Gauged Linear Sigma Model in the Geometric Phase Guangbo Xu joint work with Gang Tian Princeton University International Conference on Differential Geometry An Event In Honour of Professor Gang Tian s 60th Birthday January 2018
Outline 1 Ordinary and Gauged Gromov Witten Theory 2 Gauged Linear Sigma Model and Gauged Witten Equation 3 The Moduli Space and Invariants 4 Other Speculations
I. Ordinary and Gauged Gromov Witten Theory
Gromov Witten Theory Let (X, ω) be a symplectic manifold with a compatible a.c.s. J. Given g, n and A H 2(X ; Z), denote { } M J g,n(x, A) = (Σ; z 1,..., z n; u : Σ X ) du + J du j Σ = 0 / It has a compactification M J g,n(x, A) which is a compact Hausdorff space. Moroever, it behaves like an oriented manifold of dimension dim vir M J g,n(x, A) = (dimx 6)(1 g) + 2c 1(A). There are evaluation maps M J g,n(x, A) π M g,n The Gromov{Witten invariants of X is formally defined as GW X (β; α 1,..., α n) = [ ] ev (α 1 α n) π β T ω(a). A M J g,n (X,A) ev X n
Gromov Witten Theory Gromov introduced pseudoholomorphic curves in 1985. Floer introduced Floer homology around 1988. Witten gave the topological field theory intepretation in 1988. Ruan Tian firstly defined GW invariants for compact semi-positive symplectic manifolds, by perturbing the J-holomorphic curve equation. They also rigorously defined the quantum cohomology. Later J.Li Tian defined GW invariants for general compact symplectic manifolds via the virtual cycle construction. G. Liu Tian constructed the Hamiltonian Floer homoloyg. Independently by Fukaya Ono. The GW invariants are not random numbers. They are correlation function of two-dimensional nonlinear sigma model. They satisfy the relations called the Cohomological Field Theory, where the associativity of quantum cohomology is one of such relations.
Gauged Gromov Witten Theory Let K be a compact Lie group, acting on (X, ω) with a moment map µ : X k k. Choose a K-invariant compatible a.c.s. J. Given a Riemann surface Σ, the variables are connections A on a principal K-bundle P Σ (gauge fields); sections of the associated bundle P K X (matter fields). Choosing a volume form on Σ, the Vortex Equation reads A u = 0, F A + µ(u) = 0. The system has gauge symmetry. One can study the moduli space of gauge equivalence classes of solutions over a fixed Σ.
Hamiltonian Gromov Witten Invariants By including marked points, in principle we can obtain HGW Σ (α 1,..., α n), α 1,..., α n H K (X ). This has been rigorously done by Mundet for K = S 1, compact X and by Cieliebak Gaio Mundet Salamon for general K and aspherical X. When allow degenerations Mundet Tian has a project on constructing a cohomological field theory on H S 1(X ) for the K = S 1 case. Gaio Salamon proved (in certain special case) an equivalence between the HGW and the GW invariants of the symplectic quotient X = µ 1 (0)/K. GIT or symplectic quotient depends on a paramter (stability condition). This framework allows to unify GW for different phases.
II. Gauged Linear Sigma Model and Gauged Witten Equation
Gauged Linear Sigma Model Spaces Definition A gauged linear sigma model space is (X, G, W, µ, τ), where 1 X is a Kähler manifold with a C -action (the R-symmetry). 2 G = K C is a reductive Lie group acting on X commuting with C. 3 W is a G-invariant holomorphic function on X and homogeneous with respect to the C -action (of certain degree r 1). 4 µ is a moment map and τ Z(k ) (stability condition). They also satisfy the following conditions. 1 τ is a regular value of µ. 2 (Geometric Phase) The singular locus of CritW is τ-unstable. 3 Some other conditions on geometry at infinity.
