Eric Sharpe - Examples of homological projective duality in physics

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Decomposition conjecture For banded gerbes, K acts trivially upon so the decomposition conjecture reduces to ⎛ Ĝ ⎞ CFT(G − gerbe on Y ) = CFT (Y, B) ⎠ (Y = [X/K]) ⎝ ∐ Ĝ where the B field is determined by the image of H 2 (Y, Z(G)) Z(G)→U(1) −→ H 2 (Y, U(1))
Quick consistency check: A sheaf on a banded G-gerbe is the same thing as a twisted sheaf on the underlying space, twisted by image of an element of H 2 (X,Z(G)) This implies a decomposition of D-branes (~ sheaves), which is precisely consistent with the decomposition conjecture.
Page 1 and 2: Examples of homological projective
Page 3 and 4: What did hpd teach us? Prior to ~ 2
Page 5 and 6: Outline: * physical realization of
Page 7 and 8: Prototype for the exotic GLSM’s I
Page 9 and 10: GLSM for P 3 [2,2] (=T 2 ): Looks l
Page 11 and 12: All phases of GLSM’s can be descr
Page 13 and 14: The other phase is more exciting: V
Page 15 and 16: A more important subtlety is the fa
Page 17: General decomposition conjecture Co
Page 21 and 22: GLSM’s Let’s now return to our
Page 23 and 24: V// − C × : Double cover P 1 Ber
Page 25 and 26: Next simplest example: // + : LG on
Page 27 and 28: The next example in the pattern is
Page 29 and 30: Check that we are seeing K’s nonc
Page 31 and 32: Summary so far: CP 7 [2,2,2,2] This
Page 33 and 34: More examples: CI of 2 quadrics in
Page 35 and 36: D-brane probes of nc resolutions Le
Page 37 and 38: Our D-brane probes of this Landau-G
Page 39 and 40: When is a matrix factorization `poi
Page 41 and 42: We’re interested in Landau-Ginzbu
Page 43 and 44: Example: Family [C 2 /Z 2 ] x,y ×
Page 45 and 46: So far: When the LG model flows in
Page 47 and 48: Case of an nc resolution, cont’d:
Page 49 and 50: So far: * examples of hpd realizing
Page 51 and 52: Let’s work through this in more d
Page 53 and 54: As a warm-up, let’s outline the G
Page 55 and 56: P 7 [2,2,2,2], cont’d In principl
Page 57 and 58: P 7 [2,2,2,2], cont’d After norma
Page 59 and 60: Result for P 7 [2,2,2,2]: n N 1 512
Page 61 and 62: Applying the same method, Compare G
Page 63: Summary: * physical realization of

Eric Sharpe - Examples of homological projective duality in physics

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