Twisted Hilbert Space of 3d Supersymmetric Gauge Theories Mathew - PowerPoint PPT Presentation

Twisted Hilbert Space of 3d Supersymmetric Gauge Theories Mathew Bullimore & Andrea Ferrari Supersymmetric Theories, Dualities and Deformations, July 2018

Introduction 1 I will talk about quantum field theories with 3d N = 2 supersymmetry and an unbroken R-symmetry. C × ◮ Perform topological twist on C using R-symmetry. ◮ Supersymmetry algebra { Q, ¯ Q } = H − m f · J f where H is hamiltonian and J f is flavour charge. ◮ m f is real mass parameter for flavour symmetry.

Introduction I want to explain how to compute the ‘twisted Hilbert space’ H of supersymmetric ground states annihilated by Q , ¯ Q . C × ◮ The supersymmetric ground states are graded by ( − 1) F and J f . ◮ It should reproduce the twisted index on S 1 × C , I = Tr H ( − 1) F x J f . ◮ The latter can be computed by supersymmetric localisation. 1 1 Benini-Zaffaroni 2 , Closset-Kim

Introduction Why is the twisted Hilbert space is a richer observable? 1. There may be cancelations when computing the trace, � g � g g � � ∧ j ( C g ) ( − 1) j − → = 0 . j j =0 j =1 2. The supersymmetric ground states may exhibit ‘wall-crossing’  � ∞  x j C  | x | < 1  j =0 H = , � ∞   x − j − 1 C − | x | > 1  j =0 whereas the supersymmetric index does not  � ∞  x j  | x | < 1  1 j =0 → = . � ∞  1 − x x − j − 1  − | x | > 1  j =0

Introduction 3. Turn on a superpotential W ( u ) depending on complex parameters u . ◮ The twisted index is independent of u . ◮ The twisted Hilbert space is a holomorphic sheaf on the parameter space of u ’s. 4. Turn on holomorphic line bundle for a U (1) flavour symmetry on C . ◮ The twisted index depends only on the flux � 1 F = m ∈ Z 2 π C ◮ The twisted Hilbert space is a holomorphic sheaf on the parameter space Pic m ( C ) .

Strategy Introduce an ‘effective’ supersymmetric quantum mechanics that exactly captures supersymmetric ground states. × C R R The supersymmetric quantum mechanics is of type N = (0 , 2) . ◮ Part 1: Supersymmetric Quantum Mechanics. ◮ Part 2: Three-dimensional N = 2 Theories.

Part 1: Supersymmetric Quantum Mechanics

Supersymmetry Algebra The supersymmetry algebra is { Q , Q } = 0 { Q , ¯ Q } = H − m f · J f { ¯ Q , ¯ Q } = 0 . ◮ Flavour symmetry G f with charge operator J f . ◮ Real mass parameter m f ∈ t f . Hilbert space of supersymmetric ground states H , ¯ Q | ψ � = 0 Q | ψ � = 0 . ◮ Graded by fermion number ( − 1) F and flavour charge J f . ◮ If spectrum is gapped, equivalent to cohomology of ¯ Q .

Chiral Multiplet Chiral multiplet ( φ, ψ ) with flavour symmetry G f = U (1) and associated mass parameter m f ∈ R . � � � � − ∂ + ∂ ∂φ + m f ¯ Q = ¯ ¯ Q = ψ φ ψ φ + m f φ . ∂ ¯ ◮ The supercharges and H − m f J f are unambiguous. ◮ There is a normal ordering ambiguity, H → H + αm J f → J f + α . ◮ This is choice of background supersymmetric Chern-Simons term for G f = U (1) flavour symmetry.

Chiral Multiplet: Hilbert Space The supersymmetric ground states are c − c + m f < 0 m f > 0 e m f | φ | 2 ¯ φ j ¯ e − m f | φ | 2 φ j ψ ◮ Introduce parameter x ∈ C ∗ to keep track of flavour charge. ◮ The supersymmetric ground states depends on the chamber,  ∞ �   x α + 1 x j C  m f > 0  2  j =0 H = . ∞ �   − x α − 1 x − j C  m f < 0  2  j =0 ◮ Wall-crossing at m f = 0 where spectrum not gapped.

Chiral Multiplet: Index The supersymmetric index is  ∞ �   x α + 1 x j  m f > 0  2  j =0 I = . ∞ �   − x α − 1 x − j  m f < 0  2  j =0 ◮ Path integral construction identifies x = e − 2 πβ ( m f + iA f ) . ◮ These are expansions of the same rational function x α + 1 2 1 − x in each chamber. ◮ Simple pole at m f = 0 where spectrum not gapped.

Gauge Theory Example 2 : ◮ U (1) vectormultiplet ( A τ , σ, λ, ¯ λ, D ) . ◮ N chiral multiplets ( φ 1 , . . . , φ N ) of charge +1 . ◮ Real FI parameter ζ > 0 : L FI = − ζD . ◮ Supersymmetric Wilson line of charge q : L WL = q ( A τ + σ ) . Global anomaly cancellation: q − N 2 ∈ Z . ( I will assume q − N 2 ≥ 0 . ) Flavour symmetry G f = PSU ( N ) . 2 Hori-Kim-Yi

Gauge Theory: Sigma Model Description � � | σφ j | 2 + e 2 | φ j | 2 − ζ ) 2 Classical potential: U = 2 ( . j j Supersymmetric ground states captured by a sigma model to � N � � | φ j | 2 = ζ /U (1) = CP N − 1 . M = j =1 ◮ Supersymmetric Wilson line generates line bundle O ( q ) . ◮ Quantization of fermions contributes K 1 / 2 = O ( − N 2 ) . M ◮ Combination F = O ( q − N 2 ) . The wavefunctions are smooth sections of � Ω 0 , ∗ ( M ) ⊗ F � α, β � = α ∧ ∗ β . ¯