Main example: hypersurfaces Let Q : C N C be a homogeneous polynomial of degree r. Let X = C N+1 with coordinates (p, x 1,..., x N ). G = C acts on X by a (p, x 1,..., x N ) = ( a r p, ax 1,..., ax N ). Superpotential W (p, x 1,..., x N ) = pq(x 1,..., x N ). Moment map µ(p, x 1,..., x N ) = r p 2 + x 1 2 + + x N 2 τ. Critical locus CritW = { (p, x) p = 0, Q(x) = 0 } { (p, 0) }. When τ > 0 it is a geometric phase. In this phase the second component above is unstable. When τ < 0 it is a Landau Ginzburg phase.
Motivation 1 LG/CY correspondence Martinec, Greene Vafa Warner, Vafa Warner 2 Witten Phases of N = 2 theories in two dimensions, 1993 3 (In the above example) when τ 0 the theory converges to the GW theory of the hypersurface (with quantum correction). 4 The infrared theory (GW theory, corresponding to τ = + ) can be completed in untraviolet. Moroever, this (gauged) theory has an analytic continuation to negative Kähler classes (τ < 0). 5 For negative τ 0 the theory converges to another conformal field theory called Landau Ginzburg theory. Mathematically it is constructed by Fan Jarvis Ruan following Witten s original idea. (FJRW).
Other Examples If one has a collection of (quasi)homogeneous polynomials Q 1,..., Q s of the same degree r, then one can construct a GLSM space for which XW τ in the geometric phase is the complete intersection in a (weighted) projective space. There are also some exotic examples due to physicists, see Hori Tong, Hori Knapp, Jockers Kummar Lapan Morrison Romo, etc.. They have nonabelian gauge groups and in some cases there are only geometric phases and no LG phases. There is also an infinite dimensional example, where X is the space of G- connections on a three-manifold, and W is the complex Chern Simons functional. This GLSM is formally equivalent to a five dimensional gauge theory which is related to the Khovanov homology of knots (see the work of Witten et al).
Gauged Witten Equation To construct such a theory, first we want some equation over Σ like A u + W (u) = 0, F A + µ(u) = 0 To have W well-defined one needs an r-spin structure over Σ. We can cook up some S 1 -bundle P R Σ with a prefered connection A R. Then for another K-bundle P K Σ, define P = P K P R. Σ Define Y = P (K S 1 ) X. W is lifted to W Γ (Y, π K Σ ), which has a vertical gradient W Γ (Y, π KΣ T vert Y ).
Gauged Witten Equation continued The variables are gauge fields A K A(P K ) and matter fields u Γ (Y ). A K and A R determine a connection A on P, and we can take covariant derivative of u by A, and the (0, 1)-part A u. Take a volume form on the punctured surface Σ {z 1,..., z k } of cylindrical type. The gauged Witten equation is AK u + W (u) = 0, F AK + µ(u) τ = 0. When K is trivial, the gauged Witten equation becomes the Witten equation studied by Fan Jarvis Ruan. When W = 0, R-symmetry is no longer required. The gauged Witten equation becomes the vortex equation.
Properties of Solutions Actually one can show A u + W (u) = 0 A u = W (u) = 0. Near a puncture with cylindrical coordinate (s, t), the vortex equation is roughly F AK + (µ(u) τ)dsdt = 0. It forces solutions to converge to the level set µ = τ. A solution is then like a gauged holomorphic map into the singular subvariety CritW with well-defined limits in X τ W := CritW // τ G. So the expected invariants (correlation functions) have multilinear functions on H ( X τ W ) (ordinary, not equivariant cohomology).
III. Virtual Cycle and Invariants
The Moduli Space Over the universal curve M g,n+1 M g,n, one can choose a family of conformal metrics which are of cylindrical type near nodes and markings. The moduli of r-spin curves M r g,n is a branched cover of M g,n. (Compactness) The sequential compactness of solutions over a sequence of r-spin curves is similar to Venugopalan s result for symplectic quasimaps (i.e., W = 0 case). Indeed in the geometric phase, one can regard solutions as quasimaps into the singular target XW τ with smooth GIT quotient X W τ. In a stable map there could be rational components with two special points, but cannot be rational components with only one special points if X is assumed aspherical.
