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GAUGED LINEAR SIGMA MODEL in GEOMETRIC PHASES 10 Hence a Local Universal Unfolding Provides a Local Orbifold Chart of the Deligne–Mumford Space

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GAUGED LINEAR SIGMA MODEL in GEOMETRIC PHASES 10 Hence a Local Universal Unfolding Provides a Local Orbifold Chart of the Deligne–Mumford Space GAUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES GANG TIAN AND GUANGBO XU Abstract We construct a cohomological field theory for a gauged linear sigma model space in a geometric phase, using the method of gauge theory and differential geometry. The cohomological field theory is expected to match the Gromov–Witten theory of the classical vacuum up to a change of variable, and is expected to match various other algebraic geometric constructions. Keywords: cohomological field theory, CohFT, gauged linear sigma model, GLSM, gauged Witten equation, vortices, moduli spaces, virtual cycle Contents 1. Introduction..................................... ........................... 3 1.1. The main theorem................................. ..................... 3 1.2. Comparison with algebraic geometric approaches . ................... 6 1.3. Virtual cycle................................... ......................... 7 1.4. Outline........................................ ......................... 8 1.5. Acknowledgement................................ ....................... 8 Part I.............................................. ............................. 8 2. Moduli Spaces of Stable r-Spin Curves...................................... 8 2.1. Marked nodal curves.............................. ...................... 8 2.2. Resolution data and gluing parameters . .................... 11 2.3. r-spin curves........................................ .................... 13 2.4. Unfoldings of r-spin curves........................................ ...... 17 2.5. Universal structures............................ ......................... 19 3. Vortex Equation and Gauged Witten Equation . .................. 22 3.1. The GLSM space................................... .................... 22 3.2. Chen–Ruan cohomology............................ ..................... 24 3.3. Gauged maps and vortices .......................... .................... 25 3.4. Gauged Witten equation . ..................... 28 arXiv:1809.00424v1 [math-ph] 3 Sep 2018 3.5. Energy and bounded solutions ...................... .................... 29 3.6. The relation with vortices . ....................... 29 4. Vortices over Cylinders............................ .......................... 30 4.1. Critical loops.................................. ......................... 30 4.2. Isoperimetric inequality ........................ ......................... 31 4.3. Vortex equation................................. ........................ 33 4.4. Annulus lemma................................... ...................... 34 4.5. Asymptotic behavior............................. ....................... 34 5. Analytical Properties of Solutions . ......................... 36 5.1. Holomorphicity ................................. ........................ 36 Date: September 5, 2018. The second named author is partially supported by the Simons Foundation through the Homological Mirror Symmetry Collaboration grant and AMS-Simons Travel Grant. The majority of this paper is finished when the second named author was in Department of Mathematics, Princeton University. 1 GAUGEDLINEARSIGMAMODELINGEOMETRICPHASES 2 5.2. Asymptotic behavior............................. ....................... 40 5.3. Uniform C0 bound.............................................. ........ 42 5.4. Energy identity................................. ........................ 43 6. Moduli Space of Stable Solutions and Compactness. ................... 44 6.1. Solitons....................................... .......................... 44 6.2. Decorated dual graphs............................ ...................... 45 6.3. Stable solutions................................ ......................... 46 6.4. Topology of the moduli spaces ...................... .................... 47 Part II............................................. ............................. 51 7. Topological Virtual Orbifolds and Virtual Cycles . ...................... 51 7.1. Topological manifolds and transversality. ....................... 51 7.2. Topological orbifolds and orbibundles . ...................... 55 7.3. Virtual orbifold atlases......................... ......................... 60 7.4. Good coordinate systems.......................... ...................... 62 7.5. Shrinking good coordinate systems................. ..................... 63 7.6. Perturbations.................................. ......................... 67 7.7. Branched manifolds and cobordism.................. .................... 72 7.8. Strongly continuous maps......................... ...................... 74 8. Virtual Cycles and Cohomological Field Theory . .................... 75 8.1. Cohomological field theories . ....................... 75 8.2. Relation with orbifold Gromov–Witten theory . ................... 77 8.3. Relation with the mirror symmetry.................. .................... 79 9. Constructing the Virtual Cycle. I. Fredholm Theory. .................... 79 9.1. Moduli spaces with prescribed asymptotic constrains . ................... 79 9.2. The Banach manifold and the Banach bundle . ............... 80 9.3. Gauge fixing..................................... ....................... 83 9.4. The index formula................................ ...................... 85 10. Constructing the Virtual Cycle. II. Gluing . ...................... 89 10.1. Stabilizers of solutions . ......................... 90 10.2. Thickening data ................................ ....................... 92 10.3. Nearby solutions............................... ........................ 99 10.4. Thickened moduli space and gluing ................. ................... 103 10.5. Proof of Proposition 10.16 ............................................. 105 10.6. Last modification............................... ....................... 112 11. Constructing the Virtual Cycle. III. The Atlas . ..................... 112 11.1. The inductive construction of charts . ..................... 113 11.2. Coordinate changes............................. ....................... 116 11.3. Constructing a good coordinate system . ................... 119 11.4. The virtual cycle............................... ........................ 120 12. Properties of the Virtual Cycles.................... ........................ 121 12.1. The dimension property.......................... ...................... 121 12.2. Disconnected graphs............................ ....................... 121 12.3. Cutting edge................................... ........................ 128 12.4. Composition................................... ........................ 128 References......................................... ............................. 129 GAUGEDLINEARSIGMAMODELINGEOMETRICPHASES 3 1. Introduction The GAUGED LINEAR SIGMA MODEL (GLSM) is a two-dimensional supersymmetric quan- tum field theory introduced by Witten [Wit93] in 1993. It has stimulated many important developments in both the mathematical and the physical studies of string theory and mir- ror symmetry. For example, it plays a fundamental role in physicists argument of mirror symmetry [HV00][GS18]. Its idea is also of crucial importance in the verification of genus zero mirror symmetry for quintic threefold [Giv96][LLY97]. 1 Since 2012 the authors have initiated a project aiming at constructing a mathemati- cally rigorous theory for GLSM, using mainly the method from symplectic and differential geometry. In our previous works [TX15, TX16a, TX16b], we have constructed certain correlation function (i.e. Gromov–Witten type invariants) under certain special condi- tions: the gauge group is U 1 and the superpotential is a Lagrange multiplier. This correlation function is thoughp ratherq restricted, as for example, we do not know if they satisfy splitting axioms or not. A major difficulty in the study of the gauged Witten equation lies in the so-called BROAD case, in which the Fredholm property of the equation is problematic. This issue also causes difficulties in various algebraic approaches. Now we have realized that if we restrict to the so-called GEOMETRIC PHASE, then the issue about broad case disappears in our symplectic setting. Our construction in the geometric phase has been outlined in [TX17]. It ends up at constructing a COHOMOLOGICAL FIELD THEORY (CohFT), namely a collection of correlation functions which satisfy the splitting axioms. The detailed construction in this scenario is the main objective of the current paper. A natural question is the relation between the GLSM correlation functions and the Gromov–Witten invariants. They are both multilinear functions on the cohomology group of the classical vacuum of the GLSM space, which is a compact K¨ahler manifold or orbifold. The relation between them is a generalization of the quantum Kirwan map, proposed by D. Salamon, studied by Ziltener [Zil14], and proved by Woodward [Woo15] in the algebraic case. In short words, the two types of invariants are related by a “coor- dinate change,” where the coordinate change is defined by counting point-like instantons. The point-like instantons are solutions to the gauged Witten equation over the complex plane C, a generalization of affine vortices in the study of the quantum Kirwan map. These instantons appear as bubbles in the adiabatic limit process of the gauged Witten equation. There has also been a physics explanation by Morrison–Plesser [MP95]. In the forthcoming work [TX] we will rigorously construct such coordinate change and prove the expected relation between GLSM correlation functions
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