Gauge Theory: Hilbert Space Turning on mass parameters m f = ( m 1 , . . . , m N ) , ∂ † e − h f Q = e h f ¯ Q = e − h f ¯ ¯ ∂ e h f where h f = m f · µ f is moment map for infinitesimal G f = PSU ( N ) transformation generated by m f . ◮ Spectrum is always gapped as target space compact. ◮ Setting m f = 0 find symmetric tensor representation of G f , H = H 0 , • ∂ ( M, F ) ¯ = S q − N 2 ( x 1 C ⊕ · · · ⊕ x N C ) . ◮ Supersymmetric index is character of this representation, I = χ S q − N 2 C N ( x 1 , . . . , x N ) .

Geometric Model A massive supersymmetric sigma model specified by: ◮ A K¨ ahler manifold M with isometry group G f . ◮ A G f -equivariant Z 2 -graded hermitian vector bundle F with odd differential δ : F → F . ◮ Real mass parameters m f ∈ t f . The wavefunctions are smooth sections of Ω 0 , • ( M ) ⊗ F with hermitian inner product � � α, β � = α ∧ ∗ β . ¯ M

Geometric Model Supercharges are conjugated Dolbeault operators, Q = e h f ¯ ∂ † F e − h f + δ † Q = e − h f ¯ ∂ F e h f + δ . ¯ where h = m f · µ f is moment map for infinitesimal G f transformation generated by m f . ◮ Supersymmetric ground states, H = H 0 , • Q ( M, F ) . ¯ ◮ If M is compact, spectrum is always gapped and H independent of m f ∈ t f . ◮ If M is non-compact, H exhibits ’wall-crossing’ across loci where fixed point set of m f ∈ t f is not-compact and spectrum not gapped.

Part 2: Supersymmetric Theories in Three Dimensions

3d N = 2 Supersymmetry Supersymmetry algebra, { Q α , ¯ Q β } = P αβ + ( m f · J f ) ǫ αβ . × R C R ◮ Topological twist on C using U (1) R-symmetry. ◮ Preserves supersymmetric quantum mechanics of type N = (0 , 2) , { Q, ¯ Q } = H − m f · J f .

Chiral Multiplet Chiral multiplet ( φ, ψ α , F ) of R -charge r . Decompose into supersymmetric quantum mechanics multiplets ◮ Chiral multiplet ( φ, ψ ) in section of K r/ 2 ⊗ L f C ◮ Fermi multiplet ( η, F ) in (0 , 1) -form section of K r/ 2 ⊗ L f C where L f is a holomorphic line bundle of degree m f on C associated to U (1) f flavour symmetry. An E -term superpotential E = ¯ Dφ generates kinetic terms along C , � � η ∂E | E | 2 = � ¯ Dφ � 2 η ∧ ¯ ¯ ∂φ ψ = ¯ Dψ . C C

Chiral Multiplet: Hilbert Space Minimize classical potential: ¯ Dφ = 0 . Fluctuations: ◮ Chiral multiplets in H 0 ( K r/ 2 ⊗ L f ) : φ 1 , . . . , φ n C C ◮ Fermi multiplets in H 1 ( K r/ 2 ⊗ L f ) : η 1 , . . . , η n F C ◮ Riemann-Roch: n C − n F = ( r − 1)( g − 1) + m f Quantizing in the chamber m f > 0 , we find ∞ � x j � nC − nF S p ( C n C ) ⊗ ∧ q ( C n F ) . H = x 2 j =0 p + q = j ◮ Depends on L f through individual numbers n C and n F . ◮ ( Can be promoted to sheaf of graded vector spaces on parameter space Pic m f ( C ) of L f . )

Chiral Multiplet: Index The twisted supersymmetric index is computed from trace, � n C − n F + j − 1 � ∞ � nC − nF x j I = x 2 n C − n F j =0 � x 1 / 2 � n C − n F = , 1 − x in agreement with 1-loop determinant from supersymmetric localisation. 3 ◮ Twisted supersymmetric index depends only on the difference n C − n F = ( r − 1)( g − 1) + m f . ◮ It is therefore constant as L f varies in parameter space Pic m f ( C ) . 3 [Benini-Zaffaroni 2 ,Closset-Kim]

Vectormultiplet Three-dimensional vectormultiplet decomposes into the following 1d N = (0 , 2) supermultiplets: ◮ A vectormultiplet for the group of gauge transformations g : C → U (1) with auxiliary field D 1 d = D + ∗ C F . ◮ An adjoint chiral multiplet with complex scalar ¯ D ¯ z . In addition: ◮ A 3d FI parameter ζ contributes � � − ζ D = − ζ D 1 d + 2 πζ m C C ◮ A 3d CS term contributes a supersymmetric Wilson line � k ( σ + iA τ ) F 2 π C

Example: U (1) 1 / 2 + 1 Chiral Consider the following model: ◮ U (1) supersymmetric Chern-Simons theory at level + 1 2 ◮ Chiral multiplet φ of charge +1 and R -charge +1 U (1) T topological flavour symmetry. U (1) U (1) T U (1) R φ +1 0 +1 T 0 +1 0 ( This is mirror to single chiral multiplet - the monopole operator T . ) Important This theory has only ‘Higgs branch’ vacua.