The Virtual Cycle The compactified moduli space M g,n(d; X, W ) of stable solutions is compact and Hausdorff. One can apply Li Tian s topological construction of virtual cycles on this moduli space. The expected dimension is equal to the expected dimension of genus g, degree d curves with n markings in X τ W whose evaluation at the markings are mapped in to the corresponding twisted sectors of X τ W. On a long cylinder the gauged Witten equation plus a gauge fixing condition is of the type of a gradient flow on the loop space, so the gluing is essentially the same as the Gromov Witten theory. The rational components in a stable solution may have finite cyclic automorphisms. One uses the virtual cycle on M g,n(d; X, W ) to define correlation functions H ( X τ W ) n H (M g,n) Λ
Invariants in Geometric Phase Theorem (Tian X., in preparation) The collection of correlation functions form a cohomological field theory on the state space H ( X W ; Λ). Remark In our first few papers, we had a different setting. 1 Only for W = pq, the hypersurface model, but allows both geometric phase and Landau Ginzburg phase. 2 Use finite volume metrics on Σ and perturbations. 3 The invariants were only defined for a fixed smooth curve Σ. However we expect to prove that the invariants under the new setting, when restricting to the hypersurface model and a fixed smooth curve, give equivalent information as in the original setting.
Algebraic Approaches 1 Ciocan-Fontanie Kim s quasimap has the same spirit of GLSM in geometric phases, for X W τ being GIT quotient of affine varieties. 2 Fan Jarvis Ruan used the quasimap setting to define GLSM invariants for cohomology classes in narrow sectors and compact type classes in broad sectors. 3 Chang Li Li Liu has the mixed-spin-p field framework and proved certain LG/CY correspondence relations between GW invariants of quitincs and FJRW invariants. 4 The above are all A-model close, closed string stories. There are other works towards proving LG/CY for B-model, open string case using GLSM (e.g. Orlov s equivalence between derived category of coherent sheaves and matrix factorizations).
IV. Future Works and Speculations
Adiabatic Limit and Point-like Instantons We use Adiabatic Limit of the gauged Witten equation to study the relation between the GLSM invariants and GW invariants of X τ W. When W = 0, Gaio Salamon consider the ɛ 0 limit of A u = 0, F A + ɛ 2 µ(u) = 0. Over a surface Σ, a sequence of solutions for ɛ n 0 converges modulo bubbles to holomorphic curves in X. Bubbles corresponding to energy concentration of rate ɛ 1 are affine vortices or point-like instantons, i.e., vortices over C. It is a codimension zero phenomenon.
A Conjectural Relation When W 0, one considers a similar adiabatic limit A u + ɛ 1 W (u) = 0, F A + ɛ 2 µ(u) = 0. The bubbles are solutions of the gauged Witten equation on C. Counting these instantons defines a class c = d 1 T d #P.L. instantons of degree d QH ( X τ W ; Λ +). Conjecture For ɛ small, α 1,..., α k ɛ GLSM = n 0 1 X α1, W..., α k, c,..., c τ n! }{{}. GW n The gluing that reverse the adiabatic limit should be the hardest part. X., Gluing affine vortices, 1610.09764 (for W = 0 case).
General LG CY Correspondence In the so-called narrow case, the algebraic method is much simpler. Chang Li Li Liu use their MSP method to obtain certain relations between FJRW and GW invariants of the Fermat quintic. Using our symplectic geometry construction it should be similar to prove certain relations for more general targets. The broad case is more complicated. Algebraic method is unavailable. Using symplectic geometry, the wall-crossing at τ = 0 is the most difficult part to handle.
Real LG CY Correspondence For a quasihomogeneous polynomial W : C N C with real coefficients, one can study the Witten equation on real r-spin curves. There could be a corresponding real Landau Ginzburg theory. On the other hand, people also study the real Gromov Witten theory, for example for hypersurfaces in P N defined by polynomials with real coefficients. So it is natural to expect a real LG/CY correspondence, one can prove such a correspondence using a real GLSM theory. Similar to Chang Li Li Liu s MSP approach, one can obtain formulae relating the real Gromov Witten invariants and real FJRW invariants at least in the narrow case. This also calls for differential geometric versions of algebraic techniques, for example, that of Kiem Li s cosection localization.
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