Well, a big question is how did the universe begin. And we, cannot answer that question. Some people think that the big bang is an explanation of how the universe began, IT IS NOT. The big bang is a theory of how the universe evolved from a split second after whatever brought it into existence. And the reason why we’ve been unable to look right back at time zero, to figure out how it really began; is that conflict between Einstein’s ideas of gravity and the laws of quantum physics. So, STRING THEORY may and will be able to – it hasn’t yet; we’re working on it today – feverishly. It may be able to answer the question, how did the universe begin. And I don’t know how it’ll affect your everyday life, but to me, if we really had a sense of how the universe really began, I think that would, really, alert us to our place in the cosmos in a DEEP way. ~ Brian Greene!

I listed the 4 essential properties of D3-branes, namely: i) the **propagation** of a D3-brane through **spacetime** generates a 4-dimensional **worldvolume** that has 4-dimensional Poincaré invariance, ii) the **string worldsheet** generating the graviton via quantum fluctuation can be topologically **compactified** on the **boundary of its corresponding** **space**, iii) D3-branes have **constant axion** **and dilaton fields**, and for the purposes of this post, iv) D3-branes are **self-dual**. Thus, the gravitonic D3-brane action with a super-Lagrangian coupling can be derived as

with

However, as I showed, one must exhibit the self-duality of the D3-brane in the Hamiltonian setting. It is my aim in this post to provide the proof. One can always **lift** an **duality** to an **duality** by introducing the D3-brane **dilaton** and **axion** which are constant background fields. Then, one can re-define an duality Lagrangian as such:

with . From the above dual Lagrangian, the D3-brane Hamiltonian action can be derived as

where

and are the Pauli matrices **cohomologically acting on the supersymmetric group indices**, and is the Wess-Zumino Lagrangian satisfying the Matsubara condition, and is given by

where and are RR-2 and RR-4 **differential forms**, and represents the Kappa symmetry of the gauge bundle of the D3-brane’s **topology**. Now, let

be a canonical **conjugate** set for the super-Kahler **phase space** variables, and define the critical **3-dimensional anti-symmetric tensor**

and introduce the de Rham variables

where transforms as

One then finds that the constraints of the system to be given by

– **symmetry constraint**

–

–

and

– the p + 1 diffeomorphism **constraints**

and

– **the fermionic constraints**

with being the **spatial** part of the **metric**, and its **determinant** being . One must now show the **Poincaré invariance** of the **bosonic constraints**

and the **supersymmetric covariance of the fermionic constraint** under the transformation of and corresponding to the fermionic field rotation **holds**. Then one gets

and

with being an matrix satisfying

and transforms as

with

and

and being an transformation satisfying

and

with rotation .

To prove **D3-brane self-duality**, we must consider

while noting that all terms in the above expression and itself are invariant under Poincaré duality transformations. So, , the **conjugate** of , is also likewise invariant, hence,

reduces to

where

holds. In terms …

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