant run. No other relevant or marginal operator. This talk. Fermion taken to be in the fundamental representation. Will be able to solve theory at all λF at large N because effectively vector model (no dof in gauge field) Shiraz Minwalla Chern Simons - Bosonic matter theories The second theory we study in this talk is the Chern Simons theory coupled to bosonic matter Z ¯ µ 2 ¯ 1 ¯ 2 DµφD φ + mBφφ + b4(φφ) 2NB We will be particularly interested in a scaling limit of the bosonic theory. To explain this we rewrite the theory as Z 2! ¯ µ 2 ¯ b4 ¯ 2 NB b4 ¯ 2 DµφD φ + mBφφ + (φφ) − σ − φφ − mB 2NB 2b4 NB Z 2 2 ! ¯ µ ¯ mB σ DµφD φ + σφφ + NB σ − NB b4 2b4 (1) Shiraz Minwalla Chern Simons coupled to critical bosons The so called critical limit of the bosonic theory is defined by the limit 2 4πmB cri b4 ! 1; mB ! 1; = mB = fixed: (2) b4 In this limit Z mcri S = D φ¯Dµφ + σφφ¯ + N B σ µ B 4π Atleast in the large N limit, this theory is also conformal at cri cri mB = 0. mB = 0 is a relevant deformation of this critical theory. Shiraz Minwalla Conjectured Duality It has been conjectured that the Fermionic Chern Simons theory is dual in the large N limti to the critical boson Chern Simons theory under the duality map κF = −κB; NF = jκBj − NB; (3) λB = λF − sgn(λF ); cri mF = −mB λB: The duality map is stated in the t’ Hooft large N limit and could receive subleading corrections at finite N and κ. Note that the map for the t’ Hooft coupling is precisely that of level rank duality. Prior to this talk: three kinds of concrete evidence. Shiraz Minwalla Evidence for duality cri Specialize to mF = m = 0. Has been shown that the spectrum of single trace operators and their three point functions match between the two sides. At arbitrary mF the thermal partition function of both 2 theoriesp has been computed on an S at temperatures or order N. Highly nontrivial. Undergo two phase transitions as function of temperature. Match exactly between the two sides. Duality conjecture generalized to general theory of bosons and fermions interaction with CS. 7 continuous parameters. Duality map determined. Leaves N = 2 and N = 1 submanifolds fixed. Giveon Kutasov type (really Benini et al) duality N = 2. Continuous deformation from known susy duality to purely bose-fermi. Shiraz Minwalla Scattering amplitudes and bosonization Bose-Fermi or bosonization duality. Sounds interesting. Would like to understand in more detail. Best case: bosonisation formula like = eφ. Too much to ask for in current context as and φ are not gauge invariant. While arbitrary insertions of and φ are not meaningful, insertions that are taken to infinity along lines of equations of motion - i.e. S matrices - are well defined. For this reason the map between S matrices is a ‘poor mans bosonization’ map between these two theories. This talk: We indpependently compute the S matrix for the bosons and fermions, and examine their interrelationship under duality. Also uncover interesting structural features along the way. Shiraz Minwalla Scattering Amplitudes: Colour Kinematics In either the bosonic or fermionic theory we refer to a quantum the transforms in the fundamental of U(N) as a particle and a quantum that tranforms in the antifundamental of U(N) as an antiparticle. We study the most general 2 × 2 scattering process. There are three such processes Pi + Pm ! Pj + Pn j n Pi + A ! Pn + A (4) Aj + An ! Ai + Aj All these S matrices are contained in appropriate on shell limits of the single correlator. In e.g. the bosonic theory this is ¯j ¯n < φi φmφ φ > Shiraz Minwalla Color Kinematics It follows from U(N) invariance that ¯j ¯n j n n j < φi φmφ φ >= aδi δm + bδi δm Consequently, in each channel we need to compute 2 distinct scattering amplitudes. For particle-particle scattering it is convenient to work in the following basis for U(N) singlets ¯j ¯n j n n j j n n j < φi φmφ φ >= Tsym δi δm + δi δm + Tas δi δm − δi δm a+b TSym = 2 is the S matrix in the channel of symmetric a−b exchange, while Tas = 2 is the S matrix in the channel of antisymmetric exchange. Shiraz Minwalla 1 Colour Kinematics, N On the other hand for particle - antiparticle exchange the following basis is most useful ! δj δn δj δn < φ φ φ¯j φ¯n >= T i m + T δnδj − i m i m Sing N Adj i m N TSing = Na + b is the scattering matrix in the singlet channel, while TAdj = b is the scattering matrix in the adjoint channel. It is not difficult to convince oneself that, at leading order in 1 the N expansion 1 a ∼ b ∼ T ∼ T ∼ T ∼ sym as Adj N On the other hand SSing ∼ O(1) Shiraz Minwalla 1 Unitarity and N We have chosen the colour structures multiplying TSing etc to be simply projectors onto the exchange representations. As projectors square to unity it follows from unitarity that y y Tc − Tc = TcTc Tc is any one of the four T matrices described above. We have used the fact that e.g. 2 ! 4 production contributes 1 to unitarity only at subleading order in N . The unitarity equation is rather trivial for Tsym, Tas, Tadj , as 1 the RHS is subleading in N . However it imposes a highly nontrivial nonlinear constraint on Tsing, the most nontrivial of the four scattering matrices. Shiraz Minwalla Non Relativistic Limit The S matrix for non relativistic particles interacting via Chern Simons exchange was worked out in the early 90s, most notably by Bak, Jackiw and collaborators. The main result is strikingly simple. Consider the scattering of two particles in representations R1 and R2, in exchange channel R. It was demonstrated that the S matrix equals the scattering matrix of a U(1) charged particle of unit charge scattering off a point like flux tube of magnetic field c2(R1)+c2(R2)−c2(R) strength ν = κ . This quantum mechanical S matrix was computed originally by Aharonov and Bohm and generalized by Bak and Camillio to take account of possible point like interactions between the scattering particles. Shiraz Minwalla Non Relativistic Scattering Amplitude θ T = −16πic (cos (πν) − 1) δ(θ) + 8ic sin(πν)Pv cot NR B B 2 1 + eiπjνj ANR k 2jνj + 8cBj sin πνj ; 1 − eiπjνj ANR k 2jνj −1 2 2jνj Γ(1 + jνj) A = : NR w R Γ(1 − jνj) (5) Here wRj2νj is a measure of the strength of the Bak-Camillio contact interaction between the scattering j particles. In the limit w(kR) 2νj ! 0 ANR ! 1 and the second line of the scattering amplitude simplifies to 8cBj sin πνj; the original Aharonov Bohm result Shiraz Minwalla The δ function and its physical interpretation The non relativistic amplitude above has a very unusual feature, a piece in the scattering amplitude proportional to the δ(θ).