An Introduction to Quantum Sheaf Cohomology - Physics - Virginia ...

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(0,2) mirror symmetry is a conjectured generalization that exchanges pairs where the and the E i → X i (X 1 , E 1 ) ↔ (X 2 , E 2 ) X i are Calabi-Yau manifolds are holomorphic vector bundles Constraints: E stable, ch 2 (E) = ch 2 (T X) Reduces to ordinary mirror symmetry when E i ∼ = T X i
(0,2) mirror symmetry Instead of exchanging (p,q) forms, (0,2) mirror symmetry exchanges sheaf cohomology: H j (X 1 , Λ i E 1 ) ↔ H j (X 2 , (Λ i E 2 ) ∨ ) Note when E i ∼ = T X i this reduces to H d−1,1 (X 1 ) ↔ H 1,1 (X 2 ) (for X i Calabi-Yau)
Page 1 and 2: An Introduction to Quantum Sheaf Co
Page 3: First, recall ordinary mirror symme
Page 7 and 8: (0,2) mirror symmetry A few highlig
Page 9 and 10: Aside on lingo: The worldsheet theo
Page 11 and 12: The A/2, B/2 models: * (0,2) analog
Page 13 and 14: A/2 model g ij ∂φ i ∂φ j + ih
Page 15 and 16: A model classical correlation funct
Page 17 and 18: A model -- worldsheet instantons Mo
Page 19 and 20: A/2 model -- worldsheet instantons
Page 21 and 22: 1st, review linear sigma model (LSM
Page 23 and 24: What about induced sheaves F → M
Page 25 and 26: Corresponding to O(k) is a rk k `un
Page 27 and 28: There is also a trivial extension o
Page 29 and 30: Quantum cohomology ... is an OPE ri
Page 31 and 32: Briefly, one computes quantum (shea
Page 33 and 34: (cont’d) Briefly, the (long exact
Page 35 and 36: (cont’d) Summary: The correlation
Page 37 and 38: More generally, for ``linear’’
Page 39 and 40: So far: * outlined A/2, B/2 models
Page 41 and 42: LG B model: The states of the theor
Page 43 and 44: To A twist, need a U(1) isometry on
Page 45 and 46: A twistable example: LG A model: LG
Page 47 and 48: LG A model, cont’d 〈O 1 · ·
Page 49 and 50: Example, cont’d Q · In the A twi
Page 51 and 52: Renormalization (semi)group flow Co
Page 53 and 54: Furthermore, RG preserves TFT’s.
Page 55 and 56:
So far we’ve outlined (2,2) Landa
Page 57 and 58:
Pseudo-topological twists: * If E a
Page 59 and 60:
Adams-Basu-Sethi Example: Correspon
Page 61 and 62:
One can also compute elliptic gener
Page 63 and 64:
(cont’d) Perform A/2 twist. If re
Page 65 and 66:
Most general case: LG model on with
Page 67 and 68:
Example: NLSM on {G = 0} ⊂ W P 4
Page 69:
Strings-Math 2011 A new biennial co